Adopted isotropic linear elastic constitutive model with its small-strain regime and strain energy

Adopt the following small-displacement elastic model on reference labels, with the body and unit conventions of Continuum body data, the fixed inertial frame and the dimensions of the fields.

(i) Reference-coordinate description. Let $I\subseteq \mathbb{R}$ be a nonempty open time interval and $O\subseteq {\mathbb{R}}^3$ an open reference region. Model data include $\xi :I\times O\to {\mathbb{R}}^3$ of class $C^2$, the displacement from a homogeneous stress-free reference placement. All spatial derivatives below are with respect to $X\in O$. Write $v_{\mathrm{r}\mathrm{e}\mathrm{f}}:={\mathrm{\partial }}_t\xi$ and $a_{\mathrm{r}\mathrm{e}\mathrm{f}}:={\mathrm{\partial }}_t^2\xi$. The linear strain is $e=\frac{1}{2}(D_X\xi +D_X{\xi }^{\mathsf{T}})$, with $e_{ij}=\frac{1}{2}({\mathrm{\partial }}_j{\xi }_i+{\mathrm{\partial }}_i{\xi }_j)$ (The small-strain isotropic linear elastic constitutive law and its strain energy). The constitutive relation is used as the first-order approximation about the reference state for small displacement gradients; no quantitative strain threshold is asserted.

If ${\chi }_t(X)=X+\xi (t,X)$ is additionally a positive-Jacobian diffeomorphism, it defines a finite motion; its spatial velocity obeys $u(t,{\chi }_t(X))=v_{\mathrm{r}\mathrm{e}\mathrm{f}}(t,X)$, not an equality at the same numerical spatial and reference arguments. This additional finite-motion interpretation is not required for the linear field equations below.

(ii) Constitutive law. The stress matrix of the branch is

$\sigma =\lambda (\mathrm{tr}e)I_3+2{\mu }_{s}e,$

with Lamé parameters $\lambda ,{\mu }_s\in \mathbb{R}$ satisfying the positivity hypotheses ${\mu }_s>0$ and $\lambda +\frac{2{\mu }_s}{3}>0$ (The small-strain isotropic linear elastic constitutive law and its strain energy). Both parameters are constant in space and time in this branch. Their dimension is that of stress, $[\lambda ]=[{\mu }_s]=\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework); in SI they are measured in $\mathrm{N}{\mathrm{m}}^{-2}$.

(iii) Strain energy. The strain-energy density of the branch is $W(e)=\frac{\lambda }{2}(\mathrm{tr}e{)}^2+{\mu }_s\mathrm{tr}(e^2)$, and under the positivity hypotheses of (ii) it satisfies $W(e)\ge 0$ with equality only for $e=0$ (The isotropic elastic strain energy is $\mu \mathrm{\mid }{e}_0{\mathrm{\mid }}_F^2+(\lambda \mathrm{/}2+\mu \mathrm{/}3)(\mathrm{tr}e{)}^2$, hence positive definite under $\mu >0$, $\lambda +2\mu \mathrm{/}3>0$).

(iv) Reference density and adopted linear balance. Fix a constant reference density ${\rho }_0>0$ and a continuous reference body-force-per-mass field $b_{\mathrm{r}\mathrm{e}\mathrm{f}}:I\times O\to {\mathbb{R}}^3$. Adopt the linear momentum balance

${\rho }_0{\mathrm{\partial }}_t^2\xi ={\rho }_0{b}_{\mathrm{r}\mathrm{e}\mathrm{f}}+{\mathrm{div}}_X\sigma ,({\mathrm{div}}_X\sigma {)}_i={\sum }_{j=1}^3{\mathrm{\partial }}_{X_j}{\sigma }_{ij}\mathrm{.}$

The velocity, acceleration and density used here are reference fields. Constant ${\rho }_0$ is not an assertion of constant current density. For a finite motion with conserved mass that density instead satisfies $\rho (t,{\chi }_t(X))\det D_X{\chi }_t(X)={\rho }_0$. The displayed linear balance is an adopted approximation model, not an exact change of variables in the finite-motion balances; no error bound for discarding higher-order terms is claimed.

(v) Validity, exclusions and status. The branch is not adopted for: finite strains where the displacement gradient is not small; anisotropic materials (the law has two moduli and an isotropic form); plastic, viscous, viscoelastic, damaged or fracturing response; temperature-dependent moduli or thermoelastic coupling; and materials with internal structure. The Lamé parameters are material parameters: whenever a numerical value of $\lambda$ or ${\mu }_s$ is used in a deduction it enters as a measured empirical premise with its own conditions and uncertainty, and no value is asserted here. The constitutive law itself is adopted model structure, not a measured law and not a consequence of the balance principles.