Constitutive modeling of complex interfaces based on a differential interfacial energy function

An important theory on the dynamics of complex interfaces is the Doi and Ohta theory where the interfacial contribution to the Cauchy stress tensor is determined from an interfacial conformation tensor. For a uniform deformation field in the Eulerian framework, Doi and Ohta adopted a decoupling approximation to reduce a fourth-order tensor into two second-order tensors and derived a differential equation governing the evolution of the interfacial conformation tensor. In this paper, a different formulation is presented for establishing the Cauchy stress tensor based on a path-independent interfacial energy function. By differentiating this interfacial energy function against a Lagrangian strain tensor, a nearly closed-form solution for the stress tensor was determined, involving only elementary algebraic and matrix operations. From this process, the stress-conformation relation proposed by Doi and Ohta is also confirmed from a thermodynamic perspective. The testing cases with uniaxial elongation and simple shear further showed improved fitting to the analytical or exact solutions.     

n the fourth order tensor valued function of the stress in return map algorithm

针对各向同性材料,基于一组相互正交的基张量,建立了一套有 效的相关运算方法. 基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则 使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴. 根据张量函数表示 定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组 基所表示. 推演结果表明：这些张量之间的运算,表现为对应系数矩阵之间的简单 关系. 其中,四阶张量求逆归结为对应的3\times3系数矩阵求逆,它对二阶张量的变换 则表现为该矩阵对3times 1列阵的变换. 最后,对这些变换关系应用于返回映 射算法的迭代格式进行了相关讨论.     

关于返回映射算法中应力的四阶张量值函数

针对各向同性材料,基于一组相互正交的基张量,建立了一套有效的相关运算方法.基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴.根据张量函数表示定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组基所表示.推演结果表明:这些张量之间的运算,表现为对应系数矩阵之间的简单关系.其中,四阶张量求逆归结为对应的3×3系数矩阵求逆,它对二阶张量的变换则表现为该矩阵对3×1列阵的变换.最后,对这些变换关系应用于返回映射算法的迭代格式进行了相关讨论.     

Force flux and the peridynamic stress tensor

The peridynamic model is a framework for continuum mechanics based on the idea that pairs of particles exert forces on each other across a finite distance. The equation of motion in the peridynamic model is an integro-differential equation. In this paper, a notion of a peridynamic stress tensor derived from nonlocal interactions is defined. At any point in the body, this stress tensor is obtained from the forces within peridynamic bonds that geometrically go through the point. The peridynamic equation of motion can be expressed in terms of this stress tensor, and the result is formally identical to the Cauchy equation of motion in the classical model, even though the classical model is a local theory. We also establish that this stress tensor field is unique in a certain function space compatible with finite element approximations.     

A non-quadratic yield function for polymeric foams

The yield behavior of a closed cell polymeric foam is investigated under multiaxial loadings. A phenomenological yield function is developed to characterize the initial yield behavior of the closed cell polymeric foam under a full range of loading conditions. The principal stresses of a relative stress tensor and the second invariant of the deviatoric stress tensor are the main parameters in the yield function. The yield function is a linear combination of non-quadratic functions of the relative principal stresses and the second invariant of the deviatoric stress tensor. The convexity of the yield surface based on the non-quadratic yield function is proved. The non-quadratic yield function is shown to well characterize the yield behavior of a closed cell polymeric foam in [Deshpande, V.S., Fleck, N.A., 2001. Multi-axial yield behavior of polymer foams. Acta Materialia 49, 1859–1866] under a full range of loading conditions. Finally, a comparison of different phenomenological yield functions to characterize the yield behavior of the foam is presented.     

A novel internal dissipation inequality by isotropy and its implication for inelastic constitutive characterization

In the present work a novel inelastic deformation caused internal dissipation inequality by isotropy is revealed. This inequality has the most concise form among a variety of internal dissipation inequalities, including the one widely used in constitutive characterization of isotropic finite strain elastoplasticity and viscoelasticiy. Further, the evolution term describing the difference between the rate of deformation tensor and the “principal rate” of the elastic logarithmic strain tensor is set, according to the standard practice by isotropy, to equal a rank-two isotropic tensor function of the corresponding branch stress, with the tensor function having an eigenspace identical to the eigenspace of the branch stress tensor. Through that a general form of evolution equation for the elastic logarithmic strain is formulated and some interesting and important results are derived. Namely, by isotropy the evolution of the elastic logarithmic strain tensor is embodied separately by the evolutions of its eigenvalues and eigenprojections, with the evolution of the eigenprojections driven by the rate of deformation tensor and the evolution of the eigenvalues connected to specific material behavior. It can be proved that by isotropy the evolution term in the present dissipation inequality stands for the essential form of the evolution term in the extensively applied dissipation inequality.     

