B-17 Flying Fortress in color by Steve Birdsall

By Steve Birdsall

Publication through Birdsall, Steve

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Many examples of this are demonstrated in later chapters. Appendix A lists the complete set of elasticity field equations in cylindrical and spherical coordinates. References Chandrasekharaiah DS, and Debnath L: Continuum Mechanics, Academic Press, Boston, 1994. Fung YC: Foundations of Solid Mechanics, Prentice Hall, Englewood Cliffs, NJ, 1965. Fung YC, and Tong P: Classical and Computational Solid Mechanics, World Scientific, Singapore, 2001. Malvern LE: Introduction to the Mechanics of a Continuous Medium, Prentice Hall, Englewood Cliffs, NJ, 1969.

Y ! p=2. y B A θ x 45Њ z Deformation: Displacements and Strains 45 TLFeBOOK 2-10*. Using MATLAB, determine the principal values and directions of the following state of strain: 2 2 eij ¼ 4 À2 0 3 À2 0 À4 1 5 Â 10À3 1 6 2-11. A rectangular parallelepiped with original volume Vo is oriented such that its edges are parallel to the principal directions of strain as shown in the following figure. For small strains, show that the dilatation is given by W ¼ ekk ¼ change in volume DV ¼ original volume Vo 2 1 3 2-12.

However, since we are developing linear elasticity, which uses only small deformation theory, the distinction between undeformed and deformed configurations can be dropped. Using Cartesian coordinates, define the displacement vectors of points Po and P to be uo and u, respectively. Since P and Po are neighboring points, we can use a Taylor series expansion around point Po to express the components of u as @u @u @u rx þ ry þ rz @x @y @z @v @v @v o v ¼ v þ rx þ ry þ rz @x @y @z @w @w @w w ¼ wo þ rx þ ry þ rz @x @y @z u ¼ uo þ 28 (2:1:1) FOUNDATIONS AND ELEMENTARY APPLICATIONS TLFeBOOK Note that the higher-order terms of the expansion have been dropped since the components of r are small.

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