Generalized elastic damage theory and its application to composite plate

Based on continuum mechanics, a generalized damage theory for elastic material which can be used for anisotropic composite is presented in this paper. This theory for anisotropic elastic material has been proposed here from the stress-strain relation of the actual damaged material. Introducing a fourth order damage operator that may be formed by a symmetrical second order damage factor tensor, the constitutive equation of the damaged material has been set up. The expressions of components of both the stress tensor and the strain tensor of the damaged material and their first order invariants have been also derived. The application of this theory to the 2-dimensional composite laminate, including the technique estimating the components of the damage factor tensor and the damage variable tensor and also the practical measure technique of the damage in the whole process, have been explained in detail. Finally, the changes of the anisotropic elastic properties and the actual stress state of damaged material have been discussed and some interesting results have been obtained in this paper.     

Constitutive equation of a non-simple elastic plastic material

Summary The mechanical response predicted by the constitutive equation of a non-simple elastic material is considered in relation to the total strain behaviour of an elastic-plastic solid extensively deformed in the range of plastic strain. Both loading and unloading are considered in relation to the range of total elastic-plastic strain. In the absence of appropriate experimental studies, comparison of the predictions of the proposed constitutive equation of a non-simple elastic material, when applied to the work-hardening behaviour of the material, has been restricted to a study of the characteristic stress-strain behaviour of a strain hardening material. This has centred on the correlation of stress-strain curves characteristic of the mechanical response of a material tested in simple compression, simple torsion and pure shear with the object of obtaining a universal stress-strain curve.With 1 Figure     

An additive theory for finite elastic–plastic deformations of the micropolar continuous media

In this paper, the method of additive plasticity at finite deformations is generalized to the micropolar continuous media. It is shown that the non-symmetric rate of deformation tensor and gradient of gyration vector could be decomposed into elastic and plastic parts. For the finite elastic deformation, the micropolar hypo-elastic constitutive equations for isotropic micropolar materials are considered. Concerning the additive decomposition and the micropolar hypo-elasticity as the basic tools, an elastic–plastic formulation consisting of an arbitrary number of internal variables and arbitrary form of plastic flow rule is derived. The localization conditions for the micropolar material obeying the developed elastic–plastic constitutive equations are investigated. It is shown that in the proposed formulation, the rate of skew-symmetric part of the stress tensor does not exhibit any jump across the singular surface. As an example, a generalization of the Drucker–Prager yield criterion to the micropolar continuum through a generalized form of the J ₂-flow theory incorporating isotropic and kinematic hardenings is introduced.     

Evaluating the development of elastic anisotropy with plastic flow

It has been observed that many initially isotropic materials show the development of anisotropic elastic response after plastic flow. It is desirable to be able to model this change in the elastic properties as a function of the extent of plastic flow. This is particularly important when considering the traveling of waves in some glassy polymers that exhibit large differences in the wave moduli along the different directions resulting from unequal plastic flow in these directions. A thermodynamically based model of plasticity is developed and used to evaluate the elastic moduli associated with infinitesimal elastic deformations around the unloaded configuration. It is shown that for this model there are at least four independent material functions describing the elastic moduli of an initially isotropic material. These moduli are functions of the isotropic invariants of the right plastic Cauchy stretch tensor.     

Limitations of the constitutive equation of a simple elastic solid

Summary The constitutive equation of a simple, isotropic elastic solid can be arranged in such a form as to give rise to a fundamental identity between Lode's stress parameter and a corresponding deformation parameter. Using the concept of a stress intensity function, it is shown that at initial yield the constitutive equation of a simple, isotropic elastic solid satisfies only the von Mises yield criterion. A general form for the deformation response coefficients is obtained by way of the concept of a deformation intensity function. In general, there are two broad classes of deformation intensity function, defined in terms of whether the deformation intensity function is continuously differentiable or whether it is piece-wise linear and continuous. Use of the fundamental identity between Lode's stress parameter and the corresponding deformation parameter leads to the conclusion that the constitutive equation of the simple, isotropic elastic solid is incompatible with any form of piece-wise linear deformation intensity function. The stretching tensor has been expressed in terms of the co-rotational and convected time derivatives of the left Cauchy-Green deformation tensor and its inverse. This form of the stretching tensor is entered into a particular form of constitutive equation of the rate-type for a simple, isotropic elastic solid. By considering infinitesimal deformations from an arbitrary configuration, the constitutive equation of the rate-type is reduced to a constitutive equation of the incremental-type. In a similar way, an incremental-type constitutive equation is obtained from the constitutive equation of a simple, isotropic elastic solid. Comparison of these two incremental-type constitutive equations leads to the identification of a particular form for the material response coefficients associated with the constitutive equation of a simple elastic solid. Further limitations of the constitutive equation of a simple, isotropic elastic solid are considered in the context of two simple modes of deformation.     

