Abstract
Besides models of the differential type considered in Chap.7, in continuum mechanics there are other types of nonideal media. One widely uses models of viscoelastic materials, which are also called continua of the integral type, or hereditarily elastic continua. Models of viscoelastic continua most adequately describe the mechanical properties of polymer materials, composites based on polymers, different elastomers, rubbers and biomaterials, in particular, human muscular tissues.
Keywords
- Constitutive Equation
- Heat Conduction Equation
- Relaxation Function
- Relaxation Curve
- Dissipation Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
8.1 Viscoelastic Continua of the Integral Type
8.1.1 Definition of Viscoelastic Continua
Besides models of the differential type considered in Chap.7, in continuum mechanics there are other types of nonideal media. One widely uses models of viscoelastic materials, which are also called continua of the integral type, or hereditarily elastic continua. Models of viscoelastic continua most adequately describe the mechanical properties of polymer materials, composites based on polymers, different elastomers, rubbers and biomaterials, in particular, human muscular tissues.
Further, viscoelastic continua of the differential type will be called continua ofthe differential type, and viscoelastic continua of the integral type – simply viscoelastic materials.
Definition
A medium is called a viscoelastic continuum of the integral type(or simply a viscoelastic continuum), if any of the models A n , B n , C n or D n is assumed for the medium, and corresponding operator constitutive equations (4.158) or (4.160)–(4.163) are functionalsof time t:
i.e. values of active variables Λ(t) depend not only on values of reactive variables \(\mathcal{R}(t)\)at the same instant of time but also on their prehistory\({\mathcal{R}}^{t}(\tau ) \equiv \mathcal{R}(t - \tau )\), i.e.on their values at all preceding instants of time 0<τ≤t, starting from initial one τ=0.
Due to such specific dependence, viscoelastic continua are also called continua with memory.
For viscoelastic continua:
-
1.
“The present can depend only on the past but not on the future”; therefore, all the functionals (8.1) depend only on their previous history (prehistory) \(\mathcal{R}(t - \tau )\), 0<τ≤t.
-
2.
“The past is not infinite”, i.e. the times τ>tdo not affect the functionals (8.1). This means that
$$\mathcal{R}(\tau ) \equiv 0\ \ \text{ at}\ \ \tau < 0;\ \ \ \ \ {\mathcal{R}}^{t}(\tau ) = \mathcal{R}(t - \tau ) \equiv 0\ \ \text{ at}\ \ t >\tau.$$(8.2)
8.1.2 Tensor Functional Space
To perform operations with functionals (8.1), we need some additional knowledge of functional analysis.
Consider the set of previous histories of k-order tensors \({}^{k}{\mathbf{T}}^{t}(\tau ) {= }^{k}\mathbf{T}(t - \tau )\)(0<τ≤t), and define for two arbitrary prehistories k T 1 tand k T 2 ttheir scalar product as follows:
The function γ(τ) is called the function of memory. This function is positive, continuous, monotonically decreasing, defined within the interval [0,+∞) and squared-integrable, i.e.
Since the memory function is monotonically decreasing, quantities of the prehistory \({}^{k}{\mathbf{T}}^{t}(\tau ) {= }^{k}\mathbf{T}(t - \tau )\)at small values of τ give a greater contribution to the scalar product (8.3) than quantities of k T t(τ) at large values of τ. In other words, a continuum better remembers events having occurred at times closer to the current instant tthan the ones at more remote times. The functionals (8.1) are assumed to have this property too; therefore, viscoelastic continua of the integral type are also called continua with fading memory.
Let us consider now the set \({}^{k}{\mathcal{H}}_{t}\)of processes of changing the tensor k T(τ) (0<τ≤t) and assign to each process the pair (k T(t),k T(τ)) consisting of values of the tensor k T(t) at time tand its prehistory \({}^{k}\mathbf{T}(\tau ) {= }^{k}\mathbf{T}(t - \tau )\)(0<τ≤t). Then we can introduce the scalar product of processes k T 1(τ) and k T 2(τ) included in \({}^{k}{\mathcal{H}}_{t}\):
where
is the scalar product of k-order tensors.
The set \({}^{k}{\mathcal{H}}_{t}\)of all processes of changing the tensor k T(τ) (0≤τ≤t), for which the scalar product (8.5) exists and which at each fixed τ are elements of the tensor space \({\mathcal{T}}_{3}^{k}({\mathcal{E}}_{3})\)with the operations of addition and multiplication by a number, is called the tensor functional space \({}^{k}{\mathcal{H}}_{t}\).
The space \({}^{k}{\mathcal{H}}_{t}\)is a Hilbert space, because we can always go into a Cartesian basis, where the components \(\bar{{T}}^{{i}_{1}\ldots {i}_{k}}(\tau )\)of tensors included in \({}^{k}{\mathcal{H}}_{t}\)are squared-integrable functions, i.e.theybelongtothe function space L 2 (m)[0,t] (where m=3k), which is known as a Hilbert space.
Due to the property (8.2), the scalar product of processes included in \({}^{k}{\mathcal{H}}_{t}\)(8.5) can be written in the form
which often proves to be convenient for analysis of the models of a viscoelastic continuum.
In the space \({}^{k}{\mathcal{H}}_{t}\), there is a natural norm of the process k T(τ):
where ( ⋅) t is the scalar product (8.5).
8.1.3 Continuous and Differentiable Functionals
Using the norm (8.8), we can introduce the concept of a continuous functionalin the form (8.1)
which is considered as the mapping of the domain Ucontained in the space \({}^{k}{\mathcal{H}}_{t}\)into the domain Vof the space \({}^{m}{\mathcal{H}}_{t}\):
Definition
The functional (8.10) is called continuousin a domain \(U {\subset }^{k}{\mathcal{H}}_{t}\), if for each process k T∈Uthe following condition is satisfied: \(\forall \epsilon >0\) ∃δ>0 such that for every process \({}^{k}\widetilde{\mathbf{T}}\), where \({(}^{k}\mathbf{T} {+ }^{k}\widetilde{\mathbf{T}}) \in U\)and
values of the operators in the norm (8.8) of the space \({}^{k}{\mathcal{H}}_{t}\)are sufficiently close:
The functional (8.10) is called linearif it satisfies the two conditions
for all processes k T 1(τ) and k T 2(τ) included in \({}^{k}{\mathcal{H}}_{t}\)and for every real number s.
In space \({}^{k}{\mathcal{H}}_{t}\)we can use Riesz’s theoremthat any linear functional (8.9) can be represented as the scalar product of a fixed element from \({}^{k}{\mathcal{H}}_{t}\)and an arbitrary process k T(τ) from \({}^{k}{\mathcal{H}}_{t}\); so a scalar linear functional in \({}^{2}{\mathcal{H}}_{t}\)has the form
where \(\widetilde{\Gamma }(t,t - \tau )\)is the prehistory and \(\widetilde{\Gamma }(t,t)\)is the instantaneous value at τ=tof the fixed process \(\widetilde{\Gamma }(t,\;\tau )\)(0≤τ≤t) for the given functional f(the appearance of one more argument tfor the process \(\widetilde{\Gamma }(t,\;\tau )\)means that this process can vary with changing the time interval considered).
Replacing the variable \(t - \tau = y\)and introducing the notation \(\widetilde{{\Gamma }}^{\top }(t,y){\gamma }^{2}(\tau ) = \Gamma (t,y)\), Γ0=Γ⊤(t,t), after the reverse substitution y→τ we obtain another representation of the linear scalar functional
called Volterra’s representation.
For viscoelastic continua, it is convenient to use the Dirac δ-functionδ(t) having the following main property:
for any continuous tensor process B(t,τ).
According to (8.16), the linear functional (8.15b) can be written in the form
Definition
The functional (8.9) is called Fréchet–differentiableat point m T∈Uof a domain \(U {\subset }^{m}{\mathcal{H}}_{t}\), if there exist two functionals \(\partial \mathcal{F}\)and \(\delta \mathcal{F}\)having the following properties:
-
they are defined over the Cartesian product of the space \({\mathcal{H}}_{t}\)
$${\partial }^{m}\mathcal{F} {:\ \ \ }^{k}{\mathcal{H}}_{ t} {\times }^{k}{\mathcal{H}}_{ t} {\rightarrow }^{m}{\mathcal{H}}_{ t};\ \ \ \ \ \ {\delta }^{m}\mathcal{F} {:\ \ \ }^{k}{\mathcal{H}}_{ t} {\times }^{k}{\mathcal{H}}_{ t} {\rightarrow }^{m}{\mathcal{H}}_{ t},$$(8.17) -
they can be written in the form similar to (8.9)
$$ \begin{array}{rcl}{ }^{m}{\mathbf{P}}_{ 1}& =& {\partial }^{m} \mathcal{F}^{t}_{\tau = 0}({}^{k}\mathbf{T}(t){,\ }^{k}{\mathbf{T}}^{t}(\tau ){\big{\vert }}^{k}\widetilde{\mathbf{T}}(t)) \\ & =& \frac{\partial } {{\partial }^{k}\mathbf{T}(t)} \mathcal{F}^{t}_{\tau = 0}{(}^{k}\mathbf{T}(t){,\ }^{k}{\mathbf{T}}^{t}(\tau )) \cdot \ldots {\cdot }^{k}{\mathbf{T}}^{(k\ldots 1)}(t), \end{array}$$(8.18)$${ }^{m}{\mathbf{P}}_{ 2} = {\delta }^{m} \mathcal{F}^{t}_{\tau = 0}({}^{k}\mathbf{T}(t){,\;}^{k}{\mathbf{T}}^{t}(\tau ){\big{\vert }}^{k}\widetilde{{\mathbf{T}}}^{t}(\tau )),$$(8.19)(the vertical line separates two different arguments of the process),
-
they are linear and continuous in the second argument,
-
they satisfy the condition: \(\forall \epsilon >0\) ∃δ such that for any process \({(}^{k}\widetilde{\mathbf{T}}(t){,\ }^{k}\widetilde{{\mathbf{T}}}^{t}(\tau ))\), for which (k T 1(t),k T 1 t(τ))⊂Uand
$${\parallel }^{k}\mathbf{T} {\parallel }_{ t} < \delta,$$(8.20)the following inequality holds:
$$\parallel {\Delta }^{m}\mathcal{F}{\parallel }_{ t} \leq \epsilon {\parallel }^{k}\widetilde{\mathbf{T}} {\parallel }_{ t},$$(8.21)
where
The operator (8.19) is called the Fréchet–derivative; and the right-hand side of expression (8.18) is the partial derivative of \(\mathcal{F}\)(considered as a tensor function of T(t)) with respect to the tensor argument T(t).
If the functional (8.9) is Fréchet–differentiable, then it is continuous (see Exercise8.1.2).
The set of processes \({}^{k}\mathbf{T}(t) {\in }^{k}{\mathcal{H}}_{{t}^{{\prime}}}\)having the first and second continuous derivatives with respect to time t: \({}^{k}\dot{\mathbf{T}}(t)\)and \({}^{k}\ddot{\mathbf{T}}(t)\), which belong to \({}^{k}{\mathcal{H}}_{{t}^{{\prime}}}\), will be denoted by \({U}_{{t}^{{\prime}}}\).
Theorem
Let the functional(8.9) be Fréchet–differentiable in \({}^{k}{\mathcal{H}}_{{t}^{{\prime}}}\) , then there exists such t (t ∈ (0,t ′ )) that for all processes k T(τ) ∈ Utthe process m S(t) is differentiable with respect to t and the following rule of differentiation of the functional with respect to time holds:
Here
▾A proof of the theorem can be found in [31]. ▴
Remark
The theorem gives the possibility to calculate the Fréchet–derivatives of the operators (8.9) by evaluating the ordinary derivative of the functions S(t) with respect to taccording to formula (8.24).
Example 8.1.
Determine the Fréchet–derivatives of the linear operator (8.15b) for the case when \(\Gamma (t,\tau ) = \Gamma (t - \tau )\)and Γ(t,t)=Γ(0). According to formula (8.24), we calculate the ordinary derivative with respect to time tby the rule of differentiation of an integral with a varying upper limit:
where \({\Gamma }^{{\prime}}(y) = \partial \Gamma (y)/\partial y\). Comparing (8.25) with (8.24), we find the partial derivative and the Fréchet–derivative:
8.1.4 Axiom of Fading Memory
For viscoelastic continua, in addition we assume the following axiom.
Axiom 1 (of fading memory).
The functionals(8.1) occurring in constitutive equations of viscoelastic continua are Fréchet–differentiable and hence satisfy the rule of differentiation with respect to time(8.24):
The interconnection between Fréchet–differentiability and fading memory of functionals is clarified by the theorem on relaxation.
Let there be a process \(\mathcal{R}(\tau )\)(0≤τ≤t), which is arbitrary up to some time t 0<t, and when t 0≥τ≥tit remains constant:
Such process \(\mathcal{R}(\tau )\)is called a process with the constant extension(Fig.8.1).
For Theorem8.2
In addition, consider a static process
Then the following theorem can be formulated.
Theorem 8.2.
Let the functional f(8.1) be Fréchet–differentiable, then its partial derivative ∂f and Fréchet–derivative δf, and also the function Λ(t) for any process \(\mathcal{R}(\tau )\) with the constant extension at fixed time t 0 have the limits as t → +∞, which coincide with values of the derivatives \(\partial \mathop{f}\limits^{{_\ast}}\),\(\delta \mathop{f}\limits^{{_\ast}}\) and Λ ∗ , respectively, in corresponding static processes \(\mathcal{R}^{{_\ast}} (\tau )\) :
In simple words, for a process \(\mathcal{R}(\tau )\)such that starting from some time t 0the process reaches a constant level, in a certain time interval a viscoelastic continuum forgets the process \(\mathcal{R}(\tau )\)up to the time t 0, because the continuum response, expressed by the functionals f, ∂fand δf, at sufficiently great values of tdiffers little from its response to a static process.
▾Consider the process
Since
so we have
Due to the property (8.4) and monotone decreasing the function γ(τ), we find that \(\parallel \widetilde{\mathcal{R}}{\parallel }_{t} \rightarrow 0\)at t→+∞.
