Simulation of strain-induced anisotropy for polymers with weighting functions

The alignment of polymer chains is a well-known microstructural evolution effect due to straining of polymers. This has a drastic influence on the macroscopic properties of the initially isotropic material, such as a pronounced strength in the loading direction of stretched films. For modeling the effect of strain-induced anisotropy, a macroscopic constitutive model is developed in this paper. Within a thermodynamic framework, an additive decomposition of the logarithmic Hencky strain tensor into elastic and inelastic parts is used to formulate the constitutive equations. As a key idea, weighting functions are introduced to represent a strain-softening/hardening effect to account for induced anisotropy. These functions represent the ratio between the total strain rate (representing the actual loading direction) and a structural tensor (representing the stretched polymer chains). In this way, we introduce material parameters as a sum of weighted direction-related quantities. The numerical implementation of the resulting set of constitutive is used to identify material parameters based on experimental data, exhibiting strain-induced anisotropy. In the finite-element examples, we simulate the cold-forming of amorphous thermoplastic films below the glass transition temperature subjected to different re-loading directions.     

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

Remarks on the Behavior of Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solids

The effect of directional reinforcing in generating qualitative changes in the mechanical response of a base neo-Hookean material is examined in the context of homogenous deformation. Single axis reinforcing giving transverse isotropy is the major focus, in which case a standard reinforcing model is characterized by a single constitutive reinforcing parameter. Various qualitative changes in the mechanical response ensue as the reinforcing parameter increases from the zero-value associated with neo-Hookean response. These include (in order): the existence of a limiting contractive stretch for transverse-axis tensile load; loss of monotonicity in off-axis simple shear; loss of monotonicity in on-axis compression; loss of positivity in the stress-shear product in off-axis simple shear; and loss of monotonicity for plane strain in on-axis compression. The qualitative changes in the simple shear response are associated with stretch relaxation in the reinforcing direction due to finite rotation. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

非均匀颗粒材料的类固-液相变行为及本构方程

以非均匀颗粒介质为研究对象,采用三维离散元方法对其在不同密集度和剪切速率下的动 力过程进行了数值模拟,分析了其在由瞬时接触的快速流动向持续接触的准静态流动的转变 过程及其行为特点. 通过对不同材料性质下相变过渡区内颗粒材料的宏观应力、接触时间数、 配位数、团聚颗粒数量、有效摩擦系数等参量的计算,更加全面地描述了非均匀颗粒材料在 类固-液相变过程中的基本特征. 基于以上数值计算结果,建立了一个适用于颗粒材料 类固态、类液态以及其相变过程的本构方程,并通过剪切室实验结果验证了它的合理性.     

Study of the Turbulent Velocity Field in the Near Wake of a Bluff Body

An experimental characterization of the turbulent flow structure formed downstream of a vertically mounted circular bluff body is performed. Three components of an instantaneous velocity field are measured using the stereo particle image velocimetry technique at the symmetry plane. The average velocity and the turbulent properties are analyzed. The results indicate a recirculation zone consisting of a toroidal vortex with similar dimensions for all Reynolds numbers. The largest turbulent fluctuations are found at the stagnation point region. The observed anisotropy of the normal Reynolds stress components is associated with the stagnation point flow, whereas the cross-correlation component extreme occurs in high strain rate regions. An analysis of the Reynolds tensor anisotropy using the Lumley triangle is performed, revealing that the largest departures from isotropy occur at high shear regions and also within the vortex.     

A theory of thermo-visco-elastic-plastic materials: thermomechanical coupling in simple shear

The constitutive relations of a theory of thermo-visco-elastic-plastic continuum have been formulated in Lagrangian form. The Lagrangian strains, strain rates, temperature, temperature rate and temperature gradients are considered as the independent constitutive variables. Three internal state variables (plastic strain tensor, back strain tensor and a scalar hardening parameter) are also incorporated. The axioms of objectivity and equipresence are followed. The Clausius–Duhem inequality is taken as the second law of thermodynamics. Several special theories are deduced based on material symmetries and/or conventionally adopted assumptions. The applications to the formation of shear bands and dynamic crack propagation are discussed.     

A ferroelectric and ferroelastic 3D hexahedral curvilinear finite element

An isoparametric 3D electromechanical hexahedral finite element integrating a 3D phenomenological ferroelectric and ferroelastic constitutive law for domain switching effects is proposed. The model presents two internal variables which are the ferroelectric polarization (related to the electric field) and the ferroelastic strain (related to the mechanical stress). An implicit integration technique of the constitutive equations based on the return-mapping algorithm is developed. The mechanical strain tensor and the electric field vector are expressed in a curvilinear coordinate system in order to handle the transverse isotropy behavior of ferroelectric ceramics. The hexahedral finite element is implemented into the commercial finite element code Abaqus® via the subroutine user element. Some linear (piezoelectric) and non linear (ferroelectric and ferroelastic) benchmarks are considered as validation tests.     

