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Linear elasticity

Linear elasticity is a branch of continuum mechanics that models the deformation of solid materials subjected to small external forces, assuming that the stress within the material is linearly proportional to the resulting strain and that the material returns to its original undeformed state upon removal of the forces.^[1] This theory applies to scenarios where deformations are infinitesimal, neglecting higher-order effects such as geometric nonlinearities or material plasticity, and is foundational for analyzing structures like beams, plates, and trusses in engineering applications.^[2] The historical development of linear elasticity traces back to the 17th century, with Robert Hooke formulating the basic proportionality between force and elongation in 1678 through experiments on springs, encapsulated in his anagram "Ut tensio, sic vis" (as the extension, so the force).^[3] Key milestones followed, including Thomas Young's introduction and popularization of the modulus of elasticity in 1807, which quantifies the stiffness of materials under uniaxial stress, and Charles-Augustin de Coulomb's measurements of shear modulus in 1780.^[3] By the 19th century, the theory was formalized with contributions from Siméon Denis Poisson, who predicted a lateral contraction ratio (now Poisson's ratio) of 0.25 for isotropic materials, though experiments by Guillaume Wertheim in 1848 confirmed its variability.^[3] These advancements culminated in the general three-dimensional framework by Claude-Louis Navier and others in the early 1800s, establishing the linear stress-strain relations still used today.^[4] At its core, linear elasticity relies on the displacement field, which describes how points in the solid move from their reference positions, leading to the symmetric strain tensor that captures infinitesimal deformations.^[1] The constitutive relation, often expressed via Hooke's law in generalized form, links the Cauchy stress tensor \sigma to the strain tensor \varepsilon for isotropic materials as \sigma = \lambda (\operatorname{tr} \varepsilon) I + 2\mu \varepsilon, where \lambda and \mu are the Lamé parameters determining compressibility and shear rigidity, respectively.^[2] Equilibrium is governed by the divergence-free condition \nabla \cdot \sigma + f = 0, where f represents body forces, enabling solutions through methods like finite element analysis for complex geometries.^[1] This framework assumes material homogeneity and isotropy unless specified otherwise, with two independent constants—such as Young's modulus E (typically $10^{11} Pa for metals) and Poisson's ratio \nu (around 0.3 for many solids)—sufficient to characterize behavior.^[2]

Fundamentals

Stress and Strain Tensors

The Cauchy stress tensor, denoted by \boldsymbol{\sigma}, is a second-order tensor that describes the state of stress at a point within a deformable solid, representing the internal forces per unit area acting on an infinitesimal surface element in the current (deformed) configuration.^[5] It was introduced by Augustin-Louis Cauchy in his foundational work on continuum mechanics.^[6] The tensor relates the traction vector \mathbf{t}, which is the force per unit area on a surface with outward unit normal \mathbf{n}, via the relation \mathbf{t} = \boldsymbol{\sigma} \mathbf{n}. In component form, this is t_i = \sigma_{ij} n_j, where the indices follow the Einstein summation convention. The components \sigma_{ij} consist of normal stresses (i = j), which act perpendicular to the surface and measure direct compressive or tensile forces per unit area, and shear stresses (i \neq j), which act parallel to the surface and represent tangential forces per unit area that cause sliding between material layers.^[5] The normal stress on a surface is given by \sigma_N = \mathbf{t} \cdot \mathbf{n}, while the shear stress magnitude is \sigma_S = \sqrt{|\mathbf{t}|^2 - \sigma_N^2}.^[5] Due to the symmetry of internal forces (from angular momentum balance), the Cauchy stress tensor is symmetric, \sigma_{ij} = \sigma_{ji}, reducing the number of independent components to six in three dimensions. This tensorial representation allows for a complete description of the stress state independent of the coordinate system, essential for analyzing deformations in solids under various loading conditions. In the context of linear elasticity, the Cauchy stress tensor is evaluated in the reference configuration under the assumption of small deformations, where the distinction between reference and current areas is negligible.^[5] The infinitesimal strain tensor, denoted by \boldsymbol{\varepsilon}, quantifies the small deformations of a solid relative to its undeformed reference configuration and is derived from the displacement field \mathbf{u}. It is defined as the symmetric part of the displacement gradient:

\boldsymbol{\varepsilon} = \frac{1}{2} \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^T \right),

or in components, \varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right), where \mathbf{x} are the material coordinates in the reference state.^[1] This form arises from the linear approximation of the Green-Lagrange strain tensor for small displacements, neglecting higher-order quadratic terms in \nabla \mathbf{u} when \|\nabla \mathbf{u}\| \ll 1. The tensor is symmetric by construction, ensuring it captures only the deformative (non-rigid) part of the motion, excluding pure rotations.^[1] Geometrically, the diagonal components \varepsilon_{ii} (no sum) represent normal strains, measuring the relative extension or contraction along the coordinate directions: for instance, \varepsilon_{11} is the engineering strain in the x_1-direction, approximately half the change in squared length of a material line element aligned with that axis. The off-diagonal components \varepsilon_{ij} ( i \neq j ) describe shear strains, which quantify the change in angle between two originally orthogonal material line elements in the i-j plane; specifically, $2\varepsilon_{ij} is the decrease in the right angle between them. The trace \operatorname{tr}(\boldsymbol{\varepsilon}) = \varepsilon_{kk} = \nabla \cdot \mathbf{u} corresponds to the volumetric strain, representing the relative change in volume under small deformations. In linear elasticity, these strain components relate the deformation in material coordinates to the stress tensor through a linear constitutive relation.^[1]

