Encyclopedia of Continuum Mechanics

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Anisotropy of Linear Creep

  • Artur GanczarskiEmail author
  • Jacek Skrzypek
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_183-1



In general, two types of thermodynamic processes in materials can be distinguished: reversible or irreversible. Process is reversible if the material after unloading returns to the initial state, whereas it is irreversible if it does not return to its initial state, but to a changed state, where strains, stresses, and material properties differ from the initial ones. During irreversible processes, also called dissipative processes, the material suffers from various dissipative phenomena, such as plasticity, creep, damage, phase transformation, etc. that all result, locally or globally, in the material microstructure change (plastic microslips, or nucleation and growth of voids, or other) due to the internal energy dissipation. Only in the case of purely reversible process, after unloading, the material microstructure remains totally unchanged, what occurs in the purely linear elastic process or the nonlinear elastic one. However, usually dissipative phenomena are coupled in a different sense with various deformations, for example, viscoelastic, elastic-visco-plastic, elastic plastic, elastic damage, plastic damage, and visco-plastic damage, as schematically shown in the graphics – see Fig. 1. Additionally, various phenomena may characterize materials that belong to different groups/classes of material symmetry: isotropy, transverse isotropy (hexagonal or tetragonal type), orthotropy, and anisotropy.

Fig. 1

Map of thermodynamically, reversible or irreversible, processes, with various symmetry groups included, the enhanced source figure taken from Ganczarski and Skrzypek (2013)

Usually, dissipative processes (single or complex) should be considered as nonlinear processes, by contrast to the purely elastic processes that can be treated as linear ones (nonlinearity of differential or equivalently integral equations). However, there exists special subclass of the linear viscoelastic processes that can be reduced to a fictitious linear elastic one by the use of the correspondence principle. The procedure described allows to replace original time-dependent differential problem by a fictitious time-independent one, the solution of which is easier. Distinction between the nonlinear or linear viscoelastic material can easily be done by checking isochronous creep curves, as shown in Fig. 2. Only in the case when the isochronous curves are family of straight lines, the material can be treated as linear viscoelastic.

Straightforward application of the constitutive equations to the anisotropic linear creep materials is difficult but can be avoided by application of the correspondence principle, based on the analogy between the governing equations for linear viscoelastic and fictitious elastic problems. The above analogy is originally dedicated to the isotropic materials only Alfrey (1944, 1948), Hoff (1954), Prager (1957) and Findley et al. (1976). It allows to find the stresses and the strains of a linear viscoelastic problem if the stresses and the strains of an equivalent elastic problem are known.

Fig. 2

Isochronous creep curves

In what follows generalization of the above analogy to the broader family of anisotropic materials is presented. It is based on inclusion of a crucial information about creep anisotropy type being included into the creep compliance \([{ }^{\mathrm {ve}}\mathbb {J}]\) or relaxation \([{ }^{\mathrm {ve}}\mathbb {E}]\) matrices. Importance of this generalization is clearly visible when creep composite materials are considered. Selected composite symmetries are sketched in Fig. 3, where symmetry of constitutive matrices, which apply to both linear viscoelastic materials and corresponding fictitious ones, is shown in order to emphasize importance and necessity of such generalization. In order to explain a possible generalization of the conventional correspondence principle to the general case of anisotropic materials, it is convenient to implement a formalism used in analysis of composite materials, for which a concept of representative unit cell (RUC) is broadly applied.

In order to apply the general correspondence principle to composite materials of various material symmetries, it is necessary to apply the chosen homogenization principle, that allows to transform constitutive equations of the composite components to the homogenized equivalent materials. The aforementioned formalism will be explained in next sections.

Fig. 3

Illustration of selected composites with respect to the format of linear creep compliance \([{ }^{\mathrm {ve}} \mathbb {J}]\) or relaxation \([{ }^{\mathrm {ve}} \mathbb {E}]\) matrices: (a) anisotropic fiber arrangement after Martin-Herrero and Germain (2007), (b) orthotropic fiber arrangement, (c) tetragonal fiber arrangement, (d) hexagonal fiber arrangement after Sun and Vaidya (1996), (e) random particle arrangement after Tjong and Ma (2000), compare cf. Skrzypek and Ganczarski (2015)

Elastic-Viscoelastic Correspondence Principle for the Case of Isotropic Materials

