2.1 Vectors and Tensors

This chapter contains an overview of some of the important concepts from continuum mechanics and establishes some of the notation and terminology that will be used in the rest of this document. The section begins by introducing the important concepts of deformation, stress and strain. Next the concept of hyperelasticity is discussed. Finally the concept of virtual work is discussed. This concept will be used later to derive the nonlinear finite element equations. For a more thorough introduction to the mathematics needed for continuum mechanics, the user can consult ¹.

It is assumed that the reader is familiar with the concepts of vectors and tensors. This section summarizes the notation and some useful relations that will be used throughout the manual.

Vectors are denoted by small, bold letters, e.g. v. Their components will be denoted by \(v_{i}\), where, unless otherwise stated, Latin under scripts such as \(i\) or \(I\) will range from 1 to 3. In matrix form a vector will be represented as a column vector and its transpose as a row vector:

\[ \begin{equation} \begin{array}{cc} \mathbf{v}=\left(\begin{array}{c} v_{1}\\ v_{2}\\ v_{3} \end{array}\right),\quad & \mathbf{v}^{T}=\left(\begin{array}{ccc} v_{1}, & v_{2}, & v_{3}\end{array}\right)\end{array}.\label{eq1} \end{equation} \]

The following products are defined between vectors. Assume u, v are vectors. Also note that the Einstein summation convention is used throughout this manual ².

The dot or scalar product:

\[ \begin{equation} \mathbf{u}\cdot\mathbf{v}=u_{i}v_{i}\,.\label{eq2} \end{equation} \]

The cross product:

\[ \begin{equation} \mathbf{u}\times\mathbf{v}=\left[\begin{array}{c} u_{2}v_{3}-u_{3}v_{2}\\ u_{3}v_{1}-u_{1}v_{3}\\ u_{1}v_{2}-u_{2}v_{1} \end{array}\right]\,.\label{eq3} \end{equation} \]

The vector outer product:

\[ \begin{equation} \left(\mathbf{u}\otimes\mathbf{v}\right)_{ij}=u_{i}v_{j}\,.\label{eq4} \end{equation} \]

Note that vectors are also known as first order tensors. Scalars are known as zero order tensors. The outer product, defined by equation \eqref{eq4}, is a second order tensor.

Second order tensors are denoted by bold, capital letters, e.g. \(\mathbf{A}\). Some exceptions will be made to remain consistent with the literature. For instance, the Cauchy stress tensor is denoted by \(\mathbf{\boldsymbol{\sigma}}\). However, the nature of the objects will always be clear from the context. The following operations on tensors are defined. Assume \(\mathbf{A}\) and \(\mathbf{B}\) are second-order tensors.

The double contraction or tensor inner product is defined as:

\[ \begin{equation} \mathbf{A}:\mathbf{B}=A_{ij}B_{ij}\,.\label{eq5} \end{equation} \]

The trace is defined as:

\[ \begin{equation} \tr\mathbf{A}=\mathbf{I}:\mathbf{A}=A_{ii}\,.\label{eq6} \end{equation} \]

Here \(\mathbf{I}\) is the second order identity tensor with components \(\delta_{ij}\).

In general the components of tensors will change under a change of coordinate system. Nevertheless, certain intrinsic quantities associated with them will remain invariant under such a transformation. The scalar product between two vectors is such an example. The double contraction between two second-order tensors is another example. The following set of invariants for second-order tensors is commonly used:

\[ \begin{equation} \begin{aligned}I_{1} & =\tr\mathbf{A},\\ I_{2} & =\frac{1}{2}\left(\left(\tr\mathbf{A}\right)^{2}-\tr\mathbf{A}^{2}\right)\\ I_{3} & =\det\mathbf{A}. \end{aligned} \,.\label{eq7} \end{equation} \]

A tensor \(\mathbf{S}\) is called symmetric if it is equal to its transpose:

\[ \begin{equation} \mathbf{S}=\mathbf{S}^{T}\,.\label{eq8} \end{equation} \]

A tensor \(\mathbf{W}\) is called anti-symmetric if it is equal to the negative of its transpose:

\[ \begin{equation} \mathbf{W}=-\mathbf{W}^{T}\,.\label{eq9} \end{equation} \]

Any second order tensor \(\mathbf{A}\) can be written as the sum of a symmetric tensor \(\mathbf{S}\) and an anti-symmetric tensor \(\mathbf{W}\):

\[ \begin{equation} \mathbf{A}=\mathbf{S}+\mathbf{W},\label{eq10} \end{equation} \]

where

\[ \begin{equation} \mathbf{S}=\frac{1}{2}\left(\mathbf{A}+\mathbf{A}^{T}\right),\mbox{\thinspace and\thinspace}\mathbf{W}=\frac{1}{2}\left(\mathbf{A}-\mathbf{A}^{T}\right).\label{eq11} \end{equation} \]

Also note that for any tensor \(\mathbf{B}\) the following holds:

\[ \begin{equation} \mathbf{B}:\mathbf{A}=\mathbf{B}:\mathbf{S}\,,\quad\mathbf{B}:\mathbf{W}=\mathbf{0}\,.\label{eq12} \end{equation} \]

With any anti-symmetric tensor a dual vector \(\mathbf{w}\) can be associated such that,

\[ \begin{equation} \mathbf{\hat{w}}\cdot\mathbf{u}=\mathbf{w}\times\mathbf{u}\,,\label{eq13} \end{equation} \]

where the second order tensor \(\mathbf{\hat{w}}\) is defined as,

\[ \begin{equation} \mathbf{\hat{w}}=\left[\begin{array}{ccc} 0 & -w_{3} & w_{2}\\ w_{3} & 0 & -w_{1}\\ -w_{2} & w_{1} & 0 \end{array}\right]\,.\label{eq14} \end{equation} \]

A second order \(\mathbf{Q}\) tensor is called orthogonal if \(\mathbf{Q}^{-1}=\mathbf{Q}^{T}\).

In the implementation of the FE method it is often convenient to write symmetric second-order tensors using Voigt notation. In this notation the components of a 2\(^{\mathrm{nd}}\) order symmetric tensor \(\mathbf{A}\) are arranged as a column vector:

\[ \begin{equation} \left[\mathbf{A}\right]=\left[\begin{array}{c} A_{11}\\ A_{22}\\ A_{33}\\ A_{12}\\ A_{23}\\ A_{13} \end{array}\right]\,.\label{eq15} \end{equation} \]

Higher order tensors will be denoted by bold, capital, script symbols, e.g. \(\boldsymbol{\mathcal{A}}\). An example of a third-order tensor is the permutation tensor \(\mathcal{E}_{ijk}\), whose components are 1 for an even permutation of \(\left(1,2,3\right)\), -1 for an odd permutation of \(\left(1,2,3\right)\) and zero otherwise. The permutation symbol is useful for expressing the cross-product of two vectors in index notation:

\[ \begin{equation} \left(\mathbf{u}\times\mathbf{v}\right)_{i}=\mathcal{E}_{ijk}u_{j}v_{k}\,.\label{eq16} \end{equation} \]

An example of a fourth-order tensor is the elasticity tensor \(\boldsymbol{\mathcal{C}}\) which, in linear elasticity theory, relates the small strain tensor \(\mathbf{\boldsymbol{\varepsilon}}\) and the Cauchy stress tensor \(\boldsymbol{\sigma}=\boldsymbol{\mathcal{C}}:\boldsymbol{\varepsilon}\).