Kinematic and Constitutive Assumptions

Linear elasticity relies on several key kinematic and constitutive assumptions that simplify the mathematical description of deformable solids. The primary kinematic assumption is that deformations are infinitesimal, meaning the displacement gradients are small enough that higher-order terms can be neglected, leading to the linear strain tensor ε as the measure of deformation. This approximation holds when strains are much less than unity, typically on the order of 0.01 or smaller, ensuring geometric linearity. Constitutively, the material response is assumed to be linear, with stress σ directly proportional to strain ε, implying a reversible and path-independent deformation process without hysteresis or permanent set. For static problems, the response is time-independent, focusing on equilibrium states rather than rate-dependent effects like viscoelasticity. The constitutive relation, known as the generalized Hooke's law, links the second-order stress tensor σ to the second-order strain tensor ε through a fourth-order stiffness tensor C:

\boldsymbol{\sigma} = \mathbf{C} : \boldsymbol{\varepsilon}

This tensorial equation captures the anisotropic material behavior, where C has up to 21 independent components due to its symmetries (major symmetry from thermodynamic considerations and minor symmetries from stress-strain reciprocity). The compliance tensor S, defined as the inverse of the stiffness tensor S = C^{-1}, allows expressing strain in terms of stress:

\boldsymbol{\varepsilon} = \mathbf{S} : \boldsymbol{\sigma}

These relations assume hyperelasticity, where the work done during deformation is stored as recoverable strain energy. The strain energy density function for linear elasticity is quadratic in strain:

w = \frac{1}{2} \boldsymbol{\varepsilon} : \mathbf{C} : \boldsymbol{\varepsilon}

This form ensures the material returns to its original configuration upon unloading, with the stress derivable as the derivative of w with respect to ε. For isotropic materials, where properties are direction-independent, the stiffness tensor reduces to two independent parameters, often expressed through engineering constants. Young's modulus E quantifies uniaxial stiffness, defined as the ratio of axial stress to axial strain, while Poisson's ratio ν measures the lateral contraction under axial extension, typically ranging from 0.2 to 0.5 for most solids. The shear modulus μ (or G) relates shear stress to shear strain, and the bulk modulus K describes volumetric response. These are interrelated by:

E = 2\mu (1 + \nu), \quad K = \frac{E}{3(1 - 2\nu)}

These relations stem from the isotropic form of Hooke's law, enabling characterization with just E and ν in practice. The foundational idea of proportionality between stress and strain traces back to Robert Hooke's 1678 anagram "ut tensio, sic vis," empirically stating that extension is proportional to applied force for springs and solids. This was mathematically formalized in the 1820s: Claude-Louis Navier derived the general equations of elasticity in 1821 assuming molecular interactions, while Augustin-Louis Cauchy established the tensorial framework in 1822, introducing the stress concept and linear constitutive relations for continuous media.

General Mathematical Formulation

Tensor-Based Equations

The tensor-based formulation of linear elasticity provides a compact and coordinate-independent description of the governing equations, integrating the kinematics of deformation, the constitutive relations between stress and strain, and the balance laws of mechanics. This approach leverages direct tensor notation to express the fundamental relations in three-dimensional Euclidean space, applicable to both static and dynamic problems in continuous media. The equations assume small deformations, linear material response, and neglect higher-order effects such as inertia in static cases unless specified. The kinematic relation defines the infinitesimal strain tensor \boldsymbol{\varepsilon} as the symmetric part of the displacement gradient tensor \nabla \mathbf{u}, given by

\boldsymbol{\varepsilon} = \sym(\nabla \mathbf{u}) = \frac{1}{2} \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^T \right),

where \mathbf{u} is the displacement vector field, and the superscript T denotes the transpose. This ensures that \boldsymbol{\varepsilon} is symmetric (\boldsymbol{\varepsilon}^T = \boldsymbol{\varepsilon}) and captures the symmetric deformation components, excluding rigid-body rotations.^[7] The constitutive equation links the Cauchy stress tensor \boldsymbol{\sigma} to the strain tensor \boldsymbol{\varepsilon} through a linear relation. For general anisotropic materials, this is expressed as

\boldsymbol{\sigma} = \mathbb{C} : \boldsymbol{\varepsilon},

where \mathbb{C} is the fourth-order elasticity tensor (stiffness tensor) with up to 21 independent components due to symmetries, and the double contraction : denotes the tensor inner product. For isotropic materials, the relation simplifies to

\boldsymbol{\sigma} = \lambda (\tr \boldsymbol{\varepsilon}) \mathbf{I} + 2\mu \boldsymbol{\varepsilon},

using the Lamé constants \lambda and \mu (with \mu being the shear modulus), where \tr is the trace and \mathbf{I} is the identity tensor. This form reflects the material's invariance under rotations and decomposes the stress into volumetric and deviatoric parts.^[8]^[1] The momentum balance equation governs the equilibrium of the body under applied forces. In the static case (no acceleration), it reads

\nabla \cdot \boldsymbol{\sigma} + \mathbf{b} = \mathbf{0},

where \mathbf{b} is the body force density per unit volume, and \nabla \cdot is the divergence operator. For dynamic problems, inertia is included via

\rho \ddot{\mathbf{u}} = \nabla \cdot \boldsymbol{\sigma} + \mathbf{b},

with \rho the mass density and \ddot{\mathbf{u}} the second time derivative of displacement (acceleration). These equations, combined with the kinematic and constitutive relations, form the complete system for linear elastodynamics.^[9]^[10] For the strain field \boldsymbol{\varepsilon} to correspond to a single-valued continuous displacement \mathbf{u} in a simply connected domain, it must satisfy the compatibility condition

\nabla \times (\nabla \times \boldsymbol{\varepsilon}) = \mathbf{0}.

