# Mathematical Modeling for Thermo-Mechanical Stress Field Associated with A Propagating Crack in Homogenous Isotropic Materials

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## Date

2020-09

## Authors

## Journal Title

## Journal ISSN

## Volume Title

## Publisher

Addis Ababa University

## Abstract

The main objective of this research is to develop mathematical modeling of the thermos
mechanical stress field associated with the propagating crack in homogeneous isentropic materials
by using a temperature field that can be replicated in real world application. In most of existing
work, the temperature field is obtained by assuming the heat flux to be singular at the crack tip.
However, such assumption could lead to a theoretical solution practically generating a temperature
field where the heat flux singular at the crack tip is not attainable. The mechanical design of
engineering structure in accordance with preexisting flaws is analyzed based on mechanical
loading and mixed mode thermo-mechanical stress field by using asymptotic solution of the steady
state 2D temperature field equation. The solution is developed for the mixed mode thermomechanical
loading
condition
when
the
crack
propagates
at
a
constant
velocity
in
homogeneous
isentropic
materials using an asymptotic approach. The solution is obtained by simultaneously
solving equation of motion defined as a function of displacement potentials, and perturbation
theory for the solution of two-dimensional steady state temperature field equation. The perturbed
temperature field equation is used to derive the first three terms of thermo-mechanical stress field
equations by superimposing with the mechanical loading equation for the steady state propagating
crack. The thermo-mechanical stress field developed by the superimposition and the developed
stress fields evaluated for the crack tip introduce at the center single edge when the value of ��
expressed by the interval [−��, ��]. Around the crack tip from the crack tip plasticity approximations
theory with the distance �� = 0.002�� at �� = 0 from the crack tip the thermo-mechanical stress
field is zero. And the graph of stress intensity factors interpreted on the behalf of stress field since
the stress and stress intensity factors have direct relationships. As the temperature and the heat flux
increases the principal thermo-mechanical stress field such as ������, ������ and ������ increases and
similar trend is observed for the stress intensity factors that the material is overstressed to
propagate the initial crack.

## Description

## Keywords

Thermo-mechanical loading, Isentropic materials, Perturbation, Asymptotic approach, Stress fields