An examination of yield surface distortion and translation

Summary It is shown that the components of a sixth rank anisotropy tensor are physically significant in representing the distortion observed in both the initial and subsequent yield surface for a polycrystalline material. This feature of anisotropy does not appear in the ellipsoidal surface given by previous theories in which second and fourth rank anisotropy tensors are employed. The number of tensor components, for a series function embodying tensor terms in ascending rank, reduces to a manageable number by the imposition of symmetry and coincidence between the axes of stress and principal orthotropic directions. The identification is made between the yield limits, as found from biaxial stress experiments and tensor components in composite sum form. A one-to-one correspondence is found from a further simplification through the assumption of incompressibility. This is confirmed experimentally for an orthotropic rolled copper and copper alloy sheet.An examination is made of the physical significance of a translation tensor which appears in the subsequent yield function. It is shown that the components of this tensor can be directly identified with those of an internal stress tensor that is a consequence of the heterogenous slipped state in a deformed polycrystal.With 8 Figures     

Strongly orthotropic continuum mechanics and finite element treatment

Despite successful application to orthotropic analysis, any Lagrangian strain tensor that is symmetric can be classified as an isotropic metric, while the infinitely orthotropic case can be accurately dealt with using one‐dimensional elements, structural tensors or kinematic constraints. In this paper, we present a strongly orthotropic continuum mechanics basis that models the exact kinematic behavior of the intermediary class of materials and also show its application to multi‐axial media and treatment using the finite element method. By asserting that mechanistic strain metrics must be material property dependent and satisfy equilibrium, we are able to derive a novel orthotropic linear strain tensor that is asymmetric and thus capable of describing all levels of orthotropy, while maintaining generality to the well‐established isotropic approach. Subsequently formulated are a material principal rotation tensor, extended orthotropic compliance tensor and an extended Mohr's plot for strain relying on an additional metric denoted as aspectual strain. Using the developed finite element formulation, it is shown that identical stress results to conventional theory for an orthotropic linear problem are predicted, while offering a more informative analysis. A second numerical example demonstrates the unique capability of this approach to solve the erroneous response of strongly orthotropic materials under trellis shear as compared with a number of conventional and contemporary approaches and thus its ability to produce kinematically exact results. Copyright © 2008 John Wiley & Sons, Ltd.     

A generalized method for characterizing elastic anisotropy in solid living tissues

The microstructure of cortical bone may exhibit either transverse isotropic or orthotropic symmetry, thus requiring either five or nine independent elastic stiffness coefficients (or compliances), respectively, to describe its elastic anisotropy. Our previous analysis to describe this anisotropy in terms of two scalar quantities for the transverse isotropic case is extended here to include orthotropic symmetry. The new results for orthotropic symmetry are compared with previous calculations using the transverse isotropic analysis on the same sets of anisotropic elastic constants for bone, determined either by mechanical or by ultrasonic experiments. In addition, the orthotropic calculation has been applied to full sets of orthotropic elastic stiffness coefficients of a large variety of wood species. Although having some resemblance to plexiform bone in microstructural organization, there is a dramatic difference in both the shear and the compressive elastic anisotropy between the two materials: wood is at least one order of magnitude more anisotropic than bone.     

Fabric tensor based boundary element analysis of porous solids

In this paper, the boundary element analysis of porous solids (sintered materials, foams, etc.) is studied utilizing a fabric tensor. The fabric tensor provides a measure of anisotropy in the solid, as well as information concerning the geometry and distribution of the pores. The homogenized, orthotropic elastic properties of a porous solid can then be predicted with the fabric tensor. To illustrate the analysis, the effect of porosity on a trabecular bone-titanium bimaterial is studied. The boundary element analysis uses an anisotropic, bimaterial Green's function so the interface does not require discretization. It is shown that the anisotropic Stroh variables are independent of the structural density and dependent on the eigenvalues of the fabric tensor. An example calculation is presented where the effect of porosity on the in-plane maximum shear stress in a trabecular bone-titanium bimaterial is substantial.     

An anisotropic linear thermo-viscoelastic constitutive law

A constitutive material law for linear thermo-viscoelasticity in the time domain is presented. The time-dependent relaxation formulation is given for full anisotropy, i.e., both the elastic and the viscous properties are anisotropic. Thereby, each element of the relaxation tensor is described by its own and independent Prony series expansion. Exceeding common viscoelasticity, time-dependent thermal expansion relaxation/creep is treated as inherent material behavior. The pertinent equations are derived and an incremental, implicit time integration scheme is presented.The developments are implemented into an implicit FEM software for orthotropic material symmetry under plane stress assumption. Even if this is a reduced problem, all essential features are present and allow for the entire verification and validation of the approach. Various simulations on isotropic and orthotropic problems are carried out to demonstrate the material behavior under investigation.     

