Numerical implementation of a J_2- and J_3-dependent plasticity model based on a spectral decomposition of the stress deviator

A new plasticity model with a yield criterion that depends on the second and third invariants of the stress deviator is proposed. The model is intended to bridge the gap between von Mises’ and Tresca’s yield criteria. An associative flow rule is employed. The proposed model contains one new non-dimensional key material parameter, that quantifies the relative difference in yield strength between uniaxial tension and pure shear. The yield surface is smooth and convex. Material strain hardening can be ascertained by a standard uniaxial tensile test, whereas the new material parameter can be determined by a test in pure shear. A fully implicit backward Euler method is developed and presented for the integration of stresses with a tangent operator consistent with the stress updating scheme. The stress updating method utilizes a spectral decomposition of the deviatoric stress tensor, which leads to a stable and robust updating scheme for a yield surface that exhibits strong and rapidly changing curvature in the synoptic plane. The proposed constitutive theory is implemented in a finite element program, and the influence of the new material parameter is demonstrated in two numerical examples.     

State update algorithm for isotropic elastoplasticity by incremental energy minimization

An original state update algorithm for the numerical integration of rate independent small strain elastoplastic constitutive models, treating in a unified manner a wide class of yield functions depending on all three stress invariants, is proposed. The algorithm is based on an incremental energy minimization approach, in the framework of generalized standard materials with convex free‐energy and dissipation potential. Under the assumption of isotropic material behavior, implying coaxiality of trial stress, increment of plastic strain, and updated stress, the problem is reduced from dimension six to three. Then, exploiting the cylindrical tensor basis associated with Haigh–Westergaard coordinates, the problem is recast in terms of two nested scalar equations. The proposed algorithm (i) exhibits global convergence even for yield functions with difficult features, such as not being defined on the whole stress space, or implying high‐curvature points of the yield domain, and (ii) requires no matrix inversion. After the tensor reconstruction of the unknowns, a simple expression for the algorithmic consistent material tangent is derived. The algorithm is validated by comparison with benchmark semi‐analytic solutions. Numerical results on single material points and finite element simulations are reported for assessing its accuracy, robustness, and efficiency. A Matlab implementation is provided as supplementary material. Copyright © 2015 John Wiley & Sons, Ltd.     

Convex forms of the cubic model

In the paper a generalised model is proposed which encompasses a number of known models of the mechanical material behaviour: the normal stress hypothesis, the maximum strain hypothesis, rotationally symmetric potentials (v. MISES, BELTRAMI, cone, paraboloid- and hyperboloid potentials) and the model of SAYIR. The model gives rise to some new solutions which extend the classic hypotheses. The model is build up of the three invariants of the stress tensor and has five free parameters. Parameters are restricted to yield a convex form. Some convex forms of the model, which contains the stresses to the power three or two, are constructed in the paper.     

Simulation of strength difference in elasto‐plasticity for adhesive materials

Experimental evidence of certain adhesive materials reveals elastic strains, plastic strains and hardening. Furthermore, a pronounced strength difference effect between tension, torsion or combined loading is observed. For simulation of these phenomena, a yield function dependent on the first and second basic invariants of the related stress tensor in the framework of elasto‐plasticity is used in this work. A plastic potential with the same mathematical structure is introduced to formulate the evolution equation for the inelastic strains. Furthermore, thermodynamic consistency of the model equations is considered, thus rendering some restrictions on the material parameters. For evolution of the strain like internal variable, two cases are considered, and the consequences on the thermodynamic consistency and the numerical implementation are extensively discussed. The resulting evolution equations are integrated with an implicit Euler scheme. In particular, the reduction of the resulting local problem is performed, and for the finite‐element equilibrium iteration, the algorithmic tangent operator is derived. Two examples are presented. The first example demonstrates the capability of the model equations to simulate the yield strength difference between tension and torsion for the adhesive material Betamate 1496. A second example investigates the deformation evolution of a compact tension specimen with an adhesive zone. Copyright © 2005 John Wiley & Sons, Ltd.     

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     

Tensor representations and solutions of constitutive equations for viscoelastic fluids

An approach is developed using tensor representations to assess and characterize both the transient behavior and equilibrium states of viscoelastic fluid constitutive equations in viscometric flows. The methodology is based on the replacement of the differential constitutive equation for the deviatoric part of the viscoelastic stress tensor by an equivalent and more tractable set of differential equations for the characteristic scalar invariants. In the case of planar flows, this equivalence leads to an explicit, closed-form analytic solution for the time evolution of the extra-stress tensor that is formally expressed as a second-order fluid relation, with time-dependent coefficients. As a validation of the approach, an analysis of the transient and equilibrium system characteristics of fluid flows described by the corotational Jeffreys model and general Oldroyd-type constitutive equations is presented.     

Critical plane approach with two families of microcracks for modelling of unilateral fatigue damage

New multiaxial fatigue damage model based on the critical plane approach is proposed. Two different physical mechanisms of the fatigue damage development on each potential failure plane (critical plane) are considered. In general, each critical plane contains two families of a parallel microcracks. The proposed model reproduces simultaneously fatigue damage induced anisotropy, the influence of positive and negative mean stresses, unilateral fatigue damage, microcrack closure effect and fatigue behaviour under variable amplitude loading. The expression for the equivalent stress in the damage evolution equation includes the stress intensity for the amplitudes as well as joint invariants for the mean values of the stress tensor and for the vectors associated with the directions of microcracks. The theoretical predictions are compared with experimental data under uniaxial cyclic loading of brass specimens. The influence of positive and negative mean stresses on the fatigue life of brass is investigated.     

Stress trajectories element method for stress determination from discrete data on principal directions

This paper presents a Trefftz-element numerical method for the reconstruction of stress trajectories and the determination of full stress tensors in two-dimensional elastic bodies from discrete data on principal directions. The conventional techniques cannot be used because neither displacements nor tractions are specified on the boundary. The proposed approach involves the subdivision of the domain into smaller subdomains and the introduction of the Cauchy integrals with unknown densities on element boundaries in order to approximate complex potentials within the elements. For polynomial approximations of the densities, this leads to piecewise polynomial approximations for the complex potentials within the entire domain and, therefore, all elasticity equations are automatically satisfied as in the Trefftz method. Continuity of the complex potentials is forced at the collocation points, which forms the first group of equations. The second group is formed by satisfying the data on principal directions known in some locations. All these equations are homogeneous; therefore, it is assumed that the average value of the maximum shear stresses at data points is unity. This guarantees the existence of a non-trivial solution of the system; however it addresses the non-uniqueness of the reconstruction of the full stress tensor. The technique is validated by reconstructing stress trajectories and determining maximum shear stresses from synthetic and photoelasticity data. It has been applied to reconstruction of tectonic stresses in the Australian region and the results were compared with previous approaches.     

The stresses in the neighborhood of a crack tip under effects of electromagnetic forces

In recent work, some basic problems of the stresses in the neighborhood of a crack tip under the effects of electromagnetic forces are studied, and the basic concept of Maxwell electromagnetic stress tensor is illustrated and used to express the effects of the electromagnetic field to the solid body. The basic governing equations of elastic stress of both potential electromagnetic force and Maxwell stress tensor and the approaches for solving these equations are provided. Two kinds of affects of the electromagnetic field on stress singularity of a crack tip are presented and analyzed. Lastly, several examples about the stress analysis on the effects of the electromagnetic field are provided     

On propagation of small non-stationary disturbances in linear viscoelastic fluids

Summary A representation is given for the solutions of linearized equations describing a flow of compressible viscoelastic fluids, the spherical part of tensor of stresses corresponding to the rheological Voigt body, while the deviator part corresponds to the Maxwell, Oldroyd and Kelvin-Voigt body of any order. Two independent equations for the propagation of longitudinal and transverse disturbances are obtained.The Laplace transform is applied to study the propagation of plane non-stationary longitudinal and transverse waves in linear viscoelastic fluids, where the spherical part of the tensor of stresses corresponds to the elastic body, while the deviator part corresponds to the Maxwell body. The problem of inversion is reduced to the numerical solution of the linear homogeneous Volterra integral equation of the second kind with a discontinuous kernel.     

