## 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** (

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

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.