In the small-strain region, stress is proportional to strain, so the graph is linear and Hooke's law holds. The proportionality stops at the limit of proportionality, while complete recovery continues up to the elastic limit.

• Beyond the elastic limit, permanent deformation begins.
• The region beyond elastic limit and before fracture is the plastic range.
• Brittle materials have small plastic range, while ductile materials show large plastic deformation.

Young's modulus is the ratio of normal stress to longitudinal strain within the proportional limit, so it measures stiffness under stretching or compression of solids.

• For a wire, Y = FL/(A deltaL).
• Larger Young's modulus means smaller elongation for the same load and dimensions.
• Force constant of a wire is k = YA/L.

Work done against restoring force is stored as strain energy. For linear elastic stretching, stored energy is one-half the product of force and extension.

• Total energy in wire: U = 1/2 F deltaL.
• Energy density = 1/2 x stress x strain.
• The area under the force-extension graph equals the stored energy.

Breaking stress is the maximum stress a material can bear before fracture, whereas breaking force depends on cross-sectional area. The material constant is stress, not force.

• Breaking force = breaking stress x area.
• Breaking stress does not depend on wire length or thickness.
• Working stress is kept below breaking stress for safety.

Bulk modulus is the ratio of normal stress to volumetric strain for uniform compression. The negative sign in K = -pV/deltaV reflects that volume decreases when pressure increases.

• Compressibility is the reciprocal of bulk modulus.
• Bulk modulus applies to solids, liquids, and gases.
• For an ideal gas, isothermal bulk modulus equals p and adiabatic bulk modulus equals gamma p.

Modulus of rigidity is the ratio of shear stress to shear strain. It measures resistance to change of shape without significant change of volume.

• Only solids exhibit static shear resistance.
• For small shear angle, shear strain equals x/L or phi.
• Higher shear modulus means greater resistance to sideways distortion.

Poisson's ratio compares lateral contraction with longitudinal extension. When a bar is stretched, radius decreases while length increases, giving the negative sign in the formal definition.

• It is dimensionless and unitless.
• Practical values lie between 0 and 0.5.
• Rubber is close to 0.5, while cork is close to 0.

When a cylinder is twisted by torque, different cylindrical shells undergo shear. The twisting couple required per unit twist is the torsional constant, proportional to eta r^4/l.

• Angle of twist increases with distance from fixed end.
• Twisting couple per unit twist is C = pi eta r^4 /(2l).
• Work done in twisting through angle theta is 1/2 C theta^2.

The interatomic force constant models atomic bonds as effective springs and is related to Young's modulus. Elastic hysteresis describes the lag between loading and unloading, so the two curves enclose an energy-loss loop.

• Interatomic force constant can be written as Y times normal interatomic spacing.
• Area of hysteresis loop equals energy dissipated per unit volume in one cycle.
• Materials with larger hysteresis are better for vibration absorption than for low-heating tyres.