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Infinitesimal strain theory

Infinitesimal strain theory, also known as small deformation theory, is a mathematical framework within continuum mechanics that describes the deformation of solid bodies when displacement gradients are small in magnitude compared to unity, allowing for linear approximations of strain measures.^[1] Under these conditions, the deformation gradient \mathbf{F} is approximated as \mathbf{F} \approx \mathbf{I} + \nabla \mathbf{u}, where \mathbf{I} is the identity tensor and \mathbf{u} is the displacement field, and the infinitesimal strain tensor \boldsymbol{\epsilon} is defined as the symmetric part of the displacement gradient: \boldsymbol{\epsilon} = \frac{1}{2} \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^T \right).^[2] This tensor captures both normal strains \epsilon_{ii} = \frac{\partial u_i}{\partial x_i} (no sum) and engineering shear strains \gamma_{ij} = 2\epsilon_{ij} = \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} for i \neq j.^[3] The theory originated in the early 19th century, with Augustin-Louis Cauchy introducing the infinitesimal strain tensor in the 1820s as a cornerstone of linear elasticity, building on earlier kinematic ideas from Leonhard Euler.^[3] It assumes that deformations are small enough that higher-order terms (such as products of displacement gradients) can be neglected, the reference and deformed configurations are nearly identical, and rotations are infinitesimal, leading to a symmetric strain tensor that measures relative distortions without rigid-body motions.^[1] These assumptions simplify the balance laws of mechanics, such as the linearized momentum equation \nabla \cdot \boldsymbol{\sigma} + \rho \mathbf{b} = \rho \ddot{\mathbf{u}}, where \boldsymbol{\sigma} is the Cauchy stress tensor.^[2] In practice, infinitesimal strain theory underpins linear elasticity models, where stress is linearly related to strain via Hooke's law \boldsymbol{\sigma} = \mathbb{C} : \boldsymbol{\epsilon}, with \mathbb{C} as the fourth-order stiffness tensor, enabling solutions to problems in structural engineering like beams, plates, and trusses under moderate loads.^[1] It is particularly valuable for quasi-static analyses where geometric nonlinearities are insignificant, but it fails for large deformations or rotations, necessitating finite strain formulations in such cases.^[2] The theory's enduring influence is evident in its integration into computational methods and standard engineering curricula, providing an efficient approximation for many real-world applications.^[3]

Fundamentals

Definition and Assumptions

Infinitesimal strain theory, also known as small deformation theory, provides a mathematical framework for describing the deformation of a continuum body subjected to small displacements, where the magnitude of the displacement gradient is much less than unity, allowing the neglect of higher-order terms in the analysis. This approach approximates the deformation state through a symmetric second-order tensor, denoted as ε, which quantifies the relative changes in length and angles within the material. Unlike more general theories, it linearizes the kinematics of deformation, making it suitable for applications in linear elasticity and small-strain plasticity in engineering and physics.^[4]^[5] The theory rests on several key assumptions to ensure the validity of the linear approximation. Displacements are assumed to be small such that the norm of the displacement gradient ||∇u|| ≪ 1, implying that rotations are infinitesimal and do not significantly alter the geometry. Strains are much less than unity, typically on the order of 0.01 or smaller, and geometric linearity is invoked, meaning no distinction is made between the undeformed reference configuration and the slightly deformed current configuration for strain calculations. These assumptions simplify the governing equations but restrict the theory to scenarios where nonlinear geometric effects, such as significant stretching or buckling, are negligible.^[5]^[6]^[4] Mathematically, the infinitesimal strain tensor ε is defined as the symmetric part of the displacement gradient tensor:

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

where \mathbf{u} is the displacement vector field describing the motion of material points from their reference positions. This expression captures both normal strains (diagonal components representing extension or contraction) and engineering shear strains (off-diagonal components halved for tensor symmetry). The formulation arises from considering the change in squared length between nearby points and neglecting quadratic terms in the displacement differences.^[5]^[6] While powerful for many practical cases, infinitesimal strain theory has inherent limitations when applied beyond its assumptions. It fails to accurately model large deformations, such as those exceeding a few percent strain, where nonlinear effects become prominent and finite strain measures, like the Green-Lagrange tensor, are required to account for the distinction between reference and deformed configurations. For instance, in highly stretched materials or post-buckling scenarios, the linear approximation leads to significant errors in predicted stresses and deformations.^[4]

