Fourth-Order Tensors

Abstract

Fourth-order tensors play an important role in continuum mechanics where they appear as elasticity and compliance tensors. In this section we define fourth-order tensors and learn some basic operations with them. To this end, we consider a set $\varvec{\mathcal {L}}\text {in}^n$ of all linear mappings of one second-order tensor into another one within $\mathbf {L}\text {in}^n$. Such mappings are denoted by a colon as

$$ \mathbf {Y}=\varvec{\mathcal {A}} : \mathbf {X}, \quad \varvec{\mathcal {A}}\in \varvec{\mathcal {L}}\text {in}^n, \; \mathbf {Y}\in \mathbf {L}\text {in}^n, \; \forall \mathbf {X}\in \mathbf {L}\text {in}^n.$$

Exercises

5.1.

Prove relations (5.20) and (5.21).

5.2.

Prove relations (5.22).

5.3.

Prove relations (5.34)–(5.36)

5.4.

Prove relations (5.42) and (5.43).

5.5.

Prove relations (5.49)–(5.52).

5.6.

Prove that $\varvec{\mathcal {A}}^\text {Tt}\ne \varvec{\mathcal {A}}^\text {tT}$ for $\varvec{\mathcal {A}}=\varvec{a}\otimes \varvec{b}\otimes \varvec{c}\otimes \varvec{d}$.

5.7.

Prove identities (5.54).

5.8.

Verify relations (5.55) and (5.56).

5.9.

Prove relations (5.70) for the components of a super-symmetric fourth-order tensor using (5.51) and (5.52).

5.10.

Prove relation (5.82) using (5.16) and (5.81).

5.11.

Verify the properties of the transposition tensor (5.85).

5.12.

Prove that the fourth-order tensor of the form

$$ \varvec{\mathcal {C}}=\left( \mathbf {M}_1\otimes \mathbf {M}_2+\mathbf {M}_2\otimes \mathbf {M}_1\right) ^\text {s} $$

is super-symmetric if $\mathbf {M}_1,\mathbf {M}_2\in \mathbf {S}\text {ym}^n$.

5.13.

Calculate eigenvalues and eigentensors of the following super-symmetric fourth-order tensors for $n=3$: (a) $\varvec{\mathcal {I}}^\text {s} ($5.86), (b) $\varvec{\mathcal {P}}_\text {sph}$ (5.89)$_1$, (c) $\varvec{\mathcal {P}}_\text {dev}^{\text {s}}$ (5.89)$_2$, (d) $\varvec{\mathcal {C}}$ (5.93).