# 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 *V ^{2}u,* 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- = *x _{yz}* = r

_{xz}= 0. Eq. (4.80) will be written as follows:

Similarly, Eq. (4.81) will be written as below:

From Eq. (4.94), *т _{ху}* can be expressed as follows:

By expressing y_{u} in term of r_{vv} 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 r_{vv} = %, 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 *(f _{x}* and/,.) are constant or zero, Eq. (5.23) will be reduced to:

Under plane strain condition, *e _{:}* = y

_{v}. =

*y*

*= 0, and*

_{X7}*a*0. Eq. (4.82) will be written as follows:

_{z}*

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 *(f _{x}* 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.

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: