Stress Functions - Beltrami Stress Functions

Beltrami Stress Functions

It can be shown that a complete solution to the equilibrium equations may be written as

Using index notation:

Engineering notation
\sigma_x = \frac{\partial^2\Phi_{yy}}{\partial z \partial z} + \frac{\partial^2\Phi_{zz}}{\partial y \partial y} -2\frac{\partial^2\Phi_{yz}}{\partial y \partial z} \sigma_{xy} =-\frac{\partial^2\Phi_{xy}}{\partial z \partial z} -\frac{\partial^2\Phi_{zz}}{\partial x \partial y} +\frac{\partial^2\Phi_{yz}}{\partial x \partial z} +\frac{\partial^2\Phi_{zx}}{\partial y \partial z}
\sigma_y = \frac{\partial^2\Phi_{xx}}{\partial z \partial z} +\frac{\partial^2\Phi_{zz}}{\partial x \partial x} -2\frac{\partial^2\Phi_{zx}}{\partial z \partial x} \sigma_{yz} =-\frac{\partial^2\Phi_{yz}}{\partial x \partial x} -\frac{\partial^2\Phi_{xx}}{\partial y \partial z} +\frac{\partial^2\Phi_{zx}}{\partial y \partial x} +\frac{\partial^2\Phi_{xy}}{\partial z \partial x}
\sigma_z = \frac{\partial^2\Phi_{yy}}{\partial x \partial x} +\frac{\partial^2\Phi_{xx}}{\partial y \partial y} -2\frac{\partial^2\Phi_{xy}}{\partial x \partial y} \sigma_{zx} =-\frac{\partial^2\Phi_{zx}}{\partial y \partial y} -\frac{\partial^2\Phi_{yy}}{\partial z \partial x} +\frac{\partial^2\Phi_{xy}}{\partial z \partial y} +\frac{\partial^2\Phi_{yz}}{\partial x \partial y}

where is an arbitrary second-rank tensor field that is continuously differentiable at least four times, and is known as the Beltrami stress tensor. Its components are known as Beltrami stress functions. is the Levi-Civita pseudotensor, with all values equal to zero except those in which the indices are not repeated. For a set of non-repeating indices the component value will be +1 for even permutations of the indices, and -1 for odd permutations. And is the Nabla operator

Read more about this topic:  Stress Functions

Famous quotes containing the words stress and/or functions:

    “It is not stressful circumstances, as such, that do harm to children. Rather, it is the quality of their interpersonal relationships and their transactions with the wider social and material environment that lead to behavioral, emotional, and physical health problems. If stress matters, it is in terms of how it influences the relationships that are important to the child.”
    —Felton Earls (20th century)

    “Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
    —Ralph Waldo Emerson (1803–1882)