Elastic continuum theories of lattice defects: a review

The presently available elastic continuum theories of lattice defects are reviewed. After introducing a few elementary concepts and the basic equations of elasticity the Eshelby’s theory of misfitting inclusions and inhomogeneities is outlined. Kovács’ result that any lattice defect can be described by a surface distribution of elastic dipoles is described. The generalization of the isotropic continuum approach to anisotropic models and to Eringen’s isotropic but non-local model is discussed. Kröner’s theroy (where a defect is viewed as a lack of strain compatibility in the medium) and the elastic field equations (formulated in a way analogous to Maxwell’s field equations of magnetostatics) are described. The concept of the dislocation density tensor is introduced and the utility of higher-order dislocation density correlation tensors is discussed. The beautiful theory of the affine differential geometry of stationary lattice defects developed by Kondo and Kröner is outlined. Kosevich’s attempt to include dynamics in the elastic field equations is described. Wadati’s quantum field theory of extended objects is mentioned qualitatively. Some potential areas of research are identified.     

弹性损伤理论的几何-拓扑描述

材料内部微观几何缺陷通常是作为物理非线性问题在本构方程中考虑。针对连续介质弹性损伤理论作几何拓扑,采用非完整标架方法把材料内部微观几何缺陷转化为材料空间的弯曲,并体现在基本几何法则中。首先由连续损伤变量定义拟塑性张量,给出这些基本张量所满足的连续性方程和基本几何法则。由此建立了弹性损伤缺陷与Riemann流形的对应关系,将物理非线性问题转化为物理线性和材料所在空间的弯曲之和。最后讨论了二维情况下,各向同性晶格材料受各向异性损伤的算例。     

Hencky's elasticity model and linear stress-strain relations in isotropic finite hyperelasticity

Summary Hencky's elasticity model is an isotropic finite elasticity model assuming a linear relation between the Kirchhoff stress tensor and the Hencky or logarithmic strain tensor. It is a direct generalization of the classical Hooke's law for isotropic infinitesimal elasticity by replacing the Cauchy stress tensor and the infinitesmal strain tensor with the foregoing stress and strain tensors. A simple, straightforward proof is presented to show that Hencky's elasticity model is exactly a hyperelasticity model, derivable from a quadratic potential function of the Hencky strain tensor. Generally, Hill's isotropic linear hyperelastic relation between any given Doyle-Ericksen or Seth-Hill strain tensor and its work-conjugate stress tensor is studied. A straightforward, explicit expression of this general relation is derived in terms of the Kirchhoff stress and left Cauchy-Green strain tensors. Certain remarkable properties of Hencky's model are indicated from both theorectical and experimental points of view.Dedicated to Prof. Dr.-Ing. Otto Timme Bruhns on the occasion of his 60th birthday     

A simple orthotropic finite elasto–plasticity model based on generalized stress–strain measures

 In this paper we present a formulation of orthotropic elasto-plasticity at finite strains based on generalized stress–strain measures, which reduces for one special case to the so-called Green–Naghdi theory. The main goal is the representation of the governing constitutive equations within the invariant theory. Introducing additional argument tensors, the so-called structural tensors, the anisotropic constitutive equations, especially the free energy function, the yield criterion, the stress-response and the flow rule, are represented by scalar-valued and tensor-valued isotropic tensor functions. The proposed model is formulated in terms of generalized stress–strain measures in order to maintain the simple additive structure of the infinitesimal elasto-plasticity theory. The tensor generators for the stresses and moduli are derived in detail and some representative numerical examples are discussed. Received: 2 April 2002 / Accepted: 11 September 2002     

Refined dynamic theory of multilayer shells and plates. Report 1. Initial hypotheses and relationships of the model

The problem of constructing a refined theory of multilayer shells and plates on the basis of hypotheses that account for the effect of inertial forces is formulated. The hypotheses are obtained by integrating equations of motion of a three-dimensional medium for assigned tangential components of the stress tensor of the classical theory of multilayer shells. Relationships of a model of the stress-strain state of the refined dynamic theory are derived in conformity with the hypotheses: the components of the displacement vector and the stress and strain tensors. All components take into account the effect of inertial forces; this distinguishes them from components obtained in cases where the same hypotheses as those for the theory of statics are introduced a priori for construction of refined theories of the dynamics of shells. Translated from Problemy Prochnosti, No. 5, pp. 91–99, May, 1996.     

An analysis of the wave structure of the defect field in a viscoplastic medium

The wave structure of a defect field in a viscoplastic medium is studied in terms of equations of defect field theory. It is established that waves in the defect field characterized by tensors of the defect density and the defect flux density possess a transverse character. A relation between the two quantities in a plane harmonic wave is determined. Partial cases of media with strongly and weakly decaying waves are considered.     

