# Fundamentals Of Statistical Mechanics by B B Laud By B B Laud

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F1 (y) ∂ f2 (x) + = −A + A = 0 ∂y ∂x In other words, f1 (y) = −Ay and f1 (x) = Ax, which correspond to rigid body rotation about the z-axis. The sign is chosen to be consistent with a counter-clockwise rotation. It is clear that the constant A could not be arbitrarily neglected; it is zero only for the case of equal biaxial tension. 2 Central Crack in an Infinite Plate under a Pair of Concentrated Forces [2–4] Wedge force loading applied normally to the crack plane often occurs in many practical applications.

The stresses for mode II are given by Eqn. 30) θ KII θ 3θ τxy = √ cos 1 − sin sin 2 2 2 2πr Those for mode III are given in Eqn. 31) The remaining task is to develop stress intensity solutions for specific crack and component geometries and loading conditions. For simple cases, closed-form solutions can be obtained. ” A few simple cases are considered in the next section to illustrate the process for obtaining stress intensity factor solutions analytically. For more complex cases, the stress intensity factors may be obtained experimentally or numerically as described in Chapter 2 and references [2–4].

The Cauchy-Riemann conditions are satisfied by any analytic function and, hence, any of its successive derivatives. This property of analytic functions makes them useful in the solution of problems in two-dimensional elasticity. Considering Eqn. 20) 34 Stress Analysis of Cracks In other words, the real and imaginary parts of analytic functions are harmonic and would satisfy the biharmonic equation (see Eqn. 14)). The task then becomes one of identifying the appropriate analytic functions that can satisfy the boundary conditions of the problem.