Historical Context

Infinitesimal strain theory emerged in the 19th century as a cornerstone of linear elasticity, building on early efforts to describe small deformations in continuous media. Augustin-Louis Cauchy played a pivotal role by introducing the concept of stress at a point in 1822, establishing the framework for analyzing internal forces in elastic bodies. This work laid the groundwork for subsequent developments in elasticity, where strain measures were needed to relate deformations to stresses. Cauchy's further contributions from 1823 to 1841 developed the theory of finite strain, from which the infinitesimal approximation was derived for cases where displacements and rotations are small compared to the body's dimensions.^[7]^[8] A key milestone came in 1828 when Siméon Denis Poisson proposed that, for an isotropic elastic material under uniaxial stress, the lateral strain is a constant fraction of the axial strain, introducing what is now known as Poisson's ratio. This relation connected transverse and longitudinal responses, essential for understanding volumetric changes in small-strain regimes. In 1839, George Green formulated a deformation tensor that provided a rigorous measure of strain, initially for finite deformations but adaptable to infinitesimal cases through linearization. The theory's foundations were further strengthened in the 1860s by Adhémar Jean Claude Barré de Saint-Venant, who derived the compatibility equations ensuring that strain fields correspond to a single-valued displacement field, resolving issues in integrating strains for continuous deformations.^[9]^[10]^[11] The evolution of infinitesimal strain theory gained momentum in the late 19th century with A. E. H. Love's 1892 treatise, A Treatise on the Mathematical Theory of Elasticity, which formalized the infinitesimal approximation and integrated it into a comprehensive mathematical framework for elastic solids. This work solidified the theory's role in resolving paradoxes arising in Euler-Bernoulli beam theory, where small deflections required linear strain assumptions to avoid inconsistencies in curvature and extension predictions. By the early 20th century, Stephen Timoshenko's influential texts, including Theory of Elasticity (1934, co-authored with J. N. Goodier) and his 1953 historical account History of Strength of Materials, popularized the theory in engineering practice, emphasizing its applicability to structural analysis. Extensions to stress-based compatibility, such as Beltrami's 1892 equations for body-force-free cases and Michell's 1900 generalization, enhanced its utility in solving boundary-value problems.^[12]^[13]^[14] As a mature framework, infinitesimal strain theory saw no fundamental theoretical updates after the mid-20th century, but recent computational mechanics in the 2020s has extended its numerical implementations, incorporating machine learning for enhanced simulation accuracy in heterogeneous materials.^[15]

Infinitesimal Strain Tensor

Geometric Derivation

In infinitesimal strain theory, the deformation of a continuum body is described by considering points at original position \mathbf{x} that undergo a small displacement \mathbf{u}(\mathbf{x}), resulting in a new position \mathbf{x}' = \mathbf{x} + \mathbf{u}(\mathbf{x}). For infinitesimal displacements where |\nabla \mathbf{u}| \ll 1, the deformation gradient tensor simplifies to \mathbf{F} \approx \mathbf{I} + \nabla \mathbf{u}, neglecting higher-order terms.^[16] To derive the strain tensor geometrically, examine the change in length of an infinitesimal line element d\mathbf{x} in the body. The original squared length is ds^2 = d\mathbf{x} \cdot d\mathbf{x}, while the deformed squared length is ds'^2 = d\mathbf{x}' \cdot d\mathbf{x}', where d\mathbf{x}' = \mathbf{F} \, d\mathbf{x}. Thus, ds'^2 = d\mathbf{x} \cdot \mathbf{F}^T \mathbf{F} \, d\mathbf{x}, and the change in squared length is ds'^2 - ds^2 = d\mathbf{x} \cdot (\mathbf{F}^T \mathbf{F} - \mathbf{I}) \, d\mathbf{x}.^[5] Substituting the approximation for \mathbf{F} yields \mathbf{F}^T \mathbf{F} \approx \mathbf{I} + \nabla \mathbf{u} + (\nabla \mathbf{u})^T, so ds'^2 - ds^2 \approx d\mathbf{x} \cdot [\nabla \mathbf{u} + (\nabla \mathbf{u})^T] \, d\mathbf{x} = 2 \, d\mathbf{x} \cdot \boldsymbol{\varepsilon} \, d\mathbf{x}, where the infinitesimal strain tensor is defined as the symmetric part \boldsymbol{\varepsilon} = \frac{1}{2} [\nabla \mathbf{u} + (\nabla \mathbf{u})^T].^[16] This neglects quadratic terms in \nabla \mathbf{u}, which are second-order small. In Cartesian coordinates, the components of the infinitesimal strain tensor are given by

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

for i, j = 1, 2, 3. The diagonal components \varepsilon_{ii} (no summation) represent normal strains, while off-diagonal components \varepsilon_{ij} ( i \neq j) capture shear strains. Geometrically, the normal strain \varepsilon_{ii} corresponds to the relative elongation of a line element along the i-direction, defined as the change in length divided by the original length. For shear strain, consider two originally perpendicular line elements along the x- and y-directions; after deformation, the angle between them changes from $90^\circ to $90^\circ - \gamma, where the engineering shear strain \gamma \approx 2 \varepsilon_{xy} for small \gamma.^[5] The symmetry of the strain tensor \varepsilon_{ij} = \varepsilon_{ji} arises geometrically from the need to separate pure deformation from rigid body rotation: the antisymmetric part of \nabla \mathbf{u} accounts for infinitesimal rotation, while the symmetric part \boldsymbol{\varepsilon} isolates the deformation, ensuring that the measure of geometric change is independent of the path taken by the line element in small displacements.