Incremental constitutive equation for large strain elasto plasticity

Starting from the standard theory with intermediate configuration for finite deformations of an isotropic elasto-plastic material with isotropic hardening, rate type constitutive equations are obtained. The small elastic strain approximation is then discussed and it is shown that, in this approximation, these equations reduce to Hill's formalism of large strain elasto-plasticity obtained from the classical Prandtl-Reuss relations of infinitesimal plasticity by substituting for the infinitesimal strain rate, stress and stress rate respectively the rate of deformation tensor, the Cauchy stress tensor and the Jaumann stress rate tensor. The limiting case of perfect plasticity is also investigated.     

Reconstruction of spatially inhomogeneous dielectric tensors through optical tomography

A method to reconstruct weakly anisotropic inhomogeneous dielectric tensors inside a transparent medium is proposed. The mathematical theory of integral geometry is cast into a workable framework that allows the full determination of dielectric tensor fields by scalar Radon inversions of the polarization transformation data obtained from six planar tomographic scanning cycles. Furthermore, a careful derivation of the usual equations of integrated photoelasticity in terms of heuristic length scales of the material inhomogeneity and anisotropy is provided, resulting in a self-contained account about the reconstruction of arbitrary three-dimensional, weakly anisotropic dielectric tensor fields.     

On the duality principle of conservation laws in finite elastodynamics

The duality principle of conservation laws which holds in finite elastodynamics is studied using the two-point tensor method. Based on the general Noether's theorem, two basic equations of variational invariance are first derived, which correspond to the action integrals given, respectively, in Lagrangian and Eulerian representations for a finite motion of an elastic body. The dual relations between the conservation laws in both representations are given. The procedure for constructing these dual relations is to apply simultaneously the same infinitesimal transformation of either time or position coordinates as well as field variables to the dual equations of variational invariance, where the position coordinates could be taken either from the reference configuration or from the deformed configuration of the material body. Based on these dual relations it is shown that the conservation equations of material momentum and moment of material momentum possess the same structure as those of physical momentum and physical moment of momentum. Furthermore, three pairs of dual relations between stress tensors and material momentum tensors of various kinds are derived based on the duality principle by using the two-point tensor method. Finally, using the dual integral forms of conservation laws the concepts of dynamic material force and moment acting on defects are introduced and analyzed. The force and moment can be decomposed into a pure kinetic part and a pure deformation part, the latter corresponding to the path-independent integral as suggested in elastostatics.     

Elasto‐plastic analysis of thin‐walled tube under cyclic bending

Abstract

The elasto‐plastic endochronic theory is extended to describe the material behavior of tubes under cyclic bending. Experimental data of cyclic bending of 6061‐T6 aluminum and 1018 steel was used for comparison. It is shown that by using the complete differential equations of the theory and by assuming a linear relation between the strain and the curvature, the theory is capable of predicting the moment‐curvature response.     

Space, time and geometry

Space, time, and their pseudo-euclidian geometry have an algebraic origin from the Hamiltonian quaternions and their product systems. They have unique mathematical properties yielding also the large dimensionless fundamental constants and particle masses of physics. All natural laws must be Lorentz-invariant, also the gravitational field equations. Einstein's application of Riemannian geometry is only a calculation device valid only for very small dimensionless gravitational potentials. In this case his field equations transform asymptotically into linear field equations for tensor potentials very similar to the electromagnetic ones for vector potentials. All numerical observations known up to date only this case. Kazuo Kondo et al. used this calculation device for many physical problems related with tensors.     

On the mechanics of large inelastic deformations: kinematics and constitutive modeling

Summary In this paper we examine the complexities associated with the kinematics of finite elastoplastic deformations and other issues related to the development of constitutive equations. The decomposition of the total strain and strain rate tensors into elastic and plastic constituents is investigated by considering both a multiplicative decomposition of the deformation gradient and an additive decomposition of the deformation vector field. Physically based definitions for the elastic and plastic strain rate tensors are given and compared with other values found in the literature. Constitutive equations for the plastic flow are derived by considering both a phenomenological-energy approach and a physically motivatedmesomechanical approach based on the double-slip idealization. It is shown that by resorting to the mechanics of the double slip, specific relations for the plastic stretching and plastic spin can be rigorously derived, taking into account the effect of noncoaxiality and material rotation. Finally, the implication of such effects to large deformations is examined in connection with the localization phenomenon.     

