Assumptions in toe
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The basic assumptions in the study of TOE is presented in these slides. It also discusses "Microscopic and Macroscopic properties" www.harekrishnahub.com
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Assumptions in toe
- 2. www.harekrishnahub.com Following are the basic assumptions made in theory of elasticity: • CONTINUITY • ELASTICITY • HOMOGENEITY • ISOTROPY
- 3. www.harekrishnahub.com CONTINUITY • It is assumed that the body is continuously distributed over its volume without any void or discontinuities. • The physical quantities such as displacements, strain, and stresses are continuously distributed over the domain of the body.
- 4. www.harekrishnahub.com ELASTICITY • It is assumed that the bodies undergoing deformations are perfectly elastic. • They regain their initial shape completely after removal of the forces. • A perfectly elastic material exhibits the following: o Instantaneous deformation and recovery during loading and unloading. o No permanent set or deformation after unloading. That is it undergoes a complete recovery. o The load – deformation curve is identical during loading and unloading. o They obey Hook’s law
- 5. www.harekrishnahub.com HOMOGENEITY • It is assumed that the body is homogeneous, i.e., it has the same properties throughout its volume. • The smallest element cut from the body possesses the same specific physical properties as the body.
- 6. www.harekrishnahub.com ISOTROPY • It is assumed that the body is Isotropic, i.e., it has the same properties in all directions. • If the properties are different in different direction, such materials are called Anisotropic. • For instance, during the process of cold rolling of steel, the crystals orient along a certain direction and the elastic properties of the metal become different in different directions. In such cases, anisotropy must be considered.
- 7. www.harekrishnahub.com • Apart from the assumptions mentioned above, we also assume that the displacement components at any point within the body during deformation are very small in comparison with its original dimensions. • Hence, while deriving the equilibrium equations, the dimensions of the body before deformations are used.
- 8. www.harekrishnahub.com • In reality, no material can be considered as perfectly elastic, homogenous, continuous or isotropic. However, these assumptions hold good only at the macroscopic level.
- 9. www.harekrishnahub.com Microscopic and Macroscopic properties • A small volume of a body or structure (say steel) is made up of millions and trillions of atoms or crystals. • These crystals are very small compared to the actual size of the body and are distributed randomly over the volume of the body. • The elastic properties of a single crystal may be very different in different directions. • However, the elastic properties of large pieces of metal represent the averages of properties of the crystals.
- 10. www.harekrishnahub.com Microscopic and Macroscopic properties • Microscopic properties are the properties of material defined at the molecular level or at the crystalline level. • Macroscopic properties are the average properties of the material under normal engineering application and represent the averages of properties of the crystals.