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Tensor Notation and Mathematical Foundations

This chapter consolidates the conventions for tensor notation, index notation, and Voigt notation used in subsequent chapters of the FrontISTR Theory Manual. Only pure notation rules that do not depend on the definitions of physical quantities are described here. Notation conventions that depend on the configuration associated with continuum motion (reference/current configuration), such as the distinction between uppercase and lowercase letters and between material and spatial derivatives, are given in Motion, Deformation, and Strain, where the concept of configuration is introduced.

Tensor Notation

Boldface Vector and Tensor Notation

As a typographic convention, scalar quantities and components of vectors and tensors are written in normal type (such as \(\rho\) and \(x_i\)), while vector and tensor quantities themselves are written in boldface (such as \(\boldsymbol{a}\), \(\boldsymbol{E}\), and \(\boldsymbol{\sigma}\)).

Einstein Summation Convention

Unless otherwise stated, the Einstein summation convention is used: when the same index appears twice in the same term, summation is performed over that index. For example,

\[ a_i b_i = \sum_i a_i b_i, \quad A_{ij} B_{ij} = \sum_{i,j} A_{ij} B_{ij}, \quad \frac{\partial a_i}{\partial x_i} = \sum_i \frac{\partial a_i}{\partial x_i} \]

and the \(i,j\) component of the product \(\boldsymbol{C} = \boldsymbol{A}\boldsymbol{B}\) of tensors \(\boldsymbol{A}\) and \(\boldsymbol{B}\) is

\[ C_{ij} = A_{ik} B_{kj} \]

As an index convention, when \(i, j, k, l, \ldots\) are used as indices without further qualification, they denote indices associated with degrees of freedom (\(1, 2, 3\) in three dimensions), while lowercase Greek letters \(\alpha, \beta, \gamma, \ldots\) used as indices denote indices associated with the nodes constituting an element.

Inner Product, Transpose, and Tensor Product

The transpose of tensor \(\boldsymbol{A}\) is denoted by \(\boldsymbol{A}^T\). The inner product (double contraction) of second-order tensors \(\boldsymbol{A}\) and \(\boldsymbol{B}\) is written as

\[ \boldsymbol{A} : \boldsymbol{B} \equiv A_{ij} B_{ij} \]

The inner product of vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) is \(\boldsymbol{a} \cdot \boldsymbol{b} = a_i b_i\), and their tensor product is \((\boldsymbol{a} \otimes \boldsymbol{b})_{ij} = a_i b_j\).

Voigt Notation

Stress and strain are symmetric second-order tensors with respect to the degrees of freedom, and the coefficients representing the linear relationship between them form a fourth-order tensor. Handling these directly in a program is disadvantageous in terms of computational cost and coding (multidimensional arrays and deeply nested loops). Therefore, general finite element programs exploit symmetry and compress stress and strain into column vectors and the fourth-order elasticity tensor into a two-dimensional matrix. This is called Voigt notation.

Hereafter, the matrix or vector representation of a tensor quantity \(\boldsymbol{A}\) is denoted by \(\hat{A}\) to distinguish it from the original tensor.

Vector Representation of Stress and Strain

For the symmetric stress tensor \(\boldsymbol{\sigma}\) and strain tensor \(\boldsymbol{\varepsilon}\), in two dimensions,

\[ \hat{\sigma} = \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix}, \qquad \hat{\varepsilon} = \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ 2 \varepsilon_{12} \end{bmatrix} \]

in three dimensions,

\[ \hat{\sigma} = \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{12} \\ \sigma_{23} \\ \sigma_{31} \end{bmatrix}, \qquad \hat{\varepsilon} = \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2 \varepsilon_{12} \\ 2 \varepsilon_{23} \\ 2 \varepsilon_{31} \end{bmatrix} \]

they are handled as column vectors as shown above. The same rule applies to derived quantities such as variations and differentials. Note that the shear components on the strain side have a factor of 2 (the stress side does not). Because of this asymmetry, the tensor inner product can be written compactly as the vector inner product

\[ \boldsymbol{\sigma} : \delta\boldsymbol{\varepsilon} = \sigma_{ij}\,\delta \varepsilon_{ij} = \sum_{i=j} \sigma_{ij}\,\delta \varepsilon_{ij} + 2\sum_{i<j} \sigma_{ij}\,\delta \varepsilon_{ij} = \hat{\sigma}^T\,\delta\hat{\varepsilon} \]

In implementation, the factor of 2 on the shear strain components is easy to mishandle, so the convention "absent for stress / present for strain" must always be kept in mind. Exactly the same Voigt convention is applied to quantities in the reference configuration (the second Piola-Kirchhoff stress \(\boldsymbol{S}\) and Green-Lagrange strain \(\boldsymbol{E}\)).

Matrix Representation of a Fourth-Order Tensor

For the linear relationship between stress and strain, \(\boldsymbol{\sigma} = \boldsymbol{C} : \boldsymbol{\varepsilon}\) (in component form, \(\sigma_{ij} = C_{ijkl} \varepsilon_{kl}\)), using the symmetry of \(\boldsymbol{\varepsilon}\) gives

\[ \sigma_{ij} = \tilde{C}_{ijkl} \varepsilon_{kl}, \qquad \tilde{C}_{ijkl} = \frac{1}{2}\bigl( C_{ijkl} + C_{ijlk} \bigr) \]

In three dimensions, arranging \(\tilde{C}\) as the 6×6 material matrix \(\hat{\tilde{C}}\) allows the original tensor relation to be expressed as

\[ \hat{\sigma} = \hat{\tilde{C}}\, \hat{\varepsilon} \]

Hereafter in this manual, unless otherwise stated, the material matrix is denoted by \(D\) (or \(\hat{C}\)). For the specific components for isotropic linear elasticity, see Linear Elasticity.

Differential Operator Conventions

Symmetric Gradient Operator

The symmetric gradient operator \(\nabla_S\) for a vector field \(\boldsymbol{u}\) is defined as

\[ \nabla_S \boldsymbol{u} = \frac{1}{2}\bigl( \nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T \bigr) \]

In component form, \((\nabla_S \boldsymbol{u})_{ij} = \tfrac{1}{2}(\partial u_i / \partial x_j + \partial u_j / \partial x_i)\). The infinitesimal-deformation strain \(\boldsymbol{\varepsilon} = \nabla_S \boldsymbol{u}\) can be expressed compactly with this operator. The distinction between the configurations (reference/current) with respect to which gradients are taken in finite deformation is covered in Motion, Deformation, and Strain.

Material Time Derivative

The material time derivative of a quantity \(A\) (the time derivative following the same material point) is denoted by a superscript dot, \(\dot{A}\):

\[ \dot{A} \equiv \frac{D A}{D t}. \]

Velocity and acceleration follow this convention.