Linear Elasticity

Definition

Linear elasticity is a mathematical model of solid mechanics where the deformation (strain) of a material is directly, linearly proportional to the applied mechanical stress, and the material perfectly returns to its original shape once the load is removed. It is the three-dimensional generalization of Hooke’s Law.

$\sigma_{ij} = C_{ijkl} \epsilon_{kl}$ How to read: Sigma sub i j equals C sub i j k l times epsilon sub k l. Meaning / when to use: This is the generalized Hooke’s Law using Einstein summation convention. It states that the stress tensor $\sigma$ is a linear function of the strain tensor $\epsilon$, connected by a 4th-order stiffness tensor $C$. Used to model the structural response of materials under load.

Why It Matters

Linear elasticity is the fundamental assumption underlying the vast majority of structural engineering, from designing skyscrapers to building car chassis. It provides a simple, highly predictive model that allows engineers to ensure structures will bend slightly under expected loads but will neither permanently deform nor break. Exceeding the linear elastic limit leads to plastic deformation and catastrophic failure.

Core Concepts

• Stress and Strain: Stress is the internal force per unit area. Strain is the normalized geometric deformation.
• Young’s Modulus (E): The scalar constant representing material stiffness in 1D tension or compression.
• Poisson’s Ratio ($\nu$): The measure of how a material expands outward when compressed, or thins out when stretched.
• Elastic Limit: The threshold of stress beyond which the material stops behaving linearly and suffers permanent plastic deformation.

Connected Concepts

• Restoring Force
• Force Non Linearity
• Modeling Complexity