In a particular case of the isotropic linear viscoelastic behavior, the separation of volume change from shape change holds, in a similar fashion as in case of the isotropic elastic behavior. Analogy, between the transformed equations of isotropic linear viscoelastic materials and the corresponding equations of isotropic elasticity, leads to the solutions of viscoelasticity on the basis of a priori known fictitious coupled elastic problem by the use of the conventional elastic-viscoelastic correspondence principle, applied to isotropic material (see Findley et al. 1976). The set of equations of isotropic linear viscoelasticity is listed below (see Skrzypek 1993):
$$\displaystyle \begin{aligned} \begin{array}{l} \frac{\partial\sigma_{ij}\left(\mathbf{x},t\right)}{\partial x_i} + b_j \left(t\right) = 0 \\ \left\{ \begin{array}{l} \mathrm{P}_1 s_{ij}\left(\mathbf{x},t\right) = \mathrm{Q}_1 e_{ij}\left(\mathbf{x},t\right) \\ \mathrm{P}_2 \sigma_{kk}\left(\mathbf{x},t\right) = \mathrm{Q}_2 \varepsilon_{kk}\left(\mathbf{x},t\right) \end{array} \right. \\ \varepsilon_{ij}\left(\mathbf{x},t\right) = \frac{1}{2} \left[ \frac{\partial u_i \left(\mathbf{x},t\right)}{\partial x_j} + \frac{\partial u_j \left(\mathbf{x},t\right)}{\partial x_i}\right] \\ P_i \left(\mathbf{x},t\right) = \sigma_{ij} \left(\mathbf{x},t\right) n_j \quad \mathrm{at} \quad \varGamma_P \\ U_i \left(\mathbf{x},t\right) = u_i \left(\mathbf{x},t\right) \quad \mathrm{at} \quad \varGamma_U \end{array} {} \end{aligned} $$
When the Laplace transformation of the above set of Eqs. (1) is carried out, the “fictitious coupled elastic problem” is met:
$$\displaystyle \begin{aligned} \begin{array}{l} \frac{\partial\widehat{\sigma}_{ij}\left(\mathbf{x},s\right)}{\partial x_i} + \widehat{b}_j \left(\mathbf{x},s\right) = 0 \\ \left\{ \begin{array}{l} \widehat{s}_{ij}\left(\mathbf{x},s\right) = 2s\widehat{\mathrm{G}} \widehat{e}_{ij}\left(\mathbf{x},s\right) = \frac{\widehat{\mathrm{Q}}_1}{\widehat{\mathrm{P}}_1} \widehat{e}_{ij}\left(\mathbf{x},s\right) \\ \widehat{\sigma}_{kk}\left(\mathbf{x},s\right) = 3s\widehat{\mathrm{K}} \widehat{\varepsilon}_{kk}\left(s\right) = \frac{\widehat{\mathrm{Q}}_2}{\widehat{\mathrm{P}}_2} \widehat{\varepsilon}_{kk}\left(\mathbf{x},s\right) \end{array} \right. \\ \widehat{\varepsilon}_{ij}\left(\mathbf{x},s\right) = \frac{1}{2} \left[ \frac{\partial \widehat{u}_i \left(\mathbf{x},s\right)}{\partial x_j} + \frac{\partial \widehat{u}_j \left(\mathbf{x},s\right)}{\partial x_i}\right] \\ \widehat{P}_i \left(\mathbf{x},s\right) = \widehat{\sigma}_{ij} \left(\mathbf{x},s\right) n_j \quad \mathrm{at} \quad \varGamma_P \\ \widehat{U}_i \left(\mathbf{x},s\right) = \widehat{u}_i \left(\mathbf{x},s\right) \quad \quad \mathrm{at} \quad \varGamma_U \end{array} {} \end{aligned} $$
in which the body forces \(\widehat {b}_j\left (\mathbf {x},s\right )\), external forces \(\widehat {P}_i\left (\mathbf {x},s\right )\), displacements \(\widehat {U}_i\left (\mathbf {x},s\right )\) as well as „fictitious elastic constants” \(\widehat {\mathrm {G}}\left (s\right )\) and \(\widehat {\mathrm {K}}\left (s\right )\) are functions of the transformed variable s.
Finally, the analogy between viscoelastic and coupled elastic problems can be formulated. If a solution of coupled fictitious elastic problem is known \(\widehat {\sigma }_{ij}\left (\mathbf {x},s\right )\) and \(\widehat {u}_i\left (\mathbf {x},s\right )\), the solution of corresponding linear viscoelastic problem can be obtained by means of the inverse Laplace transformations \(\sigma _{ij}\left (\mathbf {x},t\right )\) and \(u_i\left (\mathbf {x},t\right )\), where following relations must hold
$$\displaystyle \begin{aligned} \widehat{\mathrm{G}}_{\mathrm{ve}} = \frac{\widehat{\mathrm{Q}}_1 \left(s\right)}{2s\widehat{\mathrm{P}}_1 \left(s\right)}, \qquad \widehat{\mathrm{K}}_{\mathrm{ve}} = \frac{\widehat{\mathrm{Q}}_2 \left(s\right)}{3s\widehat{\mathrm{P}}_2 \left(s\right)} {} \end{aligned} $$
The correspondence principle can be applied only to the boundary problems where the interface between the boundary ΓP (where the external forces are prescribed) and the boundary ΓU (where the surface displacements are given) is independent of time (see Findley et al. 1976). The above limitation does not hold in case of some material forming processes, for instance, rolling, where the interface between boundaries ΓP and ΓU varies with time.

Generalized Elastic-Viscoelastic Correspondence Principle for the Case of Anisotropic Materials

The elastic-viscoelastic correspondence principle applied to isotropic material is based on mathematically convenient separation of the volumetric and the shape change effects from total viscoelastic deformation. However, in case of any class of material anisotropy, such separation does not occur.

The set of equations, which governs the linear anisotropic viscoelastic materials, differs from the appropriate set of equations for linear isotropic viscoelastic materials in the constitutive equations only (see Eq. (1)2,3 versus Eq. (2)2,3).

In a general case of anisotropic linear viscoelastic material, the integral form of constitutive equations is furnished in the tensorial index notation as (see Shu and Onat 1967) or equivalently where Open image in new window defines the fourth-rank tensor of creep functions, whereas Open image in new window is the fourth-rank tensor of relaxation functions characterizing anisotropic viscoelastic properties of the material. Assuming the symmetry conditions, Open image in new window or Open image in new window, both tensors of viscoelastic anisotropy have 21 independent functions (cf. Fig. 3a).
Alternatively, when the vector-matrix notation is used, the general constitutive equations of anisotropic linear viscoelastic material take the equivalent integral format
$$\displaystyle \begin{aligned} \left\{ \varepsilon(t)\right\} = \int\limits_{0}^{t} [{}^{\mathrm{ve}}\mathrm{J}(t - \xi)] \frac{\partial}{\partial\xi}\left\{ \sigma(\xi)\right\} \mathrm{d}\xi {} \end{aligned} $$
$$\displaystyle \begin{aligned} \left\{ \sigma(t)\right\} = \int\limits_{0}^{t} [{}^{\mathrm{ve}}\mathrm{E}(t - \xi)] \frac{\partial}{\partial\xi}\left\{ \varepsilon(\xi)\right\} \mathrm{d}\xi {} \end{aligned} $$
When non-abbreviated notation of Eqs. (4) and (5) is used, the following constitutive integral equations of anisotropic linear viscoelastic material hold where [veJ]ij = [J(tξ)]ij is the creep compliance matrix, whereas [veE]ij = [E(tξ)]ij is the relaxation matrix. Both stresses and strains are functions of time veσij =veσij(t), veεij = veεij(t), in similar fashion like elements of creep compliance Open image in new window and relaxation Open image in new window matrices.
Applying the Laplace transform to Eqs. (8) and (9), the associated fictitious elastic constitutive equations in the transformed domain, in linear algebraic format, hold or
Fig. 4

Elastic-viscoelastic homogenization method based on the representative unit cell (RUC) applied to the associated elastic material by correspondence principle, compare Skrzypek and Ganczarski (2015)

The Generalized Correspondence Principle Applied to the Case of Fiber-Reinforced Composites

Application of correspondence principle occurs to be very useful since it makes possible to convert time-dependent heterogeneous viscoelastic problem to associated time-independent elastic problem, for which the homogenization tools can directly be applied.

Temporary micromechanics-based homogenization models take into account not only the volume fraction of constituents but also their configuration, geometry, and other factors, such as built-in residual stresses due to fabrication methods. Among them the following homogenization methods are frequently used: the method of concentric cylinder assembly (CCA) by Hashin and Rosen (1964), the Mori–Tanaka Method (MT) by Mori and Tanaka (1973), the generalized method of cells (GMC) by Paley and Aboudi (1992), or strain-compatible method of cells (SCMC) by Gan et al. (2000). All micromechanics-based homogenization methods assume existence of periodically repeated representative volume element (RVE) or representative unit cell (RUC), the size and geometry of which must capture the essence of the true composite behavior on the macroscale and which can be mapped into a point of a homogeneous continuum characterized by the displacement field u and the gradient ∇u.

Extensive state-of-the-art review of the micromechanics-based analysis of composite materials, enriched by numerous actual results, both in the field of homogenization techniques and its experimental validation for real long fiber-reinforced composites, is found in the recently published monograph by Aboudi et al (2013).