Nonlinear isotropic constitutive laws: choice of the three invariants, convex potentials and constitutive inequalities

The aim of this paper is a new formulation of nonlinear isotropic constitutive laws. Our main hypothesis claims that the eigenvalues of stress and strain tensors are classified in the same order (the eigenvector associated to the highest eigenvalue of the stress tensor is also associated to the highest eigenvalue of the strain tensor, etc.). Further, we assume the existence of a differentiable convex isotropic potential. By introducing three new invariants for each tensor (called X, Y, Z for the stress tensor S and x, y, z for the strain tensor E) a constitutive law is revealed to be a simple duality between the chosen invariants: (x, y, z) and (X, Y, Z) look like Cartesian coordinates of E and S. We look at several potentials chosen as polynomials of these invariants. Finally, first and third order isotropic elasticity laws are reviewed and convexity of the potentials is discussed.     

Multiple-factor dependence of the yielding behavior to isotropic ductile materials

Extensively experimental evidences have shown that the yielding behaviors for many isotropic materials exhibit both the pressure and stress-state dependence. The strength-differential in tension and compression results not only from the hydrostatic stress, but also from the loading type. Different materials often demonstrate different yielding features in stress space even though they have the same ratio of compression yield-stress to tension yield-stress. Bases on a physical hypothesis introduced herein, a yield criterion is proposed to fulfill the experimental observations. The yield criterion can be represented finally with three independent invariants of the stress tensor and the deviatoric stress tensor. The yield criterion can well predict the yielding behavior arising from the yielding mechanism of the multiple-factor dependence. The yielding analysis with the effect of the hydrostatic stress does not have a limit when regardless of the strength-differential in tension and compression. The yield criterion assures the convexity in a wider range of material properties so that it can be used as the plastic potential in the implementation of predicting the subsequent yield surface. The yield criterion reveals a clearly transforming procedure with respect to its simplified form when applied to different materials from a general compressive isotropic material to an ideal incompressive isotropic material, back to the Mises’s criterion. Divers experiments on metallic and polymeric materials are compared. The results show that the new yield criterion is in fairly good agreement with all referred experiments.     

Mathematical and numerical model for nonlinear viscoplasticity

A macroscopic model describing elastic-plastic solids is derived in a special case of the internal specific energy taken in separable form: it is the sum of a hydrodynamic part depending only on the density and entropy, and a shear part depending on other invariants of the Finger tensor. In particular, the relaxation terms are constructed compatible with the von Mises yield criteria. In addition, Maxwell-type material behaviour is shown up: the deviatoric part of the stress tensor decays during plastic deformations. Numerical examples show the ability of this model to deal with real physical phenomena.     

On the representation and implicit integration of general isotropic elastoplasticity based on a set of mutually orthogonal unit basis tensors

A simple and compact representation framework and the corresponding efficient numerical integration algorithm are developed for constitutive equations of isotropic elastoplasticity. Central to this work is the utilization of a set of mutually orthogonal unit tensor bases and the corresponding invariants. The set of bases can be regarded equivalently as a local cylindrical coordinate system in the three‐dimensional coaxial tensor subspace, namely, the principal space. The base tensors are given in the global coordinate system. Similar to the principal space approach, the proposed method reduces the problem dimension from six to three. In contrast to the conventional approach, the transformation procedure between the principal space and the general space is avoided and explicit computation of the principal axes is bypassed. With the proposed technique, the matrices, which need to be inverted during iteration, take a simple form for the great majority of constitutive equations in use. The tangent operator consistent with the proposed algorithm can be decomposed into the direct sum of two linear maps over the coaxial tensor subspace and the subspace orthogonal to it. Consequently, its closed form is derived in an extremely simple manner. Finally, numerical examples demonstrate the high quality performances of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

