On constitutive equations for fiber-reinforced nonlinearly viscoelastic solids

In this paper, we are interested in developing constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation. These invariants are analyzed, and we obtain restrictions such as positivity of some of them.     

On thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic solids with application to biomechanics

In this paper, we are interested in developing thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids, in isothermal processes. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation: 10 of them are isotropic invariants and 8 of them are associated with the deformation and the orientation of the fiber. Among the 8 anisotropic invariants just 6 are related to the viscoelastic response. The terms in the Cauchy stress tensor associated to these 6 invariants are analyzed with respect to thermodynamical consistency, and we obtain restrictions for the corresponding constitutive coefficients. This framework is applied to viscoelastic potentials within the context of biomaterials.     

A constitutive model in finite viscoelasticity

A new constitutive model is suggested for the viscoelastic behavior of rubber-like materials at finite strains. The model treats a viscoelastic medium as a system with a variable number of purely elastic links, which can arise and collapse due to micro-Brownian motion of molecules.Assuming that the processes of birth and death for elastic links are independent of stresses, we obtain operator linear constitutive equations in finite viscoelasticity. According to this model, elastic and viscous effects may be distinguished and described independently of each other by a relaxation measure and a strain energy density.The potential energy of deformations is assumed to depend on the principal invariants of the relative Finger tensor of strains. Unlike the standard approach, we do not suggest any expression for the strain energy densitya priori, but suppose that this function is presented as a sum of two functions of one variable which are found by fitting experimental data.The proposed approach allows results of several experiments (uniaxial tension, biaxial tension, and torsion) for styrene butadiene rubber and butyl rubber to be predicted correctly.     

Hyperelastic dielectrics with saturated polarization2

Variational and invariance principles of modern continuum mechanics are used to establish the field equations, boundary conditions and constitutive relations of a non-linear hyperelastic dielectric with constant magnitude ‘saturated’ polarization. Euclidean invariance places restrictions on the Lagrangian and implies the basic conservation laws. The principles of objectivity and material symmetry restrict the form of the constitutive equations. Four equivalent forms of the free energy functional are listed and for one of these forms the minimal isotropy integrity basis. consisting of eleven invariants, is constructed. The positive definiteness of the energy functional is used to derive various inequalities for the material constants of isotropic dielectrics.     

Ein nichtlineares Stoffgesetz für die ebene photoviskoelastische Spannungsanalyse

On the basis of a phenomenological theory of photoviscoelasticity which is guided by the results on the macroscopic behaviour of materials, nonlinear constitutive equations in the form of a single-integral representation are derived, being valid for homogeneous, isotropic, and non-aging, amorphous plastics, including large deformations. They yield the basic equations of two-dimensional nonlinear photoviscoelasticity which are used for the experimental analysis of plane states of stresses in polymeric structures under static or quasistatic isothermal loading. The example of an unsaturated polyester material labeled Leguval E 81 shows how to determine the material functions in the constitutive equations. The results allow us to make some simplifications which reduce the amount of numerical calculations when applying the proposed non-linear photoviscoelastic method.     

The Attainable Region of strain-invariant space for elastic materials

Within the theory of isothermal isotropic non-linear elasticity, the selection of the appropriate form for the strain energy function W in terms of the strain invariants is still an issue. The purpose of this paper is to introduce ideas and techniques which it is hoped will contribute to the task of finding an appropriate form for the strain energy. Three principal ideas are developed in this paper. Firstly, not all of invariant-space corresponds to real deformations. Constitutive equations only need to match real behaviour over a restricted part of invariant space, called the Attainable Region, bounded by states of deformation corresponding to uniaxial and equi-biaxial extension. Secondly, examples are given of how to exploit the fact that the Attainable Region is restricted. Mapping a deformation onto this region allows visualization of how close the deformation is to the well-understood uniaxial, equi-biaxial and simple shear deformations, and how this varies in space or time. Thirdly, acceptable strain invariants do not have to be obviously symmetric functions of the principal stretches. The ordered principal stretches are themselves invariants, and explicit unique algebraic expressions can be given through which the greatest, middle or least stretch can be calculated in terms of the usual invariants. Thus invariants can be chosen which are apparently non-symmetric functions of the ordered stretches.     

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.     