This Saint-Venant compatibility equation ensures integrability of the kinematic relation, preventing gaps or overlaps in the deformed configuration. In multiply connected domains, additional cut-surface conditions may apply, but the curl-curl form suffices for most applications.^[11] Boundary conditions complete the formulation by specifying either kinematic or traction constraints on the surface \partial \Omega. Essential (Dirichlet) conditions prescribe the displacement \mathbf{u} = \tilde{\mathbf{u}} on \partial \Omega_u, while natural (Neumann) conditions specify the surface traction \hat{\mathbf{t}} = \boldsymbol{\sigma} \cdot \mathbf{n} on \partial \Omega_t, where \mathbf{n} is the outward unit normal. Mixed conditions can combine both on respective partitions of the boundary, ensuring well-posedness of the boundary-value problem.^[12]

Equilibrium and Boundary Conditions

In linear elasticity, boundary value problems are formulated either in strong or weak forms. The strong form consists of the governing partial differential equations, such as the equilibrium equations and constitutive relations, along with explicit boundary conditions, requiring solutions that are sufficiently smooth (typically twice differentiable).^[13] In contrast, the weak form integrates these equations over the domain using test functions, reducing the smoothness requirements to once differentiable functions and naturally incorporating boundary conditions through integration by parts.^[13] This formulation is foundational for numerical methods like the finite element method, where the Galerkin approach approximates the solution by projecting the weak form onto a finite-dimensional subspace spanned by basis functions, leading to a system of algebraic equations.^[13] The principle of virtual work provides the weak formulation for static equilibrium in linear elasticity. It states that for a body in equilibrium under applied body forces \mathbf{b} and surface tractions \hat{\mathbf{t}}, the internal virtual work done by the stress tensor \boldsymbol{\sigma} through a compatible virtual strain \delta \boldsymbol{\varepsilon} equals the external virtual work done by the applied loads through a compatible virtual displacement \delta \mathbf{u}. Mathematically, this is expressed as

\int_V \boldsymbol{\sigma} : \delta \boldsymbol{\varepsilon} \, dV = \int_V \mathbf{b} \cdot \delta \mathbf{u} \, dV + \int_{S_t} \hat{\mathbf{t}} \cdot \delta \mathbf{u} \, dS,

where V is the volume of the body and S_t is the traction boundary.^[14] This principle derives from the conservation of energy and is equivalent to the strong form of the equilibrium equations under suitable regularity conditions.^[15] Betti's reciprocity theorem extends these ideas to linear elastic systems under multiple load sets. For two independent equilibrium states with displacements \mathbf{u}_1, \mathbf{u}_2 and corresponding body forces \mathbf{b}_1, \mathbf{b}_2 plus surface tractions \hat{\mathbf{t}}_1, \hat{\mathbf{t}}_2, the theorem asserts that the work done by the forces of the first state through the displacements of the second equals the work done by the forces of the second through the displacements of the first: \int_V \mathbf{b}_1 \cdot \mathbf{u}_2 \, dV + \int_{S_t} \hat{\mathbf{t}}_1 \cdot \mathbf{u}_2 \, dS = \int_V \mathbf{b}_2 \cdot \mathbf{u}_1 \, dV + \int_{S_t} \hat{\mathbf{t}}_2 \cdot \mathbf{u}_1 \, dS.^[16] Originally proved by Enrico Betti in 1872, this reciprocity relation holds for any linear elastic body and facilitates indirect solution methods, such as influence functions in structural analysis.^[16] Boundary conditions in linear elasticity problems are typically mixed, combining Dirichlet and Neumann types. Dirichlet (essential) boundary conditions prescribe the displacement \mathbf{u} = \bar{\mathbf{u}} on a portion S_u of the boundary, enforcing kinematic constraints like fixed supports.^[17] Neumann (natural) boundary conditions specify the traction \boldsymbol{\sigma} \cdot \mathbf{n} = \hat{\mathbf{t}} on the complementary portion S_t, where \mathbf{n} is the outward normal, representing applied forces or pressures.^[17] These are incorporated into the weak form, with Dirichlet conditions imposed on the trial functions and Neumann conditions appearing in the external work term. Uniqueness and existence of solutions for these boundary value problems are guaranteed under the positive definiteness of the elasticity tensor \mathbf{C}, which ensures the strain energy is strictly convex. Kirchhoff's theorem establishes that, for a homogeneous isotropic body with positive definite \mathbf{C}, the solution to the mixed boundary value problem is unique up to rigid body motions, which are eliminated by sufficient Dirichlet conditions.^[18] Existence follows from the Lax-Milgram theorem in the Hilbert space of admissible functions, provided the load functionals are continuous and \mathbf{C} satisfies the strong ellipticity condition \boldsymbol{\xi} : \mathbf{C} : \boldsymbol{\xi} > 0 for all nonzero symmetric tensors \boldsymbol{\xi}.^[18] These results underpin the well-posedness of linear elasticity formulations.^[18]