A model of composite laminated Reddy plate based on new modified couple stress theory

Based on new modified couple stress theory a model for composite laminated Reddy plate is developed in first time. In this theory a new curvature tensor is defined for establishing the constitutive relations of laminated plate. The characterization of anisotropy is incorporated into higher-order laminated plate theories based on the modified couple stress theory by Yang et al. in 2002. The form of new curvature tensor is asymmetric, however it can result in same as the symmetric curvature tensor in the isotropic elasticity. The present model of thick plate can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model for cross-ply composite laminated Reddy plate of couple stress theory with one material’s length constant is used to demonstrate the scale effects. Numerical results show that the present plate model can capture the scale effects of microstructure. Additionally, the present model of thick plate model can be degenerated to the model of composite cross-ply laminated Kirchhoff plate and Mindlin plate of couple stress theory.     

Damage Constitutive Equations and its Application to Fiber Reinforced Composites under Transverse Impact

Two continuous field variables, called as continuity tensor and damage variable tensor, are used to describe the anisotropic responses of an elastic-brittle material under transverse impact load. Based on the continuum damage mechanics, anisotropic damage constitutive equations in both full and incremental forms are proposed here. The expressions of effective elastic module tensor, damage variable tensor and damage propagation force tensor are further derived, and the methods for determining the tensors are explained in detail. An example of strain and damage response of a fiber reinforced composite laminated plate under transverse impact load is employed to demonstrate the application of this theory. In the example, the damage variable coupled with geometric large deformation of laminated plate is also considered. The calculating results illustrate the influence of damage on strain field in the impacted laminated plate.     

Dual orthotropic damage–effect tensors with complementary structures

This paper deals with secant constitutive relations of orthotropic elastic damage based on the so-called damage–effect tensors, namely the fourth-order operators that define the linear transformations between nominal and effective stress and strain quantities. The damage–effect tensors are expressed by orthotropic representations in terms of symmetric second-order damage tensor variables. The paper provides a set of new dual orthotropic damage–effect tensors that possess complementary structures in the dual compliance- and stiffness-based derivations. More specifically, each orthotropic damage–effect tensor of the solution set possesses an inverse (its dual counterpart) that displays the structure of the (major) transpose of the tensor obtained by replacing the adopted second-order damage tensor variables with their inverses.     

Two‐point constitutive equations and integration algorithms for isotropic‐hardening rate‐independent elastoplastic materials in large deformation

This paper presents alternative forms of hyperelastic–plastic constitutive equations and their integration algorithms for isotropic‐hardening materials at large strain, which are established in two‐point tensor field, namely between the first Piola–Kirchhoff stress tensor and deformation gradient. The eigenvalue problems for symmetric and non‐symmetric tensors are applied to kinematics of multiplicative plasticity, which imply the transformation relationships of eigenvectors in current, intermediate and initial configurations. Based on the principle of plastic maximum dissipation, the two‐point hyperelastic stress–strain relationships and the evolution equations are achieved, in which it is considered that the plastic spin vanishes for isotropic plasticity. On the computational side, the exponential algorithm is used to integrate the plastic evolution equation. The return‐mapping procedure in principal axes, with respect to logarithmic elastic strain, possesses the same structure as infinitesimal deformation theory. Then, the theory of derivatives of non‐symmetric tensor functions is applied to derive the two‐point closed‐form consistent tangent modulus, which is useful for Newton's iterative solution of boundary value problem. Finally, the numerical simulation illustrates the application of the proposed formulations. Copyright © 2008 John Wiley & Sons, Ltd.     

Macroscopic material properties from quasi-static, microscopic simulations of a two-dimensional shear-cell

One of the essential questions in the area of granular matter is, how to obtain macroscopic tensorial quantities like stress and strain from “microscopic” quantities like the contact forces in a granular assembly. Different averaging strategies are introduced, tested, and used to obtain volume fractions, coordination numbers, and fabric properties. We derive anew the non-trivial relation for the stress tensor that allows a straightforward calculation of the mean stress from discrete element simulations and comment on the applicability. Furthermore, we derive “elastic” (reversible) mean displacement gradient, based on a best-fit hypothesis. Finally, different combinations of the tensorial quantities are used to compute some material properties. The bulk modulus, i.e. the stiffness of the granulate, is a linear function of the trace of the fabric tensor which itself is proportional to the density and the coordination number. The fabric, the stress and strain tensors are not co-linear so that a more refined analysis than a classical elasticity theory is required. Received: 23 July 1999     

Distribution of directional data and fabric tensors

Distribution of directional data is characterized by what is termed fabric tensors. A formal least square approximation is applied, and three kinds of fabric tensors are defined in connection with the choice of a basis of the space of functions on a unit sphere or a unit circle. All the resulting equations are Cartesian tensor equations, and they are interpreted in terms of the representation theory of the rotation group and the potential theory in electrodynamics. It is also shown how this characterization is related to the spherical harmonics expansion or the Fourier series expansion. Finally, a method of statistical test is presented in the Cartesian tensor form to check the true form of the distribution. A physical example is also given to illustrate the proposed technique.