## Graduation Semester and Year

2021

## Language

English

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy in Aerospace Engineering

## Department

Mechanical and Aerospace Engineering

## First Advisor

Seiichi Nomura

## Abstract

A functionally graded material (FGM) is an inhomogeneous material whose material properties vary continuously. Since FGM’s material properties are continuous at the interface of the two materials, unlike conventional composites whose material properties are discontinuous at the interface, it is best suited to prevent catastrophically failure, stress concentration, and residual stresses. Literature reviews revealed that most of the research on FGMs was a 2-D analysis, especially on FGM plates and beams. This research focuses on 3-D stress analysis for an elastic medium that contains a spherical inclusion made of an FGM embedded in a matrix phase. To the best of my knowledge and literature surveys, this is the first attempt to analyze inclusion-type FGMs semi-analytically. The material properties are assumed to vary linearly, which is suitable for a solid or hollow FGM sphere, although this assumption can be relaxed to the exponential or power-law variations. I have considered three different scenarios, which are (1) 3-D thermal stress analysis of an infinitely extended medium which contains a spherical FGM inclusion with a constant heat source in the FGM inclusion, (2) 3-D thermal stress analysis of an infinitely extended medium which contains a spherical FGM inclusion subject to constant heat flux at the far fields, and (3) 3-D elastic stress analysis of an infinitely extended medium which contains a spherical FGM inclusion subject to constant strain at the far fields. Each one of them is discussed separately. The 3-D temperature distribution is sought in the heat source problem due to the constant heat source in the inclusion, and the 3-D heat conduction equation was solved analytically. The material properties of Ni-ceramic are taken as an example, and I plotted the graph of temperature distribution. It is found that the graph is smooth at the interface of FGM-matrix, unlike the two-phase conventional composites, where there is an abrupt change in the slope at the interface. Furthermore, this temperature distribution is substituted into the elasticity equation to solve Navier’s equation of displacement. The stress equilibrium equation is converted into a second order ordinary differential equation using tensor analysis. A computer algebra tool, Mathematica, was used to solve the differential equation to obtain the exact solution. However, the output was too lengthy and not practically useful. Therefore, I have adopted the least square method and plotted the results. The least square method is used to solve the differential equation because of its simplicity. The results are then compared with previously published articles, where they give an excellent comparison. It has also been shown that the von Mises stress is continuous at the interface of the FGM-matrix medium, unlike the two-phase conventional composites where there is a considerable gap at the interface. In the heat flux problem, the same approach has been used as in the heat source problem I have considered. The only difference is that the temperature distribution is not due to the heat source but due to the constant heat flux at the far fields. I obtained the exact solution of the 3-D heat conduction equation, and the result of temperature distribution is plotted using the material properties of glass-ceramic. The temperature distribution is substituted in the stress equilibrium equation, and Mathematica is employed to solve the differential equation analytically. However, the set of two simultaneous differential equations are not solvable analytically. Again, I have adopted the least square method and plotted the results. The results are then compared with the two-phase conventional composite. In the constant strain problem, the same approach is used as in the heat flux problem I have considered, but instead of considering constant heat flux at the far fields, constant strain at the far fields is considered. Here, the total strain is decomposed into the deviatoric and the hydrostatic strains. Both of them are analyzed separately. In the 3-D stress analysis due to the hydrostatic strain, I obtained the exact solution for the stress equilibrium equation, plotted the results, and compared it with the two-phase conventional composites. 3-D stress analysis due to the deviatoric strain is also performed by solving the stress equilibrium equation using the least square method. The results are shown and compared with the two-phase conventional composites. Based on the temperature field I obtained in the heat flux problem, the effective thermal conductivity is derived using the self-consistent method and compared with the Voigt-Reuss upper and lower bounds. As a result, the curve lies within the upper and lower bounds. The solution obtained in this research can be directly used to assess the performance of composites where multiple FGM-made inclusions are distributed in a matrix phase. Finite element tools such as Ansys or Abaqus are not suitable for this research because they cannot handle the continuous variation of the material properties. Also, they cannot handle an infinitely extended matrix.

## Keywords

Functionally graded materials, Thermal stress analysis, The von Mises stress, Effective thermal conductivity

## Disciplines

Aerospace Engineering | Engineering | Mechanical Engineering

## License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

## Recommended Citation

Krishna, Nilabh, "STRESS ANALYSIS OF FUNCTIONALLY GRADED MATERIALS" (2021). *Mechanical and Aerospace Engineering Dissertations*. 17.

https://mavmatrix.uta.edu/mechaerospace_dissertations/17

## Comments

Degree granted by The University of Texas at Arlington