Stress–strain measures used in the ANSYS package to solve elastoplastic problems at finite strains

It is shown that the Dienes (or the Green–McInnis–Naghdi) derivative is used in the ANSYS package rather than the Jaumann derivative as an objective derivative for the Cauchy stress tensor when solving elastoplastic problems, although the usage of the Jaumann derivative is stated in the ANSYS theory reference manual. In this manual it is also stated that, for these problems, the strain tensor is the Hencky logarithmic strain tensor; however, in reality, this strain tensor is the right nonholonomic strain tensor such that the left nonholonomic strain tensor associated with the right one is generated by the Dienes derivative.     

A rheological theory for liquid crystal thin films

A macroscopic rheological theory for compressible isothermal nematic liquid crystal films is developed and used to characterize the interfacial elastic, viscous, and viscoelastic material properties. The derived expression for the film stress tensor includes elastic and viscous components. The asymmetric film viscous stress tensor takes into account the nematic ordering and is given in terms of the film rate of deformation and the surface Jaumann derivative. The material function that describes the anisotropic viscoelasticity is the dynamic film tension, which includes the film tension and dilational viscosities. Viscous dissipation due to film compressibility is described by the anisotropic dilational viscosity. Three characteristic film shear viscosities are defined according to whether the nematic orientation is along the velocity direction, the velocity gradient, or the unit normal. In addition the dependence of the rheological functions on curvature and film thickness has been identified. The rheological theory provides a theoretical framework to future studies of thin liquid crystal film stability and hydrodynamics, and liquid crystal foam rheology. Received: 9 October 2000 Accepted: 6 April 2001     

Flows of constant stretch history for polymeric materials with power-law distributions of relaxation times

It is herein shown that for separable integral constitutive equations with power-law distributions of relaxation times, the streamlines in creeping flow are independent of flow rate.For planar flows of constant stretch history, the stress tensor is the sum of three terms, one proportional to the rate-of-deformation tensor, one to the square of this tensor, and the other to the Jaumann derivative of the rate-of-deformation tensor. The three tensors are the same as occur in the Criminale-Ericksen-Filbey Equation, but the coefficients of these tensors depend not only on the second invariant of the strain rate, but also on another invariant which is a measure of flow strength. With the power-law distribution of relaxation times, each coefficient is equal to the second invariant of the strain rate tensor raised to a power, times a function that depends only on strength of the flow. Axisymmetric flows of constant stretch history are more complicated than the planar flows, because three instead of two nonzero normal components appear in the velocity gradient tensor. For homogeneous axisymmetric flows of constant stretch history, the stress tensor is given by the sum of the same three terms. The coefficients of these terms again depend on the flow strength parameter, but in general the dependences are not the same as in planar flow.     

弹性大变形问题中的应力状态描述、奇异性问题和余能原理

对弹性大变形理论中的3方面问题进行了综述.首先,对各种应变度量的共轭应力进行综述.大变形问题引起的应力状态描述的复杂性引起了许多学者的兴趣,对这个问题的研究也促进了大变形弹性理论的发展.在各种特定问题中,人们提出了不同的应力张量来描述应力状态,如Caucby应力张量、第一类和第一二类Piola-Kirchhoff应力张...     

Analysis of simple shear endochronic equations for finite plastic deformation

IntroductionThestress_strainbehaviorofmaterialswithfiniteplasticdeformationisaninterestingissue ,onwhichsignificantprogresshasbeenmadethroughboththephenomenologicalandphysicalapproaches.Thephenomenologicalapproachisbasedoncontinuummechanicsofplasticity .Ithasitsadvantageinsolvingcomplicatedproblemsbecauseofitssimplicity .Mostofphenomenologicaltheoriesareinvolvedintheconceptofcorotationalrates.Thematerialderivativeofstresswasnotobjectiveunderfinitedeformation .TheJaumannratewasusuallyusedbefo…     

Analyse contrastive de modèles incrémentaux et hypoplastiques

In this paper we first establish two necessary and sufficient conditions in order that incremental constitutive equations expressing the strain rate tensor as a function of the Jaumann's derivative of the Cauchy's stress tensor can be inverted under the general form of hypoplastic models when the stress state is located inside the domain bounded by the limit state surface. We are then interested in the physical meaning of these conditions with regard to the incremental response of the material.     

On the asymptotic crack-tip stress fields in nonlocal orthotropic elasticity

The crack-tip stress fields in orthotropic bodies are derived within the framework of Eringen’s nonlocal elasticity via the Green’s function method. The modified Bessel function of second kind and order zero is considered as the nonlocal kernel. We demonstrate that if the localisation residuals are neglected, as originally proposed by Eringen, the asymptotic stress tensor and its normal derivative are continuous across the crack. We prove that the stresses attained at the crack tip are finite in nonlocal orthotropic continua for all the three fracture modes (I, II and III). The relative magnitudes of the stress components depend on the material orthotropy. Moreover, non-zero self-balanced tractions exist on the crack edges for both isotropic and orthotropic continua. The special case of a mode I Griffith crack in a nonlocal and orthotropic material is studied, with the inclusion of the T-stress term.     

