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Question 24.4: A thin ply consists of an isotropic material having a Young’...

A thin ply consists of an isotropic material having a Young’s modulus of 80,000 N / mm ^{2} and a Poisson’s ratio of 0.3. Determine the terms in the reduced stiffness and compliance matrices.

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The reduced stiffnesses are

 

k_{11}=k_{22}=E /\left(1-ν^{2}\right)=80,000 /\left(1-0.3^{2}\right)=87912 N / mm ^{2}

 

k_{12}=v E /\left(1-ν^{2}\right)=0.3 \times 80,000 /\left(1-0.3^{2}\right)=26374 N / mm ^{2}

 

Also k_{33}=G but since the ply is isotropic the shear modulus may be expressed in terms of Young’s modulus and Poisson’s ratio, i.e.,

 

G=E / 2(1+ν) \quad \text { (see Eq. (1.50)) }

 

\gamma=\frac{2(1+ν)}{E} \tau  (1.50)

 

so that

 

k_{33}=80,000 / 2(1+0.3)=30769 N / mm ^{2}

 

The reduced compliances are

 

s_{11}=s_{22}=1 / E=1 / 80,000=12.5 \times 10^{-6}

 

s_{12}=s_{21}=-ν / E=-0.3 / 80,000=-3.75 \times 10^{-6}

 

s_{33}=1 / G=1 / 30,769=32.5 \times 10^{-6}

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