Since a Fréchet–differentiable functional fis continuous as well as ∂fand δf, the condition (8.12) yields the inequality
this means that the first limit of (8.30) exists. In a similar way, we can prove that the second and third limits of (8.30) exist too. ▴
8.1.5 Models A n of Viscoelastic Continua
For models A n of viscoelastic continua, the free energy ψ is a functional in the form (8.1), and as reactive variables one should choose the set (4.150):
According to the rule (8.24) of differentiation of a functional, we obtain the expression for the total derivative of ψ with respect to t:
Substituting this expression into PTI (4.123) and collecting like terms, we get
When the prehistories \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}{}^{t}\), θtand the current values \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(t)\)and θ(t) are fixed, the increments \(d\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}\), dθ and dtcan vary arbitrarily, therefore the identity (8.37) holds when and only when the coefficients of these increments vanish. As a result, we obtain the equation system
that together with (8.35) is a system of constitutive equations for models A n of viscoelastic continua.
Just as for ideal continua, for viscoelastic materials it is sufficient to give only the free energy functional (8.35), then the remaining relations are determined by its differentiation according to formulae (8.38).
Notice that although relations (8.38) are formally similar to the corresponding relations (4.170) for models A n of ideal continua, they essentially differ by the fact that in (8.38) there is a functional dependence on \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}\)and θ. Moreover, viscoelastic continua are dissipative: for them the dissipation function w ∗is not identically zero.
8.1.6 Corollaries of the Principle of Material Symmetry for Models A n of Viscoelastic Continua
According to the principle of material symmetry (Axiom14), for each viscoelastic continuum there exists an undistorted reference configuration \(\widehat{\mathcal{K}}\). Just as in Sect.4.7.4, for simplicity, let the reference configuration \(\mathcal{K}^{\circ }\)be undistorted. Then for \(\mathcal{K}^{\circ }\)there is a subgroup \({\mathop{G}\limits^{\circ }}{}_{s} \subset U\)of the unimodular group Usuch that for each transformation tensor \(\mathbf{H} \in {\mathop{G}\limits^{\circ }}{}_{s}\)(\(\mathbf{H} :\mathop{ \mathcal{K}}\limits^{\circ }\rightarrow \mathcal{K}^{{_\ast}}\)) the constitutive equations (8.38) written for \(\mathcal{K}^{\circ }\)are transformed during the passage to \(\mathcal{K}^{{_\ast}}\)as follows:
For viscoelastic solids, due to H-indifference of all the tensors \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\), \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(t)\)and \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}{}^{t}(\tau ) =\mathop{ \mathbf{C}}\limits^{\mathrm{(n)}}(t - \tau )\)(see Sect.4.7.4), relations (8.39) take the forms
Theorem 8.3.
The principle of material symmetry in the form(8.40) holds for models A n of viscoelastic solids if and only if the condition(8.40b) for ψ is satisfied.
▾It is evident that the condition (8.40b) is necessary.
Prove that this condition is sufficient. Let the condition (8.40b) be satisfied. Then, since the functional \(\mathcal{F}\)is a tensor function being the derivative of ψ with respect to \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(t)\), so with the help of the method used in the proof of Theorem4.23 we can prove that the condition (8.40b) yields (8.40a). To prove Eq.(8.40c) we should use formula (8.36) for δψ and rewrite it in \(\mathcal{K}^{{_\ast}}\):
Since ψ=ψ∗and the passage from \(\mathcal{K}^{\circ }\)to \(\mathcal{K}^{{_\ast}}\)is independent of t, so \(d\psi /\mathit{\mathit{dt}}\,=\,d{\psi }^{{_\ast}}/\mathit{\mathit{dt}}\). DuetoH-indifferenceofthetensors\(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\)and\(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}\),wehavetherelation\(\mathop{\mathbf{T}}\limits^{{\mathrm{(n)}}}{}^{{_\ast}}\cdot \cdot \ (d\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}{}^{{_\ast}}\!/\mathit{\mathit{dt}})=\mathop{\mathbf{T}}\limits^{\mathrm{(n)}} \cdot \cdot \ (d\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}/\mathit{\mathit{dt}})\), and also \(\partial {\psi }^{{_\ast}}/\partial \theta \,=\,\partial \psi /\partial \theta \). Thus, the right-hand side of Eq.(8.41) coincides with δψ, and hence (δψ)∗=δψ; i.e. Eq.(8.40c) actually holds. ▴
8.1.7 General Representation of Functional of Free Energy in Models A n
Scalar functional ψ (8.35) satisfying the condition (8.40b) is called functionally indifferent relative to the group \({\mathop{G}\limits^{\circ }}{}_{s}\). Let us find a general representation of such a functional in terms of invariants of tensor \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}\)of the corresponding group \({\mathop{G}\limits^{\circ }}{}_{s}\).
In Sect.8.1.3 we have derived a general representation of a linear scalar functional in the form (8.15d). Similarly to (8.15d), define a quadratic scalar functionalas a double integral in the form
where 4 A(t,τ1,τ2) is a fixed fourth-order tensor called the core of the functional. We also define a n-fold scalar functional:
where
Here 2n A(t, τ1,…,τ m ) is the core of this functional (it is a fixed tensor of order (2n) depending on m+1 arguments).
Theorem 8.4 (Stone–Weierstrass).
Any continuous scalar functional (8.35) in space \({\mathcal{H}}_{t}\) can be uniformly approximated by n-fold scalar functionals(8.43):
where the equality means that the partial sum uniformly converges in the norm(8.8).
▾A proof of the theorem for the space \({\mathcal{H}}_{t}\)can be found in [9]. ▴
Let us consider the integrand ψ n (t,τ1,…,τ n ) of the n-fold functional (8.43). Atany fixed set of values t,τ1,…,τ m , this expression is a scalar function (but not a functional!)of ntensor arguments \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}({\tau }_{i}) \equiv \mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{i}\)(i=1,…,m):
Here while values of t, τ1,…,τ m vary, the number and the form of tensor arguments of the functions remain unchanged.
On substituting the representation (8.45) into the condition (8.40b) of functional indifference of ψ, we find that the functions \(\widetilde{{\psi }}_{m}\)(8.44) at each fixed value t,τ1,…,τ m must satisfy the condition
i.e. they must be H-indifferent scalar functions relative to the group \(\mathop{G}\limits^{ {\circ }}_{s}\).
Theorem 8.5.
Every scalar function(8.46) of m tensor arguments C 1,…,C m , which is indifferent relative to a group \(\mathop{G}\limits^{ {\circ }}_{s}\) , can be represented as a function of finite number z (z ≤ 6m) of simultaneous invariants
relative to the group \(\mathop{G}\limits^{ {\circ }}_{s}\) in the form
▾By analogy with simultaneous invariants of two tensors, which have been considered in Sect.7.1.4, we now introduce simultaneous invariants of m tensorsrelative to a group \(\mathop{G}\limits^{ {\circ }}_{s}\). The simultaneous invariants are scalar functions J γ(8.48), which are H-indifferent relative to the group \(\mathop{G}\limits^{ {\circ }}_{s}\); i.e. they satisfy the relations
Their functional basis consists of zsimultaneous invariants, where zis a finite number, which cannot exceed the total number of components of all the tensors, i.e. z≤6m. Moreover, since J γ (s)form a basis, any other H-indifferent scalar function relative to the same group \(\mathop{G}\limits^{ {\circ }}_{s}\)can be expressed in this basis. But the function \(\widetilde{\psi }\)(8.40) is just such a function due to (8.47), therefore the relation (8.49) actually holds. ▴
Substitution of the expression (8.49) into (8.43) and then into (8.45) yields the following general representation of the continuous functional (8.35), which is functionally indifferent relative to a group \(\mathop{G}\limits^{ {\circ }}_{s}\):
In deriving this formula we have used representation (8.15c) of linear functionals with the help of δ-function. Let us perform now the inverse operation: segregate the δ-type constituent from the cores \(\widetilde{{\psi }}_{m}\), that allows us to separate the instantaneous value of the tensor \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(t)\)from its prehistory. Formula (8.15d) can be generalized for functions of m+1 arguments t,τ1,…,τ m as follows:
We assume that at k=mthe argument \({\tau }_{k+1}\,=\,{\tau }_{m+1}\)does not appear among arguments of the function ψ mk : ψ mm =ψ mk (t,J γ (s)).
On substituting (8.51a) into (8.51), we get
Rearrange summands in the expression (8.52) and take into account that simultaneous invariants J γ (s)of mtensors, among which there are ktensors \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(t)\), can always be expressed in terms of simultaneous invariants of \((m - k + 1)\)tensors \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}({\tau }_{k+1}),\ldots \mathop{\mathbf{C}}\limits^{\mathrm{(n)}}({\tau }_{m})\)and \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(t)\). Then we finally obtain the general form of the functional (8.35)
which is functionally indifferent relative to the group \(\mathop{G}\limits^{ {\circ }}_{s}\). Here we have introduced the notation: φ0– the instantly elastic part and φ m – cores of the functional:
Simultaneous invariants \({J}_{\gamma }^{(s)}(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(t))\)of one tensor are simply invariants \({I}_{\gamma }^{(s)}(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(t))\)of the tensor relative to the same group \(\mathop{G}\limits^{ {\circ }}_{s}\).
The expression (8.53) is the desired general representation of the functional (8.35) for models A n .
8.1.8 Model A n of Stable Viscoelastic Continua
Definition8.4.
One can say that this is the model A n of a stable viscoelastic continuum, if the functional ψ of the model is invariant relative to a shift of the process of deforming and heating in time; i.e. if there are two processes \((\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(\tau ),\theta (\tau ))\)and \(\mathop{\widetilde{\mathbf{C}}}\limits^{\mathrm{(n)}}(\tau ),\widetilde{\theta }(\tau ))\), 0≤τ≤t 1, such that they are different from each other only by a shift in time:
then the corresponding values of the functionals ψ and \(\widetilde{\psi }\)are different only by a shift in time as well (Fig.8.2):
Remark .
Since the functional ψ for stable continua is invariant relative to a shift in time, its partial derivatives with respect to \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(t)\)and θ(t) and also the Fréchet–derivative δψ have this property; hence the constitutive equations (8.38) are invariant relative to a shift in time too. Due to the property of invariance, stable continua are also called non-aging, and their constitutive equations do not change with time themselves when there are no deformations and variations of temperature.
Let us consider two important models of stable continua.
For the definition ofa stable viscoelastic continuum
8.1.9 Model A n of a Thermorheologically Simple Viscoelastic Continuum
One can say that this is the model A n of a viscoelastic continuum with difference cores, if in the general representation of the functional ψ (8.53) there is no explicit dependence of cores φ m on the times tand τ i , but there is a dependence only on their difference t−τ i or on the temperatures θ(t) and θ(τ i ):
and φ0does not depend explicitly on temperature:
Functions φ m are assumed to satisfy the following conditions of normalization and symmetry with respect to any permutations of the first marguments:
where θ0=θ(0), θ n =θ(τ n ), \({y}_{n} = t - {\tau }_{n}\), and 1≤n, l≤m.
Simultaneous invariants J γ (s)can always be chosen to satisfy the normalization conditions
For the model with difference cores, the functional (8.53) has the form
where φ m are determined by formula (8.57).
Theorem 8.6.
The model A n of a viscoelastic continuum with difference cores is stable.
▾Let the first process be of the form \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(\tau )\), 0≤τ≤t, then the corresponding functional ψ(t) has the form (8.61). Since the second process \(\mathop{\widetilde{\mathbf{C}}}\limits^{\mathrm{(n)}}(\tau )\)when τ≤t 0is identically zero, so, due to the normalization conditions (8.59) and (8.60), we have φ m ≡0 when 0≤τ i ≤t 0(i=1,…,m); therefore, the lower limits of the integrals in the expression (8.61) for functional \(\widetilde{\psi }(t)\)can be determined as t 0:
Replacing \(\mathop{\widetilde{\mathbf{C}}}\limits^{\mathrm{(n)}}({\tau }_{i})\)by \(\mathop{\widetilde{\mathbf{C}}}\limits^{\mathrm{(n)}}({\tau }_{i} - {t}_{0})\)when τ i >t 0and then substituting \({\tau }_{i} - {t}_{0} =\widetilde{ {\tau }}_{i}\), we obtain (when t>t 0)
Comparing (8.61) and (8.63), we verify that \(\widetilde{\psi }(t)\)and ψ(t) are connected only by the time-shift t→(t−t 0), because at t=t 0
i.e. the relation (8.56) holds. ▴
On substituting the functional (8.61) into (8.38), we get the general form of constitutive equations for stable continua:
Here the cores φ m are functions in the form (8.57), and ∂φ m ∕∂tis the partial derivative of the function when its first arguments (t−τ1),…,θ(τ m ) vary and the arguments J γ (s)are fixed (i.e. there is no differentiation with respect to J γ (s)). Wehave introduced the following notation for a value of the function φ m+1(8.57) at \({\tau }_{m+1} = t\):
In deriving the expression for the dissipation function we have used the conditions(8.59).
8.1.10 Model A n of a Thermoviscoelastic Continuum
Let us consider the most widely used method to take the dependence of cores φ m (8.57) on temperature θ into account.
For the model A n of a thermoviscoelastic continuum with difference cores, temperature θ appears in simultaneous invariants J γ (s), i.e.
where
The tensor \(\epsilon ^{\circ }\)is called the tensor of heat deformation, and α– the tensor of heat expansion. Both the tensors are symmetric and H-indifferent relative to a considered group \(\mathop{G}\limits^{ {\circ }}_{s}\):
therefore, the functions J γ (s)of \({\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}}_{\theta }\)are also H-indifferent relative to the group \(\mathop{G}\limits^{ {\circ }}_{s}\).
Taking the dependence of constitutive equations upon temperature as the difference between the deformation tensor and the heat deformation tensor (8.66) is called the Duhamel–Neumann model.
In a similar way, the Duhamel–Neumann model describes the dependence of the function φ0on temperature:
Since \(\partial \mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }/\partial \theta = -\alpha \), the derivatives with respect to θ(t) in (8.64) for this model have the form
where ∂ ′∕∂θ means the derivative with respect to the second argument in formula (8.68).
Taking into account that \(\partial {I}_{\gamma }^{(s)}/\partial \mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta } = \partial {I}_{\gamma }^{(s)}/\partial \mathop{\mathbf{C}}\limits^{\mathrm{(n)}}\)and \(\partial {J}_{\gamma }^{(s)}/\partial \mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta } = \partial {J}_{\gamma }^{(s)}/\partial \mathop{\mathbf{C}}\limits^{\mathrm{(n)}}\)and substituting (8.68a) into (8.64), we obtain the expression for the specific entropy
8.1.11 Model A n of a Viscoelastic Continuum with Difference Cores
One can say that this is the model A n of a thermorheologically simple viscoelastic continuum, if the cores φ m (8.57) in constitutive equations (8.53) and (8.64) depend on temperature θ in the functional way through the so-called reduced time:
where
is the reduced time being a functional of the function a θ(θ) called the function of the temperature-time shift.
The functions φ m (8.70) and a θsatisfy the normalization conditions
If the process \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(\tau )\)is considered with respect to the reduced time
then, since dτ i ′=a θ(θ(τ i ))dτ i , the functional (8.53) with the core (8.70) can be written with respect to the reduced time
Substituting the functional (8.73) into (8.38) and using the differentiation rule (8.27), we obtain constitutive equations for a thermorheologically simple viscoelastic continuum
Here we have used that \(\partial /\partial t = {a}_{\theta }(\partial /\partial {t}^{{\prime}})\).
Notice that with the help of (8.38) and (8.74) the dissipation function w ∗can be represented in another equivalent form
which proves to be useful for cyclic loading.
Theorem 8.7.
A thermorheologically simple continuum is stable.
▾The reduced time (8.71) for the shifted process of heating \(\widetilde{\theta }(\tau ) = \theta (\tau - {t}_{0})\)with use of the normalization condition (8.72) can be represented in the form
when t 0<τ<t. The further proof is the same as the one in Theorem8.6(see Exercise8.1.1). ▴
8.3 Principal, Quadratic and Linear Models of Viscoelastic Continua
8.3.1 Principal Models A n of Viscoelastic Continua
Constitutive equations containing multiple integrals of the type (8.53), (8.61) or (8.73) are very awkward, and their application in practice is considerably difficult. Therefore, special models of viscoelastic materials, in which one may retain a finite number of integrals, are widely used.
For the principal model A n of a thermoviscoelastic continuum with difference cores, the sum (8.61) contains only one integral (m=1), i.e. ψ in this model has the form
Here φ0and φ1are functions of the arguments indicated, and the function φ1is chosen with the negative sign that can always be done by simple renaming the functions.
Constitutive equations for the continuum considered have the form (8.64), where mshould be assumed to be equal to 1:
Here we have denoted the partial derivatives of φ0and φ1
and also the partial derivative tensors of J γ (s)with respect to \({\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}}_{\theta }(t)\):
The simultaneous invariants \({J}_{\gamma }^{(s)}({\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}}_{\theta }(t),{\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}}_{\theta }(\tau ))\)can be chosen so that the first rones form a functional basis of invariants \({I}_{\gamma }^{(s)}({\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}}_{\theta }(t))\)in the same group \(\mathop{G}\limits^{ {\circ }}_{s}\). We will assume below that J γ (s)are ordered in such a way; then the following equations hold:
These equations have been used for deriving the relation (8.78).
According to (8.64) and (8.69), the dissipation function w ∗and the specific entropy η for the principal models A n have the forms
Here ∂∕∂tis the partial derivative of φ1with respect to the first argument, and ∂ ′∕∂θ is the partial derivative of φ0with respect to the second argument.
Simultaneous invariants J γ (s)of two tensors can be written by analogy with the ones for continua of the differential type (see Sect.7.1.4).
8.3.2 Principal Model A n of an Isotropic Thermoviscoelastic Continuum
For the principal model A n of an isotropic thermoviscoelastic continuum with difference cores, the functional basis of simultaneous invariants J γ (I)consists of 9 invariants, which can be chosen as follows (see (5.35)):
The derivative tensors of these invariants are calculated by the formulae (see [12])
Substituting these expressions into (8.77) and collecting terms with the same tensor powers, we obtain constitutive equations for the principal model A n of an isotropic thermoviscoelastic continuum:
Here we have denoted the functionals
Relation (8.85) is formally similar to the corresponding relation (4.325) for an isotropic elastic continuum, but in (8.85) \(\breve{{\varphi }}_{1}\), \(\breve{{\varphi }}_{2}\)and \(\breve{{\varphi }}_{3}\)are no longer functions of invariants of the tensor \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}\); they are functionals in the form (8.86).
8.3.3 Principal Model A n of a Transversely Isotropic Thermoviscoelastic Continuum
For the principal model A n of a transversely isotropic (relative to the group T 3) thermoviscoelastic continuum with difference cores, the functional basis of simultaneous invariants J γ (3)consists of 11 invariants, which can be chosen as follows (see (5.36)):
Here the invariants I γ (3)are determined by formulae (4.300). The partial derivatives J γC (3)of these invariants have the forms (see [12])
Substituting these expressions into (8.77) and rearranging the summands, we obtain constitutive equations for the principal model A n of a transversely isotropic thermoviscoelastic continuum:
Here we have denoted the functionals
8.3.4 Principal Model A n of an Orthotropic Thermoviscoelastic Continuum
For the principal model A n of an orthotropic thermoviscoelastic continuum with difference cores, the functional basis of simultaneous invariants consists of 12 invariants, which can be chosen as follows (see (5.37)):
This set should be complemented by two more invariants (being dependent) in order to obtain relations symmetric with respect to the vectors \(\widehat{{\mathbf{c}}}_{\alpha }^{2}\):
The partial derivative tensors of these invariants have the forms (see [12])
where the tensor 6 O m is determined by formula (4.319).
Substituting these expressions into (8.77) and grouping like terms, we obtain constitutive equations for the principal model A n of an orthotropic thermoviscoelastic continuum:
Here we have denoted the functionals
8.3.5 Quadratic Models A n of Thermoviscoelastic Continua
For the quadratic model A n of a thermoviscoelastic continuum with difference cores, we retain two integrals in the sum (8.61), i.e. m=1,2. A form of constitutive equations for specific symmetry groups \(\mathop{G}\limits^{ {\circ }}_{s}\)becomes considerably more complicated, because there appear double integrals and we need to consider simultaneous invariants J γ (s)of three tensors. Therefore, one usually considers the particular case of the quadratic model when m=1 and 2, but simultaneous invariants J γ (s)of only two tensors appear there just as in the principal model:
Here φ0, φ1and φ2are functions of the arguments indicated. Since the core φ2in this model is independent of \({\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}}_{\theta }(t)\), so \(\partial {\varphi }_{2}/\partial {\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}}_{\theta }(t) \equiv 0\), and constitutive equations prove to be coincident with (8.78); and hence they coincide with (8.85), (8.89) and (8.94) too. The distinction between the principal and quadratic models consists only in the forms of the functional ψ and the dissipation function w ∗. Such a situation is typical for viscoelastic continua when distinct functionals of the free energy ψ correspond to the same relations between the tensors of stresses and deformations. Comparing the principal model (8.77) with the quadratic one (8.96), we can also notice that the principal model has only one core φ1appearing also in relation (8.78), and the quadratic model has two cores φ1and φ2, one of which is not included in relation (8.78) between stresses and deformations. Thus, for the principal model we can restore the functional of the free energy ψ by Eqs.(8.78) up to the entropy term ∂φ0∕∂θ and the constant φ0(0,θ0)=ψ0.
Models of viscoelastic continua having such a property are called mechanically determinate.
The quadratic model is not mechanically determinate: due to the presence of the core φ2we cannot restore the form of ψ by relations (8.78) between stresses and deformations. Nevertheless, this model is also used in practice due to its quadratic structure being typical for thermodynamic potentials.
8.3.6 Linear Models A n of Viscoelastic Continua
The quadratic model A n of a thermoviscoelastic continuum with difference cores (8.96), where the functions φ0, φ1and φ2depend linearly upon the quadratic invariants J γ (s)and quadratically upon the linear invariants J γ (s), is called linear(cubic invariants do not occur in this model):
Here l γβ, l γγare constants; q γβ(t−τ), q γγ(t−τ) are one-instant cores (they are functions of one argument, being symmetric in γ and β); \({p}_{\gamma \beta }(t - {\tau }_{1},t - {\tau }_{2})\)and \({p}_{\gamma \gamma }(t - {\tau }_{1},t - {\tau }_{2})\)are two-instant cores (they are functions of two variables, being symmetric in γ and β, and also in t−τ1and t−τ2). We have introduced the notation
where r 1is the number of linear invariants \({I}_{\gamma }^{(s)}(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }(t))\)in group \(\mathop{G}\limits^{ {\circ }}_{s}\)(r 1≤r), and (r 2−r 1) is the number of the quadratic simultaneous invariants J γ (s)(τ1, τ2) in this group, where r 2≤z.
Since not all simultaneous invariants contained in the full basis \({J}_{\gamma }(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }({\tau }_{1}),\;\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }({\tau }_{2}))\)appear in the expression for the functional ψ in linear models, it is convenient to renumber these invariants in comparison with the bases (8.83), (8.87), and (8.91) by enumerating first the linear invariants \({I}_{\gamma }(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }(t))\)and then the quadratic simultaneous invariants \({J}_{\gamma }(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }({\tau }_{1}),\;\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }({\tau }_{2}))\).
The principal linear modelA n of viscoelastic continua can be obtained by applying similar relationships to the functions φ0and φ1of the principal model (8.77) when φ2=0.
The functions φ0γand φ1γ(8.79) for the principal linear model have the forms
As noted above, constitutive equations (8.77) for both the models coincide; and, for the linear models, they have the form
where we have denoted the linear functionals
Here we have taken into account that all the linear invariants have the form \({I}_{\gamma }^{(s)} =\mathop{ \mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta } \cdot \cdot \ {\mathbf{O}}_{\gamma }^{(s)}\), where O γ (s)are producing tensors of the group, and for the quadratic invariants,
(For isotropic continua, in order to satisfy this condition, as invariants I γ (I)one should choose the invariants I 1(C θ) and I 1(C θ 2).) Notice that when the cores q γβ(t) and q γγ(t) in (8.101) are absent, then these relations exactly coincide with relations (4.322a) of linear models A n for ideal continua if in the last ones we assume that \(\bar{{m}}_{\gamma } = 0\).
For principal linear models A n of viscoelastic continua, the dissipation function w ∗(8.82) has the form
and for linear models A n of viscoelastic continua, terms with two-instant cores should be added to the expression (8.103a):
For linear models A n , the specific entropy η, according to (8.69) and (8.82), has the form
where ψ0(θ) is a function only of the temperature. The dissipation function with the help of formula (8.75) can be represented in the form
8.3.7 Representation of Linear Models A n in the Boltzmann Form
For linear models A n of viscoelastic continua, representations of the free energy ψ in the form (8.96), (8.97), the constitutive equations in the form (8.100), (8.101) and the dissipation function w ∗in the form (8.103b) are called the model A n in the Volterra form(the name is connected to the fact that integral expressions of the form (8.101) occurring in this representation were considered for the first time by Volterra in 1909).
Let us give another equivalent representation for these models. Introduce new two-instant cores \(\bar{{\psi }}_{\gamma \beta }(y,z)\)and \(\bar{{\psi }}_{\gamma \gamma }(y,z)\)satisfying the differential equations
Then the following theorem holds.
Theorem 8.8.
Let the two-instant cores \(\bar{{\psi }}_{\gamma \beta }(y,z)\) and \(\bar{{\psi }}_{\gamma \gamma }(y,z)\)
-
1.
Be symmetric functions of their arguments:
$$\bar{{\psi }}_{\gamma \beta }(y,z) =\bar{ {\psi }}_{\gamma \beta }(z,y),\ \ \ \ \bar{{\psi }}_{\gamma \gamma }(y,z) =\bar{ {\psi }}_{\gamma \gamma }(z,y), $$(8.105a) -
2.
Be two times continuously differentiable functions of their arguments within the interval (0,t),
-
3.
Satisfy the conditions(8.105),
then we can pass from the representation of linear model A n in the Volterra form (Eqs.(8.96), (8.97), (8.100), (8.101), and(8.103b)) to an equivalent representation of the model A n in the Boltzmann form
Here we have introduced the notation
▾To prove the theorem, it is sufficient to transform the integrals in (8.106)–(8.108) as follows:
Here we have taken into account that I γ (s)(0)=0 and changed the variables
In a similar way, we can transform the integrals of \(\bar{{\psi }}_{\gamma \gamma }\)with taking into account that \({J}_{\gamma }^{(s)}(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }({\tau }_{1}),\;\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }({\tau }_{2}))\)is a linear function with respect to each tensor argument; therefore, the following relations hold:
hence
On substituting (8.110) and (8.111) into (8.106), we actually obtain the expressions (8.96) and (8.97) for ψ.
The representations (8.107) and (8.108) can be proved in a similar way (see Exercise8.2.1). ▴
Remark .
If we consider constitutive equations in the Volterra form (8.96), (8.97), and (8.100) and pass to the limit at t→0, then all the integral summands containing the cores q γβand q γγvanish. As a result, we get instantly elastic relationswhich exactly coincide with the corresponding equations (4.322a), (4.322a), and (4.322a) of models A n of elastic continua:
In order to obtain these relations from the Boltzmann form (8.106), (8.107), one should represent the deformation tensors in the form \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }(\tau )\,=\,\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }(0)h(\tau )\), where h(τ) is the Heaviside function. Then we find that I γ (s)(τ)=I γ (s)(0)h(τ) and J γ (s)(τ1,τ2)=I γ (s)(0)h(τ1)h(τ2). Substituting these expressions into (8.106) and (8.107) and using (8.105), we obtain the desired relations (8.112) as t→0+.
8.3.8 Mechanically Determinate Linear Models A n of Viscoelastic Continua
As noted in Sect.8.2.5, quadratic models A n , including the linear models (8.97), are not mechanically determinate due to the presence of the two-instant cores \({p}_{\gamma \beta }(t - {\tau }_{1},t - {\tau }_{2})\)and \({p}_{\gamma \gamma }(t - {\tau }_{1},t - {\tau }_{2})\). However, the models may become mechanically determinate after introduction of the additional assumption on a form of the two-instant cores; we assume that they depend on the sum of their arguments:
In this case, the cores \(\bar{{\psi }}_{\gamma \beta }\)and \(\bar{{\psi }}_{\gamma \gamma }\)become one-instant, and with the help of formula (8.109a) they can be uniquely expressed in terms of the cores r γβ(y) and r γγ(y) included in constitutive equations (8.107):
The functional (8.106) of the free energy takes the form
The dissipation function w ∗(8.108) in this model is also determined completely by the functional (8.107) of the constitutive equations:
The cores r γβ(y) and r γγ(y), according to (8.114) and (8.105), are connected to the cores q γβ(y) and q γγ(y) by the relations
The cores q γβ(y) and q γγ(y) are called the relaxation cores, and the cores r γβ(y) and r γγ(y) are called the relaxation functions.
8.3.9 Linear Models A n for Isotropic Viscoelastic Continua
Let us derive now constitutive equations for linear models A n of viscoelastic continua in the Volterra (8.100) and Boltzmann (8.107) forms for different symmetry groups \(\mathop{G}\limits^{ {\circ }}_{s}\).
For linear models A n of viscoelastic isotropic continua, the invariants (8.98) and the derivative tensors I γC (I)(8.102) have the forms
Then constitutive equations (8.100), (8.106), and (8.107) become
Here we have denoted the linear functionals
Thus, for an isotropic continuum, there are two independent constants l 1,l 2and two cores q γ(t−τ) connected to the cores r γ(y) by the relations (8.117)
Introducing the fourth-order tensor functional similar to the tensor \({}^{4}\mathop{\mathbf{M}}\limits^{\circ }\)(4.322a) for elastic continua:
we can represent constitutive equations (8.120) in the operator form
which is analogous to relations (4.322a) for semilinear isotropic elastic continua.
8.3.10 Linear Models A n of Transversely Isotropic Viscoelastic Continua
For linear models A n of viscoelastic transversely isotropic continua, from (8.87) and (8.88) we obtain
where 4 O 3is determined by formula (8.88). Relations (8.100) take the form
Here the linear operators \(\breve{{l}}_{\gamma \beta }{I}_{\beta }^{(3)}\)and \(\breve{{l}}_{\gamma \gamma }\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}\)are determined by expressions (8.101), which can be represented in the Boltzmann form (8.107)
For a transversely isotropic continuum, there are five independent constants l 11, l 22, l 12, l 33, l 44and five cores q γβ(t−τ) or r γβ(t−τ).
Introducing the tensor functional being analogous to the tensor of elastic moduli (4.322a):
we can also represent constitutive equations (8.126) in the form (8.124).
8.3.11 Linear Models A n of Orthotropic Viscoelastic Continua
For linear models A n of viscoelastic orthotropic continua, due to (8.91)–(8.93), the invariants (8.98) and the derivative tensors (8.102) take the forms
and the relations (8.100) can be written as follows:
Thus, there are nine independent constants l 11, l 22, l 33, l 12, l 13, l 23, l 44, l 55, l 66and nine cores q γβ(t−τ) or r γβ(t−τ).
Introducing the tensor functional being analogous to (4.322a):
we can represent constitutive equations (8.130) in the operator form (8.124).
8.3.12 The Tensor of Relaxation Functions
According to the operator form (8.124) of constitutive equations for linear models A n of viscoelastic continua, we can introduce the fourth-order tensor 4 R(t), called the tensor of relaxation functions, by the same formulae as for the elastic moduli tensor \({}^{4}\mathop{\mathbf{M}}\limits^{\circ }\)in linear models A n of elastic continua (see Sect.4.8.7) if in the corresponding formulae the elastic constants l αβare replaced by the relaxation functions r αβ(t).
For an isotropic continuum, this tensor has the form
for a transversely isotropic continuum
and for an orthotropic continuum
Then the operator relations (8.124) can be represented as follows:
where
is a tensor linear functional.
For instantaneous loading as t→0+, these relations coincide with (8.112) and with the corresponding relations (4.330a) of models A n for a linear-elastic continuum, because
The tensor
is called the tensor of relaxation cores. This tensor for different groups \(\mathop{G}\limits^{ {\circ }}_{s}\)has the same form as the tensor 4 R(t) in (8.132)–(8.134) if in these formulae the substitution r αβ(t)→q αβ(t) has been made.
According to (8.137) and (8.138), the constitutive equations (8.135) can be written in the Volterra form
that is equivalent to the form (8.100). The operator (8.115) of the free energy ψ for the mechanically determinate model A n with the help of the tensor of relaxation functions can be represented in the form (see Exercise8.2.3)
and the dissipation function (8.116) – in the form
Formula (8.141) gives the following theorem.
Theorem 8.9.
For mechanically determinate linear models A n of viscoelastic continua, the tensors of relaxation cores 4 K(t) are
-
1.
Nonnegative-definite:
$$\mathbf{h} \cdot {\cdot \ }^{4}\mathbf{K}(t) \cdot \cdot \ \mathbf{h} \geq 0,\ \ \ \ \ \forall \mathbf{h}\neq 0,\ \forall t \geq 0,$$(8.142) -
2.
Symmetric in the following combinations of indices:
$${ }^{4}\mathbf{K}(t) {= }^{4}{\mathbf{K}}^{(1243)}(t) {= }^{4}{\mathbf{K}}^{(2134)}(t) {= }^{4}{\mathbf{K}}^{(3412)}(t)\ \ \ \ \forall t \geq 0,$$(8.143)
(i.e. these tensors have the same symmetry as the elastic moduli tensor \({}^{4}\mathop{\mathbf{M}}\limits^{\circ }\) for linear models A n of elastic continua).
▾The dissipation function is always nonnegative (w ∗≥0) and vanishes for viscoelastic continua only if \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }(\tau ) \equiv 0\). Then, choosing the process of deforming in the form of a step-function:
where h(τ) is the Heaviside function, and his a symmetric non-zero constant tensor, we obtain
Substituting (8.144a) into (8.141) and using the property (8.16) of the δ-function and formula (8.138), we find that
i.e. the tensor 4 K(t) is nonnegative-definite.
The existence of the quadratic form (8.145) and the symmetry of the tensor hlead to symmetry of 4 K(t) in the first–second and third–fourth indices and also in pairs of the indices, i.e. the relations (8.143) actually hold.▴
As follows from (8.142) and (8.138), the tensor of relaxation functions 4 R(t) generates the monotonically non-increasing form
And if the elastic moduli tensor \(\mathop{\mathbf{M}}\limits^{\circ } {= }^{4}\mathbf{R}(0)\)has the symmetry (8.143), then from (8.143) it follows that the tensor 4 R(t) has the same symmetry ∀t≥0:
8.3.13 Spectral Representation of Linear Models A n of Viscoelastic Continua
Let us apply now the theory of spectral decompositions of symmetric second-order tensors (see [12]). According to this theory, for any symmetric tensors, in particular for \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }(\tau )\)and \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}(\tau )\), we can introduce their spectral decompositionsrelative to a symmetry group \(\mathop{G}\limits^{ {\circ }}_{s}\)chosen:
Here P α (T)and P α (C)(\(\alpha = 1,\ldots,\bar{n}\)) are the orthoprojectorsof the tensors \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\)and \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }\)(their number is denoted by \(\bar{n}\)), which are symmetric second-order tensors having the following properties: a) they are mutually orthogonal, b) they are linear, c) they are indifferent relative to the group \(\mathop{G}\limits^{ {\circ }}_{s}\):
Here the fourth-order tensors 4Γαare indifferent relative to the group \(\mathop{G}\limits^{ {\circ }}_{s}\), independent of \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\)and formed only by producing tensors of the group (see Sect.4.8.3). Among the tensors \({}^{4}{\Gamma }_{\alpha }\ (\alpha = 1,\ldots,\bar{n})\), there are mreducibletensors, i.e. obtained with the help of the tensor product of the second-order tensor a αbeing symmetric and indifferent relative to the same group \(\mathop{G}\limits^{ {\circ }}_{s}\):
Expressions for 4Γαand a αhave the following forms (see [12]):
for the isotropy group \(\mathop{G}\limits^{ {\circ }}_{s} = I\)
for the transverse isotropy group \(\mathop{G}\limits^{ {\circ }}_{s} = {T}_{3}\)
for the orthotropy group \(\mathop{G}\limits^{ {\circ }}_{s} = O\)
With the help of the orthoprojectors P α (T), introduce spectral invariantsof the tensor \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\)and denote them by \({Y }_{\alpha }(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}})\). For those P α (T), for which 4Γαis a reducible tensor, the invariant Y αis introduced as follows:
and called the spectral linear invariant. For the remaining P α (T)(\(\alpha = m + 1,\ldots,\bar{n}\)), these invariants are introduced by the formula
and called the spectral quadratic invariants. From (8.150) and (8.152) it followsthat
Notice that for the linear invariants (8.156a), formula (8.156b) also holds.
Due to (8.157), the spectral decomposition of the symmetric second-order tensor \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\)(8.148) can be represented in the form
For any fourth-order tensor indifferent relative to a group \(\mathop{G}\limits^{ {\circ }}_{s}\), including the tensor of relaxation functions 4 R(t), we can also introduce the spectral representation
where R αβ(t) and R αα(t) are the spectral relaxation functions expressed uniquely in terms of r αβ(t) and r αα(t) (see Exercise8.2.6).
With the help of the spectral decompositions (8.148) and (8.159) the constitutive equations (8.135) can be represented as relations between the spectral linear invariants and the orthoprojectors (see Exercise8.2.9)
where
Relations (8.160) are called the spectral representation for linear models A n of viscoelastic continua. If we introduce the spectral decomposition (8.148) also for the tensor h:
then the inequality (8.146) with use of (8.159) takes the form
Values of the spectral linear invariants Y α(h) can be assumed to be zero; then, since the spectral invariants are independent, from (8.162) we obtain the condition of monotone non-increasing the spectral relaxation functions:
With the help of the spectral relaxation functions one can formulate special cases of linear models A n for viscoelastic continua. So for the simplest linearmodel A n of an isotropic viscoelastic continuum, one of the two spectral relaxation cores is assumed to be constant:
According to the results of Exercise8.2.6and formula (8.122), this condition can be rewritten as the relation between the cores q 1(t) and q 2(t)
Constitutive equations (8.160) in this case take the form
Here we have denoted the orthoprojectors of the tensors relative to the full orthogonal group I, called the deviators
Equations (8.166) and (8.167) can be rewritten in the form
8.3.14 Exponential Relaxation Functions and Differential Form of Constitutive Equations
To solve problems analytically or numerically it is convenient to have an analytical form of the spectral relaxation functions R αβ(t). As established in the preceding section, the property of monotone non-increasing the spectral functions R αα(t) is a consequence of the dissipation inequality w ∗≥0. Functions having such a property can be approximated by the sum of exponents:
where B αβ (γ)and ταβ (γ)are the constants called the spectrum of relaxation valuesand the spectrum of relaxation times, respectively, and R αβ ∞is the limiting value of the relaxation functions:
which may be zero: R αβ ∞=0.
The constants R αβ ∞and B αβ (γ)satisfy the normalization condition at t=0:
where \(\mathop{C}\limits^{ {\circ }}_{\alpha \beta } = {R}_{\alpha \beta }(0)\)are the spectral (two-index) elastic moduli under instantaneous loading.
There are other methods of analytical approximation to the relaxation functions, however exponential functions have certain merits: (1) choosing a sufficiently large number Nof exponents in (8.169), we can approximate practically any function R αβ(t), (2) constitutive equations (8.160) and (8.161) with exponential cores admit their inversion (see Sect.8.2.15), where cores of the inverse functionals prove to be exponential as well, and (3) the cores (8.169) allow us to represent constitutive equations (8.160), (8.161) or (8.135), (8.136) in the differential form.
Indeed, performing the subsequent substitutions (8.169)→(8.159)→(8.138), we find the expression for the tensor of relaxation cores:
Introduce the second-order tensors
Differentiating W αβ (γ)with respect to tand eliminating the integral, we obtain that the tensors W αβ (γ)satisfy the first-order differential equations
Substituting (8.172) into (8.139) and using the expressions (8.173) and (8.174), we obtain the following representation of constitutive equations:
Here the spectrum ofviscous stressesis denoted by W (γ)being second-order tensors of the form
Thus, with the help of the exponential cores (8.169) the constitutive equations for the mechanically determinate model A n of viscoelastic continua (8.139) can be represented in the differential form(8.175)–(8.177). A result of the passage from integral relations to differential ones is the appearance of additional unknowns, namely the tensors W αβ (γ), for which Eqs.(8.175) have been stated.
In computations the differential form (8.175)–(8.177), as a rule, proves to be more convenient than the integral form (8.139).
Substitution of the expressions (8.172) and (8.138) into (8.141) yields
Here we have denoted the linear spectral invariants of the tensor \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }(\tau )\)(see (8.156a)):
Substituting the exponential cores (8.173) into (8.178) and modifying the double integrals
with use of the notation (8.174) we can represent (8.178) in the form
8.3.15 Inversion of Constitutive Equations for Linear Models of Viscoelastic Continua
In viscoelasticity theory one often uses constitutive equations inverse to (8.135) or (8.139). To derive the equations we should consider relationship (8.139) as a linear integral Volterra’s equation of the second kind relative to the process \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }(\tau )\),0≤τ≤t. The core K(t) of this equation is assumed to be continuously differentiable and to satisfy the conditions (8.142) and (8.143). As known from the theory of integral equations, Eq.(8.139) with such core always has a solution, and this solution is written in the same form as the initial equation:
Here 4Π is the tensor of elastic pliabilities, which is inverse of the tensor of elastic moduli \({}^{4}\mathop{\mathbf{M}}\limits^{\circ }\):
and 4 N(t) is the tensor of creep coreshaving the same form as the tensor 4 K(t) (8.172):
The functions N αβ(t) and N αα(t) are called the spectral creep cores.
To find a relation between the cores 4 N(t) and 4 K(t), we should substitute (8.139) into (8.182); as a result, we obtain the identity
Changing the integration order in the double integral:
(see Fig.8.3, where the integration domain is a shaded triangle), from (8.185) we obtain
This equation holds for any \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }(\tau )\)if and only if the expression in braces vanishes.
The integration domain in the double integral
The substitution of variables \(x = t - \tau \)in braces gives
Then the substitution of variables \(u = y + x - t\)under the integral sign, where (t−x≤y≤t) and (0≤u≤x), yields
Reverting to the initial notation of arguments x→tand u→τ, we obtain the integral relation between the tensor of relaxation cores 4 K(t) and the tensor of creep cores 4 N(t):
If the core 4 K(t) is known, then the relation (8.187) is a linear integral Volterra’s equation of the second kind for evaluation of the core 4 N(t), and vice versa.