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.     

A note on undistorted states of isotropic elastic solids

Non-homogeneous conformal deformations are shown to be possible in an isotropic elastic material in the absence of body forces if and only if the material satisfies a certain condition which renders it incapable of obeying the classical pressure-compression inequality. The undistorted states of materials in this class (which are obtained by subjecting an undistorted state called reference configuration to all possible conformal deformations) are shown to be at best neutrally stable when subject to hydrostatic loading everywhere on the boundary.     

On the stability of the flow of a viscoelastic fluid down an inclined plane

The paper presents an analysis of laminar flow of a film of viscoelastic fluid flowing under gravity down an infinite inclined plane. It is assumed that the mechanical behavior of the fluid can be represented by a generalized Maxwell model, whose constitutive equation contains a time derivative of the deviator of the stress tensor in the Jaumann sense [1. 2]. The equations of motion of the viscoelastic fluid considered here admit an exact solution for the case of rectilinear laminar flow with a plane free boundary. The stability of this flow with respect to surface waves is investigated by the method of successive approximations described in [3, 4].     

A direct approach to fiber and membrane reinforced bodies. Part II. Membrane reinforced bodies

The paper deals with membrane reinforced bodies with the membrane treated as a two-dimensional surface with concentrated material properties. The bulk response of the matrix is treated separately in two cases: (a) as a coercive nonlinear material with convex stored energy function expressed in the small strain tensor, and (b) as a no-tension material (where the coercivity assumption is not satisfied). The membrane response is assumed to be nonlinear in the surface strain tensor. For the nonlinear bulk response in Case (a), the existence of states of minimum energy is proved. Under suitable growth conditions, the equilibrium states are proved to be exactly states of minimum energy. Then, under appropriate invertibility condition of the stress function, the principle of minimum complementary energy is proved for equilibrium states. For the no-tension material in Case (b), the principle of minimum complementary energy (in the absence of the invertibility assumption) is proved. Also, a theorem is proved stating that the total energy of the system is bounded from below if and only if the loads can be equilibrated by a stress field that is statically admissible and the bulk stress is negative semidefinite. Two examples are given. In the first, we consider the elastic semi-infinite plate with attached stiffener on the boundary (Melan’s problem). In the second example, we present a stress solution for a rectangular panel with membrane occupying the main diagonal plane.     

A tensor method for the derivation of the equations of rigid body dynamics

A tensor method for the derivation of the equations of rigid body dynamics,based onthe concepts of continuum mechanics,is presented.The formula of time derivative of theinertia tensor with zero corotational rate is used to prove the equivalences of five methods,namely,Lagrange’s equations,Nielsen’s equations,Gibbs-Appell’s equations,Kane’sequations and the generalized momentum type of Kane’s equations.Some differentialidentities on angular velocity and angular acceleration are given.     

Numerical Investigation of Large Elastoplastic Strains of Three-Dimensional Bodies

A method of stress—strain analysis of elastoplastic bodies with large displacements, rotations, and finite strains is developed. The incremental loading technique is used within the framework of the arbitrary Lagrangian—Eulerian formulation. Constitutive equations are derived which relate the Jaumann derivative of the Cauchy—Euler stress tensor and the strain rate. The spatial discretization is based on the FEM and multilinear three-dimensional isoparametric approximation. An algorithm of stress—strain analysis of elastic, hyperelastic, and perfectly plastic bodies is given. Numerical examples demonstrate the capabilities of the method and its software implementation __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 36–43, June 2005.     

The stress conjugate to logarithmic strain

A stress S is said to be conjugate to a strain measure E if the inner product S · E̊ is the power per unit volume. The logarithmic strain In U, with U the right stretch tensor, has been considered an interesting strain measure because of the relationship of its material time derivative (In U) with the stretching tensor D. In a previous article (Int. J. Solids Structures 22, 1019–1032 (1986)) a formula for (In U) was obtained in direct notation for the cases where the principal stretches are repeated, as well as for the case where they are all distinct. Here the formula for (In U) and the definition of conjugate stress are used to derive an explicit, properly invariant expression for the stress conjugate to the logarithmic strain.     

On the partition of rate of deformation in crystal plasticity

Two different partitions of the rate of deformation tensor into its elastic and plastic parts are derived for elastic–plastic crystals in which crystallographic slip is the only cause of plastic deformation. One partition is associated with the Jaumann, and the other with convected rate of the Kirchhoff stress. Different expressions for the plastic part of the rate of deformation are obtained, and corresponding constitutive inequalities discussed. Relationship with the plastic part of the rate of the Lagrangian strain is also given.