A Variationally Consistent Time Modelling of Elastic-Plastic Constitutive Equations

A general energy-based time discretization method for evolutive analysis is presented. Most known time integration procedures (mid-point rule, backward difference, etc.) are shown to be particular cases of it. For space continuous systems, a sequence of weighted boundary value problems of deformation-theory plasticity are obtained, each characterizable by a number of variational principles useful for finite element discretization.

Erratum to: Letter to the Editor [Engineering Fracture Mechanics 2003 (70) 1219-21]

Erratum and Corrections

An approach to elastic shakedown based on the maximum plastic dissipation theorem

ELASTIC-PERFECTLY PLASTIC SOLID STRUCTURES are considered subjected to combined loads, superposition of permanent (mechanical) loads and cyclically variable loads, the latter being specified to within a scalar multiplier. The classical maximum dissipation theorem is used to derive known results of the shakedown theory, as well as a few apparently novel concepts: the shakedown limit load associated with a given (noninstantaneous) collapse mode, the mixed upper bound to the shakedown safety factor, and the mixed static-kinematic formulation of the shakedown safety factor problem. The shakedown load boundary surface is also investigated and a number of its notable features are pointed out. A s…

A Consistent Formulation of the BEM within Elastoplasticity

A symmetric-definite BEM formulation is derived by making alternatively use of two energy principles, i.e. the Hellinger-Reissner principle and a boundary min-max principle ad-hoc formulated. Two kinds of discretization are operated, one by boundary elements to model the system elastic properties, another by cell-elements to model the material plastic behavior. The cell yielding laws are expressed in terms of generalized variables and comply with the features of associated plasticity, due to the maximum plastic work theorem used for their derivation.

A Thermodynamic Plasticity Formulation with Local and Nonlocal Internal Variables

In order to obtain the elastic response of nonhomogeneous materials, it is often sufficient to adopt an implicit homogenization technique which allows one to treat the material as an equivalent continuum medium. For large stress concentration or for accurate small scale studies this widely applied technique may show some limit and a more refined analysis might be required involving nonlocal elastic effects, see e.g. Kroner (1967), Eringen et al. (1977).

OPTIMUM DESIGN OF REINFORCED CONCRETE STRUCTURES UNDER VARIABLE LOADINGS

Abstract This paper presents a method for optimal design of reinforced concrete (RC) structures, subjected to quasi-static variable loads and accounting for cross-sections limited ductility requirements. It consists of a very simple refinement procedure to be applied in the classical Optimal Shakedown Design, (OSD), which leads to a strengthened structure satisfying the requisite that the actual plastic relative rotations, developed at a specified set of critical sections as a result of a variable repeated loading program, do not exceed given upper limits. This strengthened design is a safe but not strictly optimal design and is obtained using the simple rule that the steel reinforcement ar…

Shakedown Problems for Material Models with Internal Variables

The classical shakedown theory is reconsidered with the objective of extending it to a quite general constitutive law for rate-insensitive elastic-plastic material models endowed with dual internal variables and thermodynamic potential. The statical and kinematical shakedown theorems, the corresponding approaches to the shakedown load multiplier problem and a deformation bounding theorem are presented and discussed with a view of further developments.

BEM Techniques in Nonlocal Elasticity

Hellinger-Reissner variational principle for stress gradient elastic bodies with embedded coherent interfaces

An Hellinger-Reissner (H-R) variational principle is proposed for stress gradient elasticity material models. Stress gradient elasticity is an emerging branch of non-simple constitutive elastic models where the infinitesimal strain tensor is linearly related to the Cauchy stress tensor and to its Laplacian. The H-R principle here proposed is particularized for a solid composed by several sub-domains connected by coherent interfaces, that is interfaces across the which both displacement and traction vectors are continuous. In view of possible stress-based finite element applications, a reduced form of the H-R principle is also proposed in which the field linear momentum balance equations are…

A Variational Formulation of the BEM for Elastic-Plastic Analysis

The quasi-static elastic perfectly plastic analysis problem is approached by the boundary element method (BEM). To this purpose, a time semidiscretization is first achieved by finite intervals (Fl) in order to transform, through a variationally consistent procedure, the evolutive problem into a discrete sequence of inelastic holonomic-type “weighted” problems for each of which a mixed boundary/domain min-max principle is established. This principle is then discretized by means of boundary elements (BE) and cell elements (CE), the latter having the only purpose of suitably interpolating the FI weighted yielding laws within the domain. The algebraic governing equations obtained show symmetry …

A Variational Approach to Boundary Element Methods

Symmetric BEM Formulations for Elastic-Damage Material Models

BEM analysis for elastic-damage materials is addressed by “undamaged” fundamental solutions. In step-by-step analysis, the actual response is obtained by an iterative procedure in which the undamaged structure is subjected to the loads and to some fictitious strains (or relaxation stresses) simulating the damage effects. Through symmetric BEM, the solution to the typical iteration problem is shown to solve a boundary/domain stationarity principle, whereas the above iterative procedure can be incorporated in a predictor/corrector scheme aimed at the integration of the damage laws. Discretization by boundary and interior elements leads to a symmetric equation system.

Optimal shakedown design of beam structures

The optimal design of plane beam structures made of elastic perfectly plastic material is studied according to the shakedown criterion. The design problem is formulated by means of a statical approach on the grounds of the shakedown lower bound theorem, and by means of a kinematical approach on the grounds of the shakedown upper bound theorem. In both cases two different types of design problems are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the design of the assigned volume whose shakedown limit load is maximum. The optimality conditions of the four problems above are found by the use of a variational approach; such…

A Boundary/Interior Element Discretization Method for the Analysis of Two- and Three-Dimensional Elastic-Plastic Structures

A coupled boundary/interior element method is presented for the analysis of elastic-plastic structures with material models endowed of dual internal variables. The domain field modelling is limited to the only plastic strains and strain-like internal variables, represented by their node values at a set of strain points in each interior element. The formulation, based on a Galerkin-type approach, is variationally consistent and leads to a fully symmetric-definite equation system. The backward difference method is adopted for the step-by-step integration procedure, and each step is addressed by an iterative predictor/corrector solution scheme. The analysis method is expected to be most approp…

A theoretical link between gradient and nonlocal elasticity models, including higher order boundary conditions

The paper presents a recently developed rational derivation of the strain gradient elasticity model from the nonlocal (or integral) model. This kind of derivations are generally recovered just by an expansion into a Taylor series of the nonlocal strain field up to a certain order, and then operating the integration (or averaging) over the spatial interaction domain. The latter procedure is fully consistent when the analysis is performed over an unbounded domain, but when a classical bounded domain is analyzed it lacks in reproducing the so-called higher-order boundary conditions. In the present contributions the complete derivation is achieved employing an extended version of the Principle …

A boundary min-max principle as a tool for boundary element formulations

Abstract A min-max principle for elastic solids, expressed in terms of the unknown boundary displacements and tractions, is presented. It is shown that its Euler-Lagrange equations coincide with the classical boundary integral equations for displacements and for tractions. This principle constitutes a suitable starting point for a symmetric sign-definite formulation of the boundary element method.

Discussion of “Shakedown under Elastic Support Conditions”

Energy-Residual-Based approach to gradient plasticity

The “energy-residual-based approach” mentioned in the title consists in a thermodynamically consistent procedure for the formulation of a phenomenological plasticity model of either strain gradient, or nonlocal (integral) type. The authors have developed this procedure on the last ten years. It seem therefore appropriate to present an update of this theory at this forum. For brevity we shell limit ourselves to strain gradient plasticity.

Bounding Techniques and Their Application to Simplified Plastic Analysis of Structures

In the framework of the simplified analysis methods for elastoplastic analysis problems, the bounding techniques possess an important role. A class of these techniques, based on the so-called perturbation method, are here presented with reference to finite element discretized structures. A general bounding principle is presented and its applications are illustrated by means of numerical examples.