3D stress -strain relationship

[Ronankeating]

εHi,

Based on that hoary schematic cube representation in 3D for stress-strain relationship. Stress tensor {σxx, σyy, σzz, σxy, σyz, σzx} and strain tensors {εxx, εyy, εzz, εxy, εyz, εzx} can be written interchangably. Let's suppose that σzz=σzx=σzy=0 then the reamining terms are written in matrix form then how that E/(1-v²) coefficient is emerging.

At first sight, is seems that its very simple to express it by multipliers but couldn't figured out .

Any help please,

 

[Studiot]

You are missing the off diagonal relationships in your tensor.

 

[Studiot]

The relationships betwenn stress and strain are

$$\begin{array}{l}<br /> {\varepsilon _x} = \frac{1}{E}\left[ {{\sigma _x} - \nu ({\sigma _y} + {\sigma _z}} \right] \\ <br /> {\varepsilon _y} = \frac{1}{E}\left[ {{\sigma _y} - \nu ({\sigma _x} + {\sigma _z}} \right] \\ <br /> {\varepsilon _z} = \frac{1}{E}\left[ {{\sigma _z} - \nu ({\sigma _x} + {\sigma _y}} \right] \\ <br /> {\gamma _{xy}} = \frac{1}{G}{\tau _{xy}} \\ <br /> {\gamma _{yz}} = \frac{1}{G}{\tau _{yz}} \\ <br /> {\gamma _{xz}} = \frac{1}{G}{\tau _{xz}} \\ <br /> \end{array}$$

Written in matrix form this becomes

$$\varepsilon = C\sigma$$

where C is the matrix

$$C = \frac{1}{E}\left[ {\begin{array}{*{20}{c}}<br /> 1 & { - v} & { - v} & 0 & 0 & 0 \\<br /> { - v} & 1 & { - v} & 0 & 0 & 0 \\<br /> { - v} & { - v} & 1 & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & {\frac{E}{G}} & 0 & 0 \\<br /> 0 & 0 & 0 & 0 & {\frac{E}{G}} & 0 \\<br /> 0 & 0 & 0 & 0 & 0 & {\frac{E}{G}} \\<br /> \end{array}} \right]$$

 

[Studiot]

(But I'm intending to write stress as function of strain)

you do this by solving (inverting) my matrix equation

ε=Cσ

this is possible and leads to

$$\begin{array}{l}<br /> {\sigma _x} = A\left[ {{\varepsilon _x} + B\left( {{\varepsilon _y} + {\varepsilon _z}} \right)} \right] \\ <br /> {\sigma _y} = A\left[ {{\varepsilon _y} + B\left( {{\varepsilon _x} + {\varepsilon _z}} \right)} \right] \\ <br /> {\sigma _z} = A\left[ {{\varepsilon _z} + B\left( {{\varepsilon _x} + {\varepsilon _y}} \right)} \right] \\ <br /> {\tau _{xy}} = G{\gamma _{xy}} \\ <br /> {\tau _{yz}} = G{\gamma _{yz}} \\ <br /> {\tau _{xz}} = G{\gamma _{xz}} \\ <br /> \end{array}$$

Also

$$\begin{array}{l}<br /> G = \frac{E}{{2\left( {1 + \nu } \right)}} \\ <br /> A = E\frac{{\left( {1 - \nu } \right)}}{{\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)}} \\ <br /> B = \frac{\nu }{{\left( {1 - v} \right)}} \\ <br /> \end{array}$$

As regards to the LaTex this site uses for formulae, I am seriously deficient in latex lore so I use MathType and copy/paste.
I wouldn't recommend MT, however as it is too expensive for what it is and does not allow the inclusion of images.

 

[Studiot]

There seems to be a missing post, however to complete a matrix expression for stress in terms of strain is

$$\sigma = D\varepsilon$$

where D is the matrix

$$D = A\left[ {\begin{array}{*{20}{c}}<br /> 1 & B & B & 0 & 0 & 0 \\<br /> B & 1 & B & 0 & 0 & 0 \\<br /> B & B & 1 & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & {\frac{G}{A}} & 0 & 0 \\<br /> 0 & 0 & 0 & 0 & {\frac{G}{A}} & 0 \\<br /> 0 & 0 & 0 & 0 & 0 & {\frac{G}{A}} \\<br /> \end{array}} \right]$$

A and B have the same meaning as before