On substituting the spectral decompositions (8.172) and (8.184) of the tensors 4 K(t), 4 N(t) and analogous decompositions of the tensors 4Π and \({}^{4}\mathop{\mathbf{M}}\limits^{\circ }\)into (8.187), due to mutual orthogonality of the tensors 4Γα, a α⊗a β(see [12]), we obtain
– the system of scalar integral equations for determining the cores N αβ(t) in terms of the cores K αβ(t) or vice versa.
By analogy with the tensor of relaxation functions 4 R(t), introduce the tensor of creep functions 4Π(t) satisfying the equation
Then the inverse constitutive equation (8.182) can be written in the Boltzmann form
The tensors of creep cores and functions have the same properties of symmetry (8.143) and (8.147) as the tensors 4 Kand 4 R(see Exercise8.2.11):
For the tensor of creep functions 4Π(t) as well as for 4 R(t), we can introduce a spectral representation by formula (8.159):
where Παα(t) and Παβ(t) are the spectral creep functions.
Theorem 8.10.
If the spectral relaxation cores K αβ (t) are exponential, i.e. have the form(8.173), then the spectral creep cores N αβ (t) are exponential too:
and vice versa.
The constants A αβ (γ)and t αβ (γ)are called the spectra of creep valuesand creep times, respectively. They, in general, are not coincident with B αβ (γ)and ταβ (γ), respectively; however the number Nin (8.192) and (8.173) is the same.
▾Show that if the cores K αβ(t) have the form (8.173), then the cores (8.193) are a solution of the integral equation (8.188). Substitution of (8.173) and (8.193) into (8.188) yields
Calculating the integral in (8.194)
and equating coefficients in (8.194) at the same exponents, we get the system
for determining the constants A αβ (γ)and t αβ (γ)in terms of the constants B βε (γ)and τβε (γ)(the constants Παβcan always be determined in terms of C αβand are assumed to be known).
At those values of B βε (γ), τβε (γ), Παβ, at which there exists a solution of the system (8.196), the exponential representation of the creep cores (8.193) exists too.▴
Formulae (8.196) give the method of calculation of the constants A αβ (γ)and t αβ (γ)in terms of B βε (γ), τβε (γ)and vice versa. From (8.188b) we can obtain simpler formulae for determining the constants A αα (γ)and t αα (γ)(\(\alpha = m + 1,\ldots,n\)):
On substituting (8.184) and (8.193) into (8.189), we find an expression for the spectral creep functions in the case of exponential cores:
where
8.4 Exercises for 8.2
8.2.1.
Using the rule of differentiation of an integral with a varying upper limit (see formulae (8.15b) and (8.25)) and calculating the derivative of the functional (8.106) with respect to t, show that PTI (4.123) actually yields formulae (8.107) and (8.108) for \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\)and w ∗.
8.2.2.
Using the definition (8.70), show that for linear models A n of thermorheologically simple viscoelastic media, relations (8.106)–(8.108) have the forms
where t ′, τ1 ′and τ2 ′are determined by (8.71).
Taking
into account, show that these relations can be represented as functions of the reduced time; in particular
8.2.3.
Using representations (8.132)–(8.134) for the tensor of relaxation functions 4 R(t), show that representations (8.140) and (8.115) for ψ are equivalent.
8.2.4.
Substituting formulae (8.132)–(8.134) for the tensor of relaxation functions 4 R(t) into the expression (8.141) for w ∗, show that formulae (8.141) exactly coincide with (8.116).
8.2.5.
Show that the condition (8.146) causes monotone non-increasing the relaxation functions
where \(\widehat{{R}}^{ijkl}(t)\)are components of the tensor 4 R(t) with respect to the basis \(\widehat{{\mathbf{c}}}_{i}\).
8.2.6.
Using representations (8.132)–(8.134) for the tensor of relaxation functions 4 R(t) and its spectral representation (8.159), and also formulae (8.152)–(8.155) for the tensors a αand 4Γα, show that the functions r αβ(t) and R αβ(t) are connected by the following relations for isotropic continua
and for transversely isotropic continua
8.2.7.
Show that for mechanically determinate linear models A n of thermorheologically simple continua, the constitutive equations obtained in Exercise8.2.2can be written in the forms (8.136), (8.140), and (8.141)
Show that the constitutive equation (8.139) for this model becomes
8.2.8.
Show that for the linear models A n of thermorheologically simple media with the exponential cores, the constitutive equations from Exercise8.2.7can be written in the form (8.175)–(8.177), (8.181)
8.2.9.
Prove that the spectral representations (8.148) and (8.159) lead to the spectral form (8.150) of constitutive equations for the linear models A n (8.135).
8.2.10.
Show that in hydrostatic compression when the Cauchy stress tensor and the deformation gradient are spherical:
a viscoelastic continuum, according to the simplest model (8.166), behaves as a purely elastic one:
i.e. there are no viscoelastic deformations in hydrostatic compression for this model. Many solids actually have such properties up to high pressures p; therefore, the simplest model (8.166) is widely used in continuum mechanics.
8.2.11.
Show that the tensors of creep cores and creep functions 4 N(t) and 4Π(t) have the same properties of symmetry (8.143), (8.147) as the tensors of relaxation cores and relaxation functions 4 K(t) and 4 R(t), and vice versa.
8.5 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids
8.5.1 Models A n of Incompressible Viscoelastic Continua
For incompressible viscoelastic continua, as usual, there is an additional condition of incompressibility, which can be written in one of the forms (4.322a)–(4.322a), and the principal thermodynamic identity takes the form (4.322a) for models A n . For models B n , its form is analogous. Substituting the functional (8.35) into (4.322a) and using the rule (8.27), we get constitutive equations for incompressible viscoelastic continua
Sinceforincompressiblecontinuathe number rof independentinvariants\({I}_{\gamma }^{(s)}(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}(t))\)is smaller by 1 than that for compressible materials, in each of the representations (8.53), (8.61), (8.73), (8.77), and (8.96) of the free energy functional the subscript γ of the function \({\varphi }_{0}({I}_{\gamma }^{(s)}(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}),\theta )\)takes on r−1 values.
In the consistent way, the number zof simultaneous invariants J γ (s)occurring in arguments of the cores φ m decreases too.
8.5.2 Principal Models A n of Incompressible Isotropic Viscoelastic Continua
For the principal models A n of incompressible isotropic viscoelastic continua, the functional ψ (8.77) depends only on five simultaneous invariants:
and the constitutive equations (8.200) become
where
and φ0γand φ1γare determined by (8.79).
For the principal linear models A n of incompressible isotropic viscoelastic continua, the functional ψ has the form (compare this with the potential for elastic incompressible materials (4.322a))
where l 1, l 2and \(\bar{m}\)are the constants, and q 1(t−τ) and q 2(t−τ) are the cores.
The corresponding constitutive equations (8.202) have the form (compare with (4.322a) for elastic continua)
Here we have denoted two linear functionals
The constants \(\bar{{\psi }}^{0}\)and p 0=p(0) are chosen from the conditions (4.322a) and (4.322a) according to formulae (4.322a), just as for elastic materials:
where p eis the constant appearing in the initial values of the stress tensors in the natural configuration \(\mathcal{K}^{e}\): \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}} = -{p}^{e}\mathbf{E}\).
8.5.3 Linear Models A n of Incompressible Isotropic Viscoelastic Continua
For linear models A n of incompressible isotropic viscoelastic continua, obtained from quadratic mechanically determinate models A n (see Sects.8.2.6–8.2.9), constitutive equations have the form similar to formulae (8.120) for compressible materials, but the functional ψ should involve the summands linear in the invariant I 1:
where \(\bar{{\psi }}^{0}\)and \(\bar{m}\)are the constants, and r 1(y) and r 2(y) are the relaxation functions.
According to formulae (8.110) and (8.111) and also Theorem8.8, the functional ψ can be written in the Volterra form
By analogy with compressible media (see Sect.8.2.13), we can consider the simplest model A n of isotropic incompressible continua, in which the creep functions q 1(y) and q 2(y) are connected by the relation (8.165):
i.e. this model involves only one core q(y).
Integrating Eq.(8.211) with respect to yand taking the initial condition (8.210) into account, we find the connection between r 1(y) and r 2(y)
Substituting (8.211) into (8.205) and (8.206) and grouping like terms, we obtain the following constitutive equation (compare with (8.168)):
where
is the deviator of the tensor \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta }\)(see (8.167)).
If q(t)≡0, then Eqs.(8.212) coincide with relations (4.322a) for isotropic incompressible elastic continua.
8.5.4 Models B n of Viscoelastic Continua
In models B n of viscoelastic continua, the free energy ψ is a functional in the form
and corresponding constitutive equations can be obtained with the help of the rule (8.27) of differentiation of a functional with respect to time; they have the form
All further constructions with functionals \(\psi ^{t}_{\tau = 0}\)and \(\mathcal{F}^{t}_{\tau = 0}\)can be performed for models B n as well.
Special models B n of viscoelastic continua can be obtained immediately from models A n , in which one should make the substitution \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}} =\mathop{ \mathbf{G}}\limits^{\mathrm{(n)}} - (1/(n -\mathrm{III}))\mathbf{E}\).
In particular, for linear mechanically determinate models B n of isotropic incompressible continua, according to the relation \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}{}^{\bullet } =\mathop{ \mathbf{G}}\limits^{{\mathrm{(n)}}}{}^{\bullet }\), (8.205) and (8.208), we obtain the constitutive equations
For the simplest models B n , the assumption (8.211a) on the functions r γ(t) yields
Since the function w ∗is nonnegative, we find that
i.e. the relaxation core q(y) is always nonnegative.
Passing to the limit as t→0, in the natural configuration \(\mathcal{K}^{e}\), where \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}(0) = -{p}^{e}\mathbf{E}\), \(\mathop{\mathbf{G}}\limits^{\mathrm{(n)}}(0) = \mathbf{E}/(n -\mathrm{III})\)and p=p 0(see Sect.4.8.6), from (8.219) we obtain the following relations between the constants ψ0, m, l 1, l 2and p 0(see (4.322a)):
Notice that relations (8.216) and (8.218) are entirely equivalent to Eqs.(8.208), (8.205); and (8.219)– to Eqs.(8.212) (the constants l 1and l 2in these equations are distinct). We can obtain new models of the class B n by taking additional assumptions on the constants m, l 1and l 2. For example, if we assume just as in the corresponding elastic models B n (see (4.322a)) that
where μ and β are two new independent constants, then from (8.219) we obtain the constitutive equations
which are not equivalent to the corresponding relations (8.212) of models A n .
Due to (8.223), relations (8.222) become
8.5.5 Models A n and B n of Viscoelastic Fluids
Equations (8.214) and (8.215) as well as (8.35) and (8.38) hold true for both solid and fluid viscoelastic continua. However, for fluids, according to the principle of material symmetry, relations (8.39) (and the analogous relations for models B n ) must be satisfied:
for any H-transformations included in the unimodular group U.
Theorem 8.11.
For models A n and B n (n =I,II,IV,V ) of viscoelastic fluids, the constitutive equations(8.214), (8.215) and(8.35), (8.38), satisfying the principle of material symmetry and being continuous functionals in space H t , can be written as follows:
▾Since ψ is assumed to be a continuous functional, we can apply Theorem8.4and expand ψ in terms of n-fold scalar functionals:
whose cores are scalar functions of mtensor arguments:
On substituting the representation (8.228) into (8.226), we get that functions \(\widetilde{{\psi }}_{m}\)must satisfy the relation
where \(\mathop{\mathbf{G}}\limits^{{\mathrm{(n)}}}_{i} \equiv \mathop{\mathbf{G}}\limits^{\mathrm{(n)}}({\tau }_{i})\), i.e. they must be H-indifferent relative to the unimodular groupU.
Applying the same reasoning as we used in proving Theorem4.31, we can show that functions of the third invariant of the tensors \(\mathop{\mathbf{G}}\limits^{{\mathrm{(n)}}}_{i}\)(or, that is the same, of values of the density ρ i =ρ(τ i ) at different times τ i ) are the only functions ensuring that the condition (8.230) is satisfied:
Separating δ-type components from the cores in this expression by analogy with (8.51a), from (8.228) we get the representation (8.227c), where the cores φ m are connected to \(\widetilde{{\psi }}_{m}\)by relations (8.54).
Substituting the functional (8.227c) into (8.215) and differentiating ψ with respect to \(\mathop{\mathbf{G}}\limits^{\mathrm{(n)}}(t)\), from (4.322a)–(4.322a) we actually obtain formulae (8.227a) and (8.227b).
Finally, using the transformations (4.322a) and (4.322a), from (8.227a) we obtain that all the relations (8.227a) for models A n and B n are equivalent to the single relation (8.227d). ▴
Notice that although Eq.(8.227d) for the Cauchy stress tensor is formally the same as the one for an ideal fluid, a viscoelastic fluid is not ideal (it is dissipative), because in this case the pressure pis a functional of the density ρ, and the dissipation function w ∗is not zero due to (8.64):
8.5.6 The Principle of Material Indifference for Models A n and B n of Viscoelastic Continua
Since all the energetic tensors \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\), \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}\)and \(\mathop{\mathbf{G}}\limits^{\mathrm{(n)}}\)are R-invariant in rigid motions, all the constitutive equations given in this section for solids and fluids, and also for incompressible continua, are the same in actual configurations \(\mathcal{K}\)and \({\mathcal{K}}^{{\prime}}\); therefore, the principle of material indifference for models A n and B n of viscoelastic media is satisfied identically.
8.6 Exercises for 8.3
8.3.1.
Using the stepwise loading (8.144) and passing to the limit as t→0+, show that the instantly elastic relations obtained from (8.224) coincide with the constitutive equations (4.322a) of the model B n of an elastic isotropic incompressible continuum.
8.3.2.
Using the method applied in Sect.8.2.14, show that for the simplest model A n of isotropic incompressible viscoelastic continua with the exponential core
the constitutive equation (8.212) takes the form
and for the simplest model B n of isotropic incompressible viscoelastic continua with the same exponential core, the constitutive equation (8.224) takes the form
8.3.3.
