Displacement, Stress, and Strain

Application of force to a solid puts the solid under stress. Stress results in strain within the solid atoms or molecules of which the solid is composed are displaced from their unstressed locations. When a solid is deformed, the displacement of each particle from its original position is represented by a displacement vector u(x,y,z,t). In general, the displacement has components, which vary continuously from point to point in the solid, in the x, y, and z directions. A plane wave generates displacements that vary harmonically in the direction of wave propagation if this is the x direction, for example, it may be represented as [1]  

Because simple translation of the entire solid is not of interest, this class of motion is eliminated to give a parameter related only to local deformations of the solid this parameter is the displacement gradient, V . The gradient of a vector field Vu is a second-rank tensor, specified by a 3 by 3 matrix. The elements of this displacement gradient matrix are given by (Vu),y = dujdxj, also denoted Uij in which i denotes the i displacement element and j denotes a derivative with respect to the y spatial coordinate, i.e. [1], 

The displacement gradient represents changes in interparticle distance as well as local rotations caused by the displacement. 

Just as the effect of simple translation was eliminated by taking the gradient of the displacement vector, the contributions due to rotations can be eliminated, resulting in a parameter that describes only the local stretching of the solid. This 

Strain is the change in length per (unstrained) unit length in the solid as a result of applied stress and can be calculated for any direction in the solid from the 

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Hooke s law is a mathematical statement of the linear relation between stress and strain and usually implies both small strains (e e) and small deformations. It is also to be noted that in general elasticity solutions in two and three dimensions, the displacement, stress and strain variables are functions of spatial position, Xj. This will be handled more explicitly in Chapter 9. 

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