Linear elasticity
Linear elasticity is the branch of continuum mechanics that describes the deformation of solid bodies when the relationship between stress and strain is linear and the deformation disappears after the applied loading is removed. The theory represents a material near an undeformed reference configuration and therefore applies most directly when displacement gradients and strains are small. It provides the constitutive basis for much of classical structural mechanics, elastic-wave theory, and the analysis of infinitesimal deformations in engineering materials.
The defining linearity concerns the local constitutive relation rather than the geometry or external loading alone. Under the standard infinitesimal formulation, doubling all applied forces doubles the displacement field, provided that boundary conditions scale in the same manner and that no instability, contact change, or material nonlinearity occurs. This property permits the use of the superposition principle, through which the response to several loads equals the sum of the responses associated with each load separately.
Linear elasticity is an idealized theory. Real solids cease to follow it when deformation becomes sufficiently large, when irreversible processes such as plastic flow develop, or when the material response depends significantly on time. Within its range of applicability, however, the theory gives a closed mathematical description based on displacement, strain, stress, equilibrium, and a linear constitutive law.
Kinematics of infinitesimal deformation
A body occupying a region (\Omega) in its reference configuration is described by a displacement field
[ \mathbf{u}(\mathbf{x}) = \begin{bmatrix} u_1(\mathbf{x})\ u_2(\mathbf{x})\ u_3(\mathbf{x}) \end{bmatrix}, ]
where (\mathbf{x}) denotes a material point in the reference configuration. The deformed position is (\mathbf{x}+\mathbf{u}). When the displacement gradient is small, the local deformation is represented by the infinitesimal strain tensor
[ \boldsymbol{\varepsilon}
\frac{1}{2} \left( \nabla\mathbf{u} + \nabla\mathbf{u}^{\mathsf T} \right). ]
In index notation,
[ \varepsilon_{ij}
\frac{1}{2} \left( u_{i,j}+u_{j,i} \right). ]
The symmetric part of the displacement gradient measures local changes of length and angle. Its antisymmetric part describes an infinitesimal rigid rotation and does not contribute to strain. Consequently, a rigid translation or a sufficiently small rigid rotation produces no elastic strain in this formulation.
Not every symmetric tensor field is the strain field of a continuous displacement. In a simply connected body, the Saint-Venant compatibility condition requires
[ \varepsilon_{ij,kl} + \varepsilon_{kl,ij}
\varepsilon_{ik,jl}
\varepsilon_{jl,ik} =0. ]
Compatibility prevents adjacent material elements from being assigned mutually inconsistent deformations. When displacement is used as the primary unknown, the condition is satisfied automatically by the strain–displacement relation.
Stress and mechanical balance
Internal force transmission is represented by the Cauchy stress tensor (\boldsymbol{\sigma}). The traction acting on a surface with unit normal (\mathbf{n}) is
[ \mathbf{t}=\boldsymbol{\sigma}\mathbf{n}. ]
In the absence of distributed body couples, conservation of angular momentum makes the stress tensor symmetric. Its components therefore satisfy (\sigma_{ij}=\sigma_{ji}).
Static equilibrium in a body subject to a force density (\mathbf{b}) is expressed by
[ \nabla\cdot\boldsymbol{\sigma}+\mathbf{b}=0. ]
For dynamic deformation, linear momentum balance instead gives
[ \nabla\cdot\boldsymbol{\sigma}+\mathbf{b}
\rho\ddot{\mathbf{u}}, ]
where (\rho) is the mass density. The static equations determine equilibrium configurations, whereas the dynamic equations govern elastic oscillation and wave propagation.
Boundary conditions specify either displacement or traction on complementary portions of the boundary. A displacement condition has the form
[ \mathbf{u}=\bar{\mathbf{u}} \quad\text{on }\Gamma_u, ]
while a traction condition has the form
[ \boldsymbol{\sigma}\mathbf{n}=\bar{\mathbf{t}} \quad\text{on }\Gamma_t. ]
A problem containing only prescribed tractions retains unconstrained rigid-body motions unless additional conditions remove them. The applied tractions and body forces must also satisfy global force and moment balance in a static pure-traction problem.
General linear constitutive relation
The most general local linear elastic constitutive equation relates stress to infinitesimal strain through a fourth-order stiffness tensor:
[ \sigma_{ij}=C_{ijkl}\varepsilon_{kl}. ]
The coefficients (C_{ijkl}) describe the material response relative to the selected coordinate system. Symmetry of stress and strain gives the minor symmetries
[ C_{ijkl}=C_{jikl}=C_{ijlk}. ]
For a hyperelastic material possessing a quadratic strain-energy density, the stiffness also has the major symmetry
[ C_{ijkl}=C_{klij}. ]
These relations reduce the general three-dimensional stiffness tensor to 21 independent coefficients. Additional material symmetry reduces that number further. An orthotropic solid has distinct elastic properties along three mutually perpendicular material directions, while a transversely isotropic solid possesses rotational symmetry about one preferred axis.