Finite element analysis of nonlinear orthotropic hyperelastic membranes

A finite element method is presented for geometrically and materially nonlinear orthotropic hyperelastic membranes. The constitutive relations are formulated in terms of the invariants of the 2D right Cauchy-Green strain tensor and the resulting system of nonlinear equations solved using a Newton-Raphson approach. Both axisymmetric and nonaxisymmetric versions of the method are developed and validated. Example problems are solved for isotropic and orthotropic membranes, and the effect of various parameters investigated. Finally, convergence studies are performed for various degrees of anisotropy.Also supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant F49620-95-1-0230.     

Boundary element analysis for effective stiffness tensors: effect of fabric tensor determination method

Second-rank fabric tensors have been extensively used to describe structural anisotropy and to predict orthotropic elastic constants. However, there are many different definitions of, and approaches to, determining the fabric tensor. Most commonly used is a fabric tensor based on mean intercept length measurements, but star volume distribution and star length distribution are commonly used, particularly in studies of trabecular bone. Here, we investigate the effect of the fabric tensor definition on elastic constant predictions using both synthetic, idealized microstructures as well as a micrograph of a porous ceramic. We use an efficient implantation of a symmetric Galerkin boundary element method to model the mechanical response of the various microstructures, and also use a boundary element approach to calculate the necessary volume averages of stress and strain to obtain the effective properties of the media.     

A generalized orthotropic hyperelastic material model with application to incompressible shells

In the present paper a new orthotropic hyperelastic constitutive model is proposed which can be applied to the numerical simulation of a wide range of anisotropic materials and particularly biological soft tissues. The model represents a non‐linear extension of the orthotropic St. Venant–Kirchhoff material and is described in each principal material direction by an arbitrary isotropic tensor function coupled with the corresponding structural tensor. In the special case of isotropy this constitutive formulation reduces to the Valanis–Landel hypothesis and may therefore be considered as its generalization to the case of orthotropy. Constitutive relations and tangent moduli of the model are expressed in terms of eigenvalue bases of the right Cauchy–Green tensor C and obtained for the case of distinct and coinciding eigenvalues as well. For the analysis of shells the model is then coupled with a six (five in incompressible case) parametric shell kinematics able to deal with large strains as well as finite rotations. The application of the developed finite shell element is finally illustrated by a number of numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.     

正交异性光弹性应力分析方法研究

本文建立了正交异性光弹性应力分析的实验边界元混合解法并编制了相应的程序系统BE-MSC.该方法仅需模型边界结点处等差线一项实验数据即可实施应力的分离。它还十分适用于存在初应力的模型材料光弹性分析问题。最后对正交异性对径受压圆盘进行了光弹性应力实例分析。      

A note on the anisotropy and fabric of highly porous materials

The dependence of the orthotropic elastic constants of a highly porous material upon the stereological parameters characterizing the anisotropy of the porous microstructure has been considered in two recent papers in this journal. In the first paper [1] dimensional arguments were employed to develop to a relationship between ratios of the orthotropic elastic constants and ratios of the mean intercept lengths for a class of cell wall bending models of highly porous materials. In the second paper [2] the general tensorial form of the relationship between the orthotropic elastic constants and the mean intercept length was described without reference to a specific form or type of porous microstructure. The purpose of this note is to observe that the particular relationships obtained from the class of cell wall bending models used in the first paper are proper special cases of the general relationships given in the second paper.     

Comparative study on orthotropic solid waves by time-domain BEM and dynamic photoelasticity

Experimental research and numerical analysis are two basic tools in the study of wave propagation problems in orthotropic media. In this paper, an experimental method, namely dynamic orthotropic photoelasticity, which studies the dynamic behavior of orthotropic materials on a macroscopic scale by employing orthotropic birefringent materials, is established. Meanwhile, a numerical method, namely time domain boundary element method (BEM) for wave propagation in orthotropic media, is also presented. The two methods are used together in the analysis of semi-infinite orthotropic plates with and without a circular hole modeled by a unidirectional fiber-reinforced composite under impact loading. The propagation, reflection and diffraction of stress waves in the orthotropic media are recorded experimentally and investigated. Time histories of birefringent fringe orders or stresses for specific points of the plates are obtained, respectively, from the two methods and compared with each other. The comparative study demonstrates the applicability and accuracy of the two methods for wave propagation problems in orthotropic media.     

Does fabric tensor exist for a fabric?

It is shown that the mean intercept length distribution for planar fibre networks or for materials composed of a set of plates is not in general elliptic and cannot be expressed analytically in terms of a second-order tensor. However, our numerical computations indicate that the polar plot of the mean intercept length at the angle of measurement may become nearly ellipsoidal as the microstructure (fibres or plates) become less discretely organized, but yet remain orthotropic. The equations presented in this study may be used to obtain fibre (plate) orientation density functions from the experimental data on mean intercept length distribution.     

子午线轮胎静负荷实验的有限元模拟

建立了一种子午线轮胎的三维非线性有限元模型,来模拟子午线轮胎的静负荷试验。模型中考虑了橡胶材料的非线性和不可压缩性、帘线-橡胶复合材料的各向异性、轮胎大变形导致的几何非线性以及轮胎与轮辋、轮胎与路面的接触非线性边界条件。提出了一种单元重叠技术,通过重叠不可压缩橡胶单元和描述帘线的单元来模拟帘线-橡胶复合材料,这种单元重叠技术产生的轮胎模型比正交各向异性模型计算过程更稳定。通过有限元分析,得到了子午线轮胎在充气压力和静负荷作用下的变形和应力分布情况以及接地区的压力分布情况,并将计算结果与试验结果进行了比较,两者吻合得很好。     

What-You-Prescribe-Is-What-You-Get orthotropic hyperelasticity

We present a model for incompressible finite strain orthotropic hyperelasticity using logarithmic strains. The model does not have a prescribed shape. Instead, the energy function shape and the material data of the model are obtained solving the equilibrium equations of the different experiments. As a result the model almost exactly replicates the given experimental data for all six tests needed to completely define our nonlinear orthotropic material. We derive the constitutive tensor and demonstrate the efficiency of the finite element implementation for complex loading situations.     

Application of generalized measures to an orthotropic finite elasto-plasticity model

A numerical study of finite orthotropic elasto-plasticity based on generalized stress–strain measures is presented. The anisotropic constitutive equations are represented by isotropic tensor functions. A simple additive decomposition of strains can be performed due to the formulation in generalized measures. Furthermore, the plasticity model does not depend on special properties of any particular measure. The required projection tensor is constructed exploiting the coaxiality of the generalized deformation tensor with the right Cauchy–Green tensor. An efficient algorithmic implementation is proposed. Finally, we discuss representative numerical examples for orthotropic elasto-plasticity, where finite deformations occur.     

Analysis of orthotropic behavior in convected coordinate frames

The aim of this contribution is to present a theoretical and numerical model applicable to large strain analysis of orthotropic bodies. This three-dimensional constitutive law which uses the concept of convected coordinate frame is devoted to materials presenting elastic orthotropic behaviors in the large deformation field such as the elastomer-fabric composite materials. The proposed model is implemented in a finite element code and numerical examples are given to demonstrate the effectiveness of the model and the numerical algorithms. Finally, the model is compared to experimental results obtained from a bulge test.     

Influence of anisotropy on a limit load of weld strength overmatched middle cracked tension specimens

ABSTRACT A plane‐strain upper bound limit load solution for weld strength overmatched middle cracked tension specimens (M(T) specimens), is found. It is assumed that the weld material is isotropic, but the base material is orthotropic and its axes of orthotropy are straight and parallel to the axes of symmetry of the specimen. A quadratic orthotropic yield criterion is adopted. The solution is based on a simple discontinuous kinematically admissible velocity field and is an extension of the corresponding solution for the specimen made of isotropic materials. These two solutions are compared to demonstrate the influence of anisotropy on the magnitude of the limit load.     

Elastic constants and their admissible values for incompressible and slightly compressible anisotropic materials

Summary Constitutive relations for incompressible (slightly compressible) anisotropic materials cannot (could hardly) be obtained through the inversion of the generalized Hooke's law since the corresponding compliance tensor becomes singular (ill-conditioned) in this case. This is due to the fact that the incompressibility (slight compressibility) condition imposes some additional constraints on the elastic constants. The problem requires a special procedure discussed in the present paper. The idea of this procedure is based on the spectral decomposition of the compliance tensor but leads to a closed formula for the elasticity tensor without explicit using the eigenvalue problem solution. The condition of nonnegative (positive) definiteness of the material tensors restricts the elastic constants to belong to an admissible value domain. For orthotropic and transversely isotropic incompressible as well as isotropically compressible materials the corresponding domains are illustrated graphically.     

Fließbedingung für Thermoplaste. Teil I: Anisotrope Fließbedingung

Yield Criterion for Thermoplastics Part I: Anisotropic Yield Criterion Starting form the theory of the plastic potential, a yield criterion is proposed for solids that are both anisotropic and hydrostatic pressure dependent in regard to macroscopic yield behaviour. A Bauschingertensor is introduced in order to describe the mechanical behaviour of materials in which the uniaxial yield strength depends on the sign of stress. Theoretical yield curves are discussed on the basis of orthotropic behaviour and are compared with experimental results. If the solid is plastically isotropic the anisotropic cofficients described by tensors of rank two and four may be taken as a linear combination of a isotropic tensor of order two. The corresponding isotropic yield condition is succesfully applied to the effect of hydrostatic pressure on the shear, compressive, and tensile yield strength of thermoplastics. In the first part of this paper the anisotropic yield behaviour is investigated.