Using the results of Exercise8.2.7and Eqs.(8.220) and (8.224), show that for the simplest linear models B n of isotropic incompressible thermorheologically simple media the following constitutive equations hold:
where t ′, τ1 ′and τ2 ′are determined by (8.71).
8.7 Statements of Problems in Viscoelasticity Theory at Large Deformations
8.7.1 Statements of Dynamic Problems in the Spatial Description
Statements of problems of viscoelasticity at large deformations can be obtained formally from the corresponding statements of problems of elasticity theory at large deformations (see Sect.6.1), if in the last ones we replace the generalized constitutive equations of elasticity \(\mathop{\mathbf{T}}\limits^{{\mathrm{(n)}}}_{G} ={ \mathcal{F}}_{G}(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{\theta },\theta )\)by relations of viscoelasticity (for a viscoelastic model chosen from the models considered in Sects.8.1–8.3).
Choosing the most general representations (8.38) and (8.215) for models A n and B n of viscoelasticity and using the method mentioned above and the statement of the dynamic θRUVF-problem of elasticity theory (see Sects.6.1.1 and 6.3.3), we obtain a statement of the dynamicθRUVF-problem of thermoviscoelasticity in the spatial description. This statement consists of the equation system in the domain V×(0,t max):
the constitutive equations in the domain \(\bar{V } \times (0,{t}_{\mathrm{max}})\):
which should be complemented by expressions (4.322a) and (4.322ac) for tensors \({}^{4}\mathop{\mathbf{E}}\limits^{{\mathrm{(n)}}}_{G}\)and \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{G}\):
the boundary conditions (5.68)–(6.76), which for the case when there are no phase transformations take the form
and also the initial conditions
On substituting the constitutive equations (8.233) and (8.234) into (8.232), we get the system of 17 scalar equations for 17 unknown scalar functions:
Since the domain V(t) in the spatial description is unknown, the system obtained should be complemented with either relation (5.89) or Eq.(5.85) for the function f(x,t) specifying a shape of the surface Σ(t) bounding the domain V(t):
In the second case the function f(x,t) appears among the unknowns (8.237).
Remark 1.
In viscoelasticity theory, in place of the energy balance equation one often uses the entropy balance equation (3.166a) (the system (8.232) has been written in this way), which contains the dissipation function w ∗explicitly. The entropy balance equation can be represented in the nondivergence form (3.166)
When the model A n of a thermoviscoelastic continuum with difference cores is considered, then η is determined by formula (8.69). If in this formula the derivative ∂ ′φ∕∂θ is assumed to depend only on θ, then after substitution of (8.69) and (8.233a) into (8.239) we obtain the following equation of heat conduction for a viscoelastic continuum in the spatial description:
Here
is the heat capacity at fixed deformations.
Remark 2.
If, for example, we consider the statement of the dynamic θRUVF-problem for mechanically determinate linear models A n of thermorheologically simple viscoelastic continua with exponential cores, then the constitutive equations (8.233) has the form derived in Exercise8.2.8:
In this case the initial conditions (8.236) are complemented by the additional conditions
and the number of unknown functions (8.237) becomes greater due to adding the functions
(here we always have W αβ (γ)≡0 when α≠β and α, β>m).
By analogy with the dynamic θRUV-, θRV- and θU-problems of thermoelasticity (see Sect.6.3.3), we can state the dynamicθRUV-problem of viscoelasticity, in which the deformation gradient Fis eliminated between the unknowns, the θUV-problem of viscoelasticity, where in addition the density ρ is eliminated, and the dynamicθU-problem of viscoelasticity, where only uand θ are unknown.
Remark 3.
Just as the statements of thermoelasticity problems in the spatial description (see Sect.6.3.3), the statements of thermoviscoelasticity problems mentioned above are strongly coupled, because they cannot be split into heat conduction problems and viscoelasticity problems even if we neglect the entropy term of the connection (i.e. the term \((\alpha /\rho ) \cdot \cdot \ \mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\)in the equation of heat conduction (8.240)).
To the six causes of the connection in thermoviscoelasticity problems, mentioned in Remark 4 of Sect.6.3.3, we should add one more factor: the presence of the dissipation function w ∗in the entropy balance equation (8.232c) or in the heat conduction equation (8.240), that is a consequence of non-ideality of viscoelastic continua. In many problems, a contribution of the dissipation function w ∗to the heat conduction equation (8.240) proves to be rather considerable and cannot be neglected in non-isothermal processes.
The effect of growing the temperature in viscoelastic materials without heat supplied to a body from the outside but only due to internal heat release in deforming (caused by the presence of the dissipation function w ∗), is called dissipative heatingof the body (see Sect.8.6).
Let us pay attention to the sixth cause of the connection mentioned in Remark 4 of Sect.6.3.3: for viscoelastic continua the dependence of the constitutive equations (8.233c) on temperature can be split into the three constituents:
-
(1)
Dependence of the heat deformation \(\epsilon ^{\circ }\)(8.66) when the Duhamel–Neumann model is used.
-
(2)
Dependence of the elastic properties on the temperature θ(t).
-
(3)
Dependence of the viscous properties, i.e. the integral part of Eqs.(8.233c), upon the temperature prehistory θt(τ).
As established in experiments, for most viscoelastic continua, the viscous properties more considerably depend on temperature than the elastic ones. Since the dissipation function w ∗depends on just the viscous properties, it also depends explicitly upon the temperature (in the model A n with the exponential cores (8.242) this dependence has the form of function a θ(θ(t))). The temperature dependence w ∗(θ) leads to the intensification of dissipative heating in viscoelastic materials and, under certain conditions, can cause the effect of heat explosion(see Sect.8.6.9).
8.7.2 Statements of Dynamic Problems in the Material Description
Using the statement of the dynamic θUVF-problem of thermoelasticity in the material description (see Sects.6.2.1 and 6.3.4) and replacing the constitutive equations (5.42) by viscoelasticity relations in the forms (8.38) and (8.215), for models A n and B n we obtain a statement of the dynamicθUVF-problem of thermoviscoelasticity in the material description. This statement consists of the equation system
defined in the domain \(\mathop{V }\limits^{ \circ }\times (0,{t}_{\mathrm{max}})\), and the constitutive equations in the same domain \(\mathop{V }\limits^{ \circ }\times (0,{t}_{\mathrm{max}})\):
which are complemented with expressions for the tensors \({}^{4}\mathop{\mathbf{E}}\limits^{{\mathrm{(n)}}}{}^{0}\)and \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{G}\):
the boundary conditions (5.77)–(5.84) taking the forms (when there are no phase transformations)
and the initial conditions
On substituting the constitutive equations (8.246) and (8.247) into (8.245), we obtain the system for 16 unknown scalar functions being components of the following vectors and tensors:
Due to the continuity equation, the density ρ can be eliminated between the unknown functions.
Just as for problems of thermoelasticity in the material description, the problem (8.245)–(8.250) is formulated for a known domain \(\mathop{V }\limits^{\circ }\), that considerably simplifies its solving.
For particular models of viscoelastic continua, formulae (8.246b)–(8.246d) are replaced by appropriate relations derived in Sects.8.1–8.3.
When models A n of viscoelastic continua with the difference cores and the Duhamel–Neumann model (8.66) are considered, the specific entropy η is determined by formula (8.69). Assuming that ∂ ′φ0∕∂θ depends only on temperature, we can rewrite the entropy balance equation of system (8.245) in the form of the heat conduction equation for a viscoelastic medium in the material description:
Here the second entropy term on the right-hand side of the equation, as a rule, can be neglected in comparison with \(\mathop{w}\limits^{ {\circ }}{}^{{_\ast}}\).
Unlike the statement of the thermoelasticity problem in the material description given in Sect.6.3.4, the thermoviscoelasticity problem (8.245)–(8.249) is strongly coupled even if there are no phase transformations, that is caused by the presence of the dissipation function \(\mathop{w}\limits^{ {\circ }}{}^{{_\ast}}\). As noted in Sect.8.4.1, in the general case of non-isothermal processes a contribution of the function \(\mathop{w}\limits^{ {\circ }}{}^{{_\ast}}\)to the heat conduction equation may be rather essential and cannot be neglected.
Using the statements of the dynamic θUV-, TθUVF- and θU-problems of thermoelasticity in the material description (see Sect.6.3.4), we can formulate the corresponding dynamic problems of thermoviscoelasticity. So the statement of the dynamicθU-problem of thermoviscoelasticity in the material descriptionconsists of the equation system (5.58):
in the domain \(\mathop{V }\limits^{ \circ }\times (0,{t}_{\mathrm{max}})\); constitutive equations (8.246); the expressions for tensors \({}^{4}\mathop{\mathbf{E}}\limits^{{\mathrm{(n)}}}{}^{0}\)and \(\mathop{\mathbf{C}}\limits^{{\mathrm{(n)}}}_{G}\)(8.247); the kinematic equation
boundary conditions (8.248) and initial conditions (8.249). The problem is solved for the four scalar functions: components of the displacement vector uand the temperature θ.
8.7.3 Statements of Quasistatic Problems of Viscoelasticity Theory in the Spatial Description
Statements of quasistatic problems of viscoelasticity theory can be obtained formally from the corresponding statements of quasistatic problems of elasticity theory at large deformations (see Sect.6.3.5) by replacing the constitutive equations of elasticity with appropriate relations of viscoelasticity derived in Sects.8.1–8.3. So the statement of the coupled quasistatic problem of thermoviscoelasticity in the spatial descriptionfor linear mechanically determinate models A n of thermorheologically simple continua has the form
Here (8.252) is the system of the equilibrium and heat conduction equations, (8.253) are the constitutive equations, (8.254) is the set of kinematic and energetic equivalence relations, (8.255) are the boundary and initial conditions, where Σσ=Σ1∪Σ2∪Σ3∪Σ4∪Σ7,Σ u =Σ5∪Σ6,Σ q =Σ1∪Σ2∪Σ3∪Σ4∪Σ6,Σθ=Σ5∪Σ7.
On substituting Eqs.(8.253) and (8.254) into (8.252), we obtain the system of four scalar equations for the four scalar unknowns: components of the displacement vector and the temperature
If the model A n with exponential cores is considered, then constitutive equations (8.253) should be replaced by relations (8.242).
In particular, one can consider isothermal processeswhen a temperature field in a body Vremains unchanged: \(\theta (\mathbf{x},t) = {\theta }_{0} = \mathrm{const}\); then the heat conduction equation can be excluded from the system (8.252). As a result, we obtain the following statement of the quasistatic problem of viscoelasticity in the spatial description for linear models A n:
to solve it for three components of the displacement vector u(x,t).
If the linear models A n (8.205) and B n (8.218) of incompressible isotropic continua are considered, then the quasistatic statement of the viscoelasticity problem in the spatial description takes the form
Remind that, according to (5.4a): \(\bar{{h}}_{G} = 1\)if G=A, and \(\bar{{h}}_{G} = 0\)if G=B. Here we have denoted the tensor \(\mathop{\mathbf{E}}\limits^{\mathrm{(n)}} {=\ }^{4}\mathop{\mathbf{E}}\limits^{\mathrm{(n)}} \cdot \cdot \mathbf{E}\).
Since the domain Vin the spatial description is unknown, so to determine it one should complement the system (8.252)–(8.255) (and also (8.4.3) and (8.258)) with Eq.(5.89).
8.7.4 Statements of Quasistatic Problems for Models of Viscoelastic Continua in the Material Description
In a similar way, we can obtain statements of quasistatic problems in the material description. So the statement of the coupled quasistatic problem of thermoviscoelasticity in the material descriptionfor linear mechanically determinate models A n of thermorheologically simple continua has the form
Substitution of constitutive equations (8.260) and kinematic relations (8.261) into (8.259) gives a system of four scalar equations of equilibrium and heat conduction with the boundary and initial conditions (8.262) for four scalar unknowns, namely three components of the displacement vector and temperature:
Considering isothermal processes when θ(X i,t)=const, from (8.259)–(8.262) we obtain the statement of the quasistatic problem of viscoelasticity in the material description for linear models A n:
to solve it for three components of the displacement vector u(X i, t).
If the models B n of incompressible isotropic continua (8.218) are considered, then the constitutive equations in (8.264) should be replaced by the relations
and the problem (8.264), (8.265) is solved for the four unknown scalar functions: u, p∥X i, t.
8.8 The Problem on Uniaxial Deforming of a Viscoelastic Beam
8.8.1 Deformation of a Viscoelastic Beam in Uniaxial Tension
As an example, consider the classical problem on a beam in uniaxial tension; this problem was studied in detail before (see Sect.6.5). A beam is assumed to be viscoelastic, isotropic and incompressible, and its constitutive equations correspond to the simplest linear models A n or B n with exponential cores (see (8.212) and (8.224) and Exercise8.3.2). A general statement of the quasistatic problem in the spatial description has the form (8.258).
The motion law for the beam in tension is independent of peculiarities of its mechanical properties; the law is the same for both elastic and viscoelastic continua and defined by formula (5.131). Hence all the kinematic characteristics, namely the tensors F, \(\mathop{\mathbf{C}}\limits^{\mathrm{(n)}}\)and \(\mathop{\mathbf{G}}\limits^{\mathrm{(n)}}\), are the same as ones for elastic continua; they are determined according to the results of Exercises2.2.1, 2.3.2, and 4.2.13:
where we have denoted the functions of k 1
and k α(t)– the elongation ratios of the beam along corresponding coordinate directions. Here, due to the incompressibility condition, \({k}_{2} = {k}_{3} = 1/\sqrt{{k}_{1}}\).
8.8.2 Viscous Stresses in Uniaxial Tension
On substituting the expression (8.267) into the differential equation of the constitutive relations for W (γ)(see Exercise8.3.2), we find that the tensors W (γ)have a diagonal form too:
Functions W α (γ)are the same for models A n and B n ; they satisfy the differential equations
which have the solution
8.8.3 Stresses in a Viscoelastic Beam in Tension
Substituting the expressions (8.267) and (8.269) into the constitutive equations from Exercise8.3.2, we find that the energetic stress tensors \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\)have the diagonal form
– for models A n and
– for models B n . As shown in Sect.6.4.3, the tensors Tand \(\mathop{\mathbf{T}}\limits^{\mathrm{(n)}}\)in this problem are connected by the relations
As follows from Sect.6.4.3, in uniaxial tension of a beam σ1≠0, and \({\sigma }_{2} = {\sigma }_{3} = 0\). Then, substituting (8.270)–(8.272) into (8.273a), we obtain the system of two equations
– for models A n , and
– for models B n .
