The small-strain isotropic linear elastic constitutive law and its strain energy

Let $B\subseteq {\mathbb{R}}^3$ be a material region (Material regions, densities and their mass) and let $O\subseteq {\mathbb{R}}^3$ be open with $\bar{B}\subseteq O$. A displacement field on $B$ is a map $\xi :O\to {\mathbb{R}}^3$ of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms); its total derivative $D\xi (X):{\mathbb{R}}^3\to {\mathbb{R}}^3$ at $X\in O$ is linear (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder) and has matrix $({\mathrm{\partial }}_j{\xi }_i(X))$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

Write $\mathrm{Sym}(3):=\{A\in M_3(\mathbb{R}):A^{\mathsf{T}}=A\}$ for the real symmetric $3\times 3$ matrices (The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations, The transpose $A^{\mathsf{T}}$ of a matrix) and $I_3$ for the identity (Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes). The linear strain of the displacement field is the symmetric-matrix field

$e(X):=\frac{D\xi (X)+D\xi (X{)}^{\mathsf{T}}}{2}\in \mathrm{Sym}(3),e_{ij}(X)=\frac{{\mathrm{\partial }}_j{\xi }_i(X)+{\mathrm{\partial }}_i{\xi }_j(X)}{2},$

the symmetric part of the Jacobian matrix, and its trace is $\mathrm{tr}e={\sum }_i{\mathrm{\partial }}_i{\xi }_i$ (The trace $\mathrm{tr}(A)$ as the sum of the diagonal entries). The Frobenius inner product of $A,C\in M_3(\mathbb{R})$ is $⟨A,C{⟩}_F:=\mathrm{tr}(A^{\mathsf{T}}C)={\sum }_{i,j}{A}_{ij}{C}_{ij}$, with squared norm $\mathrm{\mid }A{\mathrm{\mid }}_F^2=⟨A,A{⟩}_F$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$); the deviatoric part of $e$ is

$e_0:=e-\frac{\mathrm{tr}e}{3}{I}_3\mathrm{.}$

An isotropic linear elastic constitutive law is a pair $(\lambda ,\mu )\in \mathbb{R}\times \mathbb{R}$ of Lamé parameters with

$\mu >0,\lambda +\frac{2\mu }{3}>0.$

Its stress matrix field is

$\sigma (X):=\lambda (\mathrm{tr}e(X))I_3+2\mu e(X)\in \mathrm{Sym}(3),$

and its strain-energy density is the real function of $e$

$W(e):=\frac{\lambda }{2}(\mathrm{tr}e{)}^2+\mu \mathrm{tr}(e^2)\mathrm{.}$

• These data are an adopted constitutive model, not a consequence of the balance laws of this collection. The pair $(\lambda ,\mu )$ and the linear relation between $e$ and $\sigma$ are model parameters and model structure; no physical material is asserted to obey them, and no measurement of $\lambda$ or $\mu$ is asserted by this definition.

• The matrix $\sigma$ is symmetric for every displacement field because $e$ is symmetric; symmetry of the stress is thus automatic in this model branch and does not require the separate angular-momentum balance. This is a property of the model, not a general theorem about stress.

• The hypotheses $\mu >0$ and $\lambda +2\mu \mathrm{/}3>0$ are exactly the positivity hypotheses proved sufficient for $W\ge 0$ with equality only at $e=0$ in the companion item; the constitutive definition itself assumes only these inequalities and makes no further definiteness claim.

• Units are not part of the definition. A physical reading assigns length units to $\xi$ and hence dimensionless strain and a stress unit to $\sigma$; the mathematical statements below use the matrices alone, and any unit convention belongs to a consuming model.