Representations in Coordinate Systems

Cartesian Coordinates

In Cartesian coordinates, the general tensor equations of linear elasticity simplify to component forms that are particularly convenient for problems involving rectangular geometries or prismatic bodies, where the orthogonal basis aligns naturally with the domain boundaries. This representation expresses displacements, strains, and stresses in terms of partial derivatives with respect to the Cartesian variables x_1, x_2, x_3 (often denoted as x, y, z), facilitating analytical and numerical solutions for bounded domains without curvature.^[19] The infinitesimal strain tensor in Cartesian coordinates arises from the symmetric part of the displacement gradient, assuming small deformations where higher-order terms are negligible. The components are given by

\varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right),

where \mathbf{u} = (u_1, u_2, u_3) is the displacement vector. For example, the normal strain in the x-direction is \varepsilon_{xx} = \partial u_x / \partial x, while the shear strain \varepsilon_{xy} = \frac{1}{2} (\partial u_x / \partial y + \partial u_y / \partial x). This relation ensures kinematic compatibility for continuous displacements in the elastic body.^[19] For isotropic materials, the linear stress-strain relation, known as Hooke's law in component form, links the stress tensor \sigma_{ij} to the strain tensor via the Lamé constants \lambda and \mu, where \mu is the shear modulus and \lambda relates to bulk compressibility. The explicit components are

\sigma_{xx} = \lambda \theta + 2\mu \varepsilon_{xx}, \quad \sigma_{yy} = \lambda \theta + 2\mu \varepsilon_{yy}, \quad \sigma_{zz} = \lambda \theta + 2\mu \varepsilon_{zz},

\sigma_{xy} = 2\mu \varepsilon_{xy}, \quad \sigma_{yz} = 2\mu \varepsilon_{yz}, \quad \sigma_{zx} = 2\mu \varepsilon_{zx},

with the dilatation \theta = \varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz} = \nabla \cdot \mathbf{u}. These equations assume material homogeneity and isotropy, where elastic properties are direction-independent, and \lambda, \mu > 0 for stability.^[20] The equilibrium equations in Cartesian coordinates balance forces on an infinitesimal element, incorporating body forces \mathbf{b} = (b_1, b_2, b_3). For static cases, they read

\frac{\partial \sigma_{ij}}{\partial x_j} + b_i = 0, \quad i = 1,2,3,

or in expanded form, \partial \sigma_{xx}/\partial x + \partial \sigma_{xy}/\partial y + \partial \sigma_{xz}/\partial z + b_x = 0, and similarly for the other components.^[21] In dynamic cases, inertia terms are included as

\rho \ddot{u}_i = \frac{\partial \sigma_{ij}}{\partial x_j} + b_i,

where \rho is the mass density and \ddot{u}_i denotes the second time derivative of displacement. These derive from Cauchy's fundamental balance laws under linear approximations.^[21] Substituting the isotropic constitutive relations into the static equilibrium equations yields Navier's equation for the displacement field:

\mu \nabla^2 \mathbf{u} + (\lambda + \mu) \nabla (\nabla \cdot \mathbf{u}) + \mathbf{b} = 0.

In component form, for the x-direction, this becomes \mu \nabla^2 u_x + (\lambda + \mu) \partial \theta / \partial x + b_x = 0, with analogous expressions for y and z. This vector equation, originally derived by Navier in 1821, governs elastostatics in homogeneous isotropic media and simplifies boundary value problems by eliminating stress variables.^[22] A common application in Cartesian coordinates involves two-dimensional approximations for thin or long bodies. In plane strain, the displacement in the z-direction is zero (u_z = 0), implying \varepsilon_{zz} = \varepsilon_{xz} = \varepsilon_{yz} = 0 and \sigma_{zz} = \nu (\sigma_{xx} + \sigma_{yy}), suitable for thick cylinders or dams where out-of-plane strain is constrained. Conversely, plane stress assumes \sigma_{zz} = \sigma_{xz} = \sigma_{yz} = 0, leading to \varepsilon_{zz} = -\nu (\varepsilon_{xx} + \varepsilon_{yy}) / (1 - \nu), applicable to thin plates like aircraft skins where through-thickness stresses are negligible. These assumptions reduce the 3D problem to 2D, with effective moduli adjusted accordingly, but plane strain overestimates stiffness compared to plane stress for the same in-plane loading.

Cylindrical and Spherical Coordinates

In linear elasticity, cylindrical and spherical coordinate systems are particularly useful for problems exhibiting rotational symmetry, such as those involving pipes, tubes, or spherical inclusions, where the geometry aligns with the coordinate axes to simplify the governing equations compared to Cartesian coordinates.^[4] These systems incorporate scale factors arising from the curvilinear metric, which modify the expressions for strain-displacement relations and equilibrium compared to the flat Cartesian metric.^[23] For cylindrical coordinates (r, \theta, z), the infinitesimal strain components in terms of the displacement field \mathbf{u} = (u_r, u_\theta, u_z) are given by:

\begin{align} \varepsilon_{rr} &= \frac{\partial u_r}{\partial r}, \\ \varepsilon_{\theta\theta} &= \frac{1}{r} \frac{\partial u_\theta}{\partial \theta} + \frac{u_r}{r}, \\ \varepsilon_{zz} &= \frac{\partial u_z}{\partial z}, \\ \varepsilon_{r\theta} &= \frac{1}{2} \left( \frac{\partial u_\theta}{\partial r} - \frac{u_\theta}{r} + \frac{1}{r} \frac{\partial u_r}{\partial \theta} \right), \\ \varepsilon_{\theta z} &= \frac{1}{2} \left( \frac{1}{r} \frac{\partial u_z}{\partial \theta} + \frac{\partial u_\theta}{\partial z} \right), \\ \varepsilon_{zr} &= \frac{1}{2} \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right). \end{align}

