3. Definition of Common Mechanical Modulus

#mechanics/bulk-modulus#mechanics/shear-modulus

(1) Bulk Modulus

We assume that an Object is under some static hydraulic pressure $-p$ in all directions, then bulk modulus^[1] is defined as the ratio of pressure and volume contraction :

$$\begin{array}{}\text{(1.1)}& K=-V\frac{dp}{dV}=-\frac{\mathrm{\Delta }p}{\frac{\mathrm{\Delta }V}{V}}\end{array}$$

In homogenization expression :

$$\begin{array}{}\text{(1.2)}& {\bar{\sigma }}_{ij}=E_{ijkl}^H{\overline{ϵ}}_{kl}\end{array}$$

For a $d$ dimension space, we have :

$$\begin{array}{}\text{(1.3)}& {\bar{\sigma }}_{ii}=E_{iikl}{ϵ}_{kl}=-d\times pp=-\frac{1}{d}{\bar{\sigma }}_{ii}{ϵ}_v=\frac{\mathrm{\Delta }V}{V}={\overline{ϵ}}_{ii}\end{array}$$

Here $ii$ is the repeated index notation (sum over dimension), which is ${\epsilon }_{11}+{\epsilon }_{22}$ in 2D case.

From (1.2), We have ${\bar{\sigma }}_{ii}=E_{iikl}^H{\bar{\epsilon }}_{kl}$, Then the bulk modulus is defined as :

$$\begin{array}{}\text{(1.4)}& K=-\frac{dp}{d{ϵ}_v}=\frac{E_{iikl}^H{\bar{\epsilon }}_{kl}}{{\bar{\epsilon }}_v}\end{array}$$

Since the ${\epsilon }_{kl}=\frac{1}{d}{\epsilon }_v{\delta }_{kl}$, we have :

$$\begin{array}{}\text{(1.5)}& \boxed{{\displaystyle K=\frac{E_{iikk}^H{\overline{ϵ}}_v}{{ϵ}_v}=E_{iikk}^H}}\end{array}$$

We note that the strain is uniform, we have :

$$\begin{array}{}\text{(1.6)}& {\bar{\epsilon }}_{kl}=\frac{1}{d}{\epsilon }_v{\delta }_{kl}\end{array}$$

then for the 2D case, the equation for bulk modulus is :

$$\begin{array}{}\text{(1.7)}& \boxed{{\displaystyle K_{ϵ}^H=\frac{1}{d^2}\sum _{i,k}{E}_{iikk}\stackrel{d=2}{⟶}\frac{1}{4}(E_{1111}+E_{1122}+E_{2211}+E_{2222})}}\end{array}$$

Where $d$ is the dimension.

(2) Shear Modulus

The shear modulus is defined as :

$$\begin{array}{}\text{(2.1)}& G=\frac{\tau }{\gamma }\end{array}$$

We have that for a shear force $\tau$,

$$\begin{array}{}\text{(2.2)}& \gamma ={ϵ}_{ij}+{ϵ}_{ji}\end{array}$$

note the stress components are :

$$\boxed{{\displaystyle {\bar{\sigma }}_{ij}=E_{ijkl}^H{\overline{ϵ}}_{kl}\stackrel{}{⟶}{E}_{1212}^H{ϵ}_{12}+E_{1221}}}$$

And we have ${\tau }_{ij}=E_{ijkl}{ϵ}_{kl}$, for

$$\begin{array}{}\text{(2.3)}& G_{12}=2\frac{E_{ijkl}{ϵ}_{ij}{\delta }_{ik}{\delta }_{jl}}{2{ϵ}_{ij}}=E_{ijij}\end{array}$$

For 2d case, it's :

$$\begin{array}{}\text{(2.4)}& G_{12}^H=E_{1212}\end{array}$$

For the 3d case, we only consider prescribed shear strain case, and it's the same as 2d form in one direction load. (for the prescribed shear stress case, it will use compliance matrix $S$)

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1. https://en.wikipedia.org/wiki/Bulk_modulus ↩︎