On the Comparison of Two Strategies to Formulate Orthotropic Hyperelasticity

The main goal of this work is to clarify the relation between two strategies to formulate constitutive equations for orthotropic materials at large strains. On the one hand, the classical approach is based on the incorporation of structural tensors into the free energy function via an enriched set of invariants. On the other hand, a fictitious isotropic configuration is introduced which renders an anisotropic, undeformed reference configuration via an appropriate linear tangent map. This formulation results in a reduced (with respect to the more general setting based on structural tensors) but nevertheless physically motivated set of invariants which are related to the invariants defined by structural tensors. As a main conceptual advantage standard isotropic constitutive equations can be applied and moreover, due to the reduced set of physically motivated invariants, the numerical treatment within a finite element setting becomes manageable.     

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.     

准脆性材料的弹塑性各向异性损伤耦合本构模型研究

针对准脆性材料的非线性特征：强度软化和刚度退化、单边效应、侧限强化和拉压软化、不可恢复变形、剪胀及非弹性体胀,在热动力学框架内,建立了准脆性材料的弹塑性与各向异性损伤耦合的本构关系。对准脆性材料的变形机理和损伤诱发的各向异性进行了诠释,并给出了损伤构形和有效构形中各物理量之间的关系。在有效应力空间内,建立了塑性屈服准则、拉压不同的塑性随动强化法则和各向同性强化法则。在损伤构形中,采用应变能释放率,建立了拉压损伤准则、拉压不同的损伤随动强化法则和各向同性强化法则。基于塑性屈服准则和损伤准则,构建了塑性势泛函和损伤势泛函,并由正交性法则,给出了塑性和损伤强化效应内变量的演化规律,同时,联立塑性屈服面和损伤加载面,给出了塑性流动和损伤演化内变量的演化法则。将损伤力学和塑性力学结合起来,建立了应变驱动的应力-应变增量本构关系,给出了本构数值积分的要点。以单轴加载-卸载往复试验识别和校准了本构材料常数,并对单轴单调试验、单轴加载-卸载往复试验、二轴受压、二轴拉压试验和三轴受压试验进行了预测,并与试验结果作了比较,结果表明,所建本构模型对准脆性材料的非线性材料性能有良好的预测能力。     

On the lower bounds for critical loads under large deformations in non-linear hyperelastic composites with imperfect interlaminar adhesion

The present paper investigates a mechanism of compressive fracture for heterogeneous incompressible non-linear materials with special kinds of defects of interfacial adhesion under large deformations. The analysis finds the lower bounds for the critical load. In order to calculate the bounds, the problem of the internal instability is considered within the scope of the exact statement based on the application of the model of a piecewise-homogeneous medium and the equations of the 3-D stability theory. The solution of the 3-D problem is found for the most general case accounting for large deformations and the biaxiality of compressive loads. The characteristic determinants are derived for the first four modes, which are more commonly observed. Special attention is given to the calculation of critical loads for hyperelastic layers described by a simplified version of Mooney's potential, namely the neo-Hookean potential.     

An elastoplasticity model in porous media theories

In this article, porous media theories are referred to as mixture theories extended by the well-known concept of volume fractions. This approach implies the diverse field functions of both the porous solid matrix and the pore fluid to be represented by average functions of the macroscale.The present investigations are based on a binary model of incompressible constituents, solid skeleton, and pore liquid, where, in the constitutive range, use is made of the second-grade character of general heterogeneous media. Within the framework of geometrically finite theories, the paper offers a set of constitutive equations for the solid matrix, the viscous pore liquid and the different interactions between the constituents. The constitutive model applies to saturated as well as to empty solid materials, taking into account the physical nonlinearities based on elasto-plastic solid deformations. In particular, the constitutive model concentrates on granular materials like soil or concrete, where the elastic deformations are usually small and the plastic range is governed by kinematically hardening properties.     

Limiting Chain Extensibility Constitutive Models of Valanis–Landel Type

A new general constitutive model in terms of the principal stretches is proposed to reflect limiting chain extensibility resulting in severe strain-stiffening for incompressible, isotropic, homogeneous elastic materials. The strain-energy density involves the logarithm function and has the general Valanis–Landel form. For specific functions in the Valanis–Landel representation, we obtain particular strain-energies, some of which have been proposed in the recent literature. The stress–stretch response in some basic homogeneous deformations is described for these particular strain-energy densities. It is shown that the stress response in these deformations is similar to that predicted by the Gent model involving the first invariant of the Cauchy–Green tensor. The models discussed here depend on both the first and second invariants.      

Restrictions on nonlinear constitutive equations for elastic shells

Constitutive equations for the resultant forces and moments applied to a shell-like body necessarily couple the influences of the shell geometry and the constitutive nature of the three-dimensional material from which the shell is constructed. Consequently, even when the nonlinear constitutive equation of the three-dimensional material is known, the complicated influence of the shell geometry on the constitutive response of the shell is not known. The main objective of this paper is to develop restrictions on the constitutive equations of nonlinear elastic shells which ensure that exact solutions of the shell equations are consistent with exact nonlinear solutions of the three-dimensional equations for homogeneous deformations. Since these restrictions are nonlinear in nature they provide valuable general theoretical guidance for specific constitutive assumptions about the coupling of material and geometric properties of shells. Examples of the linear theories of a plate and a spherical shell are considered.     

Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

Micromechanical analysis of the finite elastic–viscoplastic response of multiphase composites

Nonlinear thermoelastic–viscoplastic constitutive equations for large deformations with isotropic and directional hardening, are incorporated into a micromechanical finite strain analysis. As a result of this analysis, which is based on the homogenization technique for periodic microstructures, a global thermoinelastic constitutive law is established that governs the overall response of multiphase materials under finite deformations. This constitutive law is expressed in terms of the instantaneous effective mechanical and thermal stress tangent tensors together with the instantaneous global inelastic stress tensor that represents the viscoplastic effects. Results for a thermoinelastic matrix reinforced by a hyperelastic compressible material are given that illustrate the response of fibrous and particulate composites to various types of loading.     

Restrictions on nonlinear constitutive equations for elastic rods

Constitutive equations for the resultant forces and moments applied to a rod-like body necessarily couple the influences of the rod geometry and the constitutive nature of the three-dimensional material from which the rod was constructed. Consequently, even when the nonlinear constitutive equation of the three-dimensional material is known, the influence of the rod geometry on the constitutive response of the rod is not known. The main objective of this paper is to develop restrictions on the constitutive equations of nonlinear elastic rods which ensure that exact solutions of the rod equations are consistent with exact nonlinear solutions of the three-dimensional equations for all homogeneous deformations. Since these restrictions are nonlinear in nature they provide valuable general theoretical guidance for specific constitutive assumptions about the coupling of material and geometric properties of rods. Also, an example of a straight beam clamped at one end and subjected to a shear force at the other end is used to examine the validity of the proposed value for the transverse shear deformation coefficient.     

A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point

The theory of a Cosserat point has been used to formulate a new 3-D finite element for the numerical analysis of dynamic problems in nonlinear elasticity. The kinematics of this element are consistent with the standard tri-linear approximation in an eight node brick-element. Specifically, the Cosserat point is characterized by eight director vectors which are determined by balance laws and constitutive equations. For hyperelastic response, the constitutive equations for the director couples are determined by derivatives of a strain energy function. Restrictions are imposed on the strain energy function which ensure that the element satisfies a nonlinear version of the patch test. It is shown that the Cosserat balance laws are in one-to-one correspondence with those obtained using a Bubnov–Galerkin formulation. Nevertheless, there is an essential difference between the two approaches in the procedure for obtaining the strain energy function. Specifically, the Cosserat approach determines the constitutive coefficients for inhomogeneous deformations by comparison with exact solutions or experimental data. In contrast, the Bubnov–Galerkin approach determines these constitutive coefficients by integrating the 3-D strain energy function using the kinematic approximation. It is shown that the resulting Cosserat equations eliminate unphysical locking, and hourglassing in large compression without the need for using assumed enhanced strains or special weighting functions.     

Large-strain constitutive laws for the cylindrical deformation of shells

Expressions are developed for the strain energy and stress resultants in the case of large strain-large bending cylindrical deformations of shells of non-linear materials. The deformation variables are the extension and curvature of the material reference surface. The Mooney material is used as an example where appropriate.The influence of the large deformations on the constitutive laws is evaluated. Of special interest is the effect of cross terms and of the ability to express the stress resultants as partial derivatives of the strain energy with respect to appropriate deformation variables.