Elasticity and Strength of Materials cheatsheet

Elasticity

Stress and Strain

Normal Stress ()

Stress measures the intensity of the internal force acting on a specific area within a material due to an external load.

Where:

• is the applied force (N).
• is the cross-sectional area (m²).

Axial Strain ()

Axial strain measures the relative change in length of a material under load:

Where:

• is the change in length (m).
• is the original length (m).

Hooke’s Law for Axial Stresses (Elastic Materials)

In the elastic region of a material, stress is proportional to strain:

Where:

• is the Young’s modulus or modulus of elasticity (Pa).
• is the unit strain (dimensionless).

Lateral Strain

The Poisson’s ratio () describes the relationship between axial and lateral strain in a material subjected to uniaxial tension or compression:

Where:

• is the Poisson’s ratio (dimensionless).
• is the lateral strain.
• is the axial strain.

Volumetric Strain ()

Volumetric strain describes the relative change in volume of a body due to applied pressure or force:

Where:

• is the change in volume.
• is the original volume.

Elasticity Moduli

Young’s Modulus (E)

Young’s modulus represents the stiffness of a material against axial deformation:

Where:

• is the stress (N/m² or Pa).
• is the unit strain (dimensionless).

Shear Modulus (G)

The shear modulus measures the stiffness against deformations in a plane, such as in torsion or shear:

Where:

• is the shear stress (N/m² or Pa).
• is the angular deformation (radians, dimensionless).

Relationship between,, and Poisson’s Ratio ()

For isotropic elastic materials, the relationship between,, and Poisson’s ratio is:

Where:

• is the Poisson’s ratio, which relates lateral strain to axial strain.

Bulk Modulus (K)

The bulk modulus or compressive modulus measures resistance to volumetric changes under uniform pressure:

Where:

• is the applied pressure (Pa).
• is the change in volume.
• is the original volume.

Strength of Materials

The strength of materials deals with the ability of materials to withstand stresses and strains without failing.

Internal Stresses

Normal Stress ()

Normal stress is the force per unit area acting perpendicular to the cross-section of a body:

Where:

• = applied force (N)
• = cross-sectional area (m²)

Shear Stress ()

Shear stress measures the intensity of forces acting tangentially on the cross-section of a body:

Where:

• is the applied shear force (N).
• is the cross-sectional area (m²).

Strength Criteria

Von Mises Criterion (Equivalent Stress)

This criterion is used to determine if a ductile material will fail under a combined stress state. The Von Mises equivalent stress is defined as:

Three-Axis Criterion

For a material subjected to stresses in three principal directions:

The maximum stress is determined by the formula:

Tensile and Compressive Strength

Yield Load ()

The yield load is the stress at which a material begins to deform plastically:

Where:

• is the applied load (N).
• is the cross-sectional area (m²).

Rupture Load ()

The rupture stress is the stress at which a material fractures:

Moment of Inertia and Torsion

Moment of Inertia ()

The moment of inertia of a section describes its resistance to bending:

For a rectangular section:

Where:

• is the width of the section (m).
• is the height of the section (m).

Shear Stress in Torsion ()

In torsion, the shear stress in a cylindrical shaft is:

Where:

• is the torsional moment (N·m).
• is the radius of the section (m).
• is the polar moment of inertia (m⁴).

Material Fatigue

Fatigue is the process of weakening a material under cyclic loads.

Fatigue Limit ()

The fatigue limit is the maximum value of cyclic stress that a material can withstand indefinitely without failing:

Number of Cycles to Fatigue ()

The number of cycles before fatigue is related to the applied stress range:

Where:

• is the difference between the maximum and minimum stress in a cycle.