Tresca-type plastic materials in the theory of hypoelasticity II. Optical constitutive equations and birefringence in simple shear

Behavior of a Tresca type plastic dielectric is investigated theoretically from a continuum mechanical point of view. The optical constitutive equations are defined as special cases of a hypo-elastic dielectric of grade two. The singularity condition of the constitutive equations satisfies the Tresca yield criterion. The index deviator tensor is proportional to the stress deviator tensor and, then, the birefringence and the extinction angle are expressed by the stress deviator. Their numerical variations with the angle of shear in simple shear deformation are shown.     

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.      

Comment on: I-Shih Liu: Constitutive theory of anisotropic rigid heat conductors

In I-Shih Liu’s paper (J Math Phys 50:083506, 2009), the compatibility of anisotropy and material frame indifference of a rigid heat conductor is investigated. For this purpose, the deformation gradient is introduced into the domain of the constitutive mapping. Because of the presupposed rigidity, the deformation gradient is here represented by an orthogonal tensor. The statement, that the usual procedure—not to introduce the deformation gradient into the state space of rigid heat conductors—causes isotropy because of material frame indifference, is misleading.     

Propagation of Lamb waves in transversely isotropic thermoelastic diffusive plate

The constitutive relations and field equations for anisotropic generalized thermoelastic diffusion are derived and deduced for a particular type of anisotropy, i.e. transverse isotropy. Green and Lindsay (GL) theory, in which, thermodiffusion and thermodiffusion–mechanical relaxations are governed by four different time constants, is selected for study. The propagation of plane harmonic thermoelastic diffusive waves in a homogeneous, transversely isotropic, elastic plate of finite width is studied, in the context of generalized theory of thermoelastic diffusion. According to the characteristic equation, three quasi-longitudinal waves namely, quasi-elastodiffusive (QED-mode), quasi-massdiffusive (QMD-mode) and quasi-thermodiffusive (QTD-mode) can propagate in addition to quasi-transverse waves (QSV-mode) and the purely quasi-transverse motion (QSH-mode), which is not affected by thermal and diffusion vibrations, gets decoupled from the rest of the motion of wave propagation. The secular equations corresponding to the symmetric and skew symmetric modes of the plate are derived. The amplitudes of displacements, temperature change and concentration for symmetric and skew symmetric modes of vibration of plate are computed numerically. Anisotropy and diffusion effects on the phase velocity, attenuation coefficient and amplitudes of wave propagation are presented graphically in order to illustrate and compare the analytically results. Some special cases of frequency equation are also deduced from the existing results.     

ANISOTROPIC STRENGTH CRITERION BASED ON T CRITERION AND THE TRANSFORMATION STRESS METHOD 1)

岩土材料通常呈现出成层水平分布特点, 即可将其视为横观各向同性材料, 横观各向同性对于岩土材料的变形以及强度值都会产生显著的影响. 基于已提出的t强度准则, t强度准则是基于各向同性单元体中存在有效滑移面来构建的, 并根据该空间有效滑移面上主剪应力与主法向应力的比值达到一定阈值为破坏条件. 在空间中存在有效滑移面与物理沉积面, 基于上述两个面在空间的位置关系, 用两面夹角作为表征横观各向同性对剪切强度影响程度的参量, 并假定当该夹角值越大, 则各向异性对强度贡献程度越大, 对应更大的应力比强度值, 反之, 则对应更小的应力比强度值. 基于上述思路并类比将其推广为正交三维各向异性准则, 基于三维各向异性材料的三维沉积面, 提出了三维特征沉积面的概念, 并基于空间滑移面与三维特征沉积面之间的夹角作为度量各向异性程度的变量, 提出了基于两面角作为参量考虑原生各向异性的应力比强度公式, 并利用该应力比强度公式来修正已提出的t强度准则, 最终建立了考虑各向异性影响的t准则公式. 在上述准则基础上, 考虑将各向异性应力空间转换为各向同性应力空间的思路, 在各向异性t准则基础上, 推导得到了基于各向异性强度t准则的变换应力公式, 利用变换应力公式可以将传统的以p, q为变量的各向同性本构模型转变为可考虑各向异性的三维本构模型. 通过对岩土材料的强度以及真三轴条件下的应力应变关系试验数据预测, 验证了所提的各向异性t准则及其变换应力公式的有效性及适用性.     

Bedding plane-emb e dde d augmente d virtual internal bonds for fracture propagation simulation in shale

To effectively simulate the fracture propagation in shale, the bedding plane(BP) effect is incorporated into the augmented virtual internal bond(AVIB) constitutive relation through BP tensor. Comparing the BP-embedded AVIB with the theory of transverse isotropy, it is found the approach can represent the anisotropic properties induced by parallel BPs. Through the simulation example, it is found that this method can simulate the stiffness anisotropy of shale and can represent the effect of BPs on hydraulic fracture propagation direction. Compared with the BP-embedded virtual internal bond(VIB), this method can account for the various Poisson's ratio. It provides a feasible approach to simulate the fracture propagation in shale.     

