Chapter 1 Introduction to Vectors and Tensors
Reference: Holzapfel, Gerhard A. "Nonlinear solid mechanics: a continuum approach for engineering science." (2002): 489-490.
1.1 Algebra of Vectors
Kronecker delta \(\,\delta_{ij} = \boldsymbol{e}_i\cdot\boldsymbol{e}_j\)
Cross product \(\,\boldsymbol{u}\times\boldsymbol{v}\,\) or \(\,\boldsymbol{u}\wedge \boldsymbol{v}\,\)
\(\epsilon_{ijk}\)
1.2 Algebra of Tensors
second-order tensor
\(\boldsymbol{v} = \boldsymbol{A}\boldsymbol{u}\) \(\rightarrow\) linear transformation
Tensor product/dyad
tensor product (or direct or matrix) or the dyad of vectors \(\boldsymbol{u}\) and \(\boldsymbol{v}\) \(\rightarrow\) \(\boldsymbol{u}\otimes\boldsymbol{v}\) or \(\boldsymbol{u}\boldsymbol{v}\)
\((\boldsymbol{u}\otimes\boldsymbol{v})\boldsymbol{w} = \boldsymbol{u}(\boldsymbol{v}\cdot\boldsymbol{w})\)
\((\boldsymbol{u}\otimes\boldsymbol{v})(\boldsymbol{w}\otimes\boldsymbol{x}) = (\boldsymbol{v}\cdot\boldsymbol{w})\boldsymbol{u}\otimes\boldsymbol{x}\)
dyadic
linear combination of dyads with scalar coefficients
\(\boldsymbol{A} = A_{ij}\boldsymbol{e}_i\otimes\boldsymbol{e}_j\)
or with matrix notation
\([\boldsymbol{A}] = \begin{bmatrix} A_{11} && A_{12} && A_{13} \\ A_{21} && A_{22} && A_{23} \\ A_{31} && A_{32} && A_{33} \\ \end{bmatrix}\)
\(A_{ij} = \boldsymbol{e}_i\cdot\boldsymbol{A}\boldsymbol{e}_j\)
dot product：
dot product of tensors \(\boldsymbol{AB}\)
\((\boldsymbol{AB})_{ij} = A_{ik}B_{kj}\)
\(\boldsymbol{A}^2 = \boldsymbol{A}\boldsymbol{A}\)
Tranpose of \(\boldsymbol{A}\):
\(\boldsymbol{v}\cdot\boldsymbol{A}^T\boldsymbol{u}=\boldsymbol{u}\cdot\boldsymbol{A}\boldsymbol{v} = \boldsymbol{A}\boldsymbol{v}\cdot\boldsymbol{u}\)
\((\boldsymbol{AB})^T = \boldsymbol{B}^T\boldsymbol{A}^T\)
\((\boldsymbol{u}\otimes\boldsymbol{v})^T = \boldsymbol{v}\otimes\boldsymbol{u}\)
\((\boldsymbol{A}^T)_{ij} = A_{ji}\)
Trace and contraction
\(tr(\boldsymbol{u}\otimes\boldsymbol{v}) = \boldsymbol{u}\cdot\boldsymbol{v} = u_iv_i\)
\(tr(\boldsymbol{A}) = A_{ij}tr(\boldsymbol{e}_i\otimes\boldsymbol{e}_j)= A_{ii}\)
\(tr(\boldsymbol{AB}) = tr(\boldsymbol{BA})\)
contraction
Identify two indices and sum over them as dummy indices
\(\boldsymbol{A}:\boldsymbol{B} = tr(\boldsymbol{A}^T\boldsymbol{B}) = A_{ij}B_{ij} = \boldsymbol{B}:\boldsymbol{A}\)
\(\boldsymbol{A}:(\boldsymbol{BC}) = (\boldsymbol{B}^T\boldsymbol{A}):\boldsymbol{C} = (\boldsymbol{A}\boldsymbol{C}^T):\boldsymbol{B}\)
\((\boldsymbol{u}\otimes\boldsymbol{v}):(\boldsymbol{w}\otimes\boldsymbol{x}) = (\boldsymbol{u}\cdot\boldsymbol{w})(\boldsymbol{v}\cdot\boldsymbol{x})\)
norm of the tensor:
\(|A| = (A:A)^{\frac{1}{2}} = (A_{ij}A_{ij})^{\frac{1}{2}}\geq 0\)
Determinant and inverse of a tensor
\(det\boldsymbol{A}\) = \(det [\boldsymbol{A}]\)
\(det(\boldsymbol{AB}) = det\boldsymbol{A}det\boldsymbol{B}\)
\(det(\boldsymbol{A}^T) = det(\boldsymbol{A})\)
singular \(\rightarrow\) \(det(\boldsymbol{A}) = 0\)
\((\boldsymbol{AB})^{-1} = \boldsymbol{B}^{-1}\boldsymbol{A}^{-1}\)
\((\alpha\boldsymbol{A})^{-1} = \frac{1}{\alpha}\boldsymbol{A}^{-1}\)
\((\boldsymbol{A}^{-1})^T = (\boldsymbol{A}^T)^{-1} = \boldsymbol{A}^{-T}\)
\(\boldsymbol{A}^{-2} = \boldsymbol{A}^{-1}\boldsymbol{A}^{-1}\)
\(det(\boldsymbol{A}^{-1}) = (det\boldsymbol{A})^{-1}\)
Orthogonal tensor
\(\boldsymbol{Q}\boldsymbol{u}\cdot\boldsymbol{Q}\boldsymbol{v}=\boldsymbol{u}\cdot\boldsymbol{v}\)
Properties
\(\boldsymbol{Q}^T\boldsymbol{Q}=\boldsymbol{Q}\boldsymbol{Q}^T=\boldsymbol{I}\)
\(\boldsymbol{Q}^T=\boldsymbol{Q}^{-1}\)
\(det(\boldsymbol{Q}^T\boldsymbol{Q})=(det\boldsymbol{Q})^2=1\)
\(det\boldsymbol{Q}=+1\)\(\quad\rightarrow\quad\) proper orthogonal \(\rightarrow\) rotation
\(det\boldsymbol{Q}=-1\)\(\quad\rightarrow\quad\) improper orthogonal \(\rightarrow\) reflection
Symmetric and skew tensors
Any tensor \(\boldsymbol{A}\) can be decomposed into a symmetric tensor \(\boldsymbol{S}\) and a skew/antisymmetric tensor \(\boldsymbol{W}\)
\(\boldsymbol{A} = \boldsymbol{S}+\boldsymbol{W}\)
\(\boldsymbol{S} = \displaystyle\frac{1}{2}(\boldsymbol{A}+\boldsymbol{A}^T)\)
\(\boldsymbol{W} = \displaystyle\frac{1}{2}(\boldsymbol{A}-\boldsymbol{A}^T)\)
some properties:
\(\boldsymbol{S}:\boldsymbol{W}=0\)
\(\boldsymbol{W}\boldsymbol{u}= \boldsymbol{w} \times \boldsymbol{u}\)
\(|w|=\displaystyle\frac{1}{\sqrt{2}}|\boldsymbol{W}|\)
where \(\boldsymbol{w}=-\displaystyle\frac{1}{2}\varepsilon_{ijk}W_{ij}\boldsymbol{e}_k\)
Projection, spherical and deviatoric tensors
project tensor
which applied to any vector \(\boldsymbol{u}\) and map it into the direction of \(\boldsymbol{e}\); or onto the plane normal to \(\boldsymbol{e}\)
\(\boldsymbol{u}_{||}=(\boldsymbol{u}\cdot\boldsymbol{e})\boldsymbol{e} = (\boldsymbol{e}\otimes\boldsymbol{e})\boldsymbol{u}= \underbrace{\boldsymbol{P}^{||}_e}_{project\,tensor}\boldsymbol{u}\)
\(\boldsymbol{u}_{\bot}=\boldsymbol{u}- \boldsymbol{u}_{||}= (\boldsymbol{I}-\boldsymbol{e}\otimes\boldsymbol{e})\boldsymbol{u}= \underbrace{\boldsymbol{P}^{\bot}_e}_{project\,tensor}\boldsymbol{u}\)
some properties
\(\boldsymbol{P}=\boldsymbol{P}^n\)
spherical part and deviatoric part
\(\boldsymbol{A}=\underbrace{\alpha\boldsymbol{I}}_{spherical}+\underbrace{dev\boldsymbol{A}}_{deviatoric}\)
where:
\(\alpha = \displaystyle\frac{1}{3}tr\boldsymbol{A}= \displaystyle\frac{1}{3}A_{ii}\)
\(dev\boldsymbol{A} = \boldsymbol{A}-\frac{1}{3}tr\boldsymbol{A}\boldsymbol{I}\)
\(tr(dev\boldsymbol{A})=0\)