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Notice that the Laplacian of a vector does not simply pass through and operate on each of the individual components as in the Cartesian case. Additional terms are generated because of the curvature of the particular coordinate system. Similar relations can be developed for cylindrical and spherical coordinate systems (see Exercises 1-15 and 1-16). The material reviewed in this chapter is used in many places for formulation development of elasticity theory. Throughout the entire text, notation uses scalar, vector, and tensor formats depending on the appropriateness to the topic under discussion.

2-17. Consider the plane deformation of the differential element ABCD defined by polar coordinates r, y as shown in the following figure. 2, investigate the changes in line lengths and angles associated with the deformation to a configuration A0 B0 C0 D0 , and develop the strain-displacement relations er ¼     @ur 1 @uy 1 1 @ur @uy uy , ery ¼ ur þ , ey ¼ þ À r 2 r @y @r @y @r r D′ D C′ B′ C A′ B rdq dr dq A Deformation: Displacements and Strains 47 TLFeBOOK This page intentionally left blank TLFeBOOK 3 Stress and Equilibrium The previous chapter investigated the kinematics of deformation without regard to the force or stress distribution within the elastic solid.

1) become e0x ¼ ex l21 þ ey m21 þ ez n21 þ 2(exy l1 m1 þ eyz m1 n1 þ ezx n1 l1 ) e0y ¼ ex l22 þ ey m22 þ ez n22 þ 2(exy l2 m2 þ eyz m2 n2 þ ezx n2 l2 ) e0z ¼ ex l23 þ ey m23 þ ez n23 þ 2(exy l3 m3 þ eyz m3 n3 þ ezx n3 l3 ) e0xy ¼ ex l1 l2 þ ey m1 m2 þ ez n1 n2 þ exy (l1 m2 þ m1 l2 ) þ eyz (m1 n2 þ n1 m2 ) þ ezx (n1 l2 þ l1 n2 ) (2:3:3) e0yz ¼ ex l2 l3 þ ey m2 m3 þ ez n2 n3 þ exy (l2 m3 þ m2 l3 ) þ eyz (m2 n3 þ n2 m3 ) þ ezx (n2 l3 þ l2 n3 ) e0zx ¼ ex l3 l1 þ ey m3 m1 þ ez n3 n1 þ exy (l3 m1 þ m3 l1 ) þ eyz (m3 n1 þ n3 m1 ) þ ezx (n3 l1 þ l3 n1 ) For the two-dimensional case shown in Figure 2-6, the transformation matrix can be expressed as 2 cos y Qij ¼ 4 À sin y 0 34 sin y cos y 0 3 0 05 1 (2:3:4) FOUNDATIONS AND ELEMENTARY APPLICATIONS TLFeBOOK y y′ x′ q q x FIGURE 2-6 Two-dimensional rotational transformation.

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