1 Definition and purpose
Voigt notation is a compact indexing scheme used to rewrite tensor quantities, especially symmetric tensors, in vector or matrix form. By replacing paired tensor indices with a single index, it reduces notational complexity and makes calculations, tabulation, and numerical implementation more convenient. The method is common in continuum mechanics, materials science, and engineering, where stress, strain, and elasticity tensors are frequently symmetric.
The main purpose of the notation is efficiency. It allows tensor equations to be expressed in a reduced form without listing every component separately. This is particularly useful when a tensor has repeated or redundant entries due to symmetry. In practice, Voigt notation is a bookkeeping device rather than a change in the underlying mathematics.
1.1 Historical development
The notation is named after Woldemar Voigt, whose work in crystallography and elasticity helped standardize compact tensor representation in scientific writing. Its adoption grew with the development of classical elasticity theory and later with computational mechanics, where compact arrays were easier to manipulate than full tensor component lists. Over time, the convention became common in engineering texts and software, although not always with identical ordering or scaling choices.
1.2 Motivation for compact tensor representation
Many tensors of interest in mechanics are symmetric, meaning that several components are equal by definition. A second-order symmetric tensor in three dimensions has six independent components rather than nine. Voigt notation collects those independent components into a six-component vector or a 6 × 6 matrix for fourth-order tensors. This reduction simplifies algebra, especially when writing constitutive laws, solving boundary-value problems, or storing data in code.
1.3 Scope of application
Voigt notation is used wherever symmetric tensor quantities are handled in a reduced coordinate form. Typical applications include stress analysis, strain analysis, elastic moduli, anisotropic material models, and finite element computation. It is less suitable for contexts where full tensor transformation properties must be kept explicit or where antisymmetric components are important.
2 Index mapping
Voigt notation assigns a single index to each ordered pair of tensor indices. The mapping is not universal, and different disciplines may adopt different component orders. As a result, correct interpretation depends on the convention used in the source material or software package.
2.1 Mapping of second-order tensors
A second-order tensor with indices i and j can be reduced to a one-index form when the tensor is symmetric. The six independent components in three dimensions, or three in two dimensions, are listed in a fixed sequence. The choice of order is conventional, but it must be applied consistently.
2.1.1 Three-dimensional convention
In a common three-dimensional scheme, the normal components are listed first, followed by shear components. One frequent mapping is 11 → 1, 22 → 2, 33 → 3, 23 or 32 → 4, 13 or 31 → 5, and 12 or 21 → 6. Under this arrangement, a symmetric tensor becomes a six-component vector.
2.1.2 Two-dimensional convention
In two dimensions, only three independent components are needed for a symmetric second-order tensor. A standard mapping is 11 → 1, 22 → 2, and 12 or 21 → 3. This reduced form is often used in plane stress and plane strain calculations.
2.2 Mapping of fourth-order tensors
Fourth-order tensors, such as stiffness and compliance tensors, can be represented as matrices in Voigt form. Each pair of tensor indices is converted to a single matrix index, producing a 6 × 6 array in three dimensions. This representation is especially useful for linear elasticity, where the relation between stress and strain becomes a matrix equation.
2.3 Common component ordering schemes
Several ordering schemes exist for the Voigt indices. Some place the shear terms after the normal terms, while others preserve a different geometric pattern or follow software-specific traditions. The most important practical rule is consistency, since changing the order without adjusting the matrix entries leads to incorrect results. Care is also needed because some conventions modify shear scaling, which affects matrix values even when the order is the same.
3 Voigt notation for stress and strain
Stress and strain are the most familiar tensor quantities written in Voigt form. Since the stress tensor is symmetric in classical continuum mechanics and the small-strain tensor is symmetric by definition, both fit naturally into compact representation. This makes them central examples in textbooks and computational routines.
3.1 Stress tensor representation
The Cauchy stress tensor is commonly written as a six-component Voigt vector in three dimensions. The normal stresses occupy the first entries, and the shear stresses occupy the remaining ones. In this form, the tensor can be inserted into matrix equations for constitutive laws or equilibrium calculations with minimal notation.
3.2 Strain tensor representation
The strain tensor may also be written in Voigt form, but the treatment of shear terms requires special attention. Depending on the convention, the shear components may be stored as tensorial shear strains or engineering shear strains. The difference affects factors of two in formulas and matrix coefficients.
3.2.1 Engineering strain versus tensorial strain
Tensorial shear strain uses the symmetric off-diagonal tensor component directly, while engineering shear strain doubles that quantity. Engineering strain is common in applied mechanics because it aligns with displacement gradients used in elementary formulas. In Voigt notation, this can lead to either a direct component mapping or a scaled mapping, so the context must be checked carefully.
3.2.2 Shear component conventions
Shear components are the main source of ambiguity in Voigt notation. Some authors define the shear entries in the strain vector as 2ε12, 2ε23, and 2ε13, while the stress vector is left unscaled. Other approaches use unscaled shear strains but compensate through the stiffness matrix. These choices preserve the physical relationship when used consistently, but mixed conventions can produce errors.
