Elasticity tensor
The elasticity tensor, also known as the elastic stiffness tensor, is a fourth-order tensor that characterizes the linear elastic response of a material by relating the stress tensor to the infinitesimal strain tensor in the constitutive equation \sigma_{ij} = C_{ijkl} \epsilon_{kl}, where \sigma_{ij} and \epsilon_{kl} are the components of the symmetric second-order stress and strain tensors, respectively, and C_{ijkl} denotes the tensor components.[1][2] In three dimensions, this tensor has 81 potential components in its most general anisotropic form, but symmetries arising from the symmetry of the stress and strain tensors (minor symmetries: C_{ijkl} = C_{jikl} = C_{ijlk}) and the major symmetry from the existence of a strain energy potential (C_{ijkl} = C_{klij}) reduce the number of independent components to 21, with the requirement of a positive-definite strain energy density function ensuring thermodynamic stability.[1][2] For materials exhibiting higher degrees of symmetry, the elasticity tensor simplifies significantly; isotropic materials, for instance, are described by just two independent constants, commonly the Lamé parameters \lambda and \mu, or equivalently Young's modulus E and Poisson's ratio \nu, leading to the form C_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}).[1][2] In orthotropic materials, such as wood or composites, the tensor requires nine independent constants, while cubic crystals need only three.[1] The inverse relation, involving the compliance tensor S_{ijkl}, allows strain to be expressed as a function of stress: \epsilon_{ij} = S_{ijkl} \sigma_{kl}, with S being the inverse of C.[1] To facilitate computations, the elasticity tensor is often represented in Voigt notation as a 6×6 matrix, mapping the six independent components of stress and strain vectors, which preserves the major symmetries and enables efficient numerical analysis in finite element methods and other simulations of solid mechanics problems.[1] The tensor's components are material properties determined experimentally through techniques like ultrasonic wave propagation, resonant ultrasound spectroscopy, or static loading tests, and they must satisfy thermodynamic stability conditions, such as positive definiteness, to ensure the material's elastic behavior is physically realistic.[1][2] In applications ranging from structural engineering to geophysics, the elasticity tensor underpins the prediction of deformation, wave propagation, and failure in anisotropic solids like crystals, composites, and biological tissues.[2]Fundamentals
Definition
In linear elasticity, the fundamental constitutive relation links the stress tensor to the strain tensor through a linear mapping. The stress tensor \sigma_{ij}, a second-rank tensor, represents the internal forces per unit area acting across an infinitesimal surface element within a deformable continuum.[3] The strain tensor \varepsilon_{kl}, a symmetric second-rank tensor, measures the relative deformation or displacement gradients in the material.[4] Under the assumption of small deformations and linear material response, known as Hooke's law in its generalized tensorial form, the components of the stress tensor are related to those of the strain tensor by \sigma_{ij} = C_{ijkl} \varepsilon_{kl}, where summation over repeated indices k and l is implied, and C_{ijkl} denotes the components of the elasticity tensor. This fourth-rank tensor C_{ijkl} fully characterizes the material's elastic behavior by specifying how applied strains produce corresponding stresses. The elasticity tensor is a fourth-rank tensor in three-dimensional Euclidean space, possessing $3 \times 3 \times 3 \times 3 = 81 components in its most general form. Physically, C_{ijkl} quantifies the directional stiffness of the material, determining the resistance to deformation along specific axes and the coupling between different deformation modes.[5] This tensorial framework generalizes the scalar Hooke's law for uniaxial loading to arbitrary three-dimensional states, originating from Augustin-Louis Cauchy's foundational work in 1828 on the molecular theory of elasticity.[6]Notation Conventions
The elasticity tensor, denoted as C_{ijkl}, is a fourth-order tensor that relates the second-order stress tensor \sigma_{ij} to the second-order infinitesimal strain tensor \epsilon_{kl} through the constitutive equation \sigma_{ij} = C_{ijkl} \epsilon_{kl}, where the Einstein summation convention is implied over repeated indices k and l.[7] In this full tensor notation, the components C_{ijkl} are defined with respect to a Cartesian coordinate system, and the tensor possesses 81 components in general, though symmetries reduce the number of independent components in practical cases, such as to 21 for materials without additional symmetry assumptions.[8] To facilitate computational and engineering applications, the elasticity tensor is often represented in contracted forms. The Voigt notation maps the fourth-order tensor to a 6×6 matrix C_{\alpha\beta}, where the indices \alpha, \beta = 1, \dots, 6 correspond to specific pairings of the original tensor indices: $11 \to 1, $22 \to 2, $33 \to 3, $23 \to 4 (or $32 \to 4), $13 \to 5 (or $31 \to 5), and $12 \to 6 (or $21 \to 6).[](https://dspace.mit.edu/bitstream/handle/1721.1/105251/12665_2016_Article_5429.pdf?sequence=1&isAllowed=y) In this scheme, the [stress](/page/Stress) components are vectorized as \boldsymbol{\sigma} = [\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{23}, \sigma_{13}, \sigma_{12}]^T, while the strain vector incorporates a factor of 2 for [shear](/page/Shear) components to preserve the work conjugacy in the inner product: \boldsymbol{\epsilon} = [\epsilon_{11}, \epsilon_{22}, \epsilon_{33}, 2\epsilon_{23}, 2\epsilon_{13}, 2\epsilon_{12}]^T.[8] This results in the [matrix](/page/Matrix) relation \boldsymbol{\sigma} = \mathbf{C} \boldsymbol{\epsilon}, where \mathbf{C}$ is the elasticity matrix used extensively in finite element analysis and engineering simulations.[9] An alternative to Voigt notation is the Kelvin notation, which also employs a 6×6 matrix representation but vectorizes both stress and strain without the factor of 2 on shear strains, instead using \sqrt{2} factors to maintain tensorial properties and simplify transformations.[9] Specifically, the vectors are \boldsymbol{\sigma} = [\sigma_{11}, \sigma_{22}, \sigma_{33}, \sqrt{2}\sigma_{23}, \sqrt{2}\sigma_{13}, \sqrt{2}\sigma_{12}]^T and \boldsymbol{\epsilon} = [\epsilon_{11}, \epsilon_{22}, \epsilon_{33}, \sqrt{2}\epsilon_{23}, \sqrt{2}\epsilon_{13}, \sqrt{2}\epsilon_{12}]^T, ensuring that the matrix \mathbf{C} preserves the major and minor symmetries of the original tensor more naturally in numerical implementations.[8] This notation, originally proposed by Lord Kelvin in 1856, is particularly advantageous in contexts requiring invariant formulations, such as crystal physics.[9] The compliance tensor, denoted S_{ijkl}, is the inverse of the elasticity tensor, satisfying S_{ijkl} C_{klmn} = \delta_{im} \delta_{jn}, where \delta is the Kronecker delta, and it relates strain to stress via \epsilon_{ij} = S_{ijkl} \sigma_{kl}.[7] In matrix form, whether Voigt or Kelvin, the compliance matrix \mathbf{S} = \mathbf{C}^{-1} follows analogous index mappings, with adjustments for shear factors to ensure consistency in engineering applications.[8] For instance, in the general case without symmetries, the full S_{ijkl} has 81 components, mirroring the structure of C_{ijkl}, but reduces similarly under symmetry constraints.Symmetries
Intrinsic Symmetries
