For a 2D or 3D state of stress, each of the stress components is taken to be a linear function of the components of strain within the linear elastic range. This assumption usually predicts the behavior of engineering materials with good accuracy. In addition, the principle of superposition applies under multi axial loading, since strain components are small quantities. In the following development, we rely on certain experimental evidence to derive the stress–strain relations for linearly elastic isotropic materials: a normal stress creates no shear strain whatsoever, and shear stress produces only shear strain. Consider now an element of unit thickness subjected to a biaxial state of stress (Figure 2.8). Under the action of the stress σx, not only would the direct strain σx/E occur, but a y contraction as well, −νσx/E. Likewise, were σy to act only, an x contraction −νσy/E and a y strain σy/E would result. Therefore, simultaneous action of both stresses σx and σy results in the following strains in the x and y directions: So, for an isotropic material, there are only two independent elastic constants. The values of E and G are determined experimentally for a given material, and ν can be found from the preceding basic relationship. Since the value of Poisson’s ratio for ordinary materials is between 0 and 1/2, we observe from Equation 2.9 that G must be between (1/3)E and (1/2)E. Thermal Stress–Strain Relations. When displacements of a heated isotropic member are prevented, thermal stresses occur. The effects of such stresses can be severe, particularly since the most adverse thermal environments are frequently associated with design requirements dealing with unusually stringent constraints as to weight and volume. The foregoing is especially true in aerospace and machine design (e.g., engine, power plant, and industrial process) applications. The total strains are obtained by adding thermal strains of the type described by Equation 1.21 and the strains owing to the stress resulting from mechanical loads. In doing so, for instance, referring to Equation 2.6 for 2D stress, The quantities T and α represent the temperature change and the coefficient of expansion, respectively. Equations for 3D stress may be readily expressed in a like manner. Note that because free thermal expansion causes no distortion in an isotropic material, the shear strain is unaffected, as shown in the preceding expressions. The differential equations of equilibrium are based on purely mechanical considerations and unchanged for thermo-elasticity. The same is true of the strain–displacement relations and hence the conditions of compatibility, which are geometrical in character (see Section 3.17). Thermo- elasticity and ordinary elasticity therefore differ only to the extent of Hooke’s law. Solutions for the problems in the former are usually harder to obtain than solutions for the problems in the latter. In statically determinate structures, a uniform temperature change will not cause any stresses, as thermal deformations are permitted to occur freely. On the other hand, a temperature change in a structure supported in a statically indeterminate manner induces stresses in the members. Detailed discussions and illustrations of thermal loads and stresses in components and assemblies.
This post was written by: Author Name
Your description comes here!
Get Free Email Updates to your Inbox!
Post a Comment