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- An Overview of Stress-Strain Analysis for Elasticity Equations;
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On simplifying Eq. The body forces F x , F y , F z acting along x, y, z direction respectively. Then the stress equilibrium relation or equation of motion in terms of stress components are given by. While defining a stress it was pointed out that stress is an abstract quantity which cannot be seen and is generally measured indirectly. Strain differs in this respect from stress. It is a complete quantity that can be seen and generally measured directly as a relative change of length or shape.
In generally, stress is the ratio of change in original dimension and the original dimension. It is the dimensionless constant quantity. Strain may be classified into three types; normal strain, shear strain and volumetric strain. The normal strain is the relative change in length whether shearing strain relative change in shape. The volumetric strain is defined by the relative change in volume.
By the definition of normal strain. This partial derivative is a displacement gradient , a measure of how rapid the displacement changes through the material, and is the strain at x , y. Physically, it represents the approximate unit change in length of a line element.
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Similarly, by considering a line element initially lying in the y- direction, the strain in the y- direction can be expressed as. The particles A and B in Figure 6 also undergo displacements in the y- direction and this is shown in Figure 7 a. In this case, we have. A similar relation can be derived by considering a line element initially lying in the y- direction. From the Figure 7 b , we have. In similar manner, the strain-displacement relation for three dimensional body is given by. As seen in the previous section, there are three strain-displacement relations Eqs. This implies that the strains are not independent but are related in some way.
The relations between the strains are called compatibility conditions.
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Then u,v is a displacement corresponding to the point P. The variable u and v are the functions of x and y. In the previous section, the state of stress at a point was characterized by six components of stress, and the internal stresses and the applied forces are accompanied with the three equilibrium equation.
These equations are applicable to all types of materials as the relationships are independent of the deformations strains and the material behavior.
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Also, the state of strain at a point was defined in terms of six components of strain. The strains and the displacements are related uniquely by the derivation of six strain-displacement relations and compatibility equations. These equations are also applicable to all materials as they are independent of the stresses and the material behavior and hence. Irrespective of the independent nature of the equilibrium equations and strain-displacement relations, usually, it is essential to study the general behavior of materials under applied loads including these relations. Strains will be developed in a body due to the application of a load, stresses and deformations and hence it is become necessary to study the behavior of different types of materials.
In a general three-dimensional system, there will be 15 unknowns namely 3 displacements, 6 strains and 6 stresses.
But we have only 9 equations such as 3 equilibrium equations and 6 strain-displacement equations to achieve these 15 unknowns. It is important to note that the compatibility conditions are not useful for the determination of either the displacements or strains. Hence the additional six equations relating six stresses and six strains will be developed. The law can be used to predict the deformations used in a given material by a combination of stresses. In general, each strain is dependent on each stress.
Similarly, stresses can be expressed in terms of strains which state that at each point in a material, each stress component is linearly related to all the strain components. For the most general case of three-dimensional state of stress, Eq. The stress-strain relation for triclinic material will consist 21 elastic constants which is given by. The stress-strain relation for monoclinic material will consist 13 elastic constants which is given by. A material that exhibits symmetry with respect to three mutually orthogonal planes is called an orthotropic material.
The stress-strain relation for orthotropic material will consist 9 elastic constants which is given by. Transversely isotropic material exhibits a rationally elastic symmetry about one of the coordinate axes x, y and z. In such case, the material constants reduce to 5 as shown below.
Indices take the value 1, 2 and 3, and the repeated suffix summation convention is adopted. For a material whose elastic properties are not a function of direction at all, only two independent elastic material constants are sufficient to describe its behavior completely. This material is called isotropic linear elastic. The stress-strain relationship for this material is written as. Also, from the above relation some important terms are induced which are as follow 1 Bulk modulus: Bulk modulus is the relative change in the volume of a body produced by a unit compressive or tensile stress acting uniformly over its surface.