Elasticity and Stress-Strain
Elasticity describes how solids deform under load and recover their shape, relating the internal stress tensor to the strain tensor through the material's elastic constants.
Definition
Elasticity is the continuum theory of reversible deformation of solids, in which the stress tensor describing internal forces is linearly related, for small deformations, to the strain tensor describing the deformation through the material's elastic moduli.
Scope
This topic covers the stress and strain tensors of a deformable solid, the generalized Hooke's law relating them, the elastic moduli (Young's modulus, shear modulus, bulk modulus, Poisson's ratio) for isotropic materials, the equations of elastic equilibrium, and the elastic energy stored in a deformed body. It is the continuum-mechanics description of small reversible deformation.
Core questions
- How do the stress and strain tensors describe the state of a deformed solid?
- What does the generalized Hooke's law relate, and through which moduli?
- How is the elastic energy of a deformed body expressed?
Key concepts
- Stress tensor
- Strain tensor
- Young's modulus and Poisson's ratio
- Shear and bulk moduli
- Elastic energy
- Equilibrium equations
Key theories
- Generalized Hooke's law
- For small deformations the stress tensor is a linear function of the strain tensor; for an isotropic material this reduces to two independent elastic constants relating stress and strain.
- Equations of elastic equilibrium
- Balancing internal stresses against applied body forces yields the equilibrium equations whose solution gives the deformation field of a loaded elastic body subject to its boundary conditions.
Clinical relevance
Elasticity theory is the basis of structural and mechanical engineering analysis, governing the design of beams, columns, pressure vessels, and machine parts, the prediction of deflection and failure under load, and the modeling of elastic biological tissues in biomechanics.
History
Hooke's seventeenth-century law that extension is proportional to force began the study of elasticity, which Navier and Cauchy turned into a continuum theory in the 1820s with the introduction of the stress tensor and elastic constants. Green and others placed the elastic energy on a firm thermodynamic footing, and the theory became central to nineteenth-century engineering.
Key figures
- Robert Hooke
- Augustin-Louis Cauchy
- Claude-Louis Navier
- George Green
Related topics
Seminal works
- landauelasticity1986
- timoshenko1970
Frequently asked questions
- What is the difference between stress and strain?
- Strain is the dimensionless measure of how much a material deforms, the relative change in length or shape, while stress is the internal force per unit area that the material develops in response; elasticity relates the two.
- Why are two elastic constants enough for an isotropic material?
- Isotropy means the material responds identically in all directions, which constrains the general elastic tensor down to two independent constants, commonly taken as Young's modulus and Poisson's ratio or the shear and bulk moduli.