# A numerical analysis for some non - linear constitutive problems in solid mechanics

## The paper deals with the application FEM for solving nonlinear constitutive problems in solid mechanics. The basis equations and algorithms of iterative processing are presented. Some programs written by languages Gibian and special operators in Castem 2000 are established. | Vietnam Journal of Mechanics, VAST, Vol. 26, 2004, No 2 (93 - 102) A NUMERICAL ANALYSIS FOR SOME NON-LINEAR CONSTITUTIVE PROBLEMS IN SOLID MECHANICS NGO HUONG NHU Institute of Mechanics ABSTRACT. The paper deals with the application FEM for solving nonlinear constitutive problems in solid mechanics. The basis equations and algorithms of iterative processing are presented. Some programs written by languages Gibian and special operators in Castem 2000 are established. The problem for the spherical shell made of elasto-plastic "material subjected to monotone increasing pressures is solved and calculated results are compared with the theoretical solution and give a good agreement. The influence of the pressure values on the plastic regions of sphere is investigated. The stress , displacement and plastic deformation states for spherical shell and plate with hollow acted on by complex cyclic loads are considered. These given programs can be applied in other problem with more complex geometry, load and material conditions. 1 The finite-element formulation The governing equation of the finite-element method for small-deformation analysis is represented as [1 J: l[B]T{a}dV =ls {N}T {T}ds + l {N}T{q}dV, () or l[BJT{a}dV = {R} , () where {T} and {q} are surface and body forces, {R} is the equivalent external force acting on the nodal point, [BJ is the strain-displacement matrix and [NJ is the matrix of the displacement interpolation function . In an elastic-plastic problem, the constitutive relation depends on deformation history, an incremental analysis should be used and the total load { R} acting on a structure is added in increments step by step. To solve () for displacements {U} corresponding to a given set of external forces the iterative methods are usually employed. Equilibrium iterative methods The load at the (m + 1)-th step can be expressed as m+l{R} = m{R} + m+ l{~R} then equation () becomes the equilibrium of the internal force m+

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