3 Stability of the critical point
In this chapter, we discuss the stability of the critical point \((u_0, \lambda_0)\). To this end, we evaluate the potential energy in a neighboring state \(u_0 + u\), where \(u \in U\) is “small”. We have, to the fourth order \[ \begin{aligned}[b] \E(u_0 + u, \lambda_0) - \E(u_0, \lambda_0) ={} &\tfrac{1}{2} \E_2(u, u) + \tfrac{1}{6} \E_3(u, u, u)\\ &+ \tfrac{1}{24} \E_4(u, u, u, u) + o(\langle u , u \rangle^2), \end{aligned} \] where the linear term has been omitted, \(u_0\) being a critical point of the energy. We now expand \(u\) as \(u = \xi \, v + \eta \, w\), with \(\xi, \eta \in \reals\) and \(v \in V\) and \(w \in W\) are fixed, orthogonal directions. Since \(v \in V\), we have \(\E_2(\lambda_0; v, \bullet) = 0\). Owing to the multi-linearity and symmetry of the successive differential of \(\E\), the above expression expands as follows \[ \begin{aligned}[b] \E(u_0 + u, \lambda_0) - \E(u_0, \lambda_0) ={} & \tfrac{1}{2} \eta^2 \, \E_2(w, w