Load-induced oriented damage and anisotropy of rock-like materials

Theoretical model for deformability of brittle rock-like materials in the presence of an oriented damage of their internal structure is formulated and verified experimentally. This model is based on the assumption that non-linearity of the stress–strain curves of these materials is a result of irreversible process of oriented damage growth. It was also assumed that a material response, represented by the strain tensor, is a function of two tensorial variables: the stress tensor and the damage effect tensor that is responsible for the current state of the internal structure of the material. The explicit form of the respective non-linear stress–strain relations that account for the appropriate damage evolution equation was obtained by employing the theory of tensor function representations and by using the results of own experiments on damage growth. Such an oriented damage that grows in the material, described by the second order symmetric damage effect tensor, results in gradual development of the material anisotropy. The validity of the constitutive equations proposed was verified by using the available experimental results for concrete subjected to the plane state of stress. The relevant experimental data for sandstone and concrete subjected to tri-axial state of stress were also used.     

Coaxiality of stress and strain in anisotropic no-tension materials

For an anisotropic no-tension material there exist at least two rotations such that stress and strain become coaxial. The same result holds for any hyperelastic material whose response is expressed in terms of the small strain tensor and whose stress function is a continuous positively homogeneous degree 1 function.     

Analyse contrastive de modèles incrémentaux et hypoplastiques

In this paper we first establish two necessary and sufficient conditions in order that incremental constitutive equations expressing the strain rate tensor as a function of the Jaumann's derivative of the Cauchy's stress tensor can be inverted under the general form of hypoplastic models when the stress state is located inside the domain bounded by the limit state surface. We are then interested in the physical meaning of these conditions with regard to the incremental response of the material.     

Convergence of Peridynamics to Classical Elasticity Theory

The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.      

Hypoelastic form of equations in nonlinear elasticity theory

It is shown that the complete system of equations of elasticity theory for an isotropic medium admits a unique representation in the hypoelastic form (the tensor of the rate of change of stresses is a linear function of the tensor of strain rates with coefficients depending on the invariants of the stress tensor). It is necessary to this end that the hypothesis be satisfied on the determination of strains by stresses which are unknown. Any arbitrariness in the choice of the coefficients of the hypoelastic relation may result in the thermodynamic identity being infringed.     

Stress and fabric in granular material

It has been well recognized that, due to anisotropic packing structure of granular material, the true stress in a specimen is different from the applied stress. However, very few research efforts have been focused on quantifying the relationship between the true stress and applied stress. In this paper, we derive an explicit relationship among applied stress tensor, material-fabric tensor, and force-fabric tensor; and we propose a relationship between the true stress tensor and the applied stress tensor. The validity of this derived relationship is examined by using the discrete element simulation results for granular material under biaxial and triaxial loading conditions.     

Relation between the stress and couple-stress tensors in the microcontinuum theory of elasticity

The stress tensor is expressed in terms of an arbitrary symmetric tensor field of second rank and the couple-stress tensor. The stress and couple-stress tensors are represented by arbitrary tensor fields satisfying the homogeneous equilibrium equations. These tensors are also given in the form of the expressions satisfying the inhomogeneous equilibrium equations used in the microcontinuum theory of elasticity. The stress tensor functions are considered.     

Representations for isotropic thermoelastic dielectrics

Using Chakrabarti and Wainwright's method, an integrity basis consisting of eighteen invariants is constructed for the energy function of deformation and polarization of Isotropic thermoelastic dielectrics. The minimality of the integrity basis is verified by group theoretic methods and this minimal integrity basis is employed to construct closed form irreducible polynomial representations for the electric tensor, the stress tensor and the electric and heat flux vectors.     

