 # Navier equations

By substituting Eqs. (5.2), (5.5) and (5.7) into the stress equilibrium equation as shown in Eq. (2.5), we can get the following: By simplifying the equation above, we get this: The following equation is obtained after expansion and rearrangement of the equation above: Simplify the equation above by factorise it with common terms yields the following: In Eq. (2.45), hydrostatic strain can be obtained by replacing the stress components with corresponding strain components: By applying the relationships in Eqs. (3.2), (3.3), (3.4) to equation above yields the follows: Equation below is obtained by substituting the equation above to Eq. (5.8): Laplace operation is defined as follows: Э“м В ~u B~u

^-4 + ^4 + ^4 can be described using V2u, and thus Eq. (5.9) can be simplified and written as: By substituting Eq. (4.90) into the term A + G, we can get: By substituting the relationship above into Eq. (5.10) leads to the follows: Similar expressions can be written for у and z directions:  Eqs. (5.11), (5.12) and (5.13) are known as Navier equations for each axis.

# Beltami-Michell stress compatibility equations

Under plane stress condition, g- = xyz = rxz = 0. Eq. (4.80) will be written as follows: Similarly, Eq. (4.81) will be written as below: From Eq. (4.94), тху can be expressed as follows: By expressing yu in term of rvv yields the following equation: Equation below is obtained by substituting the relationships in Eqs. (5.14), (5.15) and (5.16) into Eq. (3.20): Remove the common term - from both sides gives us the follows:

e ° By expanding the equation above leads to this: Under plane stress condition, Eqs. (2.5) and (2.6) are written as follow: Differentiate the terms in Eq. (5.18) with respect to Л' and we obtain this: Also, differentiate the terms in Eq. (5.19) with respect to у yields the following equation: By adding Eqs. (5.20) to (5.21), and noting that rvv = %, gives the following result: By rearranging the equation above the following is obtained: By substituting Eq. (5.22) into Eq. (5.17) yields the follows: Equation below is obtained through expansion of equation above: Eliminating the common terms on both sides yields the follows: Rearrange the equation above and we get: Through simplification of equation above, the following is obtained: Further simplifying the equation above by introducing Laplace operator yields this: Eq. (5.23) is the Beltrami-Michell stress compatibility equation for plane stress condition. Under the special case where body forces (fx and/,.) are constant or zero, Eq. (5.23) will be reduced to: Under plane strain condition, e: = yv. = yX7 = 0, and az * 0. Eq. (4.82) will be written as follows: Express the equation above with a: in terms of other parameters gives us the following: By substituting Eq. (5.24) into Eq. (4.80) produces the following: Simplify the equation above yields the following: Similarly, for у-axis, its normal strain can be written as: The following results through substitution of Eqs. (5.25), (5.26) and (5.16) into Eq. (3.20): After removal of common term - from both sides, we get: Expansion of equation above provides us the following equation: After rearranging the equation above, the following is obtained: Eliminating the term (1 + v) from both sides yields the following: By substituting Eq. (5.22) into the equation above we can get: Expanding and rearranging the equation above gives us the following: Simplification of the equation above results in this: Further simplifying the equation above by introducing Laplace operator yields the following: Eq. (5.27) is the Beltrami-Michell stress compatibility equation for plane strain condition. Under the special case where body forces (fx and/,.) are constant or zero, Eq. (5.27) will be reduced to:

# Airy stress function

When at rest, a body possesses potential energy, say xp. Body force can thus be expressed in term of such potential energy: Also, the stress component can be expressed in term of stress function. For a 2-D scenario, Under plane stress condition, the equilibrium equation can be written as follows by substituting relationships in Eq. (5.28) into Eq. (5.18): Simplify the equation above yields the following equation: Similarly, the following equation can be derived from Eq. (5.19): We get the equation below by substituting relationships in Eqs. (5.28) and (5.29) into Eq. (5.23): With simplification, the equation below is obtained: By introducing Laplace operator to the equation, it can be transformed to this form: Expand the equation above gives us the follows: By rearranging the equation above leads to the follows: Simplify the equation above yields the follows: Eq. (5.32) is a biharmonic equation for plane stress condition. Under a special case where no body force presents, Eq. (5.32) will be reduced to: Under plane stress condition, the equilibrium equation can be written as follows by substituting relationships in Eqs. (5.28) and (5.29) into Eq. (5.27): Simplify the equation above gives us: By introducing the Laplace operator to the equation leads to the follows: Expand the equation above yields the following equation: Rearrange the equation above results in the follows: Simplify the equation above gives us the following expression: Eq. (5.33) is a biharmonic equation for plane strain condition. Under a special case where no body force presents, Eq. (5.33) will be reduced to: 