When the elastic-viscoelastic analogy is applied to anisotropic composites at the level of RUC, the application of the Laplace transform allows to reduce the integral equation of homogenized material (4) or (5) to a coupled set of equations of a fictitious elastic problem in the space of the transformed variable s (the variable s is not denoted by bar symbol)
$$\displaystyle \begin{aligned} ^{\mathrm{e}}\widehat{\overline{\varepsilon}}_{ij}(s) = s\widehat{\overline{J}}_{ijkl}(s) ^{\mathrm{e}}\widehat{\overline{\sigma}}_{kl}(s) {} \end{aligned} $$
$$\displaystyle \begin{aligned} ^{\mathrm{e}}\widehat{\overline{\sigma}}_{ij}(s) = s\widehat{\overline{E}}_{ijkl}(s) ^{\mathrm{e}}\widehat{\overline{\varepsilon}}_{kl}(s) {} \end{aligned} $$
Finally, solution of the anisotropic linear viscoelastic problem can be obtained by the use of inverse Laplace transformation from the transformed domain \(^{\mathrm {e}}\widehat {\overline {\varepsilon }}_{ij}(s)\), \(^{\mathrm {e}}\widehat {\overline {\sigma }}_{ij}(s)\) to the physical domain \(^{\mathrm {ve}}\overline {\varepsilon }_{ij}(t), ^{\mathrm {ve}}\overline {\sigma }_{ij}(t)\). When absolute notation is used, the following holds (see Haasemann and Ulbricht 2010) Recall definition of the Laplace transform of the function f(t) (t > 0) into the function of transformed variable \(\widehat {f}(s)\)
$$\displaystyle \begin{aligned} \mathscr{L}\left\{ f(t)\right\} = \widehat{f}(s) \stackrel{\mathrm{def}}{=} \int\limits_{0}^{\infty} f(t)e^{-st}\mathrm{d}t {} \end{aligned} $$
and definition of the convolution of two functions
$$\displaystyle \begin{aligned} f(t) \stackrel{\mathrm{def}}{=} \int\limits_{0}^{t} f_1(t-\xi)f_2(\xi)\mathrm{d}\xi \equiv f_1(t)\ast f_2(t) {} \end{aligned} $$
Applying the Laplace transform (15) to the convolution integral (16), the convolution theorem is furnished (see Findley et al. 1976) Taking next the Laplace transform of the integral form of constitutive equation of the anisotropic linear viscoelasticity (14), the equivalent transformed algebraic equation of anisotropic linear elasticity is achieved, according to the scheme defined by a function of the transformed variable s. The transformed matrix of anisotropic fictitious elasticity \(s\widehat {\overline {\mathbb {E}}}(s)\) is defined at the level of RUC of considered composite. The above matrix is obtained by the homogenization of the transformed isotropic local matrices \(s\widehat {\overline {\mathbb {E}}}^{(\beta \gamma )}(s)\) for the constituent materials at the level of composite microstructure (sub-cells). The procedure described above is sketched by the following scheme shown in Fig. 4. By contrast, there is no unique and direct homogenization procedure to yield the effective creep compliance \({{ }^{\mathrm {ve}}J}_{ijkl}^{(\beta \gamma )}(t)\) and relaxation \({{ }^{\mathrm {ve}}E}_{ijkl}^{(\beta \gamma )}(t)\) tensors (e.g., Haasemann and Ulbricht 2010). Hence, to overcome this deficiency, the suggested scheme is as follows: step 1, apply the Laplace transform at the level of sub-cell in order to eliminate physical time (left path in Fig. 4); step 2, use a homogenization method in order to reach the RUC level for the fictitious elastic RUC of composite material; and step 3, apply the inverse Laplace transform to reach physical viscoelastic RUC level (right back path in Fig. 4).

Usually for the sake of simplicity of further applications, the transversely isotropic effective relaxation matrix \(s\widehat {\overline {\mathbb {E}}}(s)\) at the level of RUC is sufficient, whereas at the microlevel (sub-cell), the isotropic matrices for the constituents (f) fiber and (m) composite matrix \(s\widehat {\overline {\mathbb {E}}}^{\mathrm {(f)}}(s)\) and \(s\widehat {\overline {\mathbb {E}}}^{\mathrm {(m)}}(s)\) are usually accepted.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Cracow University of TechnologyKrakówPoland

Section editors and affiliations

  • BłażCej Skoczeń
    • 1
  1. 1.Institute of Applied Mechanics, Faculty of Mechanical EngineeringCracow University of Technology (CUT)CracowPoland