Linear elasticity: Difference between revisions

Revision as of 08:23, 13 May 2022

Revision as of 08:25, 13 May 2022

Line 120:

===Elastostatics===

Elastostatics is the study of linear elasticity under the conditions of equilibrium, in which all forces on the elastic body sum to zero, and the displacements are not a function of time. The [[Momentum#Linear momentum for a system|equilibrium equations]] are then <math display="block"> \sigma_{ji,j} + F_i = 0.</math>

!EngineeringIn engineering notation (with tau isas [[shear stress]]),

|* <math>\frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = 0\,\!</math>

:{| class="collapsible collapsed" width="30%" style="text-align:left"

*<math>\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y = 0\,\!</math>

!Engineering notation (tau is [[shear stress]])

*<math>\frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_z}{\partial z} + F_z = 0\,\!</math>

|-

|<math>\frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = 0\,\!</math>

<math>\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y = 0\,\!</math>

<math>\frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_z}{\partial z} + F_z = 0\,\!</math>

|}

This section will discuss only the isotropic homogeneous case.

Line 174 ⟶ 168:

===== The biharmonic equation =====

The elastostatic equation may be written:

<math display="block">(\alpha^2-\beta^2) u_{j,ij} + \beta^2 u_{i,mm} = -F_i.</math>

\beta^2 u_{i,mm} = -F_i.</math>

Taking the [[divergence]] of both sides of the elastostatic equation and assuming the body forces has zero divergence (homogeneous in ___domain) (<math>F_{i,i}=0\,\!</math>) we have

<math display="block">(\alpha^2-\beta^2) u_{j,iij} + \beta^2u_{i,imm} = 0.</math>

Noting that summed indices need not match, and that the partial derivatives commute, the two differential terms are seen to be the same and we have: <math display="block">\alpha^2 u_{j,iij} = 0</math> from which we conclude that: <math display="block">u_{j,iij} = 0.</math>

<math display="block">\alpha^2u_{j,iij} = 0</math>

from which we conclude that:

<math display="block">u_{j,iij} = 0.</math>

Taking the [[Laplacian]] of both sides of the elastostatic equation, and assuming in addition <math>F_{i,kk}=0\,\!</math>, we have