8.8.4 Resolving Relation \( \sigma_{1} (k_{1}) \) for a Viscoelastic Beam
Eliminating pamong these systems, we obtain the relations between σ1and k 1:
– for models A n ,
– for models B n . Here we have denoted the following functions of k 1:
Here we have taken into account that \({W}_{2}^{(\gamma )} = -(1/2)\ {W}_{1}^{(\gamma )}\)according to (8.268) and (8.272).
When W 1 (γ)≡0, the elasticity relations (5.166) follow from (8.275b).
8.8.5 Method of Calculating the Constants \( B^{(\gamma)} \) and \( \tau^{(\gamma)} \)
Consider a beam under stepwise deforming, when the elongation ratio k 1(t) is given in the form
Substitution of (8.277) into (8.270b) yields
Taking (8.278) into account, from (8.275) we find the expression for the relaxation stresses
for all the models A n and B n , where r(t) is the relaxation function, which, according to (8.169), has the exponential form
Equation (8.279) rewritten in the form
can be used for determining the constants B (γ)and τ(γ). If the experimental relaxation curve Δσ({ ex})(t) obtained at some fixed value of k 1 0is known, then with the help of relation (8.281) this curve can be approximated by choosing the parameters B (γ)and τ(γ)satisfied the condition of the minimum distance between the functions Δσ(t) and Δσ({ ex})(t) at Kpoints t i (i=1,…,K):
The functions \(\mathop{L}\limits^{\mathrm{(n)}}({k}_{1}^{0})\)and \(\mathop{f}\limits^{{\mathrm{(n)}}}_{1}({k}_{1}^{0})\), according to (8.268) and (8.276), include no material constants; therefore, at given k 1 0their values are known. Sometimes to improve the convergence of the iterative procedure for minimizing the functional (8.282), the parameters τ(γ)should be given a priori, for example, in the form τ(γ)=t (γ), where t (γ)are some instants of time. Substituting experimental values of the relaxation stresses Δσ({ ex})(t γ) at the times t γinto Eq.(8.281), we obtain a system of linear algebraic equations for determining the constants B (γ). This system can be solved numerically, for example, by Holetskii’s method.
We seek further values of the parameters t γ, for which the functional Δ (8.282) reaches its minimum value. Values B (γ)at such t γare unknown.
Table8.1shows values of the parameters τ(γ)and B (γ)obtained by the method mentioned above for rubber and polyurethane elastomer. The values of τ(γ)and B (γ)are the same for corresponding models A n and B n but are different for the models with different subscripts n.
For polyurethane, in approximating by the models B IVand B Vwe used the relaxation curve at higher values of the extension deformation δ1=80% than for the models B Iand B II(δ1=8.3%), that improved the quality of further simulation of the viscoelastic properties by these models.
Figures 8.4and 8.5show graphs of relaxation curves \({\sigma }_{1}(t) = {\sigma }_{1}(0) - \Delta \sigma (t)\), where the value of Δσ(t) has been determined by (8.281) with the optimum constants B (γ)and τ(γ)for rubber and polyurethane, and also their corresponding experimental relaxation curves \({\sigma }_{1}^{(\text{ ex})}(t) = {\sigma }_{1}(0) - \Delta {\sigma }^{(\text{ ex})}(t)\). The accuracy of approximation to the experimental curves with the help of the exponential cores (8.281) is sufficiently high: the mean-square deviation does not exceed 1%. Notice that the longer is a time interval of relaxation considered, the greater number Nof exponents in the relaxation spectrum we need to reach a high accuracy of the approximation.
Graph of stress-relaxation for rubber at deformation 10% and its approximation with the help of the exponential cores (8.281)
Graph of stress-relaxation for polyurethane at deformation 8.3% and its approximation with the help of the exponential cores (8.281)
For rubber, the relaxation curve was approximated within the interval from 0 to 1min. To do this, we used three exponents; and for approximation of the relaxation curve for polyurethane within the interval up to one hour we used five exponents.
8.8.6 Method for Evaluating the Constants \( \bar{m},\,l_{1},\,l_{2} \) and β, m
After calculation of the material constants B (γ), τ(γ), we can determine values of the constants \(\bar{m}\), l 1, l 2or β, mby Eq.(8.275) and by experimental diagrams of deforming σ1=σ1 ({ ex})(k 1) obtained, for example, in the case of deforming the beam with a constant rate b:
The function W 1(t) in these computations is known and determined by (8.269a):
The constants l 1, l 2, \(\bar{m}\)or m, β have been found from the condition of the best approximation to the function σ1 ({ ex})(k 1) by σ1(k 1), determined according to (8.275), by minimizing the functional of a mean-square error:
For models B n , the space of the optimization parameters μ and β is two-dimensional; and for models A n , the space of the parameters l 1, l 2and \(\bar{m}\)is three-dimensional. To accelerate the process of solving the optimization problem, we have used the method of gradient descent with fitting an initial point of the minimizing procedure. Values of the constants μ and β obtained for rubber and polyurethane are given in Table8.2, and l 1, l 2and \(\bar{m}\)– in Table8.3.
Figures 8.6and 8.7show experimental diagrams of deforming σ1 ({ ex})(δ1) for rubber and polyurethane in tension, and also approximations to the diagrams with the help of models A n and B n of incompressible viscoelastic continua. For the materials considered, the models A Iand B Iexhibit the best approximation to the** experimental data. For rubber, the model A Iapproximates accurately enough the diagram σ1 ({ ex})(δ1) within the whole interval of deforming, including the domain of maximum deformations (higher than 100%), while the other models give a considerable error in this domain.
Approximation of the diagrams of deforming for rubber (a) and polyurethane (b)intension by linear models B n of viscoelastic continua
Approximation of the diagrams of deforming for rubber (a) and polyurethane (b)intension by linear models A n of incompressible viscoelastic continua
8.8.7 Computations of Relaxation Curves
After evaluation of all the constants B (γ), τ(γ)and l 1, l 2, \(\bar{m}\)or m, β, we can verify the models A n and B n , for example, by performing computations of relaxation curves at different values of the parameter k 1 0with use of Eq.(8.275) for the stepwise process (8.277):
– for models A n , or
– for models B n .
Figures 8.8and 8.9show the relaxation curves σ1(t) obtained in experiments and in computations by the method mentioned above for polyurethane elastomer.
Computed and experimental relaxation curves for polyurethane at different values of the extension deformation; computations for different models B n of viscoelastic continua
Computed and experimental relaxation curves for polyurethane at different values of the extension deformation; computations for different models A n of viscoelastic continua
To verify the models A n and B n we use the relaxation curves when k 1 0=1.23 (δ1=23%), k 1 0=1.49 (δ1=49%) and k 1 0=1.8 (δ1=80%). The computations performed show that for the considered material the models A Iand A IIare of the best accuracy, while the model A Vgives the worst result of description of the relaxation curves.
8.8.8 Cyclic Deforming of a Beam
Assume that all material constants appearing in models A n or B n of viscoelastic continua are known, for example, they are determined by the method mentioned above. Consider a cyclic regime of deforming, which is widely used in practice. In this case, the function k(t) first grows with a constant rate band then diminishes with the same rate, then grows again etc. (the saw-tooth regime, Fig.8.10). Such a function k(t) can be written in an analytical form with the help of the Heaviside functions:
Here bis the rate of deforming; h(t) is the Heaviside function; a 0=1, \({a}_{m} = 2{(-1)}^{m}\)and m≥1 are the constants; and t 0is the time interval of monotonicity of the function (the semiperiod of the cycle).
The cyclic process of deforming with a constant rate
Substituting (8.286) into Eq.(8.284) and then solving this equation numerically, for example, with the help of the implicit difference approximation
we can find values of the functions W 1 (γ)(t) for the cyclic process of deforming. Here we have introduced the notation: t j – instants of time (nodes), W 1,j (γ)=W 1 (γ)(t j )– values of functions in the nodes, Δt– the step in time.
The stresses σ1in cyclic deforming are determined by the general formula (8.275a) for models A n , or by (8.275b)– for models B n .
If all material constants of the models are known, then, according to these formulae, we can calculate the dependences of cyclically varying stresses upon time σ1(t), and also plot the cyclic diagrams of deforming (σ1(t), δ1(t)), where \({\delta }_{1}(t) = {k}_{1}(t) - 1\)is the relative elongation.
Figure 8.11shows cyclic diagrams of deforming for polyurethane, which have been obtained by the method mentioned above for models B n of viscoelastic continua. Unlike ideal elastic media, for viscoelastic continua the cyclic diagrams of deforming do not coincide at the stages of loading (m=0,2,4,…) and unloading (m=1,3,5,…) but exhibit characteristic loops.
Cyclic diagrams of deforming of polyurethane for different models B n of viscoelastic continua
8.9 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming
8.9.1 The Problem on Dissipative Heating of a Beam Under Cyclic Deforming
Let us remind now that viscoelastic continua are not ideal, and for them the dissipation function w ∗is different from identically zero. Therefore, in general, at any regime of deforming of viscoelastic continua, even under constant loads, there occurs an internal heat release in these materials, i.e. dissipative heating due to the presence of the dissipation function w ∗in the heat conduction equation. However, at such small-intensive regimes, dissipative heating is rather small (the temperature changes by some fractions of the Celsius degree due to w ∗), and it can usually be neglected (although for some special problems these values can prove to be considerable). For cyclic regimes of loading when the number of cycles Mis rather great, the situation is different. The temperature of dissipative heating at such regimes can reach 100∘C and higher, and even prove to be the cause of heat destruction of the material.
Consider a computational method for dissipative heating of a beam in longitudinal quasistatic cyclic tension by an arbitrary periodic law
Here k 1(t) is the elongation ratio (see Sects.6.5 and 8.5.1), t 0, as before, is the cycle period (Fig.8.12), and the number of cycles is assumed to be large: M≫1. In this case one can say that the multicycle regime of deforming is considered.
The cyclic regime of deforming
Then the functions W 1 (γ)(t) can be evaluated with the help of formula (8.284) and difference approximation (8.287), and the stresses σ1(t)– with the help of formula (8.275).
8.9.2 Fast and Slow Times in Multicycle Deforming
For the multicycle regime of deforming, we introduce the small parameter \(\varkappa = 1/M \ll 1\), the new dimensionless variable \(\xi = t/{t}_{0}\)having the sense of a counter of the number of oscillation cycles and called the fast time, and also the slow time \(\bar{t} = t/{t}_{1}\)(where t 1=Mt 0), which takes on values \(0 \leq \bar{ t} \leq 1\)within the whole interval of cyclic deforming.
The periodic functions (8.288) can be considered as functions of the fast time with the period equal to 1:
For example, for harmonic oscillations we have
because the oscillation period in this case is equal to \({t}_{0} = 2\pi /\omega \). Here \(\bar{{k}}_{1}\)is the mean value of the function k 1, and k 1 0is the oscillation amplitude (both the values are constants).
The relaxation core q(t) is considered, on the contrary, as a function of the slow time, because we assume that during one period of oscillations its changes are negligible:
for example, for the exponential core of relaxation (8.280)
The temperature θ of a viscoelastic continuum can be considered as a quasiperiodic functionof both the fast and slow times:
and is a singly-periodic function of ξ. The arguments ξ and \(\bar{t}\)for such a function are assumed to be independent.
8.9.3 Differentiation and Integration of Quasiperiodic Functions
Differentiation of quasiperiodic functions is performed by the rule of differentiation of composite functions:
Integration of quasiperiodic functions \(b(\tau ) = b(\bar{\tau },\;\xi )\), where \(\bar{\tau } = \tau /{t}_{1}\), is realized as follows:
Here we have denoted the average value of a quasiperiodic function over the oscillation cycle, which is a function only of the slow time \(\bar{\tau }\), by
and the value O(1) contains the terms comparable in magnitude with the first term \({t}_{1}{ \int \nolimits \nolimits }_{0}^{\bar{t}}\langle b\rangle d\bar{\tau }\)of formula (8.295).
8.9.4 Heat Conduction Equation for a Thin Viscoelastic Beam
Consider the simplest linear models B n of isotropic incompressible thermorheologically simple continua, whose constitutive equations have been derived in Exercise8.3.3. A statement of the coupled quasistatic problem of thermoviscoelasticity in the material description has the form (8.259), (8.261), (8.262), and the heat conduction equation in this system becomes
Here and below for simplicity we assume that q m =0.
Integrating this equation over \(\mathop{V }\limits^{\circ }\)and then using the Gauss–Ostrogradskii formula and the boundary conditions (8.262), we obtain
Suppose that a beam considered is thin, i.e. its width h 2 0 and height h 3 0 are much smaller than its length h 1 0, so changes in the temperature θ along the coordinates \(\mathop{x}\limits^{ {\circ }}{}^{2} \)and \(\mathop{x}\limits^{ {\circ }}{}^{3}\)can be neglected. The beam ends \(\mathop{x}\limits^{ {\circ }}{}^{1} = 0\)and \(\mathop{x}\limits^{ {\circ }}{}^{1} = {h}_{1}^{0}\)are assumed to be heat-insulated (for them \(\mathop{q}\limits^{ {\circ }}_{e} = 0\)), and on the lateral surface of the beam the conditions of convective heat transfer are given:
where α T is the specific heat transfer coefficient, and θ e is the temperature of the surroundings (being a constant). Since in cyclic tension of the beam \(\mathop{w}\limits^{ {\circ }}{}^{{_\ast}}\)is independent of coordinates, Eq.(8.298) takes the form
where
is the integral coefficient of heat transfer.
