New explicit expressions of the Hill polarization tensor for general anisotropic elastic solids

Except for particular cases, the classical expressions of the Eshelby or Hill polarization tensors, depend, respectively, on a simple or double integral for a fully anisotropic two-dimensional or three-dimensional elastic body. When the body is two-dimensional, we take advantage of Cauchy’s theory of residues to derive a new explicit expression which depends on the two pairs of complex conjugate roots of a quartic equation. If the body exhibits orthotropic symmetry, these roots are explicitly given as a function of the independent components of the elasticity tensor. Similarly, the double integral is reduced to a simple one when the body is three-dimensional. The corresponding integrand depends on the three pairs of complex conjugate roots of a sextic equation which reduces to a cubic one for orthotropic symmetry. This new expression improves significantly the computation times when the degree of anisotropy is high. For both two and three-dimensional bodies, degenerate cases are also studied to yield valid expressions in any events.     

On the construction of the green tensor for an orthotropic elastic medium

The Lifshits–Rosenzweig method is used to obtain the general solution for the Fourier amplitudes and find the general form of the coefficients of a sixth-order equation in the case of an orthotropic elastic medium and also compute the Green tensor for the special case of an orthotropic medium, namely, hard pine (Pinus australis).     

An ‘extended’ volumetric/deviatoric formulation of anisotropic damage based on a pseudo-log rate

Following a framework of elastic degradation and damage previously proposed by the authors, an ‘extended’ formulation of orthotropic damage in initially isotropic materials, based on volumetric/deviatoric decomposition, is presented. The formulation is founded on the concept of energy equivalence and makes use of second-order symmetric tensor damage variables. It is characterized by fourth-order damage-effect tensors (relating nominal to effective stresses and strains) built from the underlying second-order damage tensors and decomposed in product-form in isotropic and anisotropic parts. The formulation is developed in two steps. First, secant relations are established. In the isotropic case, the model embeds a path parameter allowing to range between pure volumetric to pure deviatoric damage. With the two undamaged material constants this makes a total of three constant parameters plus an evolving scalar damage variable, giving rise to a four-parameter model with two varying isotropic material coefficients. In the anisotropic case, the model is still characterized by the same three material constants plus three evolving variables which are the principal values of a second-order damage tensor. This leads to a six-parameter restricted form of orthotropic damage. In the second step, damage evolution rules are formulated in terms of a pseudo-logarithmic rate of damage. This allows to define meaningful conjugate forces that constitute a feasible space in which loading functions and damage evolution rules can be defined. The present ‘extended’ formulation is closed by the derivation of the tangent stiffness.     

Simple Torsion of Isotropic, Hyperelastic, Incompressible Materials with Limiting Chain Extensibility

The purpose of this research is to investigate the simple torsion problem for a solid circular cylinder composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Three popular models that account for hardening at large deformations are examined. These models involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The main mechanical quantities of interest in the torsion problem are obtained in closed form. In this way, it is shown that the torsional response of all three materials is similar. While the predictions of the models agree qualitatively with experimental data, the quantitative agreement is poor as is the case for the neo-Hookean material. In fact, by using a global universal relation, it is shown that the experimental data cannot be predicted quantitatively by any strain-energy density which depends solely on the first invariant. It is shown that a modification of the strain energies to include a term linear in the second invariant can be used to remedy this defect. Whether the modified strain-energies, which reflect material hardening, are a feasible alternative to the classic Mooney–Rivlin model remains an open question which can be resolved only by large strain experiments. This revised version was published online in August 2006 with corrections to the Cover Date.     

In-plane/out-of-plane concentrated forces and moments on composite laminates with elliptical elastic inclusions

The problems of composite laminates containing elliptical elastic inclusions subjected to concentrated forces and moments are considered in this paper. By employing Stroh-like formalism for the coupled stretching–bending analysis, analytical closed form solutions are obtained explicitly. The generality of the solutions provided in this paper can be shown as follows: (1) The laminates include any kinds of laminate lay-ups, symmetric or unsymmetric, which allow the stretching and bending deformations couple each other. (2) The concentrated forces and moments can be applied in in-plane and/or out-of-plane directions, located inside and/or outside the inclusions. (3) The elliptical elastic inclusions can be any kinds of elastic materials including the limiting cases such as holes, rigid inclusions, cracks, line inclusions, etc. Since no such general solution has been found in the literature, the solutions are checked and verified by the special cases that no inclusions are embedded in the laminates, and that the inclusions are replaced by holes. Moreover, with various hardness ratios of inclusion and matrix some numerical examples showing the stress resultants along the interface are presented. Like the Green’s functions for the infinite laminates and those containing holes/cracks, the present solutions associated with the in-plane concentrated forces and out-of-plane concentrated moments have exactly the same mathematical form as those of the corresponding two-dimensional problems, in which the only difference is the contents of the symbols. While for the other loading cases, new types of solutions are obtained explicitly.     