Internal stresses in inelastic BEM using complex-variable differentiation

 A new approach is proposed for nonlinear boundary element methods in computing internal stresses accurately using a complex-variable formulation. In this approach, the internal stresses are obtained from the numerical derivatives of the displacement integral equations that involve only weakly singular integrals. The collocation points in the displacement integral equations are defined as complex variables whose imaginary part is a small step size for numerical derivatives. Unlike the finite difference method whose solution accuracy is step-size dependent, the complex-variable technique can provide “numerically-exact” derivatives of complicated functions, which is step-size independent in the small asymptotic limit. Meanwhile, it also circumvents the tedious analytical differentiation in the process. Consequently, the evaluation of the nonlinear stress increment only deals with kernels no more singular than that of the displacement increment. In addition, this technique can yield more accurate stresses for nodes that are near the boundary. Three examples are presented to demonstrate the robustness of this method. Received 9 March 2001     

Equations in stresses for the three-, two-, and one- dimensional dynamic problems of thermoelasticity

Source systems of differential equations for the six components of the tensor of dynamic stresses are presented in Cartesian and cylindrical coordinates by using the equations of motion, Hooke’s law, Cauchy formulas, and the Saint-Venant compatibility conditions for strains. Without introducing auxiliary potential functions, these systems of interrelated equations are reduced to systems of hierarchically connected wave equations for the key function, i.e., the first invariant and unknown components of the stress tensor. We also present the systems of key equations for the solution of two-dimensional dynamic problems in stresses (the plane problem in Cartesian coordinates and the plane and axially symmetric problems in cylindrical coordinates) and the wave equations for the one-dimensional problems of evaluation of the normal and tangential components of stresses in the same coordinate systems. This enables us to use standard methods of mathematical physics for the solution of the equations obtained as a result. If, in the quasistatic problem, the influence of temperature and bulk forces is absent, then the proposed equations coincide with the equations known from the literature. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 36, No. 2, pp. 20–26, March-April, 2000.     

Invarianten‐basierte Mehrachsigkeitshypothese zur Anwendung bei Schwingbeanspruchung

Multiaxial hypothesis based on invariants for the application of fatigue loading Increasing deviations of experimentally determined and calculated fatigue lives can be observed for multiaxially loaded specimens with increasing contribution of shear stresses. An improvement of this situation can be gained by linking the calculation procedure to both constant amplitude life curves for pure push/pull and shear loading. A hypothesis is presented in this paper which is formulated strictly using only the invariants of the stress tensor to interpolate between the border cases. A modification of this hypothesis is able to take nonproportional loading due to a phase shift between the stress components into account.     

Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery

A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded cylindrical shells subjected to mechanical loadings. Eight types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded cylindrical shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.     

A hierarchic 3D finite element for laminated composites

In this paper a new 3D multilayer element is presented for analysis of thick‐walled laminated composites. This element uses two steps to calculate the full stress tensor. In the first step the in‐plane stresses are computed from the material law using a displacement approximation, and then the transverse stresses are calculated from the 3D equilibrium equations. Since the 3D equilibrium equations require high‐order interpolation functions, a hierarchic interpolation of displacements is used. The new element is compared with existing ones, e.g. from MSC.MARC. Copyright © 2004 John Wiley & Sons, Ltd.     

正交异性材料在R.Hill屈服准则下平面应力问题的特征场(h>2)

大多数金属材料在承载时产生了各向异性强化的特点,因此,对这一类材料特性的深入研究显得十分必要,正确建立及分析理论模型将对在实际工程中的应用提供很好的理论依据。本文在R.Hill屈服准则下提出了无量纲化的解析应力解,讨论了屈服参数2>h时,正交异性材料平面应力问题的应力场及应力特征场,利用应力微分平衡方程得到了决定特征曲线特征的参数。结果说明特征线一般互不正交。特别地分析了平头冲压边界附近的情况,得到了各应力状态不同区域的应力特征场,应用数值分析的方法对过渡区进行了计算。