Strain and Hooke's Law

Case Study: Consider a rectangular block of rubber subjected to different types of loading. In the first scenario, a tensile force is applied perpendicular to its face. In the second scenario, a force is applied parallel to its top face while the bottom face is fixed. Differentiate the resulting strains.

Case Study: A steel specimen is loaded in a tensile testing machine. The load is gradually increased, and the corresponding deformation is recorded. At a certain point, the load is removed. Explain how Hooke's Law applies during this process and what happens if the proportional limit is exceeded.

Case Study: A cylindrical metal rod is subjected to an axial compressive force. Describe the deformation in both the axial and transverse directions, and explain how Poisson's ratio relates them.

Case Study: A structural element in a submarine pressure hull is subjected to triaxial compressive stresses due to hydrostatic pressure. Explain why calculating the strain in one direction requires considering the stresses in all three orthogonal directions.

Problem: A structural steel tie rod has an original length of $3.50 \text{ m}$. When subjected to a tensile force, it elongates by $2.10 \text{ mm}$. Calculate the normal strain in the rod.

Problem: An aluminum alloy rod ($E = 70.0 \text{ GPa}$) has a diameter of $12.0 \text{ mm}$ and a length of $600 \text{ mm}$. If it is subjected to a tensile force of $8.00 \text{ kN}$, determine the normal stress and the total elongation of the rod.

Problem: A rectangular rubber block with a height of $50.0 \text{ mm}$ is subjected to a shear force parallel to its top face. The top face displaces horizontally by $1.50 \text{ mm}$ relative to the fixed bottom face. Calculate the shear strain in the block.

Problem: A steel block ($G = 75.0 \text{ GPa}$) is subjected to a shear stress of $45.0 \text{ MPa}$. Determine the resulting shear strain in the material.

Problem: A standard structural steel rod ($E = 200 \text{ GPa}$, $\nu = 0.300$) is $2.50 \text{ m}$ long and has a diameter of $25.0 \text{ mm}$. It is subjected to an axial tensile load of $85.0 \text{ kN}$. Calculate the change in its length and the change in its diameter.

Problem: A composite rod consists of a solid brass segment ($E_b = 100 \text{ GPa}$) with length $L_1 = 400 \text{ mm}$ and area $A_1 = 500 \text{ mm}^2$, and a solid aluminum segment ($E_a = 70.0 \text{ GPa}$) with length $L_2 = 300 \text{ mm}$ and area $A_2 = 300 \text{ mm}^2$. An axial tensile load of $30.0 \text{ kN}$ is applied at the end of the aluminum segment. Calculate the total elongation of the rod.

Problem: During a tensile test, a steel specimen with a cross-sectional area of $200 \text{ mm}^2$ and a gauge length of $50.0 \text{ mm}$ experiences an elongation of $0.0350 \text{ mm}$ when an axial load of $28.0 \text{ kN}$ is applied. Assuming the material remains within the linear elastic region, calculate the modulus of elasticity of the steel.

Problem: A thin plate is subjected to biaxial stresses $\sigma_x = 80.0 \text{ MPa}$ and $\sigma_y = 50.0 \text{ MPa}$. The material properties are $E = 200 \text{ GPa}$ and $\nu = 0.300$. Determine the principal strains $\epsilon_x$ and $\epsilon_y$ in the plate.