Components and Physical Interpretation

In three-dimensional Cartesian coordinates, the infinitesimal strain tensor \boldsymbol{\varepsilon} is a symmetric second-order tensor expressed as a matrix with components:

\boldsymbol{\varepsilon} = \begin{pmatrix} \varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\ \varepsilon_{xy} & \varepsilon_{yy} & \varepsilon_{yz} \\ \varepsilon_{xz} & \varepsilon_{yz} & \varepsilon_{zz} \end{pmatrix},

where the diagonal elements \varepsilon_{xx}, \varepsilon_{yy}, and \varepsilon_{zz} denote the normal strain components, and the off-diagonal elements \varepsilon_{xy}, \varepsilon_{xz}, and \varepsilon_{yz} represent the shear strain components.^[16]^[5] The normal strain components measure the relative elongation or shortening of material line elements aligned with the coordinate axes; for instance, \varepsilon_{xx} = \partial u_x / \partial x, where \mathbf{u} = (u_x, u_y, u_z) is the displacement vector, quantifies the longitudinal stretch (positive for extension, negative for compression) in the x-direction per unit original length.^[17]^[18] Similarly, \varepsilon_{yy} = \partial u_y / \partial y and \varepsilon_{zz} = \partial u_z / \partial z capture extensions along the y- and z-directions, respectively.^[16] The shear strain components describe the average angular distortion in the respective planes; for example, \varepsilon_{xy} = \frac{1}{2} \left( \frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x} \right) represents half the change in the right angle between line elements originally aligned with the x- and y-axes, with the engineering shear strain \gamma_{xy} = 2 \varepsilon_{xy} denoting the total angle change.^[19]^[5] Analogous interpretations apply to \varepsilon_{xz} and \varepsilon_{yz}, which govern shearing in the xz- and yz-planes.^[18] Physically, these components can be visualized by considering the deformation of an infinitesimal cubic element with edges aligned to the axes: positive \varepsilon_{xx} elongates the edges parallel to x while compressing those in the transverse directions if Poisson effects are present, whereas \varepsilon_{xy} distorts the cube into a parallelepiped by shearing the faces in the xy-plane, altering the 90° angles to approximately $90^\circ - \gamma_{xy}.^[19]^[18] The trace of the tensor, \operatorname{tr}(\boldsymbol{\varepsilon}) = \varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz}, approximates the relative volumetric change \Delta V / V for small deformations, indicating uniform dilation (positive trace) or contraction (negative trace).^[17] In the infinitesimal strain framework, the tensor components depend solely on the first-order displacement gradients and are thus path-independent, meaning the resulting strain at a point is determined uniquely by the final displacement field regardless of the deformation history, in contrast to finite strain measures that incorporate nonlinear terms sensitive to loading paths.^[17]

Properties of the Strain Tensor

Transformation Rules

The infinitesimal strain tensor, being a second-order Cartesian tensor, transforms under a rotation of the coordinate system according to the rule \boldsymbol{\varepsilon}' = \mathbf{R} \boldsymbol{\varepsilon} \mathbf{R}^T, where \mathbf{R} is the orthogonal rotation matrix satisfying \mathbf{R}^T \mathbf{R} = \mathbf{I} and \det \mathbf{R} = 1, \boldsymbol{\varepsilon} is the strain tensor in the original coordinates, and \boldsymbol{\varepsilon}' is the tensor in the rotated frame.^[16] This transformation law ensures that the strain tensor components adjust to reflect the change in basis while preserving its symmetric nature, as the symmetry of \boldsymbol{\varepsilon} implies the symmetry of \boldsymbol{\varepsilon}'.^[20] Additionally, the trace of the strain tensor, which represents volumetric strain, remains invariant under this transformation because \operatorname{tr}(\boldsymbol{\varepsilon}') = \operatorname{tr}(\mathbf{R} \boldsymbol{\varepsilon} \mathbf{R}^T) = \operatorname{tr}(\boldsymbol{\varepsilon}).^[16] The derivation of this transformation follows from the corresponding rule for the displacement gradient tensor. In the rotated coordinate system, the displacement gradient transforms as \nabla \mathbf{u}' = \mathbf{R} (\nabla \mathbf{u}) \mathbf{R}^T, where \nabla \mathbf{u} is the original displacement gradient.^[20] Since the infinitesimal strain tensor is the symmetric part of the displacement gradient, \boldsymbol{\varepsilon} = \frac{1}{2} \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^T \right), applying the same operation to the transformed gradient yields \boldsymbol{\varepsilon}' = \frac{1}{2} \left( \nabla \mathbf{u}' + (\nabla \mathbf{u}')^T \right). Substituting the transformation for \nabla \mathbf{u}' and using the orthogonality of \mathbf{R} (so \mathbf{R}^T = \mathbf{R}^{-1}) confirms that \boldsymbol{\varepsilon}' = \mathbf{R} \boldsymbol{\varepsilon} \mathbf{R}^T.^[16] In component form, for a rotation defined by direction cosines a_{ij}, the transformed components are \varepsilon'_{ij} = a_{ik} a_{jl} \varepsilon_{kl}, which is the standard tensor transformation rule for second-order quantities.^[20] This distinguishes the strain tensor from engineering strain measures, such as normal and shear strain vectors, which do not transform as simple tensors and require additional rules to maintain physical consistency across coordinate systems.^[16] As an illustrative example in two dimensions, consider a rotation by an angle \theta about the out-of-plane axis, with \mathbf{R} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}. For a strain tensor \boldsymbol{\varepsilon} = \begin{pmatrix} \varepsilon_{xx} & \varepsilon_{xy} \\ \varepsilon_{xy} & \varepsilon_{yy} \end{pmatrix}, the transformed normal strain in the x'-direction is \varepsilon'_{xx} = \varepsilon_{xx} \cos^2 \theta + \varepsilon_{yy} \sin^2 \theta + 2 \varepsilon_{xy} \sin \theta \cos \theta.^[20] This expression highlights how the components mix under rotation, providing a basis for graphical representations like Mohr's circle without altering the underlying tensor properties.^[16]

Invariants and Principal Strains

The infinitesimal strain tensor \boldsymbol{\varepsilon} possesses three scalar invariants, I_1, I_2, and I_3, which remain unchanged under orthogonal transformations of the coordinate system, as established by the transformation rules for second-order tensors.^[16] These invariants provide a basis-independent characterization of the strain state and are fundamental for deriving scalar measures in constitutive relations.^[2] The first invariant is the trace of the strain tensor, I_1 = \operatorname{tr}(\boldsymbol{\varepsilon}) = \varepsilon_{kk} = \varepsilon_{11} + \varepsilon_{22} + \varepsilon_{33}, representing the volumetric strain associated with the dilatation of the material.^[16] The second invariant is given by I_2 = \frac{1}{2} \left[ (\operatorname{tr}(\boldsymbol{\varepsilon}))^2 - \operatorname{tr}(\boldsymbol{\varepsilon}^2) \right], which captures interactions between the principal strain components.^[16] The third invariant is the determinant, I_3 = \det(\boldsymbol{\varepsilon}), reflecting the overall signed volume change.^[16] The principal strains are the eigenvalues \lambda_1 \geq \lambda_2 \geq \lambda_3 of \boldsymbol{\varepsilon}, obtained by solving the characteristic equation \lambda^3 - I_1 \lambda^2 + I_2 \lambda - I_3 = 0, with corresponding principal directions given by the eigenvectors.^[16] These eigenvalues represent the maximum, intermediate, and minimum normal strains, occurring on planes free of shear strain.^[21] The maximum shear strain component is then (\lambda_1 - \lambda_3)/2.^[22] To determine the principal strains, one solves the eigenvalue problem \det(\boldsymbol{\varepsilon} - \lambda \mathbf{I}) = 0. In two dimensions, for the strain components \varepsilon_{xx}, \varepsilon_{yy}, and \varepsilon_{xy}, the principal strains simplify to

\lambda = \frac{\varepsilon_{xx} + \varepsilon_{yy}}{2} \pm \sqrt{ \left( \frac{\varepsilon_{xx} - \varepsilon_{yy}}{2} \right)^2 + \varepsilon_{xy}^2 }.

^[21] This quadratic solution highlights the role of the invariants in plane strain analysis. Due to their coordinate independence, the invariants I_1, I_2, and I_3 are essential in formulating material failure criteria, such as the von Mises equivalent strain, which assesses distortion without reference to specific stress components.^[23]

Volumetric and Deviatoric Components

In infinitesimal strain theory, the strain tensor \boldsymbol{\varepsilon} can be decomposed into a volumetric (or hydrostatic) component and a deviatoric component, allowing separate analysis of volume change and shape distortion. The volumetric part is given by \boldsymbol{\varepsilon}_\text{vol} = \frac{\text{tr}(\boldsymbol{\varepsilon})}{3} \mathbf{I}, where \text{tr}(\boldsymbol{\varepsilon}) is the trace of the strain tensor and \mathbf{I} is the identity tensor, representing isotropic expansion or contraction. The deviatoric part is then \boldsymbol{\varepsilon}_\text{dev} = \boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}_\text{vol}, which is traceless (\text{tr}(\boldsymbol{\varepsilon}_\text{dev}) = 0) and captures pure shear deformation without altering volume. In component form, this decomposition is expressed as \varepsilon_{ij} = \frac{1}{3} \varepsilon_{kk} \delta_{ij} + \varepsilon'_{ij}, where \varepsilon'_{ij} = \varepsilon_{ij} - \frac{1}{3} \varepsilon_{kk} \delta_{ij} and \delta_{ij} is the Kronecker delta.^[24]^[23] The volumetric strain e_v = \text{tr}(\boldsymbol{\varepsilon}) = \varepsilon_{kk} quantifies the relative volume change \Delta V / V \approx e_v under small deformations, where \varepsilon_m = e_v / 3 is the mean normal strain. This scalar measure arises from the first invariant of the strain tensor and physically corresponds to dilatation, such as uniform compression or expansion in isotropic materials. For instance, in elastic solids, it relates to the bulk modulus via \sigma_{kk} = 3K e_v, decoupling volume effects from shear.^[24]^[23] The deviatoric component \boldsymbol{\varepsilon}_\text{dev} isolates distortion, measuring changes in shape without volume alteration, which is crucial for understanding yielding and plastic flow in metals. A key scalar invariant for the deviatoric strain is the second invariant J_2 = \frac{1}{2} \text{tr}(\boldsymbol{\varepsilon}_\text{dev}^2), often used in criteria like von Mises for effective strain magnitude, given by \sqrt{\frac{4}{3} J_2} in principal strain terms as \frac{\sqrt{2}}{3} \sqrt{(\varepsilon_1 - \varepsilon_2)^2 + (\varepsilon_2 - \varepsilon_3)^2 + (\varepsilon_3 - \varepsilon_1)^2}, where \varepsilon_1, \varepsilon_2, \varepsilon_3 are principal strains.^[25] For incompressible materials, such as ideal fluids or rubbers under small strains, the volumetric strain vanishes (e_v = \text{tr}(\boldsymbol{\varepsilon}) = 0), implying \boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}_\text{dev} and pure deviatoric deformation, which simplifies analysis in hydrodynamics and elastomers by enforcing \varepsilon_{kk} = 0. This condition is enforced in constitutive models like neo-Hookean hyperelasticity approximated for small strains.^[24]^[23]

Octahedral and Equivalent Strains

In infinitesimal strain theory, scalar measures such as octahedral and equivalent strains provide useful summaries of the deviatoric components of the strain tensor, facilitating analyses of distortion and shear effects in material behavior. These quantities are particularly valuable for evaluating energy dissipation and failure criteria in engineering applications, where the full tensor may be cumbersome. The octahedral planes are those whose normal vector is given by \mathbf{n} = \frac{1}{\sqrt{3}} (1, 1, 1), equally inclined to the three principal axes of the strain tensor. The normal strain on such a plane, known as the octahedral normal strain, is \varepsilon_\mathrm{oct} = \frac{1}{3} \mathrm{tr}(\boldsymbol{\varepsilon}), representing the average normal strain across all directions.^[26] This measure captures the volumetric component of deformation independently of shear effects. The octahedral shear strain quantifies the shearing distortion on the octahedral plane and is defined as \gamma_\mathrm{oct} = \frac{2}{3} \sqrt{6 J_2}, where J_2 = \frac{1}{2} \boldsymbol{\varepsilon}_\mathrm{dev} : \boldsymbol{\varepsilon}_\mathrm{dev} is the second invariant of the deviatoric strain tensor \boldsymbol{\varepsilon}_\mathrm{dev}.^[6] This scalar relates directly to the magnitude of shear acting on the octahedral planes, serving as an indicator of plastic deformation progression under multiaxial loading. The equivalent strain, or von Mises strain, offers a comprehensive scalar representation of the overall deviatoric strain intensity, defined as \varepsilon_\mathrm{eq} = \sqrt{\frac{2}{3} \boldsymbol{\varepsilon}_\mathrm{dev} : \boldsymbol{\varepsilon}_\mathrm{dev}}. In terms of the principal strains \varepsilon_1, \varepsilon_2, \varepsilon_3, it takes the form \varepsilon_\mathrm{eq} = \frac{\sqrt{2}}{3} \sqrt{ (\varepsilon_1 - \varepsilon_2)^2 + (\varepsilon_2 - \varepsilon_3)^2 + (\varepsilon_3 - \varepsilon_1)^2 }, ensuring equivalence to the axial strain in uniaxial tension for consistent energy calculations. These strain measures underpin the distortion energy theory, originally formulated for stress but extended analogously to strains in plasticity models, where \gamma_\mathrm{oct} correlates with the peak shear distortion on octahedral planes to predict yielding and fracture. The deviatoric strain tensor, from which they are derived, isolates shape-changing deformations excluding pure volume change.

Strain Compatibility

Compatibility Equations

The compatibility equations, also known as Saint-Venant's compatibility conditions, provide the necessary and sufficient mathematical constraints that a given infinitesimal strain field must satisfy to be integrable into a single-valued, continuous displacement field in a simply connected domain. These equations ensure the existence of a displacement vector \mathbf{u} such that the symmetric strain tensor \boldsymbol{\varepsilon} = \frac{1}{2} (\nabla \mathbf{u} + (\nabla \mathbf{u})^T) holds everywhere. They were first fully formulated by Barré de Saint-Venant in 1864.^[11] The derivation stems from the definition of the strain tensor and the equality of mixed partial derivatives for the smooth displacement components u_i. Specifically, the second derivatives \frac{\partial^2 u_i}{\partial x_j \partial x_k} = \frac{\partial^2 u_i}{\partial x_k \partial x_j} must commute. Expressing these derivatives in terms of the strain components and their first derivatives leads to relations among the second derivatives of \varepsilon_{ij}. An equivalent vectorial form arises by considering the vanishing curl of the displacement gradient: \nabla \times (\nabla \mathbf{u}) = \mathbf{0}. Applying the curl operator again yields \nabla \times (\nabla \times (\nabla \mathbf{u})) = \mathbf{0}, which, when symmetrized and substituted using the strain-displacement relation, results in the incompatibility tensor \mathrm{inc}(\boldsymbol{\varepsilon}) = \nabla \times (\nabla \times \boldsymbol{\varepsilon}) = \mathbf{0} (in the appropriate tensor sense).^[27] In three dimensions, the Saint-Venant compatibility equations consist of six independent conditions out of the full set of 81 possible combinations, accounting for the symmetries of the strain tensor:

\frac{\partial^2 \varepsilon_{ij}}{\partial x_k \partial x_l} + \frac{\partial^2 \varepsilon_{kl}}{\partial x_i \partial x_j} - \frac{\partial^2 \varepsilon_{il}}{\partial x_j \partial x_k} - \frac{\partial^2 \varepsilon_{jk}}{\partial x_i \partial x_l} = 0

for i,j,k,l = 1,2,3, with the independent equations corresponding to the unique pairs (ij,kl) under permutation symmetry (e.g., \frac{\partial^2 \varepsilon_{11}}{\partial x_2^2} + \frac{\partial^2 \varepsilon_{22}}{\partial x_1^2} = 2 \frac{\partial^2 \varepsilon_{12}}{\partial x_1 \partial x_2}). This form is equivalent to the vanishing of the incompatibility tensor components.^[27] In two dimensions (e.g., plane strain, where \varepsilon_{13} = \varepsilon_{23} = \varepsilon_{33} = 0), the conditions simplify to a single equation:

\frac{\partial^2 \varepsilon_{xx}}{\partial y^2} + \frac{\partial^2 \varepsilon_{yy}}{\partial x^2} = 2 \frac{\partial^2 \varepsilon_{xy}}{\partial x \partial y},

reflecting the reduction from the full 3D set by assuming no variation in the third direction. These equations are essential for ensuring the strain field is admissible in boundary value problems of linear elasticity, as incompatible strains cannot arise from any continuous deformation; for instance, they are violated across discontinuities such as cracks where the displacement field jumps.^[27]

Physical Implications

The compatibility equations in infinitesimal strain theory ensure that a given strain field \epsilon corresponds to a continuous and single-valued displacement field u, thereby guaranteeing the existence of a physically realizable deformation without gaps or overlaps in the material body. This condition is crucial for maintaining the integrity of the continuum, as it prevents discontinuities that would otherwise arise if the strains were not derivable from a smooth displacement vector field, analogous to ensuring pieces of a jigsaw puzzle fit together seamlessly after deformation.^[28] In particular, these equations are essential for inverse problems, where the goal is to recover the displacements from measured or assumed strains, which is uniquely determined up to rigid body motions.^[28] In the context of linear elasticity, the compatibility equations, combined with the equilibrium equations and constitutive relations (such as Hooke's law), form a complete set for solving boundary value problems, enabling the prediction of both stress and displacement fields under applied loads. The Beltrami-Michell equations further integrate these compatibility conditions directly into the stress formulation, providing a stress-based approach to satisfy the necessary conditions for a compatible deformation while accounting for body forces and equilibrium.^[27] This integration is particularly valuable for problems where displacements are not directly observable, allowing solutions in terms of stresses alone.^[29] A notable aspect in reduced dimensions is that, for 2D plane strain problems, the six compatibility equations of 3D infinitesimal strain theory condense to a single independent equation relating the in-plane strain components, reflecting the fewer degrees of freedom in displacement. Failure to satisfy this condition indicates the presence of plastic zones, defects, or dislocations, where the strain field is incompatible due to irreversible deformations or microstructural irregularities.^[28]^[30] Practically, compatibility equations find application in experimental techniques like photoelasticity, where measured fringe patterns reveal full-field strain distributions that can be checked against compatibility to validate the elastic assumption and verify the integrity of the strain field in prototypes or models.^[31]

Special Cases

Plane Strain

Plane strain refers to a simplified state of deformation in infinitesimal strain theory where the strain components perpendicular to a designated plane, typically the x-y plane, vanish. Specifically, \epsilon_{zz} = 0, \gamma_{xz} = 0, and \gamma_{yz} = 0, assuming all derivatives with respect to the z-coordinate are zero (\partial / \partial z = 0). This condition arises from the displacement field where the axial displacement u_z = 0 and the out-of-plane variations of the in-plane displacements are absent: \partial u_x / \partial z = 0 and \partial u_y / \partial z = 0.^[32]^[33] Under plane strain assumptions, the infinitesimal strain tensor reduces to a two-dimensional form, retaining only the in-plane components \epsilon_{xx}, \epsilon_{yy}, and \epsilon_{xy}. These are defined as

\begin{align} \epsilon_{xx} &= \frac{\partial u_x}{\partial x}, \\ \epsilon_{yy} &= \frac{\partial u_y}{\partial y}, \\ \epsilon_{xy} &= \frac{1}{2} \left( \frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x} \right), \end{align}

with all other components zero. The compatibility requirement for these strains simplifies to a single governing equation in two dimensions:

\frac{\partial^2 \epsilon_{xx}}{\partial y^2} + \frac{\partial^2 \epsilon_{yy}}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{xy}}{\partial x \partial y}.

This equation ensures the existence of a continuous displacement field consistent with the strains.^[33]^[28] Physically, plane strain applies to scenarios involving long prismatic bodies or structures where deformation is uniform along the length (z-direction), preventing axial extension or contraction. A representative example is a thick-walled cylinder under internal pressure, where the length far exceeds the diameter, rendering end effects negligible and justifying the plane strain approximation. In such cases, the zero axial strain \epsilon_{zz} = 0 is enforced by constraints or geometry, but the corresponding axial stress \sigma_{zz} remains nonzero due to the Poisson effect, calculated as \sigma_{zz} = \nu (\sigma_{xx} + \sigma_{yy}), where \nu is the Poisson's ratio. This arises from the constitutive relations in linear elasticity, which couple the stresses to counteract the Poisson-induced axial contraction from in-plane loading.^[34] Although plane strain effectively poses a two-dimensional problem for strain analysis, the associated stress state is fully three-dimensional, with \sigma_{zz} contributing to the overall equilibrium. This distinguishes it from plane stress conditions, where \sigma_{zz} = 0 but \epsilon_{zz} \neq 0, as in thin plates free to expand laterally. The plane strain formulation thus facilitates efficient modeling of confined deformations while accounting for the full elastic response.^[32]^[34]

Antiplane Strain

Antiplane strain represents a specialized case within infinitesimal strain theory where the deformation involves solely out-of-plane displacement perpendicular to the reference plane, with no in-plane motion. The displacement field is defined by u_x = 0, u_y = 0, and u_z = w(x,y), where w(x,y) denotes the axial displacement varying only with the in-plane coordinates x and y. This configuration simplifies the analysis to a two-dimensional problem in the xy-plane.^[35] The resulting infinitesimal strain tensor exhibits pure shear characteristics, with the only non-zero components being the off-diagonal shear strains \varepsilon_{xz} = \frac{1}{2} \frac{\partial w}{\partial x} and \varepsilon_{yz} = \frac{1}{2} \frac{\partial w}{\partial y}; all normal and other shear components vanish. The compatibility equations for this strain state are automatically satisfied due to the displacement-based formulation. However, the equilibrium equations in the absence of body forces and for constant material properties require w to be a harmonic function, satisfying the two-dimensional Laplace equation \nabla^2 w = \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} = 0. The magnitude of the shear strain in this setup is given by \sqrt{\varepsilon_{xz}^2 + \varepsilon_{yz}^2}, which quantifies the local deformation intensity.^[35]^[36] Physically, antiplane strain models scenarios such as the torsion of prismatic shafts under Saint-Venant's semi-inverse assumptions, where the cross-section undergoes longitudinal shear without axial extension or in-plane distortion. It also describes Mode III loading in fracture mechanics, involving antiplane shear around cracks, where the displacement is perpendicular to the crack plane and loading direction. These applications highlight its utility in simplifying complex elastic problems to scalar potential formulations.^[37]^[38]

Decomposition and Rotation

Infinitesimal Rotation Tensor

In infinitesimal strain theory, the displacement gradient tensor \mathbf{H} = \nabla \mathbf{u} decomposes additively into the symmetric infinitesimal strain tensor \boldsymbol{\varepsilon} and the antisymmetric infinitesimal rotation tensor \boldsymbol{\omega}, such that \mathbf{H} = \boldsymbol{\varepsilon} + \boldsymbol{\omega}, where \boldsymbol{\omega} = \frac{1}{2} (\nabla \mathbf{u} - (\nabla \mathbf{u})^T).^[16] This decomposition separates the deformative and rigid-body components of the displacement field under the assumption of small gradients.^[23] In three dimensions, the infinitesimal rotation tensor takes the matrix form

\boldsymbol{\omega} = \begin{bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{bmatrix},

with components defined as \omega_x = \frac{1}{2} \left( \frac{\partial u_z}{\partial y} - \frac{\partial u_y}{\partial z} \right), \omega_y = \frac{1}{2} \left( \frac{\partial u_x}{\partial z} - \frac{\partial u_z}{\partial x} \right), and \omega_z = \frac{1}{2} \left( \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \right), where \mathbf{u} = (u_x, u_y, u_z) are the displacement components.^[23] These components arise directly from the antisymmetric nature of \boldsymbol{\omega}, ensuring only three independent values.^[16] Physically, the infinitesimal rotation tensor \boldsymbol{\omega} characterizes the rigid-body rotation of material elements in the linear approximation of small deformations.^[23] For infinitesimal rotations, it acts on any vector \mathbf{v} equivalently to the cross product with the associated rotation vector \boldsymbol{\Omega}, such that \boldsymbol{\omega} \mathbf{v} = \boldsymbol{\Omega} \times \mathbf{v}, where \boldsymbol{\Omega} = (\omega_x, \omega_y, \omega_z) represents the axis and magnitude of the rotation. As a skew-symmetric tensor, \boldsymbol{\omega} has zero trace, \operatorname{tr}(\boldsymbol{\omega}) = 0, and in the linear theory, it decouples from the strain tensor, meaning constitutive relations for stress depend solely on \boldsymbol{\varepsilon} without influence from \boldsymbol{\omega}.^[16]

Relations to Strain and Axial Vector

In infinitesimal strain theory, the displacement gradient tensor \nabla \mathbf{u} decomposes additively into the symmetric infinitesimal strain tensor \boldsymbol{\varepsilon} and the antisymmetric infinitesimal rotation tensor \boldsymbol{\omega}, expressed as \nabla \mathbf{u} = \boldsymbol{\varepsilon} + \boldsymbol{\omega}. This decomposition isolates the deformative (symmetric) kinematics in \boldsymbol{\varepsilon}, which quantifies stretching and shearing, from the rigid-body rotational (antisymmetric) kinematics in \boldsymbol{\omega}.^[39] The infinitesimal rotation tensor \boldsymbol{\omega} is conveniently represented by its associated axial vector \mathbf{\Omega} = (\Omega_x, \Omega_y, \Omega_z), defined componentwise as \Omega_x = -\omega_{yz}, \Omega_y = -\omega_{zx}, \Omega_z = -\omega_{xy}. The magnitude |\mathbf{\Omega}| corresponds to the infinitesimal rotation angle, while the direction of \mathbf{\Omega} aligns with the axis of rotation. In index notation, the relation between the tensor and its axial vector is given by

\omega_{ij} = -\varepsilon_{ijk} \Omega_k,

where \varepsilon_{ijk} is the Levi-Civita symbol.^[40] The axial vector \mathbf{\Omega} relates directly to the displacement field via \mathbf{\Omega} = \frac{1}{2} \nabla \times \mathbf{u}, linking the local rotation to the curl of the displacement. For irrotational displacements, where \boldsymbol{\omega} = \mathbf{0}, the motion consists of pure strain without rotation. This concept extends analogously to fluid mechanics, where the vorticity \boldsymbol{\zeta} = \nabla \times \mathbf{v} = 2 \mathbf{\Omega} (with \mathbf{v} as velocity) measures twice the rotation rate of fluid elements.^[40]

Non-Cartesian Formulations

Cylindrical Coordinates

In cylindrical coordinates (r, \theta, z), the displacement vector is \mathbf{u} = u_r \mathbf{e}_r + u_\theta \mathbf{e}_\theta + u_z \mathbf{e}_z, where \mathbf{e}_r, \mathbf{e}_\theta, and \mathbf{e}_z are the orthonormal basis vectors. The infinitesimal strain tensor \boldsymbol{\epsilon} captures the symmetric part of the displacement gradient in this curvilinear system, adapted to account for the varying geometry.^[33] The normal strain components represent extensions along the coordinate directions. The radial normal strain is the partial derivative of the radial displacement with respect to radius:

\epsilon_{rr} = \frac{\partial u_r}{\partial r}.

The axial normal strain along the z-direction is similarly straightforward:

\epsilon_{zz} = \frac{\partial u_z}{\partial z}.

However, the circumferential or hoop normal strain \epsilon_{\theta\theta} includes both the angular variation of the tangential displacement and a geometric contribution from radial displacement, reflecting the curvature of cylindrical surfaces:

\epsilon_{\theta\theta} = \frac{1}{r} \frac{\partial u_\theta}{\partial \theta} + \frac{u_r}{r}.

This term u_r / r arises because radial expansion alters the circumference proportionally to the radius, providing a key distinction from Cartesian formulations.^[33]^[41] The shear strain components describe angular distortions between coordinate planes. The in-plane shear strain between radial and circumferential directions incorporates a correction for the coordinate system's geometry:

\epsilon_{r\theta} = \frac{1}{2} \left[ \frac{1}{r} \frac{\partial u_r}{\partial \theta} + \frac{\partial u_\theta}{\partial r} - \frac{u_\theta}{r} \right].

The radial-axial shear strain is:

\epsilon_{rz} = \frac{1}{2} \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right),

and the circumferential-axial shear strain is:

\epsilon_{\theta z} = \frac{1}{2} \left[ \frac{1}{r} \frac{\partial u_z}{\partial \theta} + \frac{\partial u_\theta}{\partial z} \right].

These shear terms average the relevant displacement gradients, with factors of $1/r adjusting for the scaling in angular directions.^[33]^[41] The expressions for \boldsymbol{\epsilon} in cylindrical coordinates are derived by linearizing the change in the squared line element ds^2 = dr^2 + r^2 d\theta^2 + dz^2 for small displacements, which corresponds to the metric tensor of the coordinate system. This approach ensures the strain measures the relative deformation of material lines while neglecting higher-order terms in the displacement gradients.^[41]

Spherical Coordinates

In spherical coordinates (r, \theta, \phi), the infinitesimal strain tensor components are derived from the displacement field \mathbf{u} = u_r \mathbf{e}_r + u_\theta \mathbf{e}_\theta + u_\phi \mathbf{e}_\phi, where \mathbf{e}_r, \mathbf{e}_\theta, \mathbf{e}_\phi are the orthonormal basis vectors. The metric of the space is given by the line element ds^2 = dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta \, d\phi^2, which accounts for the curvature effects in the deformation measures under the infinitesimal approximation, where higher-order terms in the displacement gradient are neglected.^[41] The normal strain components are expressed as follows:

\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}{r} + \frac{u_\theta}{r} \cot \theta

The \varepsilon_{\phi\phi} component includes the \cot \theta term, which arises from the convergence of the meridional lines toward the poles, reflecting the geometry of spherical symmetry.^[41] The shear strain components are symmetric and given by:

\varepsilon_{r\theta} = \frac{1}{2} \left[ \frac{1}{r} \frac{\partial u_r}{\partial \theta} + \frac{\partial u_\theta}{\partial r} - \frac{u_\theta}{r} \right]

\varepsilon_{r\phi} = \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]

\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}{r} \cot \theta \right]

These expressions are particularly useful for problems involving radial and spherical symmetry, such as point loads or expansions in spherical geometries, where the coordinate system's alignment with the symmetry simplifies the tensor representation.^[41]

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