# Introduction to Elasticity/Sample midterm 2

## Sample Midterm Problem 2

Given:

A strain gage rosette provides the following data

$\varepsilon _{1}=0.01;~~\varepsilon _{2}=0.02;~~\varepsilon _{30^{o}}=0$ where the $X_{1}$ and $X_{2}$ directions are perpendicular to each other and $\varepsilon _{30^{o}}$ is the extensional strain of a line element at an angle of $30^{o}$ to the $X_{1}$ axis (in the counterclockwise direction).

Find:

• (a) Compute $\varepsilon _{60^{o}}$ .
• (b) Is the result valid if the material is anisotropic ?

## Solution

### Part (a)

From the previous problem, for an angle of rotation of 30$^{o}$ , the rotation matrix $\left[L\right]$ is

$l_{ij}=\left[L\right]={\begin{bmatrix}{\sqrt {3}}/2&1/2&0\\-1/2&{\sqrt {3}}/2&0\\0&0&1\end{bmatrix}}$ Therefore, the components of strain in the rotated co-ordinate system are given by

$\left[{\boldsymbol {\varepsilon }}\right]^{'}=\left[L\right]\left[{\boldsymbol {\varepsilon }}\right]\left[L\right]^{T}~~{\text{or,}}~~\varepsilon _{ij}^{'}=l_{ip}l_{jq}\varepsilon _{pq}$ Since we are given $\varepsilon _{30^{o}}=\varepsilon _{11}^{'}$ , we will calculate the value of this strain in terms of the original components of strain. Thus,

{\begin{aligned}\varepsilon _{11}^{'}=&l_{1p}l_{1q}\varepsilon _{pq}\\=&l_{11}l_{11}\varepsilon _{11}+l_{12}l_{11}\varepsilon _{21}+l_{13}l_{11}\varepsilon _{31}+l_{11}l_{12}\varepsilon _{12}+l_{12}l_{12}\varepsilon _{22}+l_{13}l_{12}\varepsilon _{32}+\\&l_{11}l_{13}\varepsilon _{13}+l_{12}l_{13}\varepsilon _{23}+l_{13}l_{13}\varepsilon _{33}\\=&l_{11}(l_{11}\varepsilon _{11}+l_{12}\varepsilon _{12}+l_{13}\varepsilon _{13})+l_{12}(l_{11}\varepsilon _{21}+l_{12}\varepsilon _{22}+l_{13}\varepsilon _{23})+\\&l_{13}(l_{11}\varepsilon _{31}+l_{12}\varepsilon _{32}+l_{13}\varepsilon _{33})\\=&({\frac {\sqrt {3}}{2}})\left[({\frac {\sqrt {3}}{2}})(0.01)+({\frac {1}{2}})\varepsilon _{12}\right]+({\frac {1}{2}})\left[({\frac {\sqrt {3}}{2}})\varepsilon _{12}+({\frac {1}{2}})(0.02)\right]\\=&(3/4)(0.01)+({\sqrt {3}}/2)\varepsilon _{12}+(1/4)(0.02)\\=&(5/4)(0.01)+({\sqrt {3}}/2)\varepsilon _{12}\end{aligned}} Therefore,

$(5/4)(0.01)+({\sqrt {3}}/2)\varepsilon _{12}=\varepsilon _{30^{o}}=0$ Hence,

$\varepsilon _{12}=-(2.5)(0.01)/{\sqrt {3}}$ Next, for an angle of rotation of 60$^{o}$ , the matrix $\left[L\right]$ is

{\begin{aligned}\left[L\right]&={\begin{bmatrix}\cos(60^{o})&\sin(60^{o})&\cos(90^{o})\\-\sin(60^{o})&\cos(60^{o})&\cos(90^{o})\\\cos(90^{o})&\cos(90^{o})&\cos(0^{o})\end{bmatrix}}\\&={\begin{bmatrix}1/2&{\sqrt {3}}/2&0\\-{\sqrt {3}}/2&1/2&0\\0&0&1\end{bmatrix}}\end{aligned}} Therefore, $\varepsilon _{60^{o}}=\varepsilon _{11}^{'}$ , is given by

{\begin{aligned}\varepsilon _{11}^{'}=&l_{1p}l_{1q}\varepsilon _{pq}\\=&l_{11}l_{11}\varepsilon _{11}+l_{12}l_{11}\varepsilon _{21}+l_{13}l_{11}\varepsilon _{31}+l_{11}l_{12}\varepsilon _{12}+l_{12}l_{12}\varepsilon _{22}+l_{13}l_{12}\varepsilon _{32}+\\&l_{11}l_{13}\varepsilon _{13}+l_{12}l_{13}\varepsilon _{23}+l_{13}l_{13}\varepsilon _{33}\\=&l_{11}(l_{11}\varepsilon _{11}+l_{12}\varepsilon _{12}+l_{13}\varepsilon _{13})+l_{12}(l_{11}\varepsilon _{21}+l_{12}\varepsilon _{22}+l_{13}\varepsilon _{23})+\\&l_{13}(l_{11}\varepsilon _{31}+l_{12}\varepsilon _{32}+l_{13}\varepsilon _{33})\\=&({\frac {1}{2}})\left[({\frac {1}{2}})(0.01)+({\frac {\sqrt {3}}{2}})\varepsilon _{12}\right]+({\frac {\sqrt {3}}{2}})\left[({\frac {1}{2}})\varepsilon _{12}+({\frac {\sqrt {3}}{2}})(0.02)\right]\\=&(1/4)(0.01)+({\sqrt {3}}/2)\varepsilon _{12}+(3/4)(0.02)\\=&(7/4)(0.01)+({\sqrt {3}}/2)(-(2.5)(0.01)/{\sqrt {3}})\\=&(7/4)(0.01)-(5/4)(0.01)=(1/2)(0.01)=0.005\\\end{aligned}} Therefore,

${\varepsilon _{60^{o}}=0.005}$ ### Part (b)

${\text{The result is valid for all materials.}}$ 