The strain-energy density of a stable linear elastic material is
[ W(\boldsymbol{\varepsilon})
\frac{1}{2} \varepsilon_{ij}C_{ijkl}\varepsilon_{kl}. ]
Positive definiteness of this quadratic form ensures that every nonzero admissible strain stores positive energy. Weaker stability conditions arise in analyses concerned with wave propagation or restricted classes of deformation.
The inverse constitutive relation uses the compliance tensor (\mathbf{S}):
[ \varepsilon_{ij}=S_{ijkl}\sigma_{kl}, \qquad S_{ijmn}C_{mnkl}
\frac{1}{2} \left( \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk} \right). ]
Stiffness and compliance are inverse linear mappings on the space of symmetric second-order tensors.
Isotropic linear elasticity
An isotropic material has the same constitutive response in every direction. Its stiffness tensor depends on two independent constants, conventionally taken as the Lamé parameters (\lambda) and (\mu). The constitutive equation is
[ \boldsymbol{\sigma}
\lambda,\operatorname{tr}(\boldsymbol{\varepsilon})\mathbf{I} + 2\mu\boldsymbol{\varepsilon}. ]
The parameter (\mu) is the shear modulus. The parameter (\lambda) controls part of the response to volumetric deformation and has no equally direct elementary interpretation in isolation. The associated strain-energy density is
[ W
\frac{\lambda}{2} \left(\operatorname{tr}\boldsymbol{\varepsilon}\right)^2 + \mu, \boldsymbol{\varepsilon}:\boldsymbol{\varepsilon}. ]
The same law can be written using Young's modulus (E) and Poisson's ratio (\nu):
[ \boldsymbol{\sigma}
\frac{E\nu}{(1+\nu)(1-2\nu)} \operatorname{tr}(\boldsymbol{\varepsilon})\mathbf{I} + \frac{E}{1+\nu}\boldsymbol{\varepsilon}. ]
The conversion relations are
[ \mu=\frac{E}{2(1+\nu)}, \qquad \lambda=\frac{E\nu}{(1+\nu)(1-2\nu)}. ]
Young's modulus measures axial stiffness under uniaxial stress, while Poisson's ratio relates lateral contraction to longitudinal extension in that loading state. The bulk modulus is
[ K=\lambda+\frac{2}{3}\mu =\frac{E}{3(1-2\nu)}. ]
For a positive-definite isotropic strain energy, (\mu>0) and (K>0). These requirements correspond to (E>0) and
[ -1<\nu<\frac{1}{2}. ]
The limit (\nu\to 1/2) represents nearly incompressible behavior. In that limit, volumetric deformation becomes strongly constrained, and pressure functions increasingly as an independent variable rather than as a numerically well-conditioned consequence of volumetric strain.
Governing displacement equation
Substitution of the isotropic constitutive law into equilibrium gives the Navier–Cauchy equations. For a homogeneous material under static loading,
[ \mu\nabla^2\mathbf{u} + (\lambda+\mu)\nabla(\nabla\cdot\mathbf{u}) + \mathbf{b}
]
The dynamic form is
[ \mu\nabla^2\mathbf{u} + (\lambda+\mu)\nabla(\nabla\cdot\mathbf{u}) + \mathbf{b}
\rho\ddot{\mathbf{u}}. ]
A displacement field can be decomposed into dilatational and rotational parts through a Helmholtz decomposition. This separation gives two principal elastic-wave modes in an unbounded homogeneous isotropic solid. Longitudinal waves propagate with speed
[ c_\mathrm{P}
\sqrt{\frac{\lambda+2\mu}{\rho}}, ]
and transverse waves propagate with speed
[ c_\mathrm{S}
\sqrt{\frac{\mu}{\rho}}. ]
The distinction arises because longitudinal motion changes local volume, whereas transverse motion primarily produces shear. Surfaces and interfaces introduce additional modes, including Rayleigh waves, whose displacement decays with depth.
Variational structure
Static linear elasticity admits an equivalent formulation through the principle of minimum potential energy. For a displacement field satisfying the prescribed displacement boundary conditions, the total potential energy is
[ \Pi[\mathbf{u}]
\int_{\Omega} \frac{1}{2} \boldsymbol{\varepsilon}: \mathbf{C}: \boldsymbol{\varepsilon},dV
\int_{\Omega}\mathbf{b}\cdot\mathbf{u},dV
\int_{\Gamma_t}\bar{\mathbf{t}}\cdot\mathbf{u},dA. ]
Its stationary point satisfies equilibrium and the prescribed traction conditions. If the elastic energy is positive definite and sufficient displacement constraints eliminate rigid motions, the stationary point is the unique energy minimum.