The elasticity tensor C_{ijkl}, which relates the stress tensor \sigma_{ij} to the strain tensor \varepsilon_{kl} via \sigma_{ij} = C_{ijkl} \varepsilon_{kl}, possesses intrinsic symmetries that stem from fundamental properties of the stress and strain tensors as well as the thermodynamic framework of linear elasticity.[10] The minor symmetries arise directly from the symmetry of the stress and strain tensors. Specifically, since the stress tensor is symmetric (\sigma_{ij} = \sigma_{ji}), it follows that C_{ijkl} = C_{jikl}; similarly, the symmetry of the strain tensor (\varepsilon_{kl} = \varepsilon_{lk}) implies C_{ijkl} = C_{ijlk}. These relations reduce the number of independent components of the fourth-rank tensor from 81 to 36, as the tensor can then be represented by a 6×6 matrix in Voigt notation with row and column symmetries.[10][2] The major symmetry, C_{ijkl} = C_{klij}, originates from the existence of a strain energy potential in hyperelastic materials, where the elastic energy density is given by the quadratic form W = \frac{1}{2} C_{ijkl} \varepsilon_{ij} \varepsilon_{kl}. This symmetry ensures that W is a scalar invariant under index permutation, and the positive definiteness of C_{ijkl} (\delta W > 0 for nonzero \varepsilon_{ij}) guarantees material stability under small deformations.[10] The major symmetry is thermodynamically grounded in the requirement that the stress derives from the potential via \sigma_{ij} = \frac{\partial W}{\partial \varepsilon_{ij}}, enforcing symmetry in the response functions.[11] This major symmetry is closely related to Onsager reciprocity principles, which stem from time-reversal invariance in non-dissipative thermodynamic systems; in elasticity, it manifests as the symmetry of the stiffness tensor, ensuring reciprocal relations between applied strains and resulting stresses.[12] When combined with the minor symmetries, these intrinsic properties further constrain the elasticity tensor to 21 independent components, forming a subspace of fully symmetric fourth-rank tensors in three dimensions.[10]Resulting Constraints
The intrinsic symmetries of the elasticity tensor significantly reduce the number of independent components required to describe linear elastic behavior. Without symmetries, the fourth-rank tensor has 81 components. The minor symmetries, stemming from the symmetry of the stress and strain tensors (\sigma_{ij} = \sigma_{ji} and \varepsilon_{ij} = \varepsilon_{ji}), reduce this to 36 independent components by enforcing C_{ijkl} = C_{jikl} = C_{ijlk}. The major symmetry, arising from the existence of a strain energy potential (ensuring thermodynamic consistency), further imposes C_{ijkl} = C_{klij}, yielding 21 independent components for the most general (triclinic) case.[13] In Voigt notation, which maps the tensor to a 6×6 matrix for computational convenience (as briefly referenced in standard notation conventions), these symmetries manifest as a symmetric matrix structure with 21 independent entries. The generic form for a triclinic material, where no additional crystal symmetries apply, is: \begin{pmatrix} C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\ C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\ C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\ C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \end{pmatrix} This matrix enforces the required symmetries, with zeros absent only due to higher material symmetries (e.g., in cubic cases). All off-diagonal elements are independent except for the inherent symmetry C = C^T.[14] For mechanical stability, the elasticity tensor must ensure positive strain energy for any non-zero deformation, expressed as the quadratic form W = \frac{1}{2} \varepsilon^T C \varepsilon > 0 for \varepsilon \neq 0, where W is the strain energy density, \varepsilon is the Voigt strain vector, and C is the stiffness matrix. This requires C to be positive definite, meaning all its eigenvalues are positive. Equivalently, Sylvester's criterion applies: all leading principal minors of the 6×6 symmetric matrix C must be positive.[1][13] These positive definiteness conditions translate to explicit numerical inequalities that depend on material symmetry, generalizing simpler 2D cases to 3D. For example, in 2D orthotropic materials (reducing to a 3×3 matrix), stability requires C_{11} > 0, C_{22} > 0, C_{66} > 0, and C_{11}C_{22} - C_{12}^2 > 0, with conditions like C_{12} + 2C_{66} > 0 emerging in isotropic limits where C_{66} = (C_{11} - C_{12})/2. In 3D, for cubic symmetry (3 independent components), the generalized Born stability criteria are C_{11} > |C_{12}|, C_{44} > 0, and C_{11} + 2C_{12} > 0, ensuring no imaginary phonon frequencies or structural instabilities. For lower symmetries like triclinic, the full 21 conditions revert to checking the 6 leading principal minors or eigenvalues numerically. These constraints not only enforce stability but also bound the feasible parameter space for experimental or computational determination of elastic constants.[13][15]Special Material Cases
Isotropic Materials
Isotropic materials exhibit elastic properties that are independent of direction, resulting in the elasticity tensor possessing only two independent constants, typically the Lamé parameters λ and μ. The fourth-rank elasticity tensor for such materials takes the form C_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}), where δ denotes the Kronecker delta, λ governs volumetric response, and μ represents the shear modulus.[16] This expression ensures that the material responds uniformly to applied stresses regardless of orientation, simplifying the general 21-component tensor to a structure with maximal symmetry.[16] In Voigt notation, which reduces the tensor to a 6×6 stiffness matrix by mapping indices (11→1, 22→2, 33→3, 23→4, 13→5, 12→6), the isotropic form exhibits a distinct pattern: the normal components are equal, the cross-normal terms are identical, and the shear components are uniform. The matrix is| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | λ+2μ | λ | λ | 0 | 0 | 0 |
| 2 | λ | λ+2μ | λ | 0 | 0 | 0 |
| 3 | λ | λ | λ+2μ | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | μ | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | μ | 0 |
| 6 | 0 | 0 | 0 | 0 | 0 | μ |
Cubic and Other Crystal Symmetries
In crystals exhibiting cubic symmetry, the elasticity tensor possesses the highest level of rotational invariance among anisotropic materials, resulting in only three independent elastic constants in Voigt notation: C_{11}, C_{12}, and C_{44}. The Voigt matrix for cubic crystals takes the form where C_{11} = C_{22} = C_{33}, C_{12} = C_{13} = C_{23}, C_{44} = C_{55} = C_{66}, and all other components are zero: \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} [18] This structure arises from the point group symmetries of the cubic system, including fourfold rotation axes along \langle 100 \rangle, threefold axes along \langle 111 \rangle, and mirror planes, which require the tensor to remain invariant under these operations, enforcing the equalities among components.[19] Face-centered cubic (FCC) metals, such as aluminum, exemplify this symmetry; for aluminum, typical values are C_{11} = 107 \, \mathrm{GPa}, C_{12} = 60 \, \mathrm{GPa}, and C_{44} = 28 \, \mathrm{GPa}.[20] The degree of elastic anisotropy in cubic crystals is often quantified by the Zener anisotropy factor A = \frac{2 C_{44}}{C_{11} - C_{12}}, which equals 1 for isotropic behavior and deviates from 1 to indicate directional variations in stiffness; for aluminum, A \approx 1.22. For crystals with lower symmetries, the number of independent elastic constants increases as fewer constraints are imposed by the point group operations. These operations—such as twofold rotations, mirrors, and glide planes—generate additional equalities or leave certain C_{ijkl} components distinct, reducing the full 21 independent components of the triclinic case stepwise.[21] The 32 crystal point groups, classified into 11 Laue classes for tensor properties (as inversion symmetry does not affect the elasticity tensor), yield the following independent constants for major symmetry classes:| Crystal System | Number of Independent Constants | Example Point Groups |
|---|---|---|
| Triclinic | 21 | 1, \bar{1} |
| Monoclinic | 13 | 2, m, 2/m |
| Orthorhombic | 9 | 222, mm2, mmm |
| Tetragonal | 6 or 7 | 4, \bar{4}, 4/m; 422, 4mm, \bar{4}2m, 4/mmm |
| Trigonal | 6 or 7 | 3, \bar{3}; 32, 3m, \bar{3}m |
| Hexagonal | 5 | 6, \bar{6}, 6/m; 622, 6mm, \bar{6}m2, 6/mmm |
| Cubic | 3 | 23, m\bar{3}; 432, \bar{4}3m, m\bar{3}m |