Analysis of the axisymmetric thermoelastoplastic state of branched laminated transversally isotropic shells

A technique is developed for determination of the axisymmetric thermoelastoplastic stress-strain state of branched laminated transversally isotropic shells of revolution under loads that cause the meridional stress state and torsion. The method is based on the rectilinear-element hypotheses for the whole package of layers. To describe the processes of active elastoplastic deformation of a transversally isotropic material, deformation-type equations, which are constructed without recourse to the plastic-potential existence condition, are used. The scalar functions in the constitutive equations depend on the shear-strain rate and temperature. The solution of the problem is reduced to numerical integration of systems of differential equations. An example of determination of the elastoplastic state of a two-layer cylindrical shell stiffened with a rigid ring support is presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 125–131, April, 2000.     

Determining the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator

A technique to determine the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator is developed. The technique is based on the theory of thin shells that takes into account transverse shear and torsional strains. Plastic equations that relate the components of the stress tensor in Eulerian coordinates with the linear components of the finite-strain tensor are used as constitutive equations. The nonlinear scalar functions in the constitutive equations are found from base tests on tubular specimens under proportional loading for different stress modes. The boundary-value problem is solved by numerically integrating a system of ordinary differential equations     

The Thermoviscoelastoplastic Axisymmetric Stress–Strain State of Laminated Flexible Orthotropic Shells

A technique is proposed to determine the thermoviscoelastoplastic axisymmetric stress–strain state of laminated shells made of isotropic and orthotropic materials. The paper deals with processes of shell loading such that both instantaneous elastoplastic and creep strains occur in isotropic materials and elastic and creep strains in orthotropic materials. The technique is developed within the framework of the Kirchhoff–Love hypotheses for a stack of layers with the use of the equations of the geometrically nonlinear theory of shells in a quadratic approximation. The deformation of isotropic materials is described by the equations of the theory of deformation along slightly curved trajectories, while the deformation of orthotropic materials is described by Hooke's law with additional terms allowing for creep. A numerical example is given     

Axisymmetric Thermoelastoplastic Stress State of Isotropic Solids of Revolution Under Impulsive Loading

Consideration is given to the solution of a dynamic problem for a solid of revolution with an arbitrary meridional section under impulsive thermomechanical loading inducing elastoplastic strains. The theory of small elastoplastic deformations is used. The constitutive equations are linearized by the variable-parameter method. The unloading process is described by a linear law. The solution technique involves the finite-element approximation in spatial coordinates and the finite-difference representation of time derivatives. Based on the principle of linear summation, recurrent relations are derived for successive evaluation of nodal displacements by an explicit scheme in time. Solution for cylinders and disks are presented to illustrate the influence of elastoplastic deformations on wave processes     

Method of successive approximations for solving boundary-value problems of plasticity with allowance for the stress mode

A method for solving boundary-value problems of plasticity with allowance for the stress mode is developed. To describe the elastoplastic deformation of an isotropic material, use is made of constitutive equations that include two nonlinear functions dependent on the stress mode and determined experimentally. The elastoplastic state of a thin cylindrical shell under axisymmetric loading is calculated as a numerical example. The numerical results demonstrate good convergence of the method. The effect of the stress mode on the strain distribution in a cylindrical shell is assessed     

Determination of the ultimate states of elastoplastic bodies

The equations of quasistatic deformation of elastoplastic bodies are considered in a geometrical linear formulation. After discretization of the equations with respect to spatial variables by the finite-element method, the problem of determining equilibrium onfigurations reduces to integration of a system of nonlinear ordinary differential equations. In the ultimate state of a body of an ideal elastoplastic material, the matrix of the system degenerates and the problem becomes singular. A regularization algorithm for determining solutions of the problems for the ultimate states of bodies is proposed. Numerical solutions of test problems of determining the ultimate loads and equilibrium configurations for ideal elastoplastic bodies confirm the reliability of the regularization algorithm proposed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 196–204, September–October, 2000.     

The Elastoplastic Axisymmetric Stress–Strain State of Flexible Laminated Shells Exposed to Radiation

A procedure is proposed to numerically study the thermoelastoplastic axisymmetric stress–strain state of laminated flexible shells exposed to radiation. The equations of thermoradiation plasticity describing simple processes are used. Results of an analysis of the elastoplastic state of a three-layer shell with regard for radiation effects are presented     

Determination of the Axisymmetric Geometrically Nonlinear Thermoviscoelastoplastic State of Laminated Medium-Thickness Shells