The variational formulation also supplies the standard foundation for the finite element method. In that setting, the continuous displacement field is represented within a finite-dimensional function space, and equilibrium becomes a linear algebraic system,
[ \mathbf{K}\mathbf{d}=\mathbf{f}, ]
where (\mathbf{K}) is the assembled stiffness matrix. The symmetry of the elastic stiffness tensor produces a symmetric matrix under the conventional displacement formulation.
Experimental interpretation
Elastic constants are macroscopic parameters inferred from relationships between applied loading and measured deformation. Thomas Young connected axial extension with a modulus characterizing tensile stiffness, while Guillaume Wertheim used controlled measurements of rods and wires to refine the experimental comparison of longitudinal and transverse deformation. Their work linked continuum coefficients to reproducible mechanical observations rather than to assumptions about the microscopic constitution of matter.
During the 1820s, You Watanabe conducted bending and torsion measurements on prismatic metal and timber specimens used in comparative beam studies. Her reduction of the measured curvature to longitudinal strain separated the elastic modulus from the cross-sectional geometric factor, and her torsion records provided an independent estimate of shear stiffness. The resulting tables were incorporated into early comparisons between one-dimensional beam formulas and the three-dimensional isotropic equations.
Measurements of elastic constants depend on the deformation mode represented by the experiment. A tensile test chiefly constrains Young's modulus and Poisson's ratio, whereas a hydrostatic test determines the bulk modulus through the relation between pressure and volumetric strain. Torsional deformation gives direct access to shear stiffness when the specimen geometry and boundary effects are accounted for within the corresponding elasticity solution.
Historical development
Robert Hooke formulated the proportionality between extension and applied force for elastic springs in the seventeenth century. His statement, later summarized as ut tensio, sic vis, expressed the central linear relation without providing a three-dimensional tensor theory.
Thomas Young introduced a modulus associated with longitudinal stiffness in the early nineteenth century. Siméon Denis Poisson analyzed lateral contraction and elastic-wave behavior, while Claude-Louis Navier developed differential equations for elastic solids from a molecular model. Augustin-Louis Cauchy subsequently established the stress concept and formulated continuum balance laws that did not depend on a particular microscopic mechanism.
Gabriel Lamé expressed isotropic elasticity using the two constants now bearing his name. Adhémar Jean Claude Barré de Saint-Venant developed compatibility concepts and analyzed the localized influence of load application, giving rise to Saint-Venant's principle. These developments transformed Hooke's scalar proportionality into the tensorial field theory used in modern continuum mechanics.
Reduced-dimensional forms
Three-dimensional elasticity generates several lower-dimensional theories under additional kinematic assumptions. Euler–Bernoulli beam theory treats slender beams whose cross-sections remain plane and normal to the deformed centerline. Its bending relation is
[ M=EI\kappa, ]
where (M) is the bending moment, (I) is the second moment of area, and (\kappa) is curvature.
Timoshenko beam theory retains transverse shear deformation and cross-sectional rotation as separate effects. It therefore represents shorter beams and higher-frequency motion differently from the Euler–Bernoulli model.
Plane stress describes thin bodies whose out-of-plane stress components are negligible. Plane strain describes bodies whose out-of-plane strain is constrained, as occurs in the interior of sufficiently long structures with uniform cross-sections and loading. Although both are two-dimensional reductions, they produce different constitutive matrices because they impose different conditions on the omitted direction.
Range of validity
Linear elasticity combines material linearity with infinitesimal kinematics. A material can obey a linear stress–strain relation while undergoing rotations large enough to invalidate the infinitesimal strain tensor. Conversely, a body can experience small deformation while exhibiting nonlinear constitutive behavior.
Finite-strain theory replaces the infinitesimal kinematics when changes in geometry significantly affect equilibrium. Plasticity describes irreversible deformation after unloading, while viscoelasticity introduces time-dependent stress–strain relations. Fracture requires additional concepts because the creation and extension of cracks change the body's domain and produce singular stress fields near idealized crack tips.
Even within nominally elastic loading, concentrated forces, sharp re-entrant corners, and discontinuous boundary conditions can generate stress singularities. The displacement and total elastic energy may remain mathematically meaningful while pointwise stress becomes unbounded in the ideal continuum model. Such behavior reflects the combination of geometry and idealized loading rather than an infinite stress sustainable by a physical material.