The equilibrium equations, in the absence of body forces, take the form:

\begin{align} \frac{\partial \sigma_{rr}}{\partial r} + \frac{1}{r} \frac{\partial \sigma_{r\theta}}{\partial \theta} + \frac{\partial \sigma_{rz}}{\partial z} + \frac{\sigma_{rr} - \sigma_{\theta\theta}}{r} &= 0, \\ \frac{\partial \sigma_{r\theta}}{\partial r} + \frac{1}{r} \frac{\partial \sigma_{\theta\theta}}{\partial \theta} + \frac{\partial \sigma_{\theta z}}{\partial z} + \frac{2 \sigma_{r\theta}}{r} &= 0, \\ \frac{\partial \sigma_{rz}}{\partial r} + \frac{1}{r} \frac{\partial \sigma_{\theta z}}{\partial \theta} + \frac{\partial \sigma_{zz}}{\partial z} + \frac{\sigma_{rz}}{r} &= 0. \end{align}

For isotropic materials, the constitutive relations remain \sigma_{ij} = \lambda \varepsilon_{kk} \delta_{ij} + 2\mu \varepsilon_{ij}, where [\lambda](/page/Lambda) and [\mu](/page/MU) are the Lamé constants, unchanged from the tensor form but expressed in the local basis.^[24]^[4] These equations derive from tensor transformations of the Cartesian components, ensuring consistency across coordinate systems.^[23] In spherical coordinates (r, \theta, \phi), the strain-displacement relations for the displacement \mathbf{u} = (u_r, u_\theta, u_\phi) are:

\begin{align} \varepsilon_{rr} &= \frac{\partial u_r}{\partial r}, \\ \varepsilon_{\theta\theta} &= \frac{1}{r} \frac{\partial u_\theta}{\partial \theta} + \frac{u_r}{r}, \\ \varepsilon_{\phi\phi} &= \frac{1}{r \sin \theta} \frac{\partial u_\phi}{\partial \phi} + \frac{u_r + u_\theta \cot \theta}{r}, \\ \varepsilon_{r\theta} &= \frac{1}{2} \left( \frac{\partial u_\theta}{\partial r} - \frac{u_\theta}{r} + \frac{1}{r} \frac{\partial u_r}{\partial \theta} \right), \\ \varepsilon_{\theta\phi} &= \frac{1}{2} \left( \frac{1}{r \sin \theta} \frac{\partial u_\theta}{\partial \phi} + \frac{1}{r} \frac{\partial u_\phi}{\partial \theta} - \frac{u_\phi \cot \theta}{r} \right), \\ \varepsilon_{\phi r} &= \frac{1}{2} \left( \frac{1}{r \sin \theta} \frac{\partial u_r}{\partial \phi} + \frac{\partial u_\phi}{\partial r} - \frac{u_\phi}{r} \right). \end{align}

The radial equilibrium equation, neglecting body forces, simplifies to:

\frac{\partial \sigma_{rr}}{\partial r} + \frac{1}{r} \frac{\partial \sigma_{r\theta}}{\partial \theta} + \frac{1}{r \sin \theta} \frac{\partial \sigma_{r\phi}}{\partial \phi} + \frac{2\sigma_{rr} - \sigma_{\theta\theta} - \sigma_{\phi\phi}}{r} = 0,

with analogous forms for the angular directions involving additional cotangent terms.^[4] The isotropic constitutive equations again use [\lambda](/page/Lambda) and [\mu](/page/MU) in the local components, facilitating solutions for radially symmetric problems like spherical cavities or inclusions.^[24] These formulations find application in analyzing pressurized cylindrical vessels, where internal pressure induces hoop and radial stresses, and spherical inclusions in composites, which model particle-reinforced materials under uniform loading.^[25] The coordinate-adapted equations reduce computational complexity for axisymmetric cases, as established in classical treatments.

Isotropic Linear Elasticity

Elastostatics

Elastostatics addresses the static equilibrium of deformable bodies under the assumptions of linear isotropic elasticity, where time-dependent effects such as inertia are absent, and the focus is on solving boundary value problems that balance internal stresses with applied loads and body forces.^[26] These problems typically involve specifying displacements or tractions on the boundary of a domain, leading to either displacement-based or stress-based formulations that ensure compatibility and equilibrium.^[27] Analytical solutions are feasible for simple geometries and loading conditions, providing benchmarks for understanding stress and strain distributions, while more complex cases often require complementary numerical approaches.^[28] In the displacement formulation, the governing equation is Navier's equation of equilibrium, derived from the balance of linear momentum in the absence of inertial terms:

\mu \nabla^2 \mathbf{u} + (\lambda + \mu) \nabla (\nabla \cdot \mathbf{u}) + \mathbf{f} = \mathbf{0},

where \mathbf{u} is the displacement vector, \lambda and \mu are the Lamé constants, and \mathbf{f} represents body forces per unit volume.^[26] This partial differential equation is solved subject to boundary conditions, such as prescribed displacements \mathbf{u} = \overline{\mathbf{u}} on part of the boundary or tractions \boldsymbol{\sigma} \cdot \mathbf{n} = \overline{\mathbf{t}} on the remainder, where \boldsymbol{\sigma} is the Cauchy stress tensor and \mathbf{n} is the outward normal.^[26] For isotropic materials, the stresses and strains are related via the constitutive law \boldsymbol{\sigma} = \lambda (\nabla \cdot \mathbf{u}) \mathbf{I} + 2\mu \boldsymbol{\epsilon}, with \boldsymbol{\epsilon} as the infinitesimal strain tensor.^[27] The stress formulation employs the Beltrami-Michell equations, which express compatibility conditions in terms of stresses, ensuring that the strain field derived from stresses via the constitutive relations is integrable to a displacement field.^[27] In two dimensions, for plane problems without body forces, these reduce to a biharmonic equation for the Airy stress function \phi, where the stress components are \sigma_{xx} = \frac{\partial^2 \phi}{\partial y^2}, \sigma_{yy} = \frac{\partial^2 \phi}{\partial x^2}, and \sigma_{xy} = -\frac{\partial^2 \phi}{\partial x \partial y}, satisfying \nabla^4 \phi = 0.^[27] This approach is particularly useful when boundary conditions are specified in terms of tractions, as it directly incorporates equilibrium \nabla \cdot \boldsymbol{\sigma} + \mathbf{f} = \mathbf{0}.^[27] Classical analytical solutions illustrate key behaviors in elastostatics. For a uniaxial bar under axial loading, the uniform stress-strain relation holds as \sigma = E \epsilon, where E is the Young's modulus, assuming small deformations and neglecting lateral effects via Poisson's ratio in the full solution.^[26] In the thick-walled cylinder problem under internal and external pressures, Lame's solution provides the radial stress as \sigma_{rr} = A + \frac{B}{r^2} and hoop stress \sigma_{\theta\theta} = A - \frac{B}{r^2}, with constants A and B determined from boundary pressures at inner radius a and outer radius b.^[29] For torsion of a circular shaft, the Prandtl stress function \phi satisfies \nabla^2 \phi = -2\mu \theta in the cross-section, where \theta is the twist per unit length, yielding shear stresses \tau_{xz} = \frac{\partial \phi}{\partial y} and \tau_{yz} = -\frac{\partial \phi}{\partial x}, with the torque related to the integral of \phi over the domain.^[30] Saint-Venant's principle states that the effects of a self-equilibrated load applied over a small portion of the boundary diminish exponentially with distance from the loaded region in semi-infinite domains, allowing the detailed load distribution to be replaced by its resultant far away without significant error.^[31] This principle justifies approximate solutions in long bodies, such as beams or cylinders, where end effects are localized, and is rigorously supported by decay estimates in linear elasticity for both static and dynamic cases.^[31] Post-2000 developments have introduced hybrid analytical-numerical methods to handle complex geometries beyond classical solvability, combining exact solutions in subdomains with finite volume or finite element discretizations to enforce equilibrium and compatibility globally.^[28] These approaches, such as hybrid stress finite volume methods, improve accuracy and efficiency for irregular shapes by leveraging analytical insights where possible while resolving numerical challenges in heterogeneous or multiply connected domains.^[28]

Elastodynamics

Elastodynamics in isotropic linear elasticity deals with time-dependent deformations where inertial effects are significant, leading to wave propagation through the material. Unlike elastostatics, which assumes quasi-static conditions and results in elliptic partial differential equations, elastodynamics incorporates the second time derivative of displacement, yielding hyperbolic wave equations that describe how disturbances travel at finite speeds. This framework is essential for analyzing phenomena such as seismic waves, ultrasonic testing, and impact loading in solids.^[32] The fundamental governing equation in the displacement formulation, known as the Navier equation, expresses the balance between inertial forces, elastic restoring forces, and external body forces. For a homogeneous isotropic material with density \rho, Lamé parameters \lambda and \mu, displacement vector \mathbf{u}, and body force per unit volume \mathbf{b}, it is given by

\rho \ddot{\mathbf{u}} = (\lambda + 2\mu) \nabla (\nabla \cdot \mathbf{u}) - \mu \nabla \times (\nabla \times \mathbf{u}) + \mathbf{b},

