# Elasticity

2 Elasticity Real materials are not perfectly rigid. When subjected to forces, they deform. If a substance deforms when subjected to a force, but returns to its initial shape when the force is removed, the substance is elastic. Consider a material of length L and cross-sectional area A. If a force F is applied along the axis of the material, and this causes a change in length ∆L of the cylinder, then we define the following: • Stress: This is the external force acting on an object per unit cross-sectional area. The result of a stress is strain. Stress = F A • Strain: This is a measure of the degree of deformation. Strain = ∆L L In a weak material, a small stress produces a large strain. For sufficiently small stresses, stress and strain are proportional. The constant of proportionality depends on the kind of material and on the nature of the deformation. The ratio of stress to strain is the elastic modulus. Elastic modulus = Stress Stress. Suppose you pull or push on a cylinder of length L and cross-sectional area A with a force F directed along the axis. The material is subject to a tensile stress. Young modulus is defined as: Y = Tensile strength Tensile strain = F/A ∆L/L. If the stress exceeds the elastic limit, the material does not return to its original shape when the stress is removed. The shear modulus measures a material’s ability to resist changes in its shape. Suppose a piece of material in the form of a rectangular block (like a brick) has one face fixed and a face F applied to the opposite face of area A. Imagine F applied parallel to the face, like the friction force. If the two faces are separated by distance h, and the sheared face moves ∆x, the shear modulus is S = Shear stress Shear strain = F/A ∆x/h

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