Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic–plastic bar

We propose a fundamentally new concept to the treatment of material instabilities and localization phenomena based on energy minimization principles in a strain-softening elastic–plastic bar. The basis is a recently developed incremental variational formulation of the local constitutive response for generalized standard media. It provides a quasi-hyperelastic stress potential that is obtained from a local minimization of the incremental energy density with respect to the internal variables. The existence of this variational formulation induces the definition of the material stability of inelastic solids based on convexity properties in analogy to treatments in elasticity. Furthermore, localization phenomena are understood as micro-structure development associated with a non-convex incremental stress potential in analogy to phase decomposition problems in elasticity. For the one-dimensional bar considered the two-phase micro-structure can analytically be resolved by the construction of a sequentially weakly lower semicontinuous energy functional that envelops the not well-posed original problem. This relaxation procedure requires the solution of a local energy minimization problem with two variables which define the one-dimensional micro-structure developing: the volume fraction and the intensity of the micro-bifurcation. The relaxation analysis yields a well-posed boundary-value problem for an objective post-critical localization analysis. The performance of the proposed method is demonstrated for different discretizations of the elastic–plastic bar which document on the mesh-independence of the results.     

One plasticity model for problems of plastic metal working

Scalar and tensor models of plastic flow of metals extending plasticity theory are considered over a wide range of temperatures and strain rates. Equations are derived using the physico-phenomenological approach based on modern concepts and methods of the physics and mechanics of plastic deformation. For hardening and viscoplastic solids, a new mathematical formulation of the boundary-value plasticity problem taking into account loading history is obtained. Results of testing of the model are given. A numerical finite-element algorithm for the solution of applied problems is described. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 159–169, November–December, 2008.     

Behaviour of an incrementally elastic thick walled sphere under internal and external pressure

The behaviour of a thick walled sphere underinternal and external pressure is considered. The material of the sphere is assumed to obey an incrementally elastic constitutive law. There is no restriction on the size of the deformation and a solution is given in terms of special functions associated with the non-linear differential equations of the problem.As a numerical example the behaviour of a spherical shell, subjected to internal pressure, is described. It is shown that at a certain critical pressure instability of the second kind (inflation) is obtained.     

Dynamic shakedown of elastic-plastic solids for a set of alternative loading histories

The paper deals with dynamic shakedown of an elastic-perfectly plastic solid body subjected to a loading history which is unknown but is allowed to belong to a given set of loading histories. In the hypothesis of a piecewise linear convex set, a sufficient shakedown theorem is given and a bounding principle for the plastic work produced is formulated in terms of the dynamic elastic responses to a discrete set of loading histories. The solution of a minimization problem gives the most stringent bound which also proves to possess a local character, i.e., it regards the plastic work density at any point.     

A new discontinuous FE formulation for crack path prediction in brittle solids

In this paper a new finite element (FE) formulation to simulate embedded strong discontinuity for the study of the fracture process in brittle or quasi-brittle solids is presented. A homogeneous discontinuity is considered to be present in a cracked finite element with the possibility to take into account the opening and the sliding phenomena which can occur across the crack faces. In such a context a new simple stress-based implementation of the discontinuous displacement field is proposed by an appropriate stress field correction introduced at the Gauss points level in order to simulate, in a fashion typical of an elastic–plastic classical FE formulation, the mechanical effects of the bridging and friction stresses due to crack faces opening and sliding which can occur during the loading–unloading process structural component or solid being analysed. The proposed formulation does not need to introduce special or modified shape functions to reproduce discontinuous displacement field but simply relaxes the stress field in an appropriate fashion. Both linear elastic and elastic–plastic behaviour of the non-cracked material can be considered. Several 2D problems are presented and solved by the proposed procedure in order to predict load–displacement curves of brittle structures as well as crack patterns that develop during the loading process.The proposed discontinuous new FE formulation gives the advantages to be simple, computationally economic and to keep internal continuity of the numerical FE model; furthermore the developed algorithm can be easily implemented in standard FE programs as a standard plasticity model.     

