Linear Elasticity
A fundamental theory in mechanics that describes how solid objects deform and regain their original shape when subjected to small forces, assuming a linear relationship between stress and strain.
Linear Elasticity
Linear elasticity forms the cornerstone of modern structural analysis and materials science, providing a mathematical framework for understanding how materials behave under small deformations.
Fundamental Principles
The theory rests on three key assumptions:
- Small deformations (infinitesimal strain theory)
- Linear relationship between stress and strain (Hooke's Law)
- Reversible deformation behavior
Mathematical Framework
The constitutive equation in linear elasticity is expressed through:
σᵢⱼ = Cᵢⱼₖₗ εₖₗ
Where:
- σᵢⱼ represents the stress tensor
- εₖₗ represents the strain tensor
- Cᵢⱼₖₗ is the fourth-order elasticity tensor
Material Properties
Key material parameters in linear elasticity include:
- Young's Modulus (E)
- Poisson's Ratio (ν)
- Shear Modulus (G)
These properties are interrelated and fully describe the elastic behavior of isotropic materials.
Applications
Linear elasticity finds widespread use in:
- Structural engineering
- Mechanical Design
- Finite Element Analysis
- Material Characterization
Limitations
The theory becomes invalid when:
- Deformations become large (Non-linear Elasticity)
- Material exhibits plastic deformation
- Time-dependent behavior becomes significant
Historical Development
The development of linear elasticity theory is closely tied to the work of:
- Robert Hooke (17th century)
- Claude-Louis Navier (19th century)
- George Green (Continuum Mechanics)
Modern Extensions
Contemporary applications extend to:
The theory continues to serve as a fundamental building block for more complex material models and analytical approaches in engineering and materials science.