By Herbert H. Woodson, James R. Melcher
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Extra info for Electromechanical Dynamics, Part 3: Elastic and Fluid Media
4) The volume V is fixed with its center at the position (a, b, c), as shown in Fig. 1. The integration is carried out over the Lagrangian variables (,xX, xz, s). 4) can be reversed in order. 4)] at p, dV = Tinj da. 5) We consider the situation in which the dimensions of the surfaces A (shown in Fig. 1) are small compared with the radius of curvature of the boundary but large compared with the thickness A of the volume element Region (1) Instantaneous position Fig. 1 Unstrained and strained (instantaneous) interface between regions (1) and (2).
To this end, we extend the notions introduced in Chapter 9 and recognize that the vibrations of continuous media in three dimensions can also be understood in terms of waves and normal modes. We have already used simple elastic models in Chapters 9 and 10 to illustrate transverse and longitudinal motions in one and two dimensions (the thin rod and membrane). We now consider these motions in three dimensions. 35), written as 2 a" p . 1) This equation is in a particularly convenient form because it makes it possible to distinguish between two essentially different kinds of material displacement.
Physical intuition tells us that each shear strain should be proportional to the corresponding shear stress. As an example, Fig. 8 shows a block of material subject to the shear stresses Ta,and Ta,. The change in angle x3 J T13 C -- TTal -Xi Fig. 8 The shear strain e,3 results from the shear stresses Tls and T31. Introduction to the Electromechanics of Elastic Media between the originally perpendicular sides of the block is in direct proportion to the applied stress. 28) where G is an experimentally determined constant called the shear modulus.
Electromechanical Dynamics, Part 3: Elastic and Fluid Media by Herbert H. Woodson, James R. Melcher