A method is developed to determine the axisymmetric geometrically nonlinear thermoelastoviscoplastic stress–strain state of branched laminated medium-thickness shells of revolution. The method is based on the hypotheses of a rectilinear element for the whole set of layers. The shells are subject to loads that cause a meridional stress state and torsion. They can consist of isotropic layers, which deform beyond the elastic limit, and elastic orthotropic layers. The relations of thermoviscoplastic theory, which describe simple processes of loading, are employed as the equations of state for the isotropic layers. The solution of the problem is reduced to numerical integration of systems of differential equations. The geometrically nonlinear elastoplastic state of a two-layer corrugated shell of medium thickness is calculated as an example     

Transition theories of elastic-plastic deformation of metallic polycrystals

Summary A general approach to the problem of determination of elastoplastic behavior of metallic polycrystals at finite deformation is presented. The relation between moving dislocation density and global slip rate for grains is developed. Transition to grain response is obtained by introducing the hardening matrix. Field equations for heterogeneous elastoplastic metals are transformed into an integral equation, using Green functions technique. This allows to find the spin of the lattice related to texture formation.Scale transition is achieved by a self-consistent approximation of the integral equation. New results concerning BCC metals (sheet steel) are presented. They apply to tensile test, Lankford coefficient, initial and subsequent yield surfaces, and evolution of the internal state of the polycrystal: second-order residual stress, stored energy and texture evolution.     

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part II – the formulation of elastoplastic coupling at large strain

The two key phenomena occurring in the process of ceramic powder compaction are the progressive gain in cohesion and the increase of elastic stiffness, both related to the development of plastic deformation. The latter effect is an example of ‘elastoplastic coupling’, in which the plastic flow affects the elastic properties of the material, and has been so far considered only within the framework of small strain assumption (mainly to describe elastic degradation in rock-like materials), so that it remains completely unexplored for large strain. Therefore, a new finite strain generalization of elastoplastic coupling theory is given to describe the mechanical behaviour of materials evolving from a granular to a dense state.The correct account of elastoplastic coupling and of the specific characteristics of materials evolving from a loose to a dense state (for instance, nonlinear – or linear – dependence of the elastic part of the deformation on the forming pressure in the granular – or dense – state) makes the use of existing large strain formulations awkward, if even possible. Therefore, first, we have resorted to a very general setting allowing general transformations between work-conjugate stress and strain measures; second, we have introduced the multiplicative decomposition of the deformation gradient and, third, employing isotropy and hyperelasticity of elastic response, we have obtained a relation between the Biot stress and its ‘total’ and ‘plastic’ work-conjugate strain measure. This is a key result, since it allows an immediate achievement of the rate elastoplastic constitutive equations. Knowing the general form of these equations, all the specific laws governing the behaviour of ceramic powders are finally introduced as generalizations of the small strain counterparts given in Part I of this paper.     

Nonaxisymmetric Thermoelastoplastic Analysis of Branched Shells of Revolution by the Semianalytic Finite-Element Method

A technique is proposed to solve elastoplastic deformation problems for branched shells of revolution under the action of asymmetric forces and a temperature field. The kinematic equations are derived within the framework of the linear Kirchhoff–Love theory of shells and the thermoelastic relations within the framework of the theory of small elastoplastic strains. The problem is given a variational formulation based on the virtual-displacement principle and the Fourier-series expansion of the unknown functions and loads with respect to the circumferential coordinate. The additional-load method is used to solve a nonlinear problem and the finite-elements method is used to carry out a numerical analysis. As an example, an asymmetric stress–strain analysis is performed for a cylindrical shell reinforced by a ring plate.     

Describing the elastoplastic deformation of layered shells under axisymmetric loading with allowance for the third invariant of the stress deviator

A method for numerical analysis of the elastoplastic stress–strain state of thin layered shells of revolution under axisymmetric loading is proposed. Constitutive equations describing the elastoplastic deformation of isotropic materials with allowance for the stress mode are used. Numerical results are presented     

Bending of an elastoplastic Hencky bar-chain: from discrete to nonlocal continuous beam models

The static behavior of an elastoplastic one-dimensional lattice system in bending, also called a microstructured elastoplastic beam or elastoplastic Hencky bar-chain (HBC) system, is investigated. The lattice beam is loaded by concentrated or distributed transverse monotonic forces up to the complete collapse. The phenomenon of softening localization is also included. The lattice system is composed of piecewise linear hardening–softening elastoplastic hinges connected via rigid elements. This physical system can be viewed as the generalization of the elastic HBC model to the nonlinear elastoplasticity range. This lattice problem is demonstrated to be equivalent to the finite difference formulation of a continuous elastoplastic beam in bending. Solutions to the lattice problem may be obtained from the resolution of piecewise linear difference equations. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process. The new nonlocal elastoplastic theory associated with both a distributed nonlocal elastoplastic law coupled to a cohesive elastoplastic model depends on length scales calibrated from the spacing of the lattice model. Differential equations of the nonlocal engineering model are solved for the structural configurations investigated in the lattice problem. It is shown that the new micromechanics-based nonlocal elastoplastic beam model efficiently captures the scale effects of the elastoplastic lattice model, used as the reference. The hardening–softening localization process of the nonlocal continuous model strongly depends on the lattice spacing which controls the size of the nonlocal length scales.     