正交各向异性材料的弹塑性损伤本构关系

基于弹塑性力学和损伤理论,建立了一个与应力球张量有关的正交各向异性材料的混合硬化屈服准则,该准则无量纲化后与各向同性材料的Mises准则同构,在此基础上,进而建立了混合硬化正交各向异性材料的增量型弹塑性损伤本构方程,并以具局部损伤的正交各向异性矩形薄板为例,采用Galerkin法和迭代法,对其弹塑性屈曲问题进行了分析,讨论了局部损伤对正交各向异性矩形薄板弹塑性屈曲临界应力的影响.      

An anisotropic elastoplastic constitutive formulation generalised for orthotropic materials

This paper presents a finite strain constitutive model to predict a complex elastoplastic deformation behaviour that involves very high pressures and shockwaves in orthotropic materials using an anisotropic Hill’s yield criterion by means of the evolving structural tensors. The yield surface of this hyperelastic–plastic constitutive model is aligned uniquely within the principal stress space due to the combination of Mandel stress tensor and a new generalised orthotropic pressure. The formulation is developed in the isoclinic configuration and allows for a unique treatment for elastic and plastic orthotropy. An isotropic hardening is adopted to define the evolution of plastic orthotropy. The important feature of the proposed hyperelastic–plastic constitutive model is the introduction of anisotropic effect in the Mie–Gruneisen equation of state (EOS). The formulation is further combined with Grady spall failure model to predict spall failure in the materials. The proposed constitutive model is implemented as a new material model in the Lawrence Livermore National Laboratory (LLNL)-DYNA3D code of UTHM’s version, named Material Type 92 (Mat92). The combination of the proposed stress tensor decomposition and the Mie–Gruneisen EOS requires some modifications in the code to reflect the formulation of the generalised orthotropic pressure. The validation approach is also presented in this paper for guidance purpose. The \({\varvec{\psi }}\) tensor used to define the alignment of the adopted yield surface is first validated. This is continued with an internal validation related to elastic isotropic, elastic orthotropic and elastic–plastic orthotropic of the proposed formulation before a comparison against range of plate impact test data at 234, 450 and \({\mathrm {895\,ms}}^{\mathrm {-1}}\) impact velocities is performed. A good agreement is obtained in each test.     

Derivatives of the three-dimensional Green’s functions for anisotropic materials

A concise formulation is presented for the derivatives of Green’s functions of three-dimensional generally anisotropic elastic materials. Direct calculation for derivatives of the Green’s function on the Cartesian coordinate system is a common practice, which, however, usually leads to a complicated course. In this paper the Green’s function derived by Ting and Lee [Ting, T.C.T., Lee, V.G., 1997. The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids. The Quarterly Journal of Mechanics and Applied Mathematics 50 (3) 407–426] is extended to obtain the derivatives. Using a spherical coordinate system, the Green’s function can be shown as the composition of two independent functions, one depends only on the radial distance of the field point to the origin and the other is in spherical angles. The method of derivation is based on the total differential scheme and then takes its partial differentiation accordingly. With the application of Cauchy residue theorem, the contour integral can be evaluated in terms of the Stroh eigenvalues of a sextic equation. For the degenerate case, evaluation of residues at multiple poles is also given. Applications of the present result are made to examine the Green’s functions and stress components for isotropic and transversely isotropic materials. The results are in exact agreement with existing solutions.     

A 3D finite volume scheme for solving the updated Lagrangian form of hyperelasticity

The finite volume discretization of nonlinear elasticity equations seems to be a promising alternative to the traditional finite element discretization as mentioned by Lee et al. [Computers and Structures (2013)]. In this work, we propose to solve the elastic response of a solid material by using a cell‐centered finite volume Lagrangian scheme in the current configuration. The hyperelastic approach is chosen for representing elastic isotropic materials. In this way, the constitutive law is based on the principle of frame indifference and thermodynamic consistency, which are imposed by mean of the Coleman–Noll procedure. It results in defining the Cauchy stress tensor as the derivative of the free energy with respect to the left Cauchy–Green tensor. Moreover, the materials being isotropic, the free‐energy is function of the left Cauchy–Green tensor invariants, which enable the use of the neo‐Hookean model. The hyperelasticity system is discretized using the cell‐centered Lagrangian scheme from the work of Maire et al. [J. Comput. Phys. (2009)]. The 3D scheme is first order in space and time and is assessed against three test cases with both infinitesimal displacements and large deformations to show the good accordance between the numerical solutions and the analytic ones. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlocal continuum crystal plasticity with internal residual stresses

We derive a three-dimensional constitutive theory accounting for length-scale dependent internal residual stresses in crystalline materials that develop due to a non-homogeneous spatial distribution of the excess dislocation (edge and screw) density. The second-order internal stress tensor is derived using the Beltrami stress function tensor φ that is related to the Nye dislocation density tensor. The formulation is derived explicitly in a three-dimensional continuum setting for elastically isotropic materials. The internal stresses appear as additional resolved shear stresses in the crystallographic visco-plastic constitutive law for individual slip systems. Using this formulation, we investigate two boundary value problems involving single crystals under symmetric double slip. In the first problem, the response of a geometrically imperfect specimen subjected to monotonic and cyclic loading is investigated. The internal stresses affect the overall strengthening and hardening under monotonic loading, which is mediated by the severity of initial imperfections. Such imperfections are common in miniaturized specimens in the form of tapered surfaces, fillets, fabrication induced damage, etc., which may produce strong gradients in an otherwise nominally homogeneous loading condition. Under cyclic loading the asymmetry in the tensile and compressive strengths due to this internal stress is also strongly influenced by the degree of imperfection. In the second example, we consider simple shear of a single crystalline lamella from a layered specimen. The lamella exhibits strengthening with decreasing thickness and increasing lattice incompatibility with shearing direction. However, as the thickness to internal length-scale ratio becomes small the strengthening saturates due to the saturation of the internal stress.Finally, we present the extension of this approach for crystalline materials exhibiting elastic anisotropy, which essentially depends on the appropriate Green function within φ.     

Superposing scheme for the three-dimensional Green’s functions of an anisotropic half-space

This paper presents a method of superposition for the half-space Green’s functions of a generally anisotropic material subjected to an interior point loading. The mathematical concept is based on the addition of a complementary term to the Green’s function in an anisotropic infinite domain. With the two-dimensional Fourier transformation, the complementary term is derived by solving the generalized Stroh eigenrelation and satisfying the boundary conditions on the free surface with the use of Green’s functions in the full-space case. The inverse Fourier transform leads to the contour integrals, which can be evaluated with the application of Cauchy residue theorem. Application of the present results is made to obtain analytical expression for the orthotropic materials which were not reported previously. The closed-form solutions for the transversely isotropic and isotropic materials derived directly from the solutions as being a special case are also given in this paper.     

Finite strain––isotropic hyperelasticity

This paper presents a strain energy density for isotropic hyperelastic materials. The strain energy density is decomposed into a compressible and incompressible component. The incompressible component is the same as the generalized Mooney expression while the compressible component is shown to be a function of the volume invariant J only. The strain energy density proposed is used to investigate problems involving incompressible isotropic materials such as rubber under homogeneous strain, compressible isotropic materials under high hydrostatic pressure and volume change under uniaxial tension. Comparison with experimental data is good. The formulation is also used to derive a strain energy density expression for compressible isotropic neo-Hookean materials. The constitutive relationship for the second Piola–Kirchhoff stress tensor and its physical counterpart, involves the contravariant Almansi strain tensor. The stress stretch relationship comprises of a component associated with volume constrained distortion and a hydrostatic pressure which results in volumetric dilation. An important property of this constitutive relationship is that the hydrostatic pressure component of the stress vector which is associated with volumetric dilation will have no shear component on any surface in any configuration. This same property is not true for a neo-Hookean Green’s strain–second Piola–Kirchhoff stress tensor formulation.     

A constitutive model for orthotropic elasto-plasticity at large strains

Summary The paper presents a thermodynamically consistent constitutive model for elasto-plastic analysis of orthotropic materials at large strain. The elastic and plastic anisotropies are assumed to be persistent in the material but the anisotropy axes can undergo a rigid rotation due to large plastic deformations. The orthotropic yield function is formulated in terms of the generally nonsymmetric Mandel stress tensor such that its skew-symmetric part is additionally taken into account. Special attention is focused on the convexity of the yield surface resulting in the nine-dimensional stress space. Of particular interest are new convexity conditions which do not appear in the classical theory of anisotropic plasticity. They impose additional constraints on the material constants governing the plastic spin. The role of the plastic spin is further studied in simple shear accompanied by large elastic and large plastic deformations. If the plastic spin is neglected, the shear stress response is characterized by oscillations with an amplitude strictly dependent on the degree of the plastic anisotropy.accepted for publication 2 March 2004     

The Explicit Determination of the Logarithm of a Tensor and Its Derivatives

The logarithm of a tensor is often used in nonlinear constitutive relations of elastic materials. Here we show how the logarithm of an arbitrary tensor can be explicitly evaluated for any underlying space dimension n. We also present a method for the explicit evaluation of the derivatives of the logarithm of a tensor.      

On the Existence of Longitudinal Plane Waves in General Elastic Anisotropic Media

We give a new proof of Kolodner's result that longitudinal waves can propagate in at least three directions in a hyperelastic anisotropic medium. We give examples of an orthotropic hyperelastic tensor with exactly three such directions, of a monoclinic elastic (but not hyperelastic) tensor with only one, and of a monoclinic elastic (elliptic, but not uniformly elliptic) tensor with no direction for longitudinal waves. This revised version was published online in August 2006 with corrections to the Cover Date.     

A thin-walled composite beam model for light-weighted structures interacting with fluids

A thin-walled beam model is proposed for structures of variable cross-section, which can be either open or closed and includes multicellular cross-sections with either isotropic or orthotropic materials. The proposed model does not require any priori definition of cross-sectional warping which instead results from the solution of the problem. To achieve that a special deformation pattern is superimposed on the bending deformation described by Euler–Bernoulli beam theory. All sectional properties are automatically incorporated in the analysis as a result of the usual variational formulation of the system of equations. The proposed model is specifically designed to simulate the dynamics of wind/hydrokinetic turbine blade with low computational cost, especially in fluid–structure interaction (FSI) simulation. A number of test cases have been carried out to validate the proposed structural model which show good agreement between the results obtained her e and the solutions available in literature. Finally, FSI simulation of a hydrokinetic blade under field condition is carried out to illustrate the capability of the current thin-walled beam model in practice.     

具确定弱区域正交各向异性薄板的弹塑性屈曲分析

基于弹塑性力学和损伤理论,建立了一个与应力球张量有关的具损伤正交各向异性材料的混合硬化屈服准则,该准则无量纲化后与各向同性材料的Mises准则同构,在此基础上,建立了正交各向异性材料的增量型和全量型弹塑性损伤本构方程,并以具确定弱区域正交各向异性矩形薄板为例,根据屈曲时的能量准则和全量理论,以等效塑性应变为内变量,对其弹塑性屈曲问题进行了分析,讨论了几何参数和弱区域对正交各向异性薄板弹塑性屈曲临界应力的影响.     

Laws of viscoplastic fluid flow in anisotropic porous and fractured media

In invariant tensor form, the laws of viscoplastic fluid flow are formulated for capillary and fractured media with a periodic microstructure that has orthotropic and transversely isotropic symmetry in the flow properties. An analysis of the laws of viscoplastic fluid flow in transversely isotropic and orthotropic porous and fractured media shows that in formulating the equations it is necessary to distinguish between the permeability tensor and the limiting gradient tensor, which may differ in the symmetry of the flow characteristics, and that the flow law is multivariant and admits one-, two-, and three-dimensional flows.     

Elasto-plastic postbuckling of damaged orthotropic plates

Based on the elasto-plastic mechanics and continuum damage theory, a yield criterion related to spherical tensor of stress is proposed to describe the mixed hardening of damaged orthotropic materials. Its dimensionless form is isomorphic with the Mises criterion for isotropic materials. Furthermore, the incremental elasto-plastic damage constitutive equations and damage evolution equations are established. Based on the classical nonlinear plate theory, the incremental nonlinear equilibrium equations of orthotropic thin plates considering damage effect are obtained, and solved with the finite difference and iteration methods. In the numerical examples, the effects of damage evolution and initial deflection on the elasto-plastic postbuckling of orthotropic plates are discussed in detail.