1 Fundamental concepts
Linear elasticity is a framework in solid mechanics for describing how a material deforms under load and returns to its original shape after the load is removed. It applies when the deformations are small enough that geometric changes can be neglected and the material response remains proportional to the applied forces.
1.1 Stress
Stress is the internal force per unit area that develops within a body in response to external loading. It measures how strongly different parts of a material act on one another and is commonly resolved into normal and shear components. In linear elasticity, stress is treated as a continuous field that varies from point to point inside the body.
1.2 Strain
Strain describes the change in shape or size of a material caused by stress. It is a dimensionless measure of deformation, often expressed as relative extension, angular distortion, or volumetric change. For small deformations, strain can be approximated by linearized expressions that simplify analysis.
1.3 Small-deformation assumption
The small-deformation assumption means that displacements, rotations, and strains are sufficiently small that terms of higher order can be ignored. Under this assumption, the original and deformed configurations are nearly identical for analytical purposes. This simplification allows the governing equations to remain linear and widely solvable.
1.4 Elastic recovery
Elastic recovery is the return of a body to its original shape after the applied load is removed. In a purely elastic material, the deformation is reversible as long as the stress remains within the elastic range. If the loading is too large, permanent deformation may occur and the linear elastic model no longer applies well.
2 Constitutive relations
Constitutive relations connect stress and strain through material properties. They specify how a particular solid responds mechanically and are central to turning general balance laws into usable equations for a specific material.
2.1 Hooke's law
Hooke's law states that, within the elastic range, stress is proportional to strain. In its simplest form, it is the one-dimensional relation between force-induced extension and applied load. In higher dimensions, it becomes a tensor relation linking the full stress and strain fields through elastic constants.
2.2 Elastic constants
Elastic constants are numerical parameters that characterize the stiffness and deformation behavior of a material. They summarize how the material resists stretching, compression, and shear. Different constants are convenient for different loading situations, but they are mathematically related for a given material model.
2.2.1 Young's modulus
Young's modulus measures resistance to uniaxial stretching or compression. A larger value indicates a stiffer material that deforms less under the same axial stress. It is one of the most commonly used indicators of material stiffness.
2.2.2 Poisson's ratio
Poisson's ratio describes the lateral contraction that accompanies axial stretching, or the lateral expansion that accompanies compression. It expresses the coupling between longitudinal and transverse deformation. For many common solids, it lies between 0 and 0.5 in linear elastic models.
2.2.3 Shear modulus
The shear modulus quantifies resistance to shape change under shear stress. It governs how strongly a material resists sliding deformation between adjacent layers. Materials with a high shear modulus are comparatively difficult to distort angularly.
2.2.4 Bulk modulus
The bulk modulus measures resistance to uniform compression. It relates pressure to the resulting volumetric strain. A large bulk modulus indicates that the material is hard to compress.
2.3 Isotropic elasticity
Isotropic elasticity assumes that material properties are the same in every direction. This symmetry greatly reduces the number of independent constants needed to describe the material. Many engineering calculations use isotropic models because they provide a simple and effective approximation for metals, polymers, and other homogeneous solids.
2.4 Anisotropic elasticity
Anisotropic elasticity describes materials whose properties depend on direction. Such behavior is common in crystals, composites, and layered structures. The constitutive description is more complex than in isotropic elasticity because different directions can respond differently to the same loading.
3 Mathematical formulation
The mathematical formulation of linear elasticity combines kinematics, constitutive laws, and balance equations. Together, these relations determine the displacement, strain, and stress fields inside an elastic body.
3.1 Tensor notation
Tensor notation provides a compact and systematic way to express the equations of elasticity in multiple dimensions. It is especially useful for writing stress, strain, and material laws in coordinate-independent form. This notation helps capture the full structure of the theory without relying on separate component equations for each direction.
3.2 Strain-displacement relations
Strain-displacement relations connect the deformation of a body to its displacement field. Under the small-deformation assumption, strain is obtained from spatial derivatives of displacement. These relations express how relative motion between nearby points produces local stretching and shearing.
3.3 Stress-strain relations
Stress-strain relations specify how the computed strain produces stress through the material’s elastic response. In linear elasticity, these relations are linear and reversible. They form the constitutive core of the theory and are combined with equilibrium equations to solve boundary-value problems.
3.4 Equilibrium equations
Equilibrium equations represent the balance of forces and moments within the body. They require that internal stresses, body forces, and applied loads be consistent with static or dynamic balance. In the static case, these equations ensure that the material is neither accelerating nor experiencing unbalanced internal force.
3.5 Boundary conditions
Boundary conditions describe how the body interacts with its surroundings at its surface. They may prescribe displacements, applied tractions, or a combination of both. Correct boundary conditions are essential for obtaining a unique and physically meaningful solution.
4 Special cases and simplifications
Many practical problems can be reduced to simpler forms of the general theory. These special cases preserve the basic assumptions of linear elasticity while lowering the mathematical complexity.
4.1 One-dimensional elasticity
One-dimensional elasticity treats deformation along a single axis. It is useful for slender bars, rods, and simple test specimens. The governing relations reduce to straightforward proportionality between axial stress, strain, and extension.
