Stress, Strain and Elasticity Tensors

Generally, any vector of a surface force acting on a body can be decomposed into tangential (fx,fy, shear) and normal (Z, pressure) components (index j = x, y, z) as shown in Fig. 8.2a. In its turn, the element of surface A is a vector characterized by its area A and outward-directed unit vector s that has also three projections in the Cartesian laboratory frame (index /). Therefore, the second rank stress tensor [1] is defined as 

Generally has 3x3 = 9 components, illustrated by Fig. 8.2b. The diagonal components (cth, c 22, C733) corresponds to pressure, off-diagonal ones correspond to shear. Since is symmetric tensor (a,y = ay,), only six components are different. Usually, in addition to the surface forces, the volume forces like gravitation or electric force are included into the stress tensor. 

We consider a piece of soft matter in which the distortion is not uniform in space but rather local [2]. Any displacement of point p to point p caused by the stress tensor and shown in Fig. 8.3a-c can be considered as a combintion of four basic displacements. They are (1) translations, (2) rotatimi of the entire body as a solid piece without deformation, (3) shear distortion, (4) expansion or compression. 

Now we are going to discuss the components of the correspondent tensors (translation is excluded). We go back to Eq. (8.2) and instead of ujx = HL write Ux = exxX or dux = Cxxdx 

For small distortions, the coefficient Cxx = duxjdx will be a function of x, y, z, describing an extension along the x-axis. Along y- and z-axes the extensions wil be described by coefficients 

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