Higher order tensors can be constructed from second order tensors in a similar way as second order tensors can be constructed from vectors. If \(\mathbf{A}\) and \(\mathbf{B}\) are second order tensors, then the following fourth order tensors can be defind by requiring that the following must hold for any second order tensor \(\mathbf{X}\):

\[ \begin{equation} \left(\mathbf{A}\otimes\mathbf{B}\right):\mathbf{X}=\left(\mathbf{B}:\mathbf{X}\right)\mathbf{A}\,,\label{eq17} \end{equation} \]

\[ \begin{equation} \left(\mathbf{A}\oslash\mathbf{B}\right):\mathbf{X}=\mathbf{A}\cdot\mathbf{X}\cdot\mathbf{B}^{T}\,,\label{eq18} \end{equation} \]

\[ \begin{equation} \left(\mathbf{A}\obslash\mathbf{B}\right):\mathbf{X}=\mathbf{A}\cdot\mathbf{X}^{T}\cdot\mathbf{B}^{T}\,,\label{eq19} \end{equation} \]

\[ \begin{equation} \left(\mathbf{A}\odot\mathbf{B}\right):\mathbf{X}=\frac{1}{2}\left(\mathbf{A}\cdot\mathbf{X}\cdot\mathbf{B}^{T}+\mathbf{A}\cdot\mathbf{X}^{T}\cdot\mathbf{B}^{T}\right)\,.\label{eq20} \end{equation} \]

The Cartesian component forms of the operators \(\otimes\), \(\oslash\), \(\obslash\) and \(\odot\) are defined as follows:

\[ \begin{equation} \left(\mathbf{A}\otimes\mathbf{B}\right)_{ijkl}=\mathbf{A}_{ij}\mathbf{B}_{kl}\,,\label{eq21} \end{equation} \]

\[ \begin{equation} \left(\mathbf{A}\oslash\mathbf{B}\right)_{ijkl}=\mathbf{A}_{ik}\mathbf{B}_{jl}\,,\label{eq22} \end{equation} \]

\[ \begin{equation} \left(\mathbf{A}\obslash\mathbf{B}\right)_{ijkl}=A_{il}B_{jk}\,,\label{eq23} \end{equation} \]

\[ \begin{equation} \left(\mathbf{A}\odot\mathbf{B}\right)_{ijkl}=\frac{1}{2}\left(A_{ik}B_{jl}+A_{il}B_{jk}\right)\,.\label{eq24} \end{equation} \]

The fourth order identity tensors are defined as:

\[ \begin{equation} \begin{aligned}\mathbf{A} & =\underline{\mathcal{I}}:\mathbf{A}\,,\\ \mathbf{A}^{T} & =\overline{\mathcal{I}}:\mathbf{A}\,, \end{aligned} \label{eq25} \end{equation} \]

where \(\underline{\mathcal{I}}=\mathbf{I}\oslash\mathbf{I}\) and \(\overline{\boldsymbol{\mathcal{I}}}=\mathbf{I}\obslash\mathbf{I}\). The components are given by:

\[ \begin{equation} \begin{aligned}\mathcal{\underline{I}}_{ijkl} & =\delta_{ik}\delta_{jl}\,,\\ \overline{\mathcal{I}}_{ijkl} & =\delta_{il}\delta_{jk}\,. \end{aligned} \label{eq26} \end{equation} \]

We may also define \(\boldsymbol{\mathcal{I}}=\frac{1}{2}\left(\underline{\boldsymbol{\mathcal{I}}}+\overline{\boldsymbol{\mathcal{I}}}\right)\), such that \(\boldsymbol{\mathcal{I}}:\mathbf{A}=\frac{1}{2}\left(\mathbf{A}+\mathbf{A}^{T}\right)\) returns the symmetric part of \(\mathbf{A}\). In the special case when \(\mathbf{A}\) is symmetric it follows that \(\underline{\mathcal{I}}:\mathbf{A}=\boldsymbol{\mathcal{I}}:\mathbf{A}=\mathbf{A}\).

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1. Bonet, Javier; Wood, Richard D.. "Nonlinear continuum mechanics for finite element analysis." Cambridge University Press (1997). ↩

2. Lai, W. Michael; Rubin, David; Krempl, Erhard. "Introduction to continuum mechanics." Butterworth-Heinemann/Elsevier (2010). ↩