The nonaxisymmetric elastoplastic state of isotropic and cylindrically orthotropic solids of revolution under nonisothermal loading

A technique for numerical analysis of the nonaxisymmetric elastoplastic stress-strain state of laminated isotropic and cylindrically orthotropic bodies of revolution under nonisothermal loading with allowance for its history is stated. The strain of isotropic materials is described by the equations of the theory of strain along small-curvature trajectories, while Hooke's law is used to describe the strain of orthotropic materials. The necessity of accounting for the loading history is shown by an example of a nonaxisymmetrically heated sandwich solid of revolution. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 2, pp. 83–90, February, 2000.     

The nonaxisymmetric thermostressed state of laminated isotropic and rectilinearly orthotropic solids of revolution

A technique for determination of the nonaxisymmetric elastoplastic stress-strain state of laminated isotropic and rectilinearly orthotropic solids of revolution under nonisothermal loading is described. The semianalytic method of finite elements is used together with the method of successive approximations. The deformation of isotropic materials is described by the equations of the theory of deformation along trajectories of small curvature, and the deformation of orthotropic materials is described by the equations of the theory of elasticity. The thermostressed state of a three-layered solid of revolution of complex form made from isotropic and cylindrically and rectilinearly orthotropic materials subject to nonaxisymmetric heating is studied. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 88–95, April, 2000.     

An endochronic plasticity formulation for filled rubber

The nature of elastomeric material demands the consideration of finite deformations, nonlinear elasticity including damage as well as rate-dependent and rate-independent dissipative properties. While many models accounting for these effects have been refined over time to do better justice to the real behavior of rubber-like materials, the realistic simulation of the elastoplastic characteristics for filled rubber remains challenging.The classical elastic-ideal-plastic formulation exhibits a distinct yield-surface, whereas the elastoplastic material behavior of filled rubber components shows a yield-surface free plasticity. In order to describe this elastoplastic deformation of a material point adequately, a physically based endochronic plasticity model was developed and implemented into a Finite Element code. The formulation of the ground state elastic characteristics is based on Arruda and Boyce (1993) eight-chain model. The evolution of the constitutive equations for the nonlinear endochronic elastoplastic response are derived in analogy to the Bergström–Boyce finite viscoelasticity model discussed by Dal and Kaliske (2009).     

The implicit standard material theory for modelling the nonassociative behaviour of metals

Summary  Modelling the elastoplastic or elastoviscoplastic behaviour of metallic materials exhibiting strain hardening and damage leads to complex nonassociative constitutive equations, sources of many theoretical and numerical troubles. The usual modelling of a nonassociative constitutive equation leads to the loss of the interesting and very useful properties of generalised standard materials deriving from the key concepts of convexity and normality. The argument that will be developed is that the bipotential concept is an appropriate answer. In the first part, after introducing the state variables generally used to describe the behaviour of metallic materials, the constitutive equations subjected to the principles of thermodynamics are derived from two potentials. The state potential gives the state laws, and the bipotential of dissipation delivers the evolution laws for state variables, through the implicit normality assumption. The second part is devoted to several particular applications to metal elastoplasticity and elastoviscoplasticity models. Received 29 March 2000; accepted for publication 26 September 2000     

Analytical Solutions of Nonlinear Static Problems for Threads of Finite Stiffness Under Active Loading

The behavior of elastoplastic threads of finite stiffness under lateral bending is analyzed. Geometrical and physical nonlinearities are taken into account. The material is assumed to be elastoplastic. The nonlinear equations describing the stress—strain state of threads are derived using the virtual-displacement principle. Numerical results are discussed __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 121–129, June 2005.     

Simulation of three-point bending at large deflections by an elastoplastic contact analysis

Three-point bending is simulated by an elaborate numerical procedure based on an elastoplastic, large deflection, contact analysis. A minimization formulation is used, which is equivalent to the incremental form posed as partial differential equations with inequalities. A sequential quadratic programming approach based on the finite-element technique is adopted as a method of solution. To examine the validity of the simulation method, experiments are carried out for specimens that have various widths.Paper was presented at the 1988 SEM Spring Conference on Experimental Mechanics held in Portland, OR on June 5–10.     

A Geometrical Approach to Nonconservative Shocks and Elastoplastic Shocks

We study systems of conservation laws with convex inequality constraints. We analyze shock solutions for the underlying nonconservative system of partial differential equations. Then we study a model elastoplastic system of partial differential equations in the context of hypoelasticity. We show that a sonic point is necessary to construct a compression solution that begins at a constrained compressed state.