基于t准则的各向异性强度准则及变换应力法

岩土材料通常呈现出成层水平分布特点, 即可将其视为横观各向同性材料, 横观各向同性对于岩土材料的变形以及强度值都会产生显著的影响. 基于已提出的t强度准则, t强度准则是基于各向同性单元体中存在有效滑移面来构建的, 并根据该空间有效滑移面上主剪应力与主法向应力的比值达到一定阈值为破坏条件. 在空间中存在有效滑移面与物理沉积面, 基于上述两个面在空间的位置关系, 用两面夹角作为表征横观各向同性对剪切强度影响程度的参量, 并假定当该夹角值越大, 则各向异性对强度贡献程度越大, 对应更大的应力比强度值, 反之, 则对应更小的应力比强度值. 基于上述思路并类比将其推广为正交三维各向异性准则, 基于三维各向异性材料的三维沉积面, 提出了三维特征沉积面的概念, 并基于空间滑移面与三维特征沉积面之间的夹角作为度量各向异性程度的变量, 提出了基于两面角作为参量考虑原生各向异性的应力比强度公式, 并利用该应力比强度公式来修正已提出的t强度准则, 最终建立了考虑各向异性影响的t准则公式. 在上述准则基础上, 考虑将各向异性应力空间转换为各向同性应力空间的思路, 在各向异性t准则基础上, 推导得到了基于各向异性强度t准则的变换应力公式, 利用变换应力公式可以将传统的以p, q为变量的各向同性本构模型转变为可考虑各向异性的三维本构模型. 通过对岩土材料的强度以及真三轴条件下的应力应变关系试验数据预测, 验证了所提的各向异性t准则及其变换应力公式的有效性及适用性.      

Overall constitutive description of symmetric layered media based on scattering of oblique SH waves

This papers investigates the scattering of oblique shear horizontal (SH) waves off finite periodic media made of elastic and viscoelastic layers. It further considers whether a Willis-type constitutive matrix (in temporal and spatial Fourier domain) may reproduce the scattering matrix (SM) of such a system. In answering this question the procedure to determine the relevant overall constitutive parameters for such a medium is presented. To do this, first the general form of the dispersion relation and impedances for oblique SH propagation in such coupled Willis-type media are developed. The band structure and scattering of layered media are calculated using the transfer matrix (TM) method. The dispersion relation may be derived based on the eigen-solutions of an infinite periodic domain. The wave impedances associated with the exterior surfaces of a finite thickness slab are extracted from the scattering of such a system. Based on reciprocity and available symmetries of the structure and each constituent layer, the general form of the dispersion and impedances may be simplified. The overall quantities may be extracted by equating the scattering data from TM with those expected from a Willis-type medium. It becomes evident that a Willis-type coupled constitutive tensor with components that are assumed independent of wave vector is unable to reproduce all oblique scattering data. Therefore, non-unique wave vector dependent formulations are introduced, whose SM matches that of the layered media exactly. It is further shown that the dependence of the overall constitutive tensors of such systems on the wave vector is not removable even at very small frequencies and incidence angles and that analytical considerations significantly limit the potential forms of the spatially dispersive constitutive tensors.     

Stress diffusion and high order viscoelastic effects in the 3D flow past a sedimenting sphere subject to orthogonal shear

The effect of polymer stress diffusion in the unbounded flow past a sedimenting, freely rotating, rigid sphere subject to shear in a plane perpendicular to the direction of sedimentation is investigated analytically. Steady state, creeping, incompressible, and isothermal flow is assumed. For viscoelastic fluids following the Oldroyd-B constitutive model, three-dimensional results for the velocity vector, pressure, and viscoelastic extra-stress tensor are derived by including an artificial diffusion term in the constitutive equation and using regular perturbation theory with the small parameter being the Deborah number. The analytical solution reveals that the influence of the stress diffusion term on the results may be significant (and sometimes unexpected) and strongly depends on the magnitude of the dimensionless diffusion coefficient. For instance, it is shown that the critical Deborah number, below which a physical solution arises, decreases with the increase in the diffusion coefficient. Also, comparison against simulation results from the literature shows excellent agreement up to shear Weissenberg number (defined as the product of the imposed shear rate with the single relaxation time of the fluid) approximately equal to unity.     

Anisotropy Consistent with Spherical Symmetry in Continuum Mechanics

We characterize the kinds of anisotropy (besides transverse isotropy) that are compatible with spherically symmetric deformations of balls and spherical shells having nonlinear constitutive equations.