where dots denote time derivatives. This equation can be equivalently written using the Laplacian as \rho \ddot{\mathbf{u}} = (\lambda + \mu) \nabla (\nabla \cdot \mathbf{u}) + \mu \nabla^2 \mathbf{u} + \mathbf{b}, highlighting the coupling of dilatational and shear modes.^[33]^[9] In the stress-based formulation, the dynamic equilibrium equation \nabla \cdot \boldsymbol{\sigma} + \mathbf{b} = \rho \ddot{\mathbf{u}} relates the stress tensor \boldsymbol{\sigma} to acceleration, while the strain \boldsymbol{\epsilon} must satisfy compatibility conditions \nabla \times (\nabla \times \boldsymbol{\epsilon}) = 0 to ensure a single-valued displacement field exists. For isotropic materials, the constitutive relation \boldsymbol{\sigma} = \lambda (\nabla \cdot \mathbf{u}) \mathbf{I} + 2\mu \boldsymbol{\epsilon} links stress and strain, completing the system. This approach is useful in problems where stress boundary conditions predominate, such as fracture mechanics.^[34]^[32] The Navier equation decomposes into uncoupled scalar wave equations for the dilatational (P-wave) and rotational (S-wave) potentials, revealing distinct propagation speeds. The longitudinal P-wave speed is c_p = \sqrt{(\lambda + 2\mu)/\rho}, which depends on both bulk and shear moduli and represents compressional motion. The transverse S-wave speed is c_s = \sqrt{\mu/\rho}, governed solely by the shear modulus and describing shear distortions; typically, c_p > c_s, with the ratio around \sqrt{3} for many materials like steel. These speeds determine how quickly elastic disturbances propagate, influencing energy dissipation and wave arrival times in applications like geophysics.^[35]^[36] A simplified one-dimensional case arises in the longitudinal vibration of thin elastic bars, where lateral contractions are neglected, leading to the wave equation \ddot{u} = c^2 \frac{\partial^2 u}{\partial x^2}. Here, u(x,t) is the axial displacement, and the wave speed c = \sqrt{E/\rho} uses Young's modulus E = \mu (3\lambda + 2\mu)/(\lambda + \mu). This model captures pulse propagation along the bar length, with solutions illustrating dispersionless travel in the long-wavelength limit.^[37] For an infinite domain, D'Alembert's solution provides the general form for the 1D wave equation: u(x,t) = \frac{1}{2} [f(x + ct) + f(x - ct)] + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s) \, ds, where f(x) and g(x) are initial displacement and velocity, respectively. This represents right- and left-propagating waves without distortion. At finite boundaries, such as fixed or free ends, waves reflect with phase changes: a fixed end inverts the displacement, while a free end preserves it, leading to standing waves; transmission across interfaces between materials involves amplitude ratios dependent on impedance mismatches Z = \rho c.^[38]^[39] In practice, pure elastic models often overestimate wave persistence due to energy dissipation; viscoelastic extensions incorporate damping via time-dependent moduli, such as the Kelvin-Voigt model \boldsymbol{\sigma} = \lambda \operatorname{tr}(\boldsymbol{\epsilon}) \mathbf{I} + 2\mu \boldsymbol{\epsilon} + \eta (\lambda_v \operatorname{tr}(\dot{\boldsymbol{\epsilon}}) \mathbf{I} + 2\mu_v \dot{\boldsymbol{\epsilon}}), where \eta is viscosity. These are increasingly applied in 2020s seismic modeling to simulate attenuation in heterogeneous media, improving predictions of earthquake ground motion.^[40]

Anisotropic Linear Elasticity

Statics in Crystals and Composites

In anisotropic linear elasticity, static equilibrium problems for crystalline materials and fiber-reinforced composites account for direction-dependent material properties, leading to more complex stress-strain relations than in isotropic media. Crystals exhibit symmetry dictated by their lattice structures, while composites derive anisotropy from oriented reinforcements like fibers in a matrix. Solutions often leverage reduced constitutive forms and variational principles to satisfy equilibrium equations ∇·σ = 0 and boundary conditions without inertial terms. A key practical tool is Voigt notation, which contracts the fourth-order stiffness tensor C_{ijkl} into a 6×6 matrix [C_{ij}] for the relation between stress and strain vectors:

\begin{pmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{13} \\ \sigma_{12} \end{pmatrix} = \begin{pmatrix} C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ C_{21} & C_{22} & \dots & \dots & \dots & \dots \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ C_{61} & C_{62} & C_{63} & C_{64} & C_{65} & C_{66} \end{pmatrix} \begin{pmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{13} \\ 2\varepsilon_{12} \end{pmatrix},

where indices follow the mapping (1=11, 2=22, 3=33, 4=23, 5=13, 6=12), and the factor of 2 for shear components ensures invariance of the strain energy. This matrix form simplifies computations in finite element analyses and transformations under coordinate rotations.^[41] Crystal symmetries reduce the number of independent constants in the Voigt matrix from 21 in the triclinic class (lowest symmetry) to as few as 3 in cubic crystals, where the matrix takes the form

[C] = \begin{pmatrix} C_{11} & C_{12} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{12} & C_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{44} \end{pmatrix},

with C_{11} controlling longitudinal stiffness along cube edges, C_{12} coupling normal stresses, and C_{44} governing shear. These constants determine mechanical stability via Born criteria, such as C_{11} > |C_{12}| and C_{44} > 0. For composites like unidirectional fiber laminates, orthotropic symmetry (three perpendicular planes) yields nine independent constants, including distinct Young's moduli E_1, E_2, E_3 and shear moduli G_{12}, G_{13}, G_{23}.^[42] Two-dimensional static problems in anisotropic media, such as plane strain in crystal plates or composite panels, are often solved using the Lekhnitskii formalism, which represents stresses and displacements via complex potentials \Phi(z_\alpha) in transformed coordinates z_\alpha = x + \mu_\alpha y (\alpha = 1,2), where \mu_\alpha are complex roots of the characteristic equation from the compliance matrix. This approach yields biharmonic-like equations for the potentials, enabling analytic solutions for holes, cracks, or inclusions by boundary value matching, generalizing isotropic complex variable methods.^[43] Closed-form solutions for uniaxial loading in orthotropic composites illustrate directional responses; under axial stress \sigma_{11} = \sigma, the strains are \varepsilon_{11} = \sigma / E_1, \varepsilon_{22} = -\nu_{12} \sigma / E_1, \varepsilon_{33} = -\nu_{13} \sigma / E_1, with no shear if aligned with principal axes, highlighting stiffness anisotropy where E_1 \gg E_2 for fiber-dominated loading. For heterogeneous composites, effective orthotropic moduli are estimated via homogenization, with Voigt bounds assuming uniform strain (upper stiffness limit C^V = \sum v_k C^{(k)}, v_k volume fractions) and Reuss bounds uniform stress (lower limit via compliance average), providing rigorous envelopes for anisotropic phases.^[44]^[45] Variational energy methods address complex geometries in anisotropic statics by minimizing the total potential energy \Pi = \int_V \frac{1}{2} \varepsilon_{ij} C_{ijkl} \varepsilon_{kl} \, dV - \int_{S_t} \mathbf{t} \cdot \mathbf{u} \, dS - \int_V \mathbf{b} \cdot \mathbf{u} \, dV, where the first term is internal strain energy, and the others account for surface tractions and body forces; admissible displacements satisfying kinematic boundaries yield the equilibrium solution as the global minimum. This principle underpins Rayleigh-Ritz approximations for irregular anisotropic domains, such as composite structures with varying fiber orientations.^[46] Micromechanical models for fiber composites extend these frameworks post-1960s, with the Halpin-Tsai equations providing semi-empirical estimates for transverse modulus E_2 and in-plane shear G_{12}:

\frac{P_c}{P_m} = \frac{1 + \xi \eta V_f}{1 - \eta V_f}, \quad \eta = \frac{P_f / P_m - 1}{P_f / P_m + \xi},

where P denotes modulus, subscripts c,f,m for composite, fiber, matrix, V_f fiber volume fraction, and \xi a phenomenological shape parameter (e.g., \xi = 1 for circular fibers, \xi = 2 for high aspect ratios). These capture fiber-matrix interactions beyond dilute approximations, aligning well with experiments for polymer-matrix composites.^[47]

Dynamics and Wave Propagation

In anisotropic linear elasticity, dynamic phenomena arise from the time-dependent equations of motion, \rho \ddot{\mathbf{u}} = \nabla \cdot \boldsymbol{\sigma}, where \mathbf{u} is the displacement vector, \rho the density, and \boldsymbol{\sigma} the stress tensor related to strain via the stiffness tensor C_{ijkl}. This leads to wave propagation where velocities and polarizations depend on direction due to material symmetry. Plane wave ansatze of the form \mathbf{u} = \mathbf{A} \exp[i(\mathbf{k} \cdot \mathbf{x} - \omega t)] are substituted into the equations, yielding an eigenvalue problem for the phase velocity c = \omega / |\mathbf{k}| and polarization \mathbf{A}.^[48] The governing relation is the Christoffel equation:

(C_{ikjl} n_j n_l - \rho c^2 \delta_{il}) A_k = 0,

with \mathbf{n} = \mathbf{k} / |\mathbf{k}| the unit wave normal and \delta_{il} the Kronecker delta. This 3x3 matrix equation, first derived for elastic media by Christoffel in 1877, admits three real positive eigenvalues for stable materials, corresponding to three propagation modes with orthogonal polarizations.^[49]^[50] The solutions yield one quasi-longitudinal (qP) mode, where \mathbf{A} is nearly parallel to \mathbf{n}, and two quasi-shear (qS1, qS2) modes, where \mathbf{A} is nearly perpendicular to \mathbf{n}, all with direction-dependent phase velocities that deviate from isotropic constancy. These are visualized via the slowness surface, plotting slowness vectors \mathbf{s} = \mathbf{n}/c, which for transversely isotropic (TI) media—such as unidirectional composites—exhibit cylindrical symmetry around the axis of isotropy, and for orthotropic media—like orthorhombic crystals or plywood—display threefold orthogonal symmetry with rectangular cross-sections in principal planes. The surfaces often show triplications in qS branches for certain symmetries, indicating non-unique velocities for given directions.^[48]^[51] Rayleigh waves on anisotropic half-spaces are evanescent surface modes satisfying traction-free boundaries, decaying as \exp(-q z) ( z \geq 0 into the half-space). Using the Stroh formalism, their speed c_R < \min(c_{qS}) solves a sextic determinant equation derived from the surface impedance tensor, ensuring uniqueness for most orientations in stable media; pseudo-Rayleigh waves may emerge in higher symmetries.^[52] These principles apply to ultrasonic nondestructive testing of anisotropic composites, where measured qP and qS velocities along fiber directions infer stiffness constants for defect detection. In geophysics, seismic anisotropy analysis via slowness surfaces interprets directionally varying P- and S-wave speeds to map fracture orientations and stress in reservoirs.^[53] Advances since 2010 in the spectral element method (SEM) facilitate 3D simulations of anisotropic waves, employing high-order Gauss-Lobatto-Legendre basis functions for accuracy in heterogeneous media like faulted basins, with applications in full-waveform inversion for exploration. In the special case of isotropic elasticity, the Christoffel equation reduces to two decoupled modes with constant speeds.^[54]

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