8.9.5 Dissipation Function for a Viscoelastic Beam
The dissipation function \(\mathop{w}\limits^{ {\circ }}{}^{{_\ast}}\)for the beam has the form (see Exercise8.3.3)
Integrating by parts yields
8.9.6 Asymptotic Expansion in Terms of a Small Parameter
Since the function k 1(t) is periodic, any algebraic function of k 1, in particular, the function \(\mathop{f}\limits^{{\mathrm{(n)}}}_{\alpha }({k}_{1})\)determined by (8.268), will be also periodic: \(\mathop{f}\limits^{{\mathrm{(n)}}}_{\alpha } =\mathop{ f}\limits^{{\mathrm{(n)}}}_{\alpha }({k}_{1}(\xi ))\). Then the temperature θ, being a solution of Eq.(8.300), will be, in general, a quasiperiodic function. Let us seek a solution of Eq.(8.300) as an asymptotic expansion in terms of a small parameter ϰ:
The first term \(\bar{\theta }\)of this expansion depends only on the slow time \(\bar{t}\). According to the rule (8.294), the derivative ∂θ∕∂tof the expansion (8.304) is calculated as follows:
The function a θ(θ) and the reduced time t ′and τ′after substitution of the expansion (8.304) into them can also be represented in the form of asymptotic expansion:
Here, as before, O(1) means the terms comparable in magnitude with the first terms of the expansion. The functions \(\bar{{a}}_{\theta } = {a}_{\theta }(\bar{\theta })\)and \(\bar{{t}}^{{\prime}}\)are expressed by the same formulae as the input functions a θand t ′, if in them the substitution \(\theta \rightarrow \bar{ \theta }\)has been made:
Substituting the expansions (8.304) and (8.306) into formula (8.303) and using the rule (8.295) of integration of quasiperiodic functions, we find an asymptotic expansion for the dissipation function
where
In deriving this expression, we have taken into account that the integrands are products of functions of the fast time \(\mathop{f}\limits^{{\mathrm{(n)}}}_{\alpha }({k}_{1}(\xi ))\)and the slow time \(\frac{\partial } {\partial {\tau }_{1}} q(\bar{{t}}^{{\prime}}- {\tau }_{1}^{{\prime}})\), and \(\frac{{\partial }^{2}} {\partial {\tau }_{1}\partial {\tau }_{2}} q(2\bar{{t}}^{{\prime}}- {\tau }_{1}^{{\prime}}- {\tau }_{2}^{{\prime}})\); hence, the integrals with respect to ξ and \(\bar{t}\)in formula (8.303) can be calculated independently.
On substituting (8.304) and (8.308) into (8.300), we obtain the following asymptotic expansion of the heat conduction equation:
8.9.7 Averaged Heat Conduction Equation
Averaging the Eq.(8.310) over the oscillation period by (8.296) and taking periodicity of the function θ(1)into account (i.e. \(\langle \partial {\theta }^{(1)}/\partial \xi \rangle = {\theta }^{(1)}(1) - {\theta }^{(1)}(0) = 0\)), we find the final form of the heat conduction equation for the temperature \(\bar{\theta }\):
where \(\langle \mathop{w}\limits^{ {\circ }}{}^{{_\ast}(0)}\rangle \equiv \bar{ {w}}^{{_\ast}}\)is the averaged dissipation function. In deriving Eq.(8.311) we have replaced the dimensionless time by the dimensional one: \(t =\bar{ t}{t}_{1}\). Collecting terms with higher powers of ϰ(with ϰ, ϰ 2etc.) in Eq.(8.310), we obtain an equation for determining the functions θ(1), θ(2)etc.; however, a contribution of these terms to the value of the temperature θ is small due to (8.304). Therefore, for the problems investigated, we can consider only the zero approximation for determining the temperature \(\bar{\theta }\).
Averaging (8.309), we find the expression for the dissipation function \(\bar{{w}}^{{_\ast}}\)averaged over the cycle of oscillations:
If the beam is elastic, then \(q(\bar{t}) \equiv 0\)and \(\bar{{w}}^{{_\ast}}\equiv 0\), and the heat conduction problem at θ0=θ e has a trivial solution: \(\theta (t) = {\theta }_{0} = {\theta }_{e} = \mathrm{const}\); i.e. an elastic beam in cyclic deforming does not change its temperature. For a viscoelastic beam \(\bar{{w}}^{{_\ast}}\geq 0\), and in Eq.(8.311) there appears a heat source; therefore, \(\partial \bar{\theta }/\partial \bar{t} \geq 0\), i.e. a heat-insulated beam will always be heated with time (while θ0=θ e and \(\bar{{\alpha }}_{T} = 0\)). This heating is caused only by energy dissipation; therefore, it is called dissipative heating.
8.9.8 Temperature of Dissipative Heating in a Symmetric Cycle
The process of deforming is called a symmetric cycle, if the average value of function \(\mathop{f}\limits^{{\mathrm{(n)}}}_{1}({k}_{1})\)over the oscillation cycle is zero: \(\langle \mathop{f}\limits^{{\mathrm{(n)}}}_{1}({k}_{1})\rangle = 0\).
As follows from (8.268), for a symmetric cycle the conditions \(\langle \mathop{f}\limits^{{\mathrm{(n)}}}_{\alpha }({k}_{1})\rangle = 0\)(α=1,2,3) are satisfied simultaneously. And from (8.312) it follows that the dissipation function depends only on the temperature \(\bar{\theta }\):
Then the heat conduction problem (8.311) takes the form
Its solution has the form
8.9.9 Regimes of Dissipative Heating Without Heat Removal
If there is no heat removal from the beam \((\bar{{\alpha }}_{T} = 0)\), then, depending on a form of the function \({a}_{\theta }(\bar{\theta })\), two basically distinct regimes of dissipative heating are possible.
-
(1)
If the function \({a}_{\theta }(\bar{\theta })\)is such that the integral
$$H(\bar{\theta }) ={ \int \nolimits }_{{\theta }_{0}}^{\theta } \frac{\rho ^{ \circ } {c}_{\epsilon }\;d\theta } {({a}_{\theta }(\bar{\theta })q(0)\bar{{f}}^{2}}$$(8.316)becomes infinite at infinity (i.e. \(H(\bar{\theta }) \rightarrow +\infty \)as \(\bar{\theta } \rightarrow \infty \)), then the temperature of dissipative heating in the beam gradually grows with no limit (Fig.8.13, the curve 1).
Fig.8.13 
Different regimes of dissipative heating for a heat-insulated viscoelastic beam: 1 – unbounded growth of the temperature, 2 – heat explosion, 3 – heat pseudoexplosion
For many real viscoelastic materials, as the function a θ(θ) one frequently uses the Williams–Landel–Ferry dependence
$${a}_{\theta }(\theta ) = \text{ exp}\ \frac{{a}_{1}(\theta - {\theta }_{0})} {{a}_{2} + \theta - {\theta }_{0}},\ \ \ \ {a}_{1},{a}_{2} -\text{ const},$$(8.317)for which the condition \(H(\bar{\theta }) \rightarrow +\infty \)as \(\bar{\theta } \rightarrow +\infty \)is actually satisfied.
-
(2)
If the dependence a θ(θ) is such that the function \(H(\bar{\theta })\)is bounded at infinity (i.e. \(H(+\infty ) < +\infty \)), then from (8.316) it follows that the temperature \(\bar{\theta }(t)\)of dissipative heating reaches infinite values in the finite time \({t}_{{_\ast}} = H(+\infty )\); in other words, the function \(\bar{\theta }(t)\)has a vertical asymptote as t→t ∗(Fig.8.13, the curve 2). The phenomenon of sharp growth of the temperature at a certain time t ∗is called the heat explosion.
If, for example, the function a θ(θ) is exponential (that is typical for some elastomers):
$${a}_{\theta } ={ \text{ e}}^{{a}_{1}(\theta -{\theta }_{0})},\ \ \ \ {a}_{ 1} >0,$$(8.318)then, calculating the integral in (8.316), we obtain the following expression for the temperature of dissipative heating:
$$\bar{\theta }(t) = {\theta }_{0} - \frac{1} {{a}_{1}}\mathrm{lg}\ \Biggl (1 -\frac{q(0)\bar{{f}}^{2}{a}_{1}t} \rho ^{ \circ } {c}_{\epsilon } \Biggr ),\ \ \ {t}_{{_\ast}} = \frac{\rho ^{ \circ } {c}_{\epsilon } }{q(0)\bar{{f}}^{2}{a}_{1}}.$$(8.319)From this equation it really follows that \(\bar{\theta } \rightarrow +\infty \)as t→t ∗.
-
(3)
In practice sometimes there is an intermediate situation when the condition \(H(\bar{\theta }) \rightarrow +\infty \)holds as \(\bar{\theta } \rightarrow +\infty \), but the growth of temperature \(\bar{\theta }(t)\)with time proves to be so sharp that it becomes similar to the effect of heat explosion: the temperature reaches its ultimate value θ k , at which there occurs a heat destruction of the material, in comparably small time t k (Fig.8.13, the curve3). Such regimes are called the heat pseudoexplosion.
8.9.10 Regimes of Dissipative Heating in the Presence of Heat Removal
When there is heat removal (α T >0), the character of dissipative heating of a body changes. Since both the functions \({\alpha }_{\theta }(\bar{\theta })\)and \({\alpha }_{T}(\bar{\theta } - {\theta }_{0})\)are non-negative, the denominator of the integrand in (8.315) at a certain finite value \(\bar{\theta } = {\theta }_{\infty } < +\infty \)can vanish; in this case the function \(H(\bar{\theta }) = t\)tends to infinity: \(\bar{\theta } \rightarrow {\theta }_{\infty }\)as t→+∞. Thus, when there is heat removal, the regimes 1, 2, and 3are impossible, because the temperature always remains bounded, in this case there are other four typical regimes 4, 5, 6, and 7(Fig.8.14).
Typical regimes of dissipative heating for a viscoelastic beam in oscillations with heat removal
-
(4)
The regime 4is realized when the dissipation function \(\bar{{w}}^{{_\ast}}\)is independent of temperature (a θ=1). In this case the heat conduction equation (8.314) yields \({d}^{2}\bar{\theta }/d{t}^{2} \leq 0\)(Exercise8.6.1); i.e. the dissipative heating curve is convex upwards. This is the most wide-spread type of a curve of dissipative heating for real practical problems.
At this curve there are two typical sections: the section ‘a’, where the rate of heating decreases rapidly from its maximum value \(\dot{\bar{\theta }}(0)\)to practically zero \(\dot{\bar{\theta }} \approx 0\), and the steady section ‘b’, where \(\dot{\bar{\theta }} \approx 0\).
If \({a}_{\theta }(\bar{\theta }) = 1\), then the heat conduction problem (8.314) admits a solution in the explicit form (Exercise8.6.1)
-
(5)
If \({a}_{\theta }(\bar{\theta })\)is unbounded as \(\bar{\theta } \rightarrow +\infty \)(for example, the exponential function (8.318)), then the temperature regime 5is realized when the curve of dissipative heating can be split into the three typical sections: ‘a’– initial, ‘b’– steady, where \(\dot{\theta } \approx \text{ const}\), and ‘c’– unsteady, where the function \(\bar{\theta }(\bar{t})\)is convex downwards (Fig.8.14) and unbounded as t→+∞.
-
(6)
If \({a}_{\theta }(\bar{\theta })\)depends on the temperature but is bounded as \(\bar{\theta } \rightarrow +\infty \)and has a point of inflection at \(\bar{\theta }\,=\,\widetilde{{\theta }}_{k}\)(as the function (8.317)), then the temperature regime 6of dissipative heating occurs (Fig.8.14), when there are four typical sections: initial– ‘a’, steady– ‘b’, unsteady– ‘c’, followed by steady one– ‘d’. The temperature \(\bar{\theta }(t)\)remains bounded as t→+∞.
-
(7)
The regime 7is realized under the same conditions as the regime 6, but at the unsteady section ‘c’ the temperature of dissipative heating \(\bar{\theta }(t)\)reaches an ultimate value θ k such that there occurs a heat destruction of the material (by analogy with the regime 3), and the section ‘d’ is not realizable.
8.9.11 Experimental and Computed Data on Dissipative Heating of Viscoelastic Bodies
Figure 8.15shows computed and experimental curves of dissipative heating for a polyurethane beam in cyclic deforming by the harmonic law (8.290). The change in temperature \(\Delta \theta = \theta - {\theta }_{0}\)was found by solving the problem (8.311), (8.312) with the help of the implicit difference approximation
where θi=θ(t i ) is the value of temperature in the ith node at time t i , and Δtis the step in time. Values of the relaxation core q(2t ′) and q(0) appearing in expression (8.312) for the dissipation function were determined by (8.292), and B (γ)and τ(γ)in this formula– by the relaxation curves with the help of the method given in Sect.8.5.5. Values of the constants B (γ)and τ(γ)for polyurethane are shown in Table8.1. The function a θ(θ) was approximated by formula (8.317), and values of the constants in this formula were assumed to be a 1=21,a 2=208K. The remaining constants in (8.311) take on the following values: \(\rho ^{\circ } = 1{0}^{3}\ \text{ kg}/{\text{ m}}^{3}\),\({c}_{\epsilon } = 0.8\ \text{ kJ}/(\text{ kg} \cdot \text{ K})\),\(\bar{{\alpha }}^{\top } = 10\ \text{ kWt}/({\text{ m}}^{3} \cdot \text{ K})\). The mean value \(\bar{{k}}_{1}\)and the oscillation amplitude k 1 0were expressed in terms of the minimum and maximum deformation values in the cycle (δ{ min}and δ{ max}):
Figure 8.15exhibits temperatures of dissipative heating, computed by the method mentioned above for different models B n at δ{ max}=50%and δ{ min}=34%. Under the conditions considered, the model B Igives the best approximation to experimental data. The distinction between different models B n is considerable: the models B IVand B Vlead to a stationary regime of dissipative heating according to the type (4), and the models B Iand B IIforecast the regime (5) with the presence of unsteady section.
Curves of dissipative heating for polyurethane, computed by models B n , and experimental curve of dissipative heating θ({ ex})
Figure 8.16shows curves of dissipative heating for polyurethane, computed by the model B Iat different values of δ{ min}(values of this parameter are given in Fig.8.16by numbers at curves); the value δmax=50%has been fixed. With growing the oscillation amplitude (i.e. in this case with decreasing the value δmin), the intensity of dissipative heating sharply increases.
Curves of dissipative heating for polyurethane, computed by the model B Iat different amplitudes of oscillations
Notice that the phenomenon of dissipative heating of viscoelastic materials can essentially reduce the durability of structures under cyclic deforming.
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© 2011 Springer Science+Business Media B.V.
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Dimitrienko, Y.I. (2011). Viscoelastic Continua at Large Deformations. In: Nonlinear Continuum Mechanics and Large Inelastic Deformations. Solid Mechanics and Its Applications, vol 174. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0034-5_8
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