Strain Compatibility Equation
With strong attraction force among particles, a body deforms in a stable and predictable manner. This properties of solid makes its deformation exclusive at a specified point. In other words, deformation occurs at any point on a body is dependent of other points. Mathematically, such dependency is expressed in Eqs. (3.2), (3.3), (3.4), (3.6), (3.7) and (3.8).
The change in shear strain уп. on .vv-plane can be expressed by performing partial differentiation to Eq. (3.6):
The following is produced by performing second order partial differentiation to Eqs. (3.2) and (3.3) with respect to у and x, respectively:
Substitution of the relationships above in Eq. (3.19) produces the follows:
The equation above is known as compatibility equation, and it must be satisfied to yield consistent displacements. Similarly, compatibility equations for yz and xz planes can be derived as follow:
Performing partial differentiation to Eqs. (3.2), (3.3), (3.4), (3.6), (3.7) and (3.8) and the following expressions are resulted:
By performing partial differentiation to Eq. (3.24) with respect to Л' yields the following:
Partial differentiation of Eq. (3.26) with respect to x will produce the following:
By adding and in Eqs. (3.27) and (3.28) and we get this:
oxoz ЭхЭу
The following is obtained through simplification of equation above:
Rearrangement of the equation above results the follows:
By substituting Eqs. (3.2) and (3.8) into equation above yields the following equation:
The following equation can be written by rearranging the equation above:
By performing partial differentiation to Eq. (3.24) with respect to у yields the following:
Partial differentiation of Eq. (3.25) with respect to у results the following:
By adding and in Eqs. (3.30) and (3.31) gives us the following:
ЭуЭг ЭхЭу
By simplifying the equation, we can get:
By rearranging the equation above the following is obtained:
Substitution of Eqs. (3.3) and (3.7) into equation above produces the following:
With rearrangement the following equation is resulted:
Partial differentiation of Eq. (3.24) with respect to z leads to the following:
By performing partial differentiation to Eq. (3.26) with respect to z. yields the following:
Addition of and in Eqs. (3.33) and (3.24) produces the following equation:
oyoz dxdy
Through simplification of the equation above we can get:
Rearrange the equation above yields the follows:
The equation below is obtained after substituting Eqs. (3.4) and (3.6) into equation above:
Rearrange the equation above results in the following expression:
