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

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Addis Ababa University


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.



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