# A Course in Mathematics Volume II by Frederick S. And Frederick H. Bailey Woods PDF

By Frederick S. And Frederick H. Bailey Woods

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**Example text**

5 Determine whether the following stress field is admissible in an elastic body when body forces are negligible. 6. At a given point in a body s11 ¼ s22 ¼ s33 ¼ Àp; s12 ¼ s23 ¼ s31 ¼ 0: Show that the normal stresses are equal to Àp and that the shearing stresses vanish for any other Cartesian coordinate system (from [1]). 13) for a twodimensional state of stress by assuming that the stress components in the x3-direction are zero and by considering a rotation about the x3-axis. Compare the resulting expressions with the Mohr’s circle representation [1].

2) Fig. 2 Components of stress (after Tauchert, Energy Principles in Structural Mechanics, McGraw-Hill, 1974). 3c) The coefficients s11, s12, . , s33 are known as stresses, while the entire array forms the Cauchy stress tensor when the appropriate transformation rule is verified. The subscript and sign conventions for components of stress sij are as follows: 1. The first subscript i refers to the normal ei which denotes the face on which Ti acts. 2. The second subscript j corresponds to the direction ej in which the stress acts.

9 Show that curl A ¼ = Â A ¼ eijk Aij ek References 13 References 1. Westergaard HM (1964) Theory of elasticity and plasticity. , New York 2. Tauchert TR (1974) Energy principles in structural mechanics. , New York 3. Ma Y, Desai CS (1990) Alternative definition of finite strains. J Eng Mech, ASCE 116(4):901–919 2 Traction, Stress, and Equilibrium Abstract An approach to the solution of problems in solid mechanics is to establish relationships first between applied loads and internal stresses and, subsequently, to consider deformations.

### A Course in Mathematics Volume II by Frederick S. And Frederick H. Bailey Woods

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