4.2 Plane stress
Plane stress applies to thin structures in which stresses normal to the plane are negligible. It is commonly used for plates and sheet-like components. The in-plane stress components dominate, while through-thickness stress is assumed to vanish or remain very small.
4.3 Plane strain
Plane strain describes situations where deformation in one direction is negligible compared with the other two. This approximation is appropriate for long bodies or cross-sectional analyses of structures that are effectively uniform along one axis. The strain in the suppressed direction is taken to be zero, although stress in that direction may still be present.
4.4 Axisymmetric problems
Axisymmetric problems involve bodies and loads that are symmetric around an axis of revolution. Because the geometry and loading do not vary with angular position, the three-dimensional problem can often be reduced to a two-dimensional formulation. This simplification is widely used for pressure vessels, rotating parts, and circular structures.
5 Energy methods
Energy methods provide alternative tools for analyzing elastic systems by focusing on stored mechanical energy rather than directly on force balance alone. They are useful for deriving equations, proving uniqueness, and approximating solutions.
5.1 Strain energy
Strain energy is the mechanical energy stored in a body due to elastic deformation. In linear elasticity, it is a quadratic function of strain and stress. The concept helps quantify how much work is absorbed by the material during loading.
5.2 Complementary energy
Complementary energy is an energy measure expressed in terms of stress rather than strain. It is particularly helpful when stress fields are easier to describe than displacement fields. In linear systems, complementary energy provides a dual formulation to strain energy.
5.3 Variational principles
Variational principles state that the actual elastic state of a system satisfies an extremum property of an energy functional. These principles offer elegant derivations of governing equations and support approximate methods such as finite elements. They also connect elasticity with broader methods in mathematical physics.
6 Solutions and analytical methods
Analytical methods seek closed-form or structured solutions to elasticity problems. Although many realistic geometries require numerical computation, classical techniques remain important for understanding fundamental behavior and benchmarking approximate methods.
6.1 Exact solutions
Exact solutions are fully analytical results obtained for idealized geometries, loading conditions, or symmetry classes. They provide insight into stress concentration, displacement patterns, and singular behavior. Such solutions are valuable as reference cases for more complex analyses.
6.2 Potential functions
Potential functions transform the elasticity equations into forms that are often easier to solve. By introducing auxiliary scalar or vector functions, one can reduce the number of unknowns or exploit harmonic properties. These methods are especially useful in two-dimensional and axisymmetric problems.
6.3 Fourier methods
Fourier methods represent displacement, stress, or load distributions as sums or integrals of sinusoidal components. They are effective for problems with regular geometry or periodic boundary conditions. The decomposition can simplify partial differential equations into more manageable forms.
6.4 Green's functions
Green's functions describe the response of an elastic medium to a localized unit load. Once known, they can be used to build solutions for more general force distributions by superposition. They are central to many analytical and semi-analytical approaches in elasticity.
7 Applications
Linear elasticity is a foundational tool in the analysis and design of solids and structures. Its simplicity and broad validity make it an important first model in many technical fields.
7.1 Structural analysis
In structural analysis, linear elasticity is used to estimate stresses, deflections, and load paths in beams, frames, plates, and shells. Engineers rely on it to ensure that components remain within safe deformation limits. It also provides a baseline for more advanced nonlinear assessments.
7.2 Mechanical engineering
Mechanical engineering uses linear elasticity to design machine parts, housings, fasteners, and load-bearing components. The theory helps predict stiffness, natural frequencies, and stress distribution under service loads. It is also essential in evaluating whether a part will function without excessive deformation.
7.3 Civil engineering
Civil engineering applies linear elasticity to buildings, bridges, pavements, and foundation elements. The model supports calculations of structural response under weight, wind, and other ordinary loads. It is particularly useful in preliminary design and in the analysis of serviceability.
7.4 Material characterization
Material characterization uses elastic response to estimate material properties from experiments such as tensile tests, compression tests, and vibration measurements. The resulting constants help identify stiffness, compressibility, and directional dependence. These measurements are important for quality control and model calibration.
8 Limitations and extensions
Linear elasticity is powerful but idealized. It works best for small deformations and reversible behavior, and it must be extended when real materials or loading conditions depart from those assumptions.
8.1 Nonlinear elasticity
Nonlinear elasticity applies when the stress-strain relationship is not proportional or when deformation becomes large enough that geometry changes matter. It is used for materials and situations where the linear approximation is no longer adequate. The equations are typically more difficult to solve and may require numerical methods.
8.2 Plasticity
Plasticity describes permanent deformation that remains after unloading. Once a material yields, the response is no longer fully recoverable, and linear elastic theory can only represent the initial loading stage. Plastic models are essential for metals and other materials subjected to high stresses.
8.3 Viscoelasticity
Viscoelasticity combines elastic recovery with time-dependent deformation. In such materials, the response depends not only on the current load but also on the loading history. This behavior is common in polymers, biological tissues, and other rate-sensitive solids.
8.4 Large-deformation theory
Large-deformation theory treats situations in which displacements, rotations, or strains are no longer small. The kinematics become nonlinear, and the distinction between reference and current configurations becomes important. This extension is necessary for highly flexible structures, soft matter, and strongly loaded components.