Solid Mechanics
What is Solid Mechanics?
Internal resisting force per unit area within a body subjected to external forces.
Key formula / rule: Normal Stress
Key points
- Define stress and strain.
- Differentiate between normal and shear stress/strain.
- State and apply Hooke's Law.
- Understand the concept of elastic limit and yield strength.
Common exam trap
Confusing stress and force.
Definitions
- Term
Stress
- Meaning
Internal resisting force per unit area within a body subjected to external forces.
- Term
Strain
- Meaning
A measure of deformation representing the ratio of change in dimension to the original dimension.
- Term
Young's Modulus (E)
- Meaning
The ratio of normal stress to normal strain in the elastic region; a measure of stiffness.
- Term
Shear Modulus (G)
- Meaning
The ratio of shear stress to shear strain in the elastic region; a measure of resistance to shear deformation.
- Term
Poisson's Ratio (ν)
- Meaning
The ratio of transverse strain to axial strain in the elastic region.
Learning objectives
Define stress and strain.
Differentiate between normal and shear stress/strain.
State and apply Hooke's Law.
Understand the concept of elastic limit and yield strength.
Calculate stress and strain for simple loading conditions.
Formulae
- Name
Normal Stress
- Note
P is the applied normal force, A is the cross-sectional area perpendicular to the force.
- Expression
\σ = \frac{P}{A}
- Name
Shear Stress
- Note
V is the applied shear force, A is the cross-sectional area parallel to the force.
- Expression
\τ = \frac{V}{A}
- Name
Normal Strain
- Note
\Δ L is the change in length, L0 is the original length.
- Expression
\varepsilon = \frac{\Δ L}{L0}
- Name
Shear Strain
- Note
\θ is the change in angle in radians; approximation is valid for small angles.
- Expression
\γ = \tan(\θ) \≈ \θ
- Name
Hooke's Law (Normal)
- Note
E is the Modulus of Elasticity (Young's Modulus).
- Expression
\σ = E \varepsilon
- Name
Hooke's Law (Shear)
- Note
G is the Shear Modulus (Modulus of Rigidity).
- Expression
\τ = G \γ
- Name
Poisson's Ratio
- Note
Relates lateral strain to axial strain under uniaxial stress.
- Expression
\ν = -\frac{\varepsilonlateral}{\varepsilonaxial}
Prerequisites
Basic mechanics (forces, equilibrium)
Understanding of geometry and area calculations
Units and dimensions
Common mistakes
Confusing stress and force.
Incorrectly calculating strain, especially shear strain.
Assuming linear elastic behavior beyond the material's elastic limit.
Using inconsistent units for stress and strain calculations.
Not considering the cross-sectional area correctly in stress calculations.
Keywords
Stress
Strain
Normal Stress
Shear Stress
Normal Strain
Shear Strain
Hooke's Law
Young's Modulus
Shear Modulus
Poisson's Ratio
Elastic Limit
Yield Strength
Practice preview
For a beam subjected to pure bending, the bending stress at a distance 'y' from the neutral axis is given by (where M is bending moment, I is moment of inertia, R is radius of curvature):…
medium
A steel rod of length L and cross-sectional area A is heated by a temperature T. If the coefficient of thermal expansion is α and Young's modulus is E, what is the thermal stress developed if the rod is fully restrained?…
medium
According to Hooke's Law, stress is directly proportional to strain within the:…
easy