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.     

The relationship between the elasticity tensor and the fabric tensor

An algebraic relationship between the fourth rank elasticity tensor of a porous, anisotropic, linear elastic material and the fabric tensor of the material is considered. The fabric tensor is a symmetric second rank tensor which characterizes the geometric arrangement of the porous material microstructure. In developing this result it is assumed that the matrix material of the porous elastic solid is isotropic and, thus, that the anisotropy of the porous elastic solid is determined by the fabric tensor. It is then shown that the material symmetries of orthotropy, transverse isotropy and isotropy correspond to the cases of three, two and one distinct eigenvalues of the fabric tensor, respectively.     

A continuum damage mechanics approach to crack tip shielding in brittle solids

This paper focuses firstly on the development of a comprehensive anisotropic theory of continuum damage mechanics for brittle solids suffering progressive deterioration. The basic concept of damage parameterization is re-examined and a new set of damage variables introduced yielding a new damage effect tensor through which the effective stress and strain tensors are defined. Constitutive equations of the damaged material are established incorporating a new hypothesis on equivalence between damaged and undamaged responses of the material. The model is completed by introduction of a general damage characteristic tensor which accounts for the experimentally observed fact that the rate of damage growth depends nonlinearly on applied external loads. The established damage model is next applied to investigate the crack-tip shielding effect due to anisotropic microcracking. The ratio of near-tip to remote stress intensity factors is obtained in closed form. A moderate but definite effect of anisotropy of microcracking is observed. The case of isotropic damage is found to be the least effective in screening remote external loads and is in accord with the results obtained by other researchers using different approaches.     

CONDITIONS FOR ADAPTATION OF DAMAGED ELASTIC–PLASTIC BODIES TO CYCLIC LOADING

Conditions for adaptation of isotropically damaged elastic–plastic bodies with isotropic strain hardening are investigated in the framework of the energy-based coupled elastic–plastic damage model by Ju. The yield function is assumed to be a homogeneous function of the first order in the stress tensor components. Due to this assumption, the notion of effective yield stress can be introduced. The loading program is supposed to be prescribed. Features of the stress path at the post-adaptation stage are considered, which lead to new necessary shakedown conditions expressed by a set of inequalities, and, in turn, to a problem of mathematical programming whose solution yields lower estimates for the damage and strain-hardening parameters. In the event, if the calculated value of the damage parameter is greater than its critical value, an adaptation to a given loading program is impossible. This condition is also necessary for adaptation in the case if only bounds for applied loads are prescribed. A correction of the constitutive material model is proposed which possibly could be good for ductile damage. The derived shakedown condition is not only necessary, but also sufficient for the plastic adaptation. The developed method is expounded in an example.     

Large strain elastic-plastic theory and nonlinear finite element analysis based on metric transformation tensors

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate-independent finite strain analysis of solids undergoing large elastic-plastic deformations. The formulation relies on the introduction of a mixed-variant metric transformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure whose rate is shown to be additively decomposed into elastic and plastic strain rate tensors. The mixed-variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response in the elastic-plastic solid. Additionally, the plastic material behavior is assumed to be governed by a generalized J ₂ yield criterion and rate-independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on 1st and higher order Padé approximations. Estimates of the stress and strain histories are obtained via a highly stable and accurate explicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step. The development of a consistent elastic-plastic tangent operator as well as its implementation into a nonlinear finite element program will also be discussed. Finally, the numerical solution of finite strain elastic-plastic problems is presented to demonstrate the efficiency of the algorithm. Received: 17 May 1998     

Plasticity theory for anisotropic porous metals

A plasticity theory for sintered porous metals with anisotropic pore structure is proposed. It is assumed that anisotropy of the bulk material and its symmetry group is generated by the pore structure that is described by the structural permeability second-order tensor. In the first part the development of a yield criterion accounting for the experimentally observed properties such as: plastic compressibility, rotation and translation (anisosensitivity) of a yield surface is dicussed. An evaluation is made of appearing material constants. The last part of the paper is concerned with a flow rule formulation. The plastic potential is assumed to be the same as the yield function.     

A dislocation theory of isotropic polycrystalline plasticity

A theory of polycrystalline plasticity is developed in which the polycrystalline solid is modeled as an isotropic continuum. The rate of plastic deformation tensor is shown to be a function of the mobile dislocation density and the dislocation velocity vector summed over all active glide planes. The dislocation velocity vector is expressed in terms of the stress tensor and the normal vector to the dislocation glide plane. The condition of plastic incompressibility yields the fact that the dislocation glide planes are the octahedral shear planes of the stress tensor. As a special case the rate of plastic deformation tensor reduces to a relation analogous to the Prandtl-Reuss flow rule. The theory has been implemented in a 2-dimensional finite element code and two example problems are presented.     

Nonlinear elastic equation of state of solids subjected to uniaxial homogeneous loading

Applying the finite deformation theory to a solid, which possesses either cubic or isotropic symmetry at stress-free natural state and is subsequently loaded homogeneously in uniaxial direction, one obtains a stress (or strain) dependence of the Young's modulus, Poisson's ratio, and a volume (or density) change, together with a nonlinear elastic relation between stress and strain. These are all expressed in terms of the second and third order elastic constants of the solid material. These expressions are illustrated with examples of cubic silicon crystal, isotropic carbon steel, Pyrex glass, and polystyrene at the relaxed state.     

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.     

Determination of Stress-Strain State and Strength of Structural Elements on the Basis of Strain-Hardening Analysis

We outline the fundamentals of the method whereby the stress-strain state and strength of structural elements loaded along linear paths and small-curvature paths are determined by allowing for strain hardening of material of the structure or plastic indicators attached to it. The method is based on a model of hardening which assumes that during the deformation beyond an elastic range the loading surface separating elastic and elastoplastic deformation ranges changes its shape and shifts in the direction of the vector which connects its center and an image point on the loading path. It is assumed that the material in its initial state is isotropic and the hypotheses for the unified stress-strain curve and for the proportionality of stress and strain deviators are met.__________Translated from Problemy Prochnosti, No. 2, pp. 28 – 48, March – April, 2005.     

Asymptotic analysis for a crack on interface of damaged materials

An asymptotic analysis for a crack lying on the interface of a damaged plastic material and a linear elastic material is presented in this paper. The present results show that the stress distributions along the crack tip are quite similar to those with HRR singularity field and the crack faces open obviously. Material constants n, μ and m0 are varied to examine their effects on the resulting stress distributions and displacement distributions in the damaged plastic region. It is found that the stress components σrr, σθ θ, σr θ and σe are slightly affected by the changes of material constants n, μ and m0, but the damaged plastic region are greatly disturbed by these material parameters. This revised version was published online in July 2006 with corrections to the Cover Date.     

The plastic potential in the theory of anisotropic elastic-plastic solids

In the paper the yield condition is proposed for the most general anisotropic material. It is one of the possible generalizations of the Huber-Mises-Hencky yield condition for the case of anisotropy. The body considered is anisotropic elastically as well as plastically. It is assumed that the plastic anisotropy tensor is a definite function of the elastic anisotropy tensor. The corresponding flow function is a part of the strain energy and its value remains unchanged when all normal components of stress are increased by the same value. The theory of the eigen states for fourth order tensors is used. The plastic anisotropy tensor proposed has the same deviatoric eigen states as the elastic anisotropy tensor. The proposed yield condition reduces to that of Huber-Mises-Hencky when the anisotropy is vanishingly small. The method presented in this paper can be also applied to describe other types of plastic anisotropy tensor.     

On a compressed elastic–plastic column optimized for post-buckling behaviour

A model of a column is proposed in order to analyse the post-buckling behaviour of a structural element in the elastic–plastic deformation range. The ideal two point I-section applied here simplifies the deformation analysis, that is, the problem of development of plastic zones in a section is eliminated, but still gives the possibility for qualitative analysis and optimization of the post-critical equilibrium paths. The coefficients of linear or parabolic variability of thickness of the flanges and their distance (web width) are accepted as model parameters and hence could be used for design variables in the optimization procedure. Moreover, the stiffness of an additional elastic support of the free end of the beam is also included as a parameter or design variable. A material model is employed with non-linear asymptotic isotropic hardening without the Bauschinger effect. Change of the tangent modulus is continuous and smooth during the transition from the elastic to plastic deformation range. The main goal of the analysis is to determine the values of the design variables for which the post-critical equilibrium paths are stable at least in the specified range of a generalized displacement. The constraints for the constant volume of the flanges and web material are applied. The inequality constraints are imposed on the flange thickness and web width. Various formulations of the optimization problem are proposed for all types of non-linear behaviour, including elastic or plastic buckling and elastic or elastic–plastic post-buckling deformation.