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On finite linear viscoelasticity of incompressible isotropic materials

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Summary

The assumption of incompressibility as a kinematic constraint condition leads to consequences, which are of physical interest in view of a thermodynamically consistent material modelling. Some of these consequences are discussed within the concept of finite linear viscoelasticity. We present two natural possibilities to generalise the familiar Maxwell-model to finite strains; both tensor-valued differential equations are integrated to yield the present Cauchy stress as a functional of the relative Piola or Green strain history. Both types of Maxwell-models are related to a free energy functional in the sense that the dissipation inequality is satisfied. The stress and energy functionals are generalised to incorporate arbitrary kernel functions of relaxation; the only restriction for thermodynamic consistency is that the relaxation functions have a negative slope and a positive curvature. The linear combination of the two types of energy functionals can be understood to be a generalisation of the Mooney-Rivlin model to viscoelasticity. The concrete representation of relaxation functions, motivated from a finite series of Maxwell-elements in parallel, implies a Prony series, corresponding to a discrete relaxation spectrum. A compact notation to express a continuous relaxation spectrum is provided by the concept of a derivative of fractional order. In particular, differential equations of fractional order lead to relaxation functions coming very close to the relaxation behaviour of real materials. As an example, the evolution equation of a fractional Maxwell-model is solved, leading to a relaxation function of the Mittag-Leffler type. An essential advantage is that a rather small number of material constants is involved in this model, while the accuracy of its prediction is quite good. Physically nonlinear effects, such as process-dependent viscosity and relaxation behaviour can be incorporated into finite linear viscoelasticity, if the natural time is replaced with a material-dependent time scale. The rate of the material-dependent time then depends on further internal variables to account for the influence of the process history; this kind of proceeding is fully compatible with the entropy inequality. Temperature-dependence is included, starting from a multiplicative decomposition of the deformation gradient into mechanical and thermal parts. The resulting constitutive theory has a rather simple structure and offers a considerable degree of freedom to allow for physically important phenomena. The numerical simulations illustrate a small selection of these possibilities.

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Abbreviations

1:

Unit tensor

Y−1, YT :

Inverse and transpose of a tensor Y

Y·X:

Scalar product of two tensors X and Y

a⊗b:

Dyadic tensor-product of two vectors a and b

tr(Y), det (Y):

Trace and determinant of a tensor Y

div v:

Divergence operator applied to a vector field v

I Y ,II Y :

Principal invariants of a tensor Y

λ1, λ2, λ3 :

Eigenvalues of a tensor

X, x(t):

Place of a material particle in the reference and the current configuration

dX,dx(t):

Tangent vectors of material lines in the reference and the current configuration

N, n(t):

Normal vectors of material surfaces in the reference and the current configuration

F(t), F(τ):

Deformation gradients at timest and τ

F t (τ):

Relative deformation gradient

B, C:

Right and left Cauchy-Green tensors

E, A, e, a:

Green, Almansi, Piola and Finger strain tensors

L, D:

Velocity gradient, symmetric part of the velocity gradient

ρ R , ρ:

Mass densities of the reference and the current configuration

ℓ i :

Stress power per unit mass

θ:

Thermodynamic temperature

ψ, γ:

Free energy per unit mass, rate of entropy production per unit mass

q, g:

heat flux vector, temperature gradient

T,\(\tilde T\) :

Cauchy- and Second Piola-Kirchhoff stress tensor

S,\(\tilde t\) :

Weighted Cauchy- (or Kirchhoff-) and convected stress tensor

p(X,t):

Hydrostatic pressure

S eq , S ov :

Equilibrium- and overstress tensors of the weighted Cauchy type

S E ,\(\tilde T_E ,\tilde t_E \) :

Extra stresses of the weighted Cauchy-, Second Piola-Kichhoff and convected type

σ, ɛ:

Uniaxial engineering stress and strain

Ê,μ A ,μ B :

Elasticity modulus and shear moduli

z, z A ,z B :

Relaxation times

G A (t),G B (t):

Relaxation functions

Γ(x):

Eulerian Gamma function

e x :

Exponential function

E α(x):

Mittag-Leffler function

O(x):

Landau symbol

h(s) :

Influence function

Fθ, F M :

Thermal and mechanical parts of the deformation gradient

Lθ, L M :

Thermal and mechanical velocity gradients

ρRθ :

Mass density of the thermomechanical intermediate configuration

B M , C M :

Mechanical right and left Cauchy-Green tensors

z(t) :

Process-dependent time scale

F0 :

Static predeformation

f(t):

Incremental, time-dependent deformation gradient

E L :

Linear incremental strain tensor

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  1. Institute of Mechanics, University of Kassel, Mönchebergstraße 7, Kassel, Germany

    P. Haupt & A. Lion

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  1. P. Haupt
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Haupt, P., Lion, A. On finite linear viscoelasticity of incompressible isotropic materials. Acta Mechanica 159, 87–124 (2002). https://doi.org/10.1007/BF01171450

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