General invariant representations of the constitutive equations for isotropic nonlinearly elastic materials

This paper develops general invariant representations of the constitutive equations for isotropic nonlinearly elastic materials. Different sets of mutually orthogonal unit tensor bases are constructed from the strain argument tensor by using the representation theorem and corresponding irreducible invariants are defined. Their relations and geometrical interpretations are established in three dimensional principal space. It is shown that the constitutive law linking the stress and strain tensors is revealed to be a simple relationship between two vectors in the principal space. Relative to two different sets of the basis tensors, the constitutive equations are transformed according to the transformation rule of vectors. When a potential function is assumed to exist, the vector associated with the stress tensor is expressed in terms of its gradient with respect to the vector associated with the strain tensor. The Hill’s stability condition is shown to be that the scalar product of the increment of those two vectors must be positive. When potential function exists, it becomes to be that the 3 × 3 constitutive matrix derived from its second order derivative with respect to the vector associated with the strain must be positive definite. By decomposing the second order symmetric tensor space into the direct sum of a coaxial tensor subspace and another one orthogonal to it, the closed form representations for the fourth order tangent operator and its inversion are derived in an extremely simple way.     

Flows of constant stretch history for polymeric materials with power-law distributions of relaxation times

It is herein shown that for separable integral constitutive equations with power-law distributions of relaxation times, the streamlines in creeping flow are independent of flow rate.For planar flows of constant stretch history, the stress tensor is the sum of three terms, one proportional to the rate-of-deformation tensor, one to the square of this tensor, and the other to the Jaumann derivative of the rate-of-deformation tensor. The three tensors are the same as occur in the Criminale-Ericksen-Filbey Equation, but the coefficients of these tensors depend not only on the second invariant of the strain rate, but also on another invariant which is a measure of flow strength. With the power-law distribution of relaxation times, each coefficient is equal to the second invariant of the strain rate tensor raised to a power, times a function that depends only on strength of the flow. Axisymmetric flows of constant stretch history are more complicated than the planar flows, because three instead of two nonzero normal components appear in the velocity gradient tensor. For homogeneous axisymmetric flows of constant stretch history, the stress tensor is given by the sum of the same three terms. The coefficients of these terms again depend on the flow strength parameter, but in general the dependences are not the same as in planar flow.     

Inelastic constitutive equations for complex flows

A new class of inelastic constitutive equations is presented and discussed. In addition to the rate-of-strain tensor, the stress is assumed to depend also on the relative-rate-of-rotation tensor, a frame-indifferent quantity that brings information about the nature of the flow. The material functions predicted by these constitutive equations are given for simple shear and uniaxial extension. A special case of these equations takes the Newtonian form, except that the viscosity is a function of the invariants of both kinematic tensors on which the stress depends. This simple constitutive equation has potential applications in liquid flow process simulations, since it combines simplicity with the capability of responding independently to shear and extension, as real liquids seem to do. Finally, possible forms for the new viscosity function are discussed.     

Statics and kinematics of discrete Cosserat-type granular materials

A theoretical framework is presented for the statics and kinematics of discrete Cosserat-type granular materials. In analogy to the force and moment equilibrium equations for particles, compatibility equations for closed loops are formulated in the two-dimensional case for relative displacements and relative rotations at contacts. By taking moments of the equilibrium equations, micromechanical expressions are obtained for the static quantities average Cauchy stress tensor and average couple stress tensor. In analogy, by taking moments of the compatibility equations, micromechanical expressions are obtained for the (infinitesimal) kinematic quantities average rotation gradient tensor and average Cosserat strain tensor in the two-dimensional case. Alternatively, these expressions for the average Cauchy stress tensor and the average couple stress tensor are obtained from considerations of the equivalence of the continuum force and couple traction vectors acting on a plane and the resultant of the discrete forces and couples acting on this plane. In analogy, the expressions for the average rotation gradient tensor and the average Cosserat strain tensor are obtained from considerations of the change of length and change of rotation of a line element in the two-dimensional case. It is shown that the average particle stress tensor is always symmetrical, contrary to the average stress tensor of an equivalent homogenized continuum. Finally, discrete analogues of the virtual work and complementary virtual work principles from continuum mechanics are derived.     

The solid phase stress tensor in porous media mechanics and the Hill-Mandel condition

An assessment of the stress tensors used currently for the modeling of partially saturated porous media is made which includes concepts like total stress, solid phase stress, and solid pressure. Thermodynamically constrained averaging theory is used to derive the solid phase stress tensor. It is shown that in the upscaling procedure the Hill conditions are satisfied, which is not trivial. The stress tensor is then compared to traditional stress measures. The physical meaning of two forms of solid pressure and of the Biot coefficient is clarified. Finally, a Bishop-Skempton like form of the stress tensor is obtained and a form of the total stress tensor that does not make use of the effective stress concept.