Four types of stress functions are known for solving elasticity problems: the components of the displacement vector, the components of the Galerkin vector, the Maxwell stress functions, and the Morera stress functions. For problems with stress type boundary conditions, the Maxwell stress functions are, in many respects, the simplest to use, but they lack the simple transformation properties of vectors. It was shown by C. Weber (i) 2 that the Maxwell and Morera functions supplement each other, and that together they are the components of a second order symmetric Cartesian tensor.
In this paper, the compatibility equations for an isotropic Hookean body that is subjected to boundary stresses and temperature gradients are developed in terms of the Maxwell stress functions, and their general solution is presented for steady temperature fields. It is shown that, when the complementary energy of a homogeneous body with arbitrary elastic properties is expressed in terms of the components of the Maxwell-Morera tensor, the Euler equations for the integral of the complementary energy density are the complete set of compatibility equations in terms of the stress components. The Maxwell-Morera tensor is generalized, so that it represents the general solution of the equilibrium equations in any curvilinear coordinates. As an application, the general solution of the equilibrium equations in cylindrical coordinates is derived.
MAXWELL'S STRESS FUNCTIONS
Clerk Maxwell observed (2) that, if body force is absent, the differential equations of equilibrium of a deformable body that is referred to rectangular coordinates (x, y, z) are identically satisfied by
in which A, B, C are arbitrary functions of x, y, z. The subscripts on A, B, C denote partial derivatives. Furthermore, any solution of the equilibrium equations can be represented by Eqs. 1. For, if a,, β’ β’., r,y are given functions of x, y, z that satisfy the equilibrium equations, the functions A, B, C are determined within certain arbitrary additive functions by Eqs. lb. If the shearing stresses are eliminated from the equilibrium equations by Eqs. lb, the resulting equations may be integrated, and Eqs. la are obtained, except for certain additive functions which may be set equal to zero in virtue of the arbitrariness in the functions A, B, C.