Notation system and list of symbols

Unit system

All quantities are expressed in the international system of units.

An underline, e.g. $\underline{u}$, denotes a vector
A double underline, e.g. $\underline{\underline{\sigma}}$, denotes a second-order tensor
A double-struck capital letter, e.g. $\mathbb{C}$, denotes a fourth-order tensor
A capital letter in square brackets, e.g. $[M]$, denotes a matrix
Space and time dependence of functions are omitted for brevity
An upper dot (or double dot), e.g. $\dot{f}$ or $\ddot{f}$, denotes first (respectively second) time derivative
A 'T' superscript, e.g. $[A]^T$, denotes the transposition operation
An asterisk superscript, e.g. $\theta^*$, denotes a test function

$\Omega$: 3D domain occupied by the studied solid
$\partial\Omega$: external boundary of $\Omega$
$\underline{n}$: external normal unit vector to $\partial\Omega$
$\textrm{div}(.)$: divergence operator
$\nabla (.)$: gradient operator
$\Delta (.)$: Laplacian operator
$s$: entropy per unit mass
$\underline{u}$: displacement vector
$\underline{\underline{\varepsilon}}$: linearized strain tensor
$\underline{\underline{\sigma}}$: Cauchy stress tensor
$\underline{f}_v$: body (volume) force vector per unit volume
$\theta$: temperature difference with reference temperature $T_0$ (unit: Kelvin or Celsius degrees)
$\underline{q}$: heat flux vector
$T_0$: reference temperature
$\rho$: mass density
$\lambda$: first Lamé elastic parameter
$\mu$: second Lamé elastic parameter (shear modulus)
$k$: thermal conductivity
$c$: specific heat
$\alpha$: thermal expansion coefficient
$\beta$: thermal modulus
$R$: internal heat source per unit volume