3.3 Symmetry considerations
Voigt notation depends on symmetry to eliminate redundant entries. For symmetric tensors, components such as σ12 and σ21 are identical, so only one needs to be stored. The compact form therefore captures all independent information while omitting repeated values. If a tensor is not symmetric, Voigt notation is incomplete unless additional conventions are introduced.
4 Elasticity and constitutive relations
Linear elasticity is the setting in which Voigt notation is most widely used. The stress-strain relation can be written as a matrix equation, making it easy to express anisotropic material behavior and to implement material laws numerically. The reduced form also clarifies the structure of isotropic, orthotropic, and more general stiffness models.
4.1 Hooke's law in Voigt form
Hooke's law in tensor notation relates stress to strain through a fourth-order elastic tensor. In Voigt form, the same relation becomes a matrix-vector equation. This representation is compact and computationally efficient, especially when the elastic tensor has symmetry properties that reduce the number of independent constants.
4.2 Stiffness matrix representation
The stiffness matrix is the Voigt matrix corresponding to the elastic moduli of a material. In three dimensions, it is typically a 6 × 6 matrix. Its entries reflect how each stress component depends on each strain component. Symmetry of the underlying elastic tensor often implies symmetry of the stiffness matrix as well.
4.2.1 Isotropic materials
For isotropic materials, the stiffness matrix has only two independent material constants, often expressed through the Lamé parameters, Young's modulus, and Poisson's ratio. The Voigt matrix shows a highly regular pattern with repeated diagonal and off-diagonal terms. This compact structure reflects the rotational symmetry of isotropic elasticity.
4.2.2 Orthotropic materials
Orthotropic materials have different properties along three orthogonal material axes. Their stiffness matrices contain more independent constants than isotropic materials, but the matrix still exhibits structured symmetry. Voigt notation is especially useful here because it organizes the many elastic coefficients into a standard tabular form.
4.3 Compliance matrix representation
The compliance matrix is the inverse of the stiffness matrix, when such an inverse exists. In Voigt notation, it gives strain as a function of stress. Compliance matrices are useful in experimental mechanics and in material characterization, where measured stresses are converted into predicted strains.
4.4 Conversion between tensor and matrix forms
Conversion requires a fixed index map and, in many cases, a shear scaling rule. The user must translate each tensor component into the correct vector or matrix entry and apply any necessary factors of two. When the convention is handled properly, the Voigt form is algebraically equivalent to the tensor form for symmetric quantities.
5 Dimensional variants
Voigt notation changes with the dimensional setting of the problem. Two-dimensional models use fewer components, while three-dimensional models retain the full set of independent symmetric terms. More abstract extensions exist, but they are less common in standard engineering applications.
5.1 Two-dimensional Voigt notation
In two-dimensional problems, the representation is reduced to three components for symmetric second-order tensors. This is particularly convenient in idealized plane problems, where out-of-plane behavior is either constrained or neglected. The exact form depends on whether the model is plane stress or plane strain.
5.1.1 Plane stress
Plane stress assumes that the out-of-plane stress components vanish. The remaining in-plane components can be assembled into a three-component Voigt vector. This model is widely used for thin plates and sheet-like structures.
5.1.2 Plane strain
Plane strain assumes that out-of-plane strain components are zero or negligible. This approximation is common in long bodies with uniform cross-section. The Voigt representation again uses a reduced vector, but the constitutive matrix differs from the plane stress case.
5.2 Three-dimensional Voigt notation
Three-dimensional Voigt notation is the standard form for full continuum mechanics calculations. It typically uses six components for symmetric second-order tensors and 6 × 6 matrices for fourth-order tensors. This format is central to isotropic and anisotropic elasticity, as well as to many finite element formulations.
5.3 Generalized higher-dimensional formulations
The same indexing idea can be extended beyond ordinary three-dimensional mechanics. In more abstract settings, symmetric tensors in n dimensions may be mapped to vectors whose length equals n(n + 1)/2. Such generalizations are mathematically straightforward, though they are less often called Voigt notation in everyday engineering use.
6 Alternative conventions
Several related notations have been developed to improve mathematical properties or preserve inner products more naturally. These alternatives are often discussed alongside Voigt notation because they use a similar component reduction but different scaling rules. Choosing among them depends on whether the priority is simplicity, orthonormality, or numerical convenience.
6.1 Mandel notation
Mandel notation is a scaled compact tensor representation that adjusts shear components to preserve certain inner-product properties. It is often preferred in theoretical work because it treats the reduced basis more symmetrically. Compared with Voigt notation, it uses different normalization for shear terms.
6.2 Kelvin notation
Kelvin notation also rescales tensor components, usually in a way that makes the reduced representation orthonormal under the natural tensor inner product. This is advantageous in spectral analysis and tensor algebra. Like Mandel notation, it differs from standard Voigt form by more than a mere relabeling.
6.3 Comparison with Voigt notation
Voigt notation is simpler and more common in engineering practice, while Mandel and Kelvin forms are often more mathematically elegant. The chief distinction lies in scaling of shear terms and in the resulting behavior of matrix operations. For many practical calculations, Voigt notation is adequate if conventions are handled carefully.
