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Voigt notation

Voigt notation is a contracted system for representing symmetric tensors by mapping pairs of indices to single indices, thereby reducing the dimensionality of tensor expressions from multi-dimensional arrays to vectors or matrices, which simplifies calculations in , elasticity, and crystal physics. Named after the German physicist , who introduced it in his 1910 textbook Lehrbuch der Kristallphysik, the notation leverages the of tensors—such as and in —to eliminate redundant components, for instance, converting a symmetric second-rank 3×3 tensor into a six-component vector. In Voigt notation, the indices of a tensor are contracted according to the scheme where 11 maps to 1, 22 to 2, 33 to 3, 23 (or 32) to 4, 13 (or 31) to 5, and 12 (or 21) to 6, allowing second-rank symmetric tensors like the stress tensor \boldsymbol{\sigma} to be expressed as a column vector \{\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{23}, \sigma_{13}, \sigma_{12}\}^T. For fourth-rank tensors, such as the elasticity tensor \mathbf{C} relating stress and strain via \boldsymbol{\sigma} = \mathbf{C} : \boldsymbol{\varepsilon}, this results in a 6×6 stiffness matrix, where the equation simplifies to a matrix-vector multiplication \{\boldsymbol{\sigma}\} = [\mathbf{C}] \{\boldsymbol{\varepsilon}\}. A key convention in engineering applications involves scaling shear strain components by a factor of 2 (e.g., \varepsilon_4 = 2\varepsilon_{23}) to preserve the form of the strain energy density expression \frac{1}{2} \boldsymbol{\sigma} : \boldsymbol{\varepsilon}, though this can introduce inconsistencies if not handled carefully. The notation's primary value lies in its facilitation of numerical computations and visualization in anisotropic media, where up to 21 independent elastic constants may exist for the , as opposed to the full 81 components of the unreduced fourth-rank tensor. It is widely applied in for propagation, for finite element analysis, and for modeling symmetries, though variants like Mandel notation adjust the factors to address some limitations. Despite its utility for display and matrix-based solvers, Voigt notation is not invariant under tensor transformations and serves mainly as an algebraic tool rather than a fundamental geometric representation.

Introduction

Definition and Purpose

Voigt notation provides a compact for symmetric second-order tensors, such as the stress tensor \sigma or the strain tensor \varepsilon, by mapping their components into a 6×1 column or a 6-component array. The indices are ordered as 11, 22, , 23, , 12, which correspond to positions 1 through 6, respectively, exploiting the tensor's where \varepsilon_{ij} = \varepsilon_{ji} to eliminate redundant components and reduce the nine potential entries to six independent ones. This approach assumes familiarity with second-order tensors but relies fundamentally on their for applicability. The purpose of Voigt notation is to streamline by converting complex multidimensional operations into familiar -vector manipulations, which is especially beneficial in physics and computations. In , it facilitates the application of , where the fourth-order stiffness tensor relates the stress and strain tensors; under Voigt notation, this relationship simplifies to a \boldsymbol{\sigma} = \mathbf{C} \boldsymbol{\varepsilon}, with \mathbf{C} as a 6×6 . A representative example illustrates this for the strain tensor \varepsilon, where the Voigt vector takes the form \begin{pmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{13} \\ 2\varepsilon_{12} \end{pmatrix}, with the factor of 2 introduced for the shear components to maintain consistency in energy expressions and preserve the tensor's physical norms during transformations.

Historical Background

Voigt notation was introduced by the German physicist in 1910 as part of his comprehensive study on the elastic properties of . In his seminal textbook Lehrbuch der Kristallphysik, Voigt developed this compact representation to simplify the description of anisotropic material behavior, particularly in the context of stress-strain relations for crystalline structures. This notation allowed for a more manageable handling of the fourth-rank by mapping its components into a 6×6 form, facilitating calculations in crystal physics without altering the underlying tensorial nature. The introduction of Voigt notation occurred amid rapid advancements in during the early 20th century, a period when was being formalized to address complex physical phenomena. Voigt built upon the foundational work in absolute differential calculus by , who developed the framework around 1890, and , who refined tensor notation and applications in the subsequent decades. These mathematical tools enabled precise modeling of anisotropic elasticity in crystals, where traditional scalar or vector approaches proved inadequate for capturing directional dependencies in material properties. Voigt's efforts were specifically aimed at linearizing the equations of elasticity for crystalline media, providing a practical bridge between theoretical tensor formulations and experimental observations of crystal deformations. Following its initial proposal, Voigt notation gradually gained prominence in the mid-20th century, particularly as emerged in the post-1950s era. It became standardized in literature, such as S. P. Timoshenko and J. N. Goodier's Theory of Elasticity (1951), where it was employed to streamline matrix-based solutions for problems. This adoption aligned with the rise of digital computing and finite element methods, which relied on the notation's vector-matrix efficiency to solve complex anisotropic systems before widespread computer availability. By the latter half of the century, it had become a in both theoretical and applied analyses of materials.