Viscoplastic creep of rocks around a lined tunnel

A rock-support analysis is done assuming that the rock behaves elastic/viscoplastically while the support can be any kind of nonlinear support. The case of circular tunnels is considered and the formulation of the mathematical problem is discussed. Furthermore, the creep of rock around a tunnel with nonlinear supports (called “yieldable” or “self-adapting” supports) to limit the closure of the tunnel is studied. It is shown that the ultimate ground reaction is not unique since it depends on the loading history such as excavation layout and support design. For relatively small pressures exerted by the support, it is shown that the onset of failure by dilatancy can be determined. Two kinds of solutions are given: a simplified approximate one, which is easy to apply and with which the qualitative discussion of the solution is simple and revealing, and a more rigorous general elaborate numerical solution obtained by using computer programs.     

An elastic-plastic analysis of an asymmetric cracked specimen subjected to longitudinal shear

Some recent elastic-plastic analyses of cracked specimens subjected to symmetric mode III loading are extended to include asymmetric loading and geometry. Solutions are given for arbitrary work hardening behaviour in any specimen that is amenable to a linear elastic analysis. It is shown that asymmetry has a major influence on the shape of the plastic zone, but does not affect the J-integral unil the loading is well into the large scale yielding range. In particular the “plastic zone corrected” estimate of J, obtained by elastically solving a problem for a crack longer than the actual one, is shown to remain a valid two-term asymptotic expansion in the presence of asymmetry. The general results are applied to a crack at an angle to a uniform stress field in a power law hardening material. The growth of the plastic zone is displayed graphically for various hardening exponents and crack orientations. No other asymmetric solution is available, but values of J are compared with those obtained from a fully plastic analysis in the symmetric case.     

Influence of plasticity on interface toughness in a layered solid with residual stresses

For crack growth along an interface between two adjacent elastic–plastic materials in a layered solid, the resistance curve behaviour is analysed by approximating the behaviour in terms of a bi-material interface under small scale yielding conditions. Thus, it is assumed that the layers are thick enough so that the extent of the plastic regions around the crack tip are much smaller than the thickness of the nearest layers. The focus is on the effect of initial residual stresses in the layered material, or on T-stress components induced during loading. The fracture process is represented in terms of a cohesive zone model. It is found that the value of the T-stress component in the softer material adjacent to the interface crack plays a dominant role, such that a negative value of this T-stress gives a significant increase of the interface fracture toughness, while a positive value gives a reduction of the fracture toughness.     

Effects of inertia on dynamic neck formation in tensile bars

Neck localization during high rate extension of round bars is analyzed numerically. An axisymmetric problem formulation is given and the material is described as rate independent and elastic–plastic. The time history of neck development is investigated and effects of geometry and initial thickness imperfections are visualized. Studies of both weakly and strongly developed necks are performed, revealing the occurrence of multiple necks in some loading cases. The influence on neck formation from background inertia (lateral inertia) corresponding to the inertia originating from homogeneous deformation of the bar is examined somewhat qualitatively by introducing an artificial volume load. Calculations show that in the present analysis, background inertia does not have any noticeable influence on the necking pattern unless the effect is artificially magnified by three orders of magnitude so that it becomes comparable to the yield stress.     

Generalized Inverses in Elastic-Plastic Analysis of Structures∗

This paper deals with elastic-plastic analysis of skeletal structures that are subjected to proportionally increasing loading. It is assumed that no local unloading occurs (holonomic behavior) and that yield conditions are piecewise linearized. A quadratic programming problem, which arises from the application of the minimum complementary energy principle for such structures, is shown to have an explicit form of solution. The matrix expression of this solution involves certain modifications of the Bott-Duffin generalized inverse. This inverse can be effectively calculated for a given structure and allows one to obtain a unique stress distribution under agiven load. Moreover, if the load is prescribed up to an unknown scalar factor, the ultimate value of this multiplier (collapse load) and the elastoplastic stress state at collapse can be found by the same method     

Solution of a variational problem for the flow in an MHD generator

The most complete study and construction of extremal plasma flow regimes in the channel of an MHD generator may be accomplished using the methods of variational calculus. The variational problem of conducting-gas motion in an MHD channel was first discussed in [1]. The general formulation of the problem for the MHD generator was considered in [2]. Solutions of variational problems for particular cases of extremal flows are given in [2–5].The present study obtains the solution of the variational problem of the flow of a variable conductivity plasma in an MHD generator which has maximal output power for given channel length or volume. An analysis of the solution is made, and a comparison of the extremal flows with optimized flow in a generator with constant values of the electrical efficiency and flow Mach number is carried out.     

Development of plasticity and damage in vibrating structural elements performing guided rigid-body motions

Summary  A numerical algorithm for studying the development of plastic and damaged zones in a vibrating structural element with a large, guided rigid-body motion is presented. Beam-type elements vibrating in the small-strain regime are considered. A machine element performing rotatory motions, similar to an element of a slider-crank mechanism, is treated as a benchmark problem. Microstructural changes in the deforming material are described by the mesolevel variables of plastic strain and damage, which are consistently included into a macroscopic analysis of the overall beam motion. The method is based on an eigenstrain formulation, considering plastic strain and damage to contribute to an eigenstrain loading of a linear elastic background structure. Rigid-body coordinates are incorporated into this beam-type structural formulation, and an implicit numerical scheme is presented for iterative computation of the eigenstrains from the mesolevel constitutive behavior. Owing to the eigenstrain formulation, any of the existing constitutive models with internal variables could in principle be implemented. Linear elastic/perfectly plastic behavior is exemplarily treated in a numerical study, where plastic strain is connected to the Kachanov damage parameter by a simple damage law. Inelastic effects like plastic shakedown and damage-induced low-cycle rupture are shown to occur in the examplary problems. Received 1 September 1999; accepted for publication 9 March 2000     

Determination of material properties using nanoindentation and multiple indenter tips

Nanoindentation is a useful method to probe the material properties of a solid. Its effective use lies in interpreting the data collected from a nanoindentation experiment with an associated analytical/numerical solution of the corresponding problem configuration. In this paper, a parametric finite element study has been performed to develop a new procedure for extracting elastic–plastic properties of a material through nanoindentation experiments with a substantially improved accuracy for the elastic properties of a elastic–plastic solid. The procedure involves data collected through the use of two, different, nanoindenter tips. Non-dimensional functions were constructed for two different indenter geometries to show that test results from multiple indenters, when appropriately manipulated, deliver superior results, compared to using one indenter. The material was assumed to be an isotropic elastic–plastic solid with power law hardening. Friction between the indenter and the material was included in the cases studied. The ratio of yield strength to elastic modulus was assumed to be in the range 0.0005–0.02 and the hardening coefficient was assumed to be between 0 and 0.4. Poisson’s ratio was fixed at 0.3.     

Shakedown analyses for rolling and sliding contact problems

There is a range of problems where repeated rolling and sliding contact occurs over a half space of an elastic–perfectly plastic material. For such problems shakedown and limit analysis provide significant advantages over other forms of analysis when a global understanding of deformation behaviour is required. In this paper, a recently developed numerical upper bound method, the Linear Matching Method (LMM), for shakedown analyses is applied to the solution of a problem previously considered by Ponter et al. [Ponter, A.R.S., Hearle, A.D., Johnson, K.L., 1985. J. Mech. Phys. Solids 33 (4), 339–362] for a moving Hertzian contact, with sliding friction. This semi-analytic solution is an upper bound based on certain specific kinematic assumptions. We show that the Ponter, Hearle and Johnson solution is a reasonable approximate solution for a circular contact area but is less accurate for an elliptic contact area. For an elliptic contact area LLM solutions converge to the line contact solution. The effect of the non-coincidence of the direction of travel and slide is also investigated.     

Transient thermoelastoplastic stress analysis for a hollow sphere using the incremental theory of plasticity

An analysis based on the incremental strain theory is formulated for solving the problem of an elastoplastic hollow sphere subjected to a transient temperature distribution. Thermal and material properties are assumed to be temperature dependent and the behaviour of the medium to be characterized by the Ramberg-Osgood stress-strain relation. A method of successive elastic solutions is used to obtain a numerical solution. An illustrative example shows that the effective stress is not a monotonie function of the radius, but is much dependent on the history, gradient, and distribution of the temperature in the hollow sphere. In addition, unloading in the plastically deformed region is confirmed from the detailed discussion on the distribution of strains. As a result, the analysis based on the total strain theory is not permissible for solving this kind of elastoplastic problems subjected to transient thermal loading. In the following analysis the problem is treated in a quasi-static sense and the inertia terms in the thermoelastoplastic equations are neglected.     

Multiple loading–unloading of an elastic–plastic spherical contact

A model for multiple repeated loading and unloading of an elastic–plastic sphere and a rigid flat is presented to cover a wide range of loading conditions far beyond the elastic limit. The sphere material is modeled as elastic linear isotropic hardening and follows the von Mises yield criterion. It is shown that although most of the plastic deformation occurs during the first loading, secondary plastic flow may evolve during the first unloading. The occurrence of this secondary plastic flow depends on the level of first loading and is strongly affected by the Poisson’s ratio and material hardening. The region of secondary plastic flow may propagate during the very first loading–unloading cycles, reaching a steady state after which the following loading–unloading cycles become fully elastic.     

Applied creep theory for bodies with anisotropy due to plastic prestrain

Plastic strains in structures at the stages of manufacturing, testing, and approaching the operation regime cause anisotropic variations in the mechanical properties of materials, including creep strength. We consider the following special but practically important class of loading processes for originally isotropic materials: a simple active plastic strain is followed by a long-term steady-state loading within the elastic limits. To describe the second stage, we present the creep strain deviator in the form of an additive orthogonal decomposition in the directions of the repeated loading and the vector anisotropy. The coefficients in the decomposition are material functions of time, of the intensities of the preliminary and repeated loadings, and of the angle between the directions of these loadings. We obtain conditions on the material functions under which, at any given time instant, there is a one-to-one continuous correspondence between the stress and strain tensors for the model proposed and the boundary-value problem in the generalized statement has a unique solution; we also prove the convergence of the iteration method of elastic solutions used to find this unique solution. The model is identified according to the creep diagrams (under steady-state stresses of different values) determined for the material in the original state and after the plastic prestrain at an angle (zero, extended, and intermediate) to the direction of the repeated loading. We show that our results are in good agreement with the results available in the literature concerning experiments in this class of processes for stainless steel at high temperature. We propose an engineering version of the theory in which only the experimental data for uniaxial tension are used. We discuss the versions of the model for the cases in which the plastic preloading is cyclic (one-dimensional or circular) and the repeated loading is unsteady.     

Steady crack growth in elastic–plastic fluid-saturated porous media

An asymptotic solution is obtained for stress and pore pressure fields near the tip of a crack steadily propagating in an elastic–plastic fluid-saturated porous material displaying linear isotropic hardening. Quasi-static crack growth is considered under plane strain and Mode I loading conditions. In particular, the effective stress is assumed to obey the Drucker–Prager yield condition with associative or non-associative flow-rule and linear isotropic hardening is adopted. Both permeable and impermeable crack faces are considered. As for the problem of crack propagation in poroelastic media, the behavior is asymptotically drained at the crack-tip. Plastic dilatancy is observed to have a strong effect on the distribution and intensity of pore water pressure and to increase its flux towards the crack-tip.