Question

The displacement field of a body is given by \( \vec{u} = yx \hat{i} + yz \hat{j} + (z + x^2) \hat{k} \). The shear strain \( \gamma_{xy} \) at \( (2, 1, 5) \) is:

Hint:

To calculate shear strain, use the formula \( \gamma_{xy} = \frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x} \), where \( u_x \), \( u_y \), and \( u_z \) are the displacement components in respective directions.

Answer 1

The shear strain \( \gamma_{xy} \) is given by the formula: \[ \gamma_{xy} = \frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x} \] where \( u_x = yx \), \( u_y = yz \), and \( u_z = z + x^2 \). 
Step 1: Find \( \frac{\partial u_x}{\partial y} \). \[ u_x = yx \quad \Rightarrow \quad \frac{\partial u_x}{\partial y} = x \] Substitute the values \( x = 2 \): \[ \frac{\partial u_x}{\partial y} = 2 \] Step 2: Find \( \frac{\partial u_y}{\partial x} \). \[ u_y = yz \quad \Rightarrow \quad \frac{\partial u_y}{\partial x} = 0 \] since \( u_y \) does not depend on \( x \). Step 3: Calculate the shear strain \( \gamma_{xy} \).
Now, substitute the values into the shear strain formula: \[ \gamma_{xy} = 2 + 0 = 2 \] Final Answer: The shear strain \( \gamma_{xy} \) at \( (2, 1, 5) \) is \( 2 \).