2. Linearized elasticity

We assume \(\nabla u\) is small, and replace Green’s strain tensor \(E(u) = \tfrac{1}{2} ( \nabla u + \nabla u^T + \nabla u^T \nabla u)\) by the geometrically linearized strain tensor

\[ \varepsilon(u) = \frac{1}{2} ( \nabla u + \nabla u^T ), \]

and use Hooke’s material law to obtain

\[ -\operatorname{div} D \varepsilon (u) = f \]

with the constitutive equation

\[ \sigma(\varepsilon) = D \varepsilon := 2 \mu \varepsilon + \lambda \operatorname{tr} (\varepsilon) I. \]

After geometric linearization, all stress tensors (both Piola-Kirchhff, Cauchy) coincide. The variational formulation is

\[ \int 2 \mu \varepsilon(u) : \varepsilon(v) + \lambda \operatorname{div} u \, \operatorname{div} v = \int f v \]

Instead of the so-called Lamé parameters \(\mu\) and \(\lambda\), one often uses Young modulus \(E\) and the Poisson ration \(\nu \in [0,1/2)\). The Young modulus \(E\) corresponds to the spring constant, and the \(\nu\) gives the relative contraction in cross direction when stretched in one direction. Setting

\[ 2 \mu = \frac{E}{1 + \nu} \qquad \text{and} \qquad \lambda = \frac{E \nu} { (1+\nu) (1-2\nu) } \]

relates the strain

\[\begin{split} \varepsilon = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -\nu & 0 \\ 0 & 0 & -\nu \end{array} \right) \end{split}\]

to the stress

\[\begin{split} \sigma = \left( \begin{array}{ccc} E & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right). \end{split}\]

Note that for \(\nu \approx \tfrac{1}{2}\) the linearized volume is preserved, it is called a nearly incompressible material. Then \(\lambda \gg \mu\). An example is rubber material, and also linearized problem from elasto-platicity are nearly incompressible. These problems are ill-conditioned, and need robust, mixed discretization methods.

Existence and uniqueness of the variational problem are proven by Lax-Milgram leamma. Fundamential is

Korn’s inequality

Let \(V = [H_{0,D}^1]^3 := \{ v \in [H^1]^3 : v = 0 \text{ on } \Gamma_D \}\), with \(|\Gamma_D| > 0\). Then there holds

\[ c_K \| u \|_{H^1}^2 \leq \| \varepsilon(u) \|_{L_2}^2 \]

The proof is non-trivial. It is interesting to note that 9 components of \(\nabla u\) are dominated by the 6 components of \(\varepsilon(u)\).

The bilinear-form

\[ A(u,v) := \int 2 \mu \varepsilon(u) : \varepsilon(v) + \lambda \operatorname{div} u \operatorname{div} v \]

is continuous and coercive with bounds

\[\begin{align*} A(u,v) & \leq (2 \mu + \lambda) \, \| u \|_{H^1} \, \| v \|_{H^1} \\ A(u,u) & \geq 2 \mu c_K \, \| u \|_{H^1}^2. \end{align*}\]

From Cea’s-Lemma we get the error estimate

\[ \| u - u_h \|_{H^1}^2 \leq \frac{2 \mu + \lambda}{2 \mu c_K} \inf_{v_h \in V_h} \| u - v_h \|_{H^1}^2. \]

It means the (linearized) elasticity problem is ill conditioned if the material is nearly incompressible (\(\lambda \gg \mu\)), or Korn’s constant \(c_K \ll 1\).

2.1. Domains with bad Korn constants:

Consider a thin rectangle \(\Omega = (0,1) \times (-t/2, t/2)\) with a small thickness \(t\), and the function space

\[ V = \{ v \in [H^1(\Omega)]^2 : v = 0 \text{ on } \{ 0 \} \times (-t/2, t/2) \} \]

Take the function

\[\begin{split} u(x,y) = \left( \begin{array}{c} -2xy \\ x^2 \end{array} \right) \in V, \end{split}\]

and compute

\[\begin{split} \nabla u = \left( \begin{array}{cc} -2y & -2x \\ 2 x & 0 \end{array} \right), \qquad \operatorname{sym} (\nabla u) = \left( \begin{array}{cc} -2y & 0 \\ 0 & 0 \end{array} \right) \end{split}\]

and

\[ \| \nabla u \|_{\Omega} = \tfrac{4}{3} t + \tfrac{1}{6} t^3, \qquad \| \operatorname{sym} (\nabla u) \|_{\Omega} = \tfrac{1}{6} t^3 \]

We see that Korn’s constant behaves badly on thin domains:

\[ c_K \leq O(t^2) \]

This is the motivation to study models on thin domains, as well as their limits on lower dimensional manifolds.