Core Notation

Component Mapping

In Voigt notation, a symmetric second-rank tensor T_{ij} in three dimensions is mapped to a six-component \mathbf{T}_V, where the independent components are ordered by their indices as follows: the normal components T_{11}, T_{22}, T_{33} correspond to positions 1 through 3, and the components follow in the T_{23}, T_{13}, T_{12} for positions 4 through 6. This ordering reduces the 9 elements of the full tensor to 6 unique values, exploiting the T_{ij} = T_{ji}. For stress tensors, which are stress-like (contravariant), the mapping uses direct components without scaling: \boldsymbol{\sigma}_V = [\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{23}, \sigma_{13}, \sigma_{12}]^T. In contrast, for strain tensors, which are strain-like (covariant), the shear components are scaled by a factor of 2 to align with shear strain conventions: \boldsymbol{\varepsilon}_V = [\varepsilon_{11}, \varepsilon_{22}, \varepsilon_{33}, 2\varepsilon_{23}, 2\varepsilon_{13}, 2\varepsilon_{12}]^T. This scaling ensures that the inner product of the and tensors in their original form, \boldsymbol{\sigma} : \boldsymbol{\varepsilon}, equals the of their Voigt vectors, \boldsymbol{\sigma}_V \cdot \boldsymbol{\varepsilon}_V, preserving work conjugacy and the of the . The absence of scaling for components maintains consistency with the equations, where the of the tensor directly governs balance without adjustment. As an example, consider a diagonal stress tensor with components \sigma_{11} = \sigma_x, \sigma_{22} = \sigma_y, \sigma_{33} = \sigma_z, and all off-diagonal elements zero. In Voigt notation, this maps to \boldsymbol{\sigma}_V = [\sigma_x, \sigma_y, \sigma_z, 0, 0, 0]^T, highlighting how pure stresses occupy the first three positions while terms vanish.

Mnemonic Rule

A standard mnemonic for recalling the index sequence in Voigt notation involves writing the second-order tensor in form, striking out the diagonal elements, continuing with the remaining elements in the third column, and then returning to the first off-diagonal element in the first row. This process yields the order: 11, 22, 33, 23, 13, 12. A visual mnemonic depicts the indices as the edges of a with vertices labeled 1, 2, and 3, where the connections represent 23 (between 2 and 3), 13 (between 1 and 3), and 12 (between 1 and 2) in a consistent traversal, such as clockwise starting from the base. This geometric representation exploits the inherent symmetry of the tensor to aid intuitive recall. Such mnemonics prove invaluable for engineers and physicists performing manual derivations or coding tensor operations, enabling rapid index assignment without consulting tables. The ordering originates from Woldemar Voigt's formulation, which employed cyclic permutations to capture the of cubic crystals in elasticity theory. This approach is specific to three-dimensional cases; in two dimensions, the components simplify to 11, 22, 12, omitting the third direction.

