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Summary and Conclusions7.1 Summary and conclusion1. The analytical solution in chapter 2 is based upon Hankel transformation for thermoelastic problem in solids have been developed and utilized, whereas, that of chapter 3 and 4 are based on Fourier sine and cosine transforms. Fourier and Laplace transfo2. Chapter 2 and 3 deal with two different types of crack in solids, with different boundary conditions, which are solved with two different ways, Hankel and Fourier half-line transforms, respectively. It has been found that behaviours of the temperature 3. Chapter 4 and 5 that deal with a solid weakened by a cylindrical hole and subjected to a heat source, the study was applied to copper material. Distributions of temperature and displacements components were estimated at different radial and axial dista4. In the static problem investigated in chapter 4, which studies the steady state of the effect of stresses, the displacement components u and w show different behaviours, because of the elasticity of the solid tends to resist axial displacements in the 5. In the dynamic problem presented in chapter 5, which studied the transient case of small and values of time, the displacements u and w show different trends, because of the solid’s elasticity that tends to resist axial displacements in that problem. Fu6. Stress components distribution exhibits a discontinuity due to the fact that the speeds of propagation of thermal and mechanical wave fronts are not identically equal. (This contradicts with that obtained in the previous static problem.)7. In chapter 6, Generalized thermoelasticity is applied for a layered composite material (sandwich structure) in which the Laplace transform technique is utilized, which is inverted using the numerical inversion method based on a Fourier series expansion8. Distributions of temperature, particles displacements and stress were estimated through the layers of the composite. Results obey physical reality for the behaviour of different substances as metals of different thermal conductivities and thermoelastic9. Most of the elasticity problems dealing with infinitely small values of time must be dealt with the generalized theory of thermo-elasticity, like in nuclear reactor bodies,… Elsewhere, the coupled theory may be used because both solutions meet for larg7.2 Future Work Recommendations• Many interesting topics for future investigation and continuation of this work are the investigation of dynamic cracks and spherical holes in solids and in composite materials.• These have many applications in manufacturing of modern electronic components.• This is a challenging task, but it is believed that the methods developed can be adapted to handle problems within the field of micro-structures.
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