Airy stress function

Airy Stress Eunction and the Biharmonic Equation The biharmonic equation in many instances has an analogous role in continuum mechanics to that of Laplace s equation in electrostatics. In the context of two-dimensional continuum mechanics, the biharmonic equation arises after introduction of a scalar potential known as the Airy stress function f such that... 

Given this definition of the Airy stress function, show that the equilibrium equations are satisfied. 

Compatibility conditions address the fact that the various components of the strain tensor may not be stated independently since they are all related through the fact that they are derived as gradients of the displacement fields. Using the statement of compatibility given above, the constitutive equations for an isotropic linear elastic solid and the definition of the Airy stress function show that the compatibility condition given above, when written in terms of the Airy stress function, results in the biharmonic equation = 0. 

Recall from our discussion in chap. 2 that the solution of elasticity problems of the two-dimensional variety presented by the edge dislocation is often amenable to a treatment in terms of the Airy stress function. Consultation of Hirth and Lothe (1992), for example, reveals a well-defined prescription for determining the Airy stress function. The outcome of this analysis is the recogiution that the stresses in the case of an edge dislocation are given by... 

The solution of plane (two-dimensional) elasticity problem now resides in the determination of an Airy stress function (h(x, y) that satisfies the governing fourth-order partial differential equation and the appropriate boundary conditions. Note that ... 

Based on the definition of stresses in terms of the Airy stress function, given in Eqn. (3.10), one obtains from the Westergaard function ... 

By using the Westergaard approach and the Airy stress function, the stresses near the tip of a crack may be considered (Fig. 3.2). A set of in-plane Cartesian coordinates X and y, or polar coordinates r and 0, is chosen, with the origin at the crack tip. The boundary conditions are as follows (i) stresses at the crack tip are very large and (ii) the crack surfaces are stress free. 

The Airy stress function Z(z) may be rewritten in terms of the new coordinates as follows ... 

Again, by taking z = f - - a for the right end, the Airy stress function in terms of the crack-tip coordinates is given by ... 

Thus, any solution to Eq. (4.24) that also fits the boundary conditions will be the elastic solution to the problem being sought. Conversely, mathematical functions that satisfy Eq. (4.23) are often studied to find the associated elastic problem. The function x is called the Airy stress function and Eq. (4.24) is called the Biharmonic equation This latter equation does not require any elastic constants for its solution, indicating the stress distributions are independent of the elastic properties. If, however, the strains are needed, the elastic constants appear once Hooke s Law is introduced. 

The various stress components are shown in Fig. 4.14 and the shear components can be shown to be symmetric. The equilibrium equations have a different form in the new coordinate system. This leads to new relationships between the stresses and the Airy stress function. Using the same approach as that outlined in the last section, the following revised versions of Eqs. (4.23) and (4.24) are obtained... 

The surface of a semi-infinite solid is subjected to a concentrated force tangential to the surface. An Airy stress function of the same form as a concentrated normal force can be used to solve the problem. 

What type of elastic problems are solved by the Airy stress functions ... 

If the stresses can be written down in terms of (x, y), also known as the Airy stress function, such that... 

The six independent components of the symmetric stress matrix are recorded here for later reference. The stress field arising from the edge component of the Burgers vector is conveniently represented in terms of the Airy stress function... 

For a sinusoidal perturbation of shape as given by (8.47) and for a uniform initial stress field, the additional elastic stress is also expected to be sinusoidal in x for fixed y. The stress field has the appropriate S3rmmetry if it is derived from an Airy stress function of the form... 

In order to obtain the stress field of an edge dislocation in an isotropic solid, we can use the equations of plane strain, discussed in detail in Appendix E. The geometry of Fig. 10.1 makes it clear that a single infinite edge dislocation in an isotropic solid satisfies the conditions of plane strain, with the strain along the axis of the dislocation vanishing identically. The stress components for plane strain are given in terms of the Airy stress function, A(r, 0), by Eq. (E.49). We define the function... 

Thus, the solution of two-dimensional elastostatic problems reduces to the integration of the equations of equilibrium together with the compatibility equation, and to satisfy the boundary conditions. The usual method of solution is to introduce a new function (commonly known as Airy s stress function), and is outlined in the next subsections. 

Airy s stress function 4>(jc, y) is related to the stresses as follows ... 

Based on the solution of concentrated forces in mode 111 loading [2], the following Airy s stress function is assumed ... 

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