Related Notations

Mandel Notation

Mandel notation is an alternative representation of symmetric second-order tensors, such as and , introduced by Jean Mandel in the 1960s for applications in , particularly in the analysis of plastic waves. Developed within the context of finite element methods and simulations, it maps these tensors to six-component vectors while incorporating specific scaling to facilitate computational efficiency. Unlike other notations, Mandel notation ensures in the basis, making it suitable for numerical implementations where preserving tensor invariants is crucial. The component mapping in Mandel notation follows the same index ordering as Voigt notation but applies orthogonal to the terms. For the strain tensor \boldsymbol{\varepsilon}, the Mandel is defined as \boldsymbol{\varepsilon}_M = \begin{pmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \sqrt{2} \varepsilon_{23} \\ \sqrt{2} \varepsilon_{13} \\ \sqrt{2} \varepsilon_{12} \end{pmatrix}, where the normal components remain unscaled, and the components are multiplied by \sqrt{2}. This representation extends analogously to the stress tensor. The transformation from a Voigt-like (using tensor components without engineering factors) to the Mandel involves a diagonal \mathbf{N} with entries $1, 1, 1, \sqrt{2}, \sqrt{2}, \sqrt{2}, such that \boldsymbol{\varepsilon}_M = \mathbf{N} \boldsymbol{\varepsilon}_V. The primary purpose of this scaling is to ensure that the Euclidean norm of the Mandel vector equals the of the original tensor, thereby preserving distances and inner products in the parameter space. Specifically, the squared Euclidean norm \|\boldsymbol{\varepsilon}_M\|^2 = \sum_{i=1}^6 (\varepsilon_M)_i^2 matches the \boldsymbol{\varepsilon} : \boldsymbol{\varepsilon} = \varepsilon_{11}^2 + \varepsilon_{22}^2 + \varepsilon_{33}^2 + 2\varepsilon_{12}^2 + 2\varepsilon_{13}^2 + 2\varepsilon_{23}^2, which simplifies linear algebra operations in simulations of . This property was particularly advantageous for Mandel's work on computational efficiency in three-dimensional problems.

Key Differences

Voigt and Mandel notations differ primarily in their scaling conventions for shear components, which affect how tensor invariants like and norms are preserved in form. In Voigt notation, the strain incorporates a factor of 2 for components (using engineering shear strains, \gamma_{ij} = 2\epsilon_{ij}) to ensure that the of the and strain vectors equals the tensor inner product \boldsymbol{\sigma} : \boldsymbol{\epsilon}, thereby simplifying the expression of without additional coefficients; however, the vector uses no such for components. In contrast, Mandel notation applies a factor of \sqrt{2} to the components of both the strain and vectors, which eliminates the need for separate conventions between contravariant (stress-like) and covariant (strain-like) forms while preserving the inner product. These scaling choices lead to distinct implications for vector norms. The Euclidean (L2) norm of the Voigt strain vector corresponds to \sqrt{2W} where W is the strain energy density for certain cases, but it does not directly match the Frobenius norm of the underlying tensor due to the non-orthonormal basis introduced by the factor of 2 on shear terms. In Mandel notation, the vector's L2 norm equals the tensor's Frobenius norm precisely, as the \sqrt{2} scaling creates an orthonormal basis that simplifies norm computations and maintains tensorial invariances in vector operations. Both notations employ the same component ordering for symmetric second-order tensors: positions 1 through 3 for the normal components ($11, 22, 33) and 4 through 6 for the components ($23, 13, 12). However, the differing scalings result in distinct matrix representations; for instance, the in Mandel notation transforms differently from its Voigt counterpart, often requiring adjustment factors to relate the two. Voigt notation offers simplicity for manual calculations and derivations involving energy-based constitutive laws, as its scaling aligns directly with work conjugacy without introducing irrational factors like \sqrt{2}. Mandel notation, by contrast, is favored in numerical methods such as finite element analysis and optimization algorithms, where the enhances computational efficiency, in eigensystems, and robustness against errors from dual bases. Conversion between the notations for strain vectors can be achieved via a diagonal N, such that \boldsymbol{\epsilon}_m = N \boldsymbol{\epsilon}_v, where N = \operatorname{diag}(1, 1, 1, 1/\sqrt{2}, 1/\sqrt{2}, 1/\sqrt{2}). For , the transformation uses a different diagonal matrix with \sqrt{2} on the shear entries to account for the lack of initial scaling in Voigt shear components.

Applications

In Continuum Mechanics

In continuum mechanics, Voigt notation facilitates the representation of the stress-strain relations for deformable solids by transforming the second-order stress and strain tensors into six-component vectors, denoted as \boldsymbol{\sigma}_v and \boldsymbol{\varepsilon}_v, respectively. The generalized is expressed in this form as \boldsymbol{\sigma}_v = \mathbf{C} \boldsymbol{\varepsilon}_v, where \mathbf{C} is the 6×6 . This matrix notation exploits the symmetry of the stress and strain tensors (each with six independent components) and the major and minor symmetries of the fourth-order , reducing its potential 81 components to at most 21 independent ones. The equations, which enforce balance of in the absence of body forces, are given in tensor form as \partial \sigma_{ij}/\partial x_j = 0. In Voigt notation, this divergence-free condition is recast into a equation suitable for numerical methods like finite elements, where the \boldsymbol{\sigma}_v contributes to the weak form of the through over the . This vectorized approach simplifies the assembly of global matrices in computational simulations of deformable . Kinematic relations linking strains to displacements are also expressed efficiently in Voigt notation, with the strain vector related to the displacement vector \mathbf{u} via \boldsymbol{\varepsilon}_v = \mathbf{B} \mathbf{u}, where \mathbf{B} is the strain-displacement operator matrix derived from spatial derivatives of functions. This is central to discretizing the governing equations in finite element methods for solving problems in . Voigt notation is essential in formulating and solving boundary value problems for isotropic materials, where the stiffness matrix \mathbf{C} simplifies to a form dependent on only two parameters, the Lamé constants \lambda and , as \mathbf{C} = \begin{bmatrix} \lambda + 2\mu & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda + 2\mu & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & \lambda + 2\mu & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu \end{bmatrix}. This structure enables straightforward incorporation of material into the stress-strain law for and kinematic analyses. As an illustrative example, consider uniaxial tension along the x-direction, where the tensor has only \sigma_{xx} \neq 0. In Voigt notation, the vector becomes \boldsymbol{\sigma}_v = [\sigma_{xx}, 0, 0, 0, 0, 0]^T, with the corresponding vector \boldsymbol{\varepsilon}_v computed via the to yield non-zero normal components influenced by Poisson's effect.

In Elasticity Theory

In elasticity theory, Voigt notation plays a central role in constitutive modeling by simplifying the representation of the fourth-order for anisotropic materials. This tensor, which relates the second-order and tensors and possesses 81 components in its general form, is contracted into a symmetric 6×6 \mathbf{C} with at most 21 independent components. This reduction arises from the minor symmetries of the and (reducing from 81 to 36 components) and the major symmetry \mathbf{C} = \mathbf{C}^T (further to 21), enabling efficient computation of - relations in materials lacking . Voigt introduced this notation in his seminal work on crystal physics to characterize the elastic behavior of specific crystal classes, particularly those with cubic and hexagonal symmetries, where the number of independent constants is further constrained by the material's point group. For cubic crystals, the stiffness matrix reduces to three independent parameters (C_{11}, C_{12}, C_{44}), while hexagonal symmetry yields five. The compliance matrix \mathbf{S} = \mathbf{C}^{-1} is similarly expressed in Voigt form, facilitating the inversion needed for strain-based analyses in anisotropic elasticity. The formulation ensures thermodynamic consistency in the strain energy density, expressed as U = \frac{1}{2} \boldsymbol{\varepsilon}_v^T \mathbf{C} \boldsymbol{\varepsilon}_v, where \boldsymbol{\varepsilon}_v is the six-component strain vector. The Voigt scaling—defining shear strains as \varepsilon_4 = 2\varepsilon_{23}, \varepsilon_5 = 2\varepsilon_{31}, \varepsilon_6 = 2\varepsilon_{12}—preserves the quadratic form's equivalence to the tensorial expression U = \frac{1}{2} C_{ijkl} \varepsilon_{ij} \varepsilon_{kl}, avoiding discrepancies in energy calculations for sheared states. In and , Voigt notation underpins the Voigt-Reuss bounds for polycrystal averaging, providing estimates of effective elastic moduli from single-crystal data. Voigt's 1910 uniform-strain assumption yields an upper bound on , assuming constant across grains, while Reuss's 1929 uniform-stress model provides a lower bound, assuming constant stress; these bounds bracket the true polycrystalline response and are widely used for bounding seismic velocities and mechanical properties. For orthotropic materials, which exhibit three orthogonal planes of symmetry (common in wood or composites), the Voigt stiffness matrix \mathbf{C} adopts a block-diagonal structure: \mathbf{C} = \begin{pmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ & C_{22} & C_{23} & 0 & 0 & 0 \\ & & C_{33} & 0 & 0 & 0 \\ & & & C_{44} & 0 & 0 \\ & \sym & & & C_{55} & 0 \\ & & & & & C_{66} \end{pmatrix}, with nine independent constants derived from engineering moduli E_1, E_2, E_3, shear moduli G_{12}, G_{13}, G_{23}, and Poisson's ratios \nu_{12}, \nu_{13}, \nu_{23}. This form highlights decoupled shear behaviors and is essential for modeling layered or fibrous anisotropic solids.