Mieczysław Kuczma

Modeling error and adaptivity in nonlinear continuum mechanics

In this paper, computable global bounds on errors due to the use of various mathematical models of physical phenomena are derived. The procedure involves identifying a so-called fine model among a class of models of certain events and then using that model as a datum with respect to which coarser models can be compared. The error inherent in a coarse model, compared to the fine datum, can be bounded by residual functionals unambiguously defined by solutions of the coarse model. Whenever there exist hierarchical classes of models in which levels of sophistication of various coarse models can be defined, an adaptive modeling strategy can be implemented to control modeling error. In the present work, the class of models is within those embodied in nonlinear continuum mechanics.

An adaptive algorithm for unilateral viscoelastic contact problems for beams and plates

Abstract An adaptive algorithm is given for the bending of viscoelastic beams and plates resting on viscoelastic, unilateral, frictionless foundations. An a posteriori error indicator proposed here is obtained by making use of upper and lower bounds generated by the saddle functional being a potential for the system of equations and inequalities which govern the contact problem.

Variational inequality formulation for flow theory plasticity

Abstract We consider the quasi-static deformation process of an elasto-plastic body, under the assumption of small strains. The elasto-plastic behaviour of the material ss assumed to be governed by the Huber-von Mises yield function with piecewise linear strain hardening (softening). The yield function is written in the terms of the strains, and we define the problem as the system that consists of (a) a variational equation which is the equilibrium condition for the body and (b) a variational inequality expressing the unilateral character of the plastic strain-rate multiplier. An iterative method for solving this nonlinear system, based on an incremental, implicit time integration scheme and on a finite element approximation is proposed. The variational inequality after being discretized using finite elements is solved as a linear complementarity problem. The results of some numerical experiments for a test problem are provided.

Analysis of Slackened Skeletal Structures by Mathematical Programming

Abstract The paper deals with the holonomic behavior of slackened-elastic–plastic (SEP) skeletal structures (beams, frames, trusses) using the quadratic programming formulations. The considerations are restricted to ‘frictionless’ quasi-static processes, small strains and piecewise linear approximations of the yield and clearance surfaces. The results of numerical experiments for several illustrative examples are presented, including the unilateral contact problem between an elastic-perfectly plastic beam and an elastic–perfectly plastic foundation. In the analysis the discrete FEM-oriented mathematical model [A. Gaw e cki, Elasto-plasticity of slackened systems, Arch. Mech. 1992;44:363–390 was employed.

Bending of elastic beams on winkler-type viscoelastic foundations with unilateral constraints☆

Abstract A solution algorithm is presented for the analysis of unilateral, frictionless contact between a beam and a viscoelastic foundation. The foundation is modelled as a Winkler-type viscoelastic medium, to which the time integration method [Mech. Teoret. i Stos. 22, 209–233 (1984)] was applied. The problem was formulated in the form of a variational inequality, from which, after space discretization by the finite element method, a linear complementarity problem was derived. The solution method covers the time-discontinuities of loads and both loading (reloading) and unloading processes. The applicability of the algorithm is illustrated by selected numerical solutions to some examples.

Elastic-plastic unilateral contact problems for slackened systems

Abstract The paper deals with the mathematical modelling of a class of unilateral elastic-plastic contact problems. The problem is posed in the framework of the so-called slackened systems. The considerations are confined to small strains and piecewise linear approximations of the yield surface and the clearance surface. An incremental formulation is proposed in the form of a linear complementarity problem (LCP). The formulation allows one to analyse a structural system which can be subjected to nonproportional loading paths. The included numerical results illustrate the application of this formulation to an elastic-plastic beam that comes into contact with a unilateral elastic-plastic foundation.

Composite Beams with Embedded Shape Memory Alloy

We are concerned with the bending problem of composite beams with embedded shape memory alloy in the form of fibres or strips. As a special case, the proposed formulation covers the case of a monolithic beam made of shape memory alloy. Shape memory alloys (SMAs) are materials that may undergo a temperature- or stress-induced martensitic phase transformation, which results in the shape memory effect and the pseudoelastic (superelastic) behaviour. The unique behaviour of SMAs opens new possibilities in design of adaptive structures. Herein we have formulated the quasi-static bending problem for a SMA beam in the form of a evolution variational inequality. After the finite element approximation, the variational inequality is solved incrementally as a sequence of linear complementarity problems. Finally, the predictions of the proposed model, including hysteresis loops, are illustrated with many results of numerical simulations for SMA beams.

Numerical Elastic-Plastic Model of RPC in the Plane Stress State

The work is concerned with the determination of effective material parameters of reactive powder concrete (RPC) in the range of its nonlinear response. We have used a two-scale modelling technique and carried out a series of experimental tests which allow us to validate the proposed numerical model of the considered RPC concrete. The behavior of a RPC concrete on a macro scale is described on the basis of phenomena occurring in the microstructure of material. The material microstructure is taken into account by means of a representative volume element (RVE), the structure of which is generated in a stochastic way with data from the designed recipes of RPC. It is assumed that the microstructure of RPC is composed of isotropic linear elastic—(perfectly) brittle constituents and at the macro scale the material is homogenized. This approach is a good basis for a simple modelling of microcracks that cause the nonlinear behaviour of the material at the macro level. The numerical analysis is carried out here for the plane stress state problem, and at each level of analysis the finite element method is applied.

Modeling Error and Adaptivity in Nonlinear Continuum Mechanics

In this report, computable global bounds on errors due to the use of various mathematical models of physical phenomena are derived. The procedure involves identifying a so-called fine model among a class of models of certain events and then using that model as a datum with respect to which coarser models can be compared. The error inherent in a coarse model, compared to the fine datum, can be bounded by residual functionals unambiguously defined by solutions of the coarse model. Whenever there exist hierarchical classes of models in which levels of sophistication of various coarse models can be defined, an adaptive modeling strategy can be implemented to control modeling error. In the present work, the class of models is within those embodied in nonlinear continuum mechanics.