Problem 2.8. Stiffness matrix

Young modulus, E [GPa]=

Poisson ratio, ν=

Shear modulus [GPa]=

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Step 1. Write parametrically the stiffness matrix and determine the material properties. Assume orthotropic material.

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Stiffness matrix of an orthotropic material is given by the following engineering constants:

Eq.(2-63) and Footnote k in Section 2.1.4

$\frac{1}{1–{\nu }_{xy}{\nu }_{yx}}\left[\begin{array}{ccc}{E}_x& {\nu }_{yx}{E}_x& 0\\ {\nu }_{xy}{E}_y& E_y& 0\\ 0& 0& G\left(1–{\nu }_{xy}{\nu }_{yx}\right)\end{array}\right]=\left[\begin{array}{ccc}10& 1& 0\\ 1& 10& 0\\ 0& 0& 10\end{array}\right]$

From the above equation the unknown material properties can be unambiously expressed:

${\mathit{\nu }}_{\mathbf{x}\mathbf{y}}\mathbf{=}{\mathit{\nu }}_{\mathbf{y}\mathbf{x}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{,}\mathbf{ }{\mathit{E}}_{\mathbf{x}}\mathbf{=}{\mathit{E}}_{\mathbf{y}}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{9}\frac{\mathbf{k}\mathbf{N}}{\mathbf{m}{\mathbf{m}}^{\mathbf{2}}}\mathbf{,}\mathbf{ }\mathit{G}\mathbf{=}\mathbf{10}\frac{\mathbf{k}\mathbf{N}}{\mathbf{m}{\mathbf{m}}^{\mathbf{2}}}$

Step 2. Check the Young modulus and the shear modulus to prove whether the material is stable.

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The material is expected to be stable, because both the Young and the shear moduli are positive.

See Eq.(2-80)

$\mathit{E}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{9}\mathbf{ }\frac{\mathbf{kN}}{{\mathbf{mm}}^{\mathbf{2}}}\mathbf{>}\mathbf{0}\mathbf{,}\mathbf{ }\mathit{G}\mathbf{=}\mathbf{10}\mathbf{ }\frac{\mathbf{k}\mathbf{N}}{\mathbf{m}{\mathbf{m}}^{\mathbf{2}}}\mathbf{>}\mathbf{0}$

For isotropic material the Poisson ratio must be smaller than 0.5, for anisotropic material the material is stable if the compliance (or the stiffness) matrix is positive definite. Symmetric, real matrices are positive definite when all of their eigenvalues are positive.

In Matlab $\mathrm{eig}\left(\left[\begin{array}{ccc}10& 1& 0\\ 1& 10& 0\\ 0& 0& 10\end{array}\right]\right)$ results in the following eigenvalues: 9.00, 10.00, 11.00, which are positive, hence the material is stable.

Step 3. Check isotropy.

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The material is isotropic when the following relation between the engineering constants is true

Eq.(2-62)

$G=\frac{E}{2(1+\nu )}$

Here

$G=10\frac{\mathrm{kN}}{{\mathrm{mm}}^2}\ne \frac{E}{2(1+\nu )}=\frac{9.9}{2(1+0.1)}=4.5\frac{\mathrm{kN}}{{\mathrm{mm}}^2}$

thus the material is not isotropic.

Material properties can be determined from the parametric form of the stiffness matrix. Assume orthotropic material. Stiffness matrix of an orthotropic material is given by the following engineering constants:

Eq.(2-63) and Footnote k in Section 2.1.4

From the above equation the unknown material properties can be unambiously expressed:

${\mathit{\nu }}_{\mathbf{x}\mathbf{y}}\mathbf{=}{\mathit{\nu }}_{\mathbf{y}\mathbf{x}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{,}\mathbf{ }{\mathit{E}}_{\mathbf{x}}\mathbf{=}{\mathit{E}}_{\mathbf{y}}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{9}\frac{\mathbf{k}\mathbf{N}}{\mathbf{m}{\mathbf{m}}^{\mathbf{2}}}\mathbf{,}\mathbf{ }\mathit{G}\mathbf{=}\mathbf{10}\frac{\mathbf{k}\mathbf{N}}{\mathbf{m}{\mathbf{m}}^{\mathbf{2}}}$

The material is expected to be stable, because both the Young and the shear moduli are positive.

See Eq.(2-80)

For isotropic material the Poisson ratio must be smaller than 0.5, for anisotropic material the material is stable if the compliance (or the stiffness) matrix is positive definite. Symmetric, real matrices are positive definite when all of their eigenvalues are positive.

In Matlab $\mathrm{eig}\left(\left[\begin{array}{ccc}10& 1& 0\\ 1& 10& 0\\ 0& 0& 10\end{array}\right]\right)$  results in the following eigenvalues: 9.00, 10.00, 11.00, which are positive, hence the material is stable.

The material is isotropic when the following relation between the engineering constants is true

Eq.(2-62)

$G=\frac{E}{2(1+\nu )}$

Here

$G=10\frac{\mathrm{kN}}{{\mathrm{mm}}^2}\ne \frac{E}{2(1+\nu )}=\frac{9.9}{2(1+0.1)}=4.5\frac{\mathrm{kN}}{{\mathrm{mm}}^2}$

thus the material is not isotropic.