Description of Residual Stress and Strain Fields in FGM Hollow Disc Subject to External Pressure.

Stanislav Strashnov,Lihui Lang,Sergei Alexandrov

doi:10.3390/ma12030440

Abstract

Elastic/plastic stress and strain fields are obtained in a functionally graded annular disc of constant thickness subject to external pressure, followed by unloading. The elastic modulus and tensile yield stress of the disc are assumed to vary along the radius whereas the Poisson’s ratio is kept constant. The flow theory of plasticity is employed. However, it is shown that the equations of the associated flow rule, which are originally written in terms of plastic strain rate, can be integrated with respect to the time giving the corresponding equations in terms of plastic strain. This feature of the solution significantly facilitates the solution. The general solution is given for arbitrary variations of the elastic modulus and tensile yield stress along the radial coordinate. However, it is assumed that plastic yielding is initiated at the inner radius of the disc and that no other plastic region appears in the course of deformation. The solution in the plastic region at loading reduces to two ordinary differential equations. These equations are solved one by one. Unloading is assumed to be purely elastic. This assumption should be verified a posteriori. An illustrative example demonstrates the effect of the variation of the elastic modulus and tensile yield stress along the radius on the distribution of stresses and strains at the end of loading and after unloading. In this case, it is assumed that the material properties vary according to power-law functions.

Highlights

Stress and strain analyses of solid and hollow circular discs have long been an important topic in the mechanics of solids
A remarkable feature of the strain solution is that the equations in (18), which are derived from the substituted into (23)
(18), which are solution significantly thefeature solution. Another feature the straininsolution is that derived from the associated flow rule written in terms of plastic strain rates, can be immediately all strain components are proportional to k introduced in (9)

Summary

Introduction

Stress and strain analyses of solid and hollow circular discs have long been an important topic in the mechanics of solids. In [7], the deformation theory of plasticity together with the von Mises yield criterion has been employed Another elastic/plastic plane strain solution for a functionally graded cylinder has been given in [8]. The general solution is derived under plane stress conditions assuming that the elastic modulus and tensile yield stress are arbitrary smooth functions of the radial coordinate. It is, assumed that plastic yielding initiates at the inner radius of the disc and that there is one plastic region throughout the process of deformation. The solution found can be considered as an extension of the solution provided in [1] to the plastic range

Statement of the Problem

Purely Elastic Solution

Unloading

Illustrative Example

Discussion

Conclusions