6.4 Advantages and limitations
The main advantage of Voigt notation is compactness. It reduces storage requirements and makes equations easier to display. Its limitation is that it can obscure tensor transformation rules and may introduce factor-of-two ambiguities, especially in shear terms. It also depends on convention, which can complicate communication between different fields or software tools.
7 Computational use
Voigt notation is deeply embedded in computational mechanics because it aligns well with arrays, vectors, and matrices. It simplifies code structures and enables efficient storage of material coefficients. As a result, it is a standard tool in finite element and constitutive model implementations.
7.1 Finite element methods
Finite element formulations often use Voigt notation to express strain-displacement relations and constitutive equations. The approach permits straightforward assembly of element matrices and stress updates. It is particularly valuable for materials with anisotropic elasticity, where the stiffness matrix has many nonzero entries.
7.2 Numerical implementation
In code, Voigt arrays are often used to store stress, strain, and elasticity data in fixed-size vectors or matrices. Developers must define a consistent component order and a matching shear convention. Small mismatches between mathematical formulas and software storage patterns are a common source of error.
7.3 Matrix assembly and storage efficiency
Because many tensor components are redundant under symmetry, Voigt notation reduces memory use and simplifies matrix assembly. This can improve performance in large-scale simulations. The savings are modest for a single tensor but significant across many elements and integration points.
7.4 Common software conventions
Different software packages may adopt different Voigt orderings or shear scalings. Some use tensorial shear strains, while others use engineering shear strains in the stored vector. When transferring data between programs, users must verify the convention to avoid incompatible results.
8 Mathematical properties
Voigt notation preserves the essential information of symmetric tensors, but it does not automatically preserve every tensor property in reduced form. Understanding its algebraic behavior helps prevent misuse. In particular, the treatment of inner products and transformations depends on the chosen scaling.
8.1 Symmetry and redundancy reduction
The notation exploits symmetry to remove duplicate components. A symmetric second-order tensor with six independent values in three dimensions can be represented without loss of information. This reduction is the mathematical basis for the method’s usefulness in mechanics.
8.2 Inner products and norm preservation
Standard Voigt form does not always preserve the natural tensor inner product unless special weighting is introduced. This issue arises because shear components may be counted differently in vector form than in tensor form. Scaled alternatives such as Mandel or Kelvin notation address this more directly.
8.3 Tensor transformation behavior
A full tensor transforms according to coordinate changes, whereas a Voigt vector transforms by an induced matrix rule that depends on the chosen basis and scaling. This can make the reduced form less transparent under rotations and general frame changes. For that reason, tensor notation is often preferred when geometric behavior is the focus.
9 Examples
Examples make the component mapping easier to understand. The same physical tensor can look quite different in full notation and in Voigt form, but the underlying quantities remain unchanged. The usefulness of the notation becomes clear when the conversion is carried through step by step.
9.1 Stress vector example
A symmetric stress tensor with components σ11, σ22, σ33, σ23, σ13, and σ12 can be written as a six-entry vector in Voigt form. Under a common ordering, the vector lists the normal stresses first and the shear stresses afterward. This allows the stress state to be used directly in matrix equations.
9.2 Elastic stiffness matrix example
An isotropic stiffness law can be written as a 6 × 6 matrix multiplying the strain vector. The matrix contains repeated values determined by the elastic constants. Even though the original fourth-order tensor has many index combinations, the Voigt matrix offers a compact and manageable representation.
9.3 Conversion walkthroughs
To convert from tensor notation to Voigt form, one identifies the independent symmetric components, assigns them to the chosen order, and applies any required scaling. The reverse conversion restores the full tensor by copying symmetric entries into their paired positions. The process is routine once the convention is fixed, but errors often occur when the shear terms are handled inconsistently.
10 Common pitfalls
Voigt notation is convenient, but it is also a frequent source of mistakes. Most errors arise from inconsistent conventions rather than from the notation itself. Careful reading of definitions is therefore essential.
10.1 Ordering mismatches
A common problem is assuming that all sources use the same component order. One text may list shear terms as 23, 13, 12, while another uses 23, 12, 13 or a different sequence altogether. Such differences can silently corrupt calculations if matrices are copied without reindexing.
10.2 Factor-of-two issues in shear terms
Shear components are often scaled differently in strain vectors and constitutive matrices. Missing or duplicating a factor of two can lead to incorrect stresses, strains, or energy values. This issue is especially common when translating between engineering formulas and tensor notation.
10.3 Confusion between engineering and tensorial strain
Engineering strain and tensorial strain are closely related but not identical in shear terms. If one convention is used in the strain vector and another is assumed in the stiffness matrix, the resulting equations will not match. Clear labeling of the strain definition is therefore important.
10.4 Mixing conventions across disciplines
Different fields, textbooks, and software packages may apply Voigt notation differently. A formula copied from materials science may not match the convention used in finite element software or in a crystallography reference. The safest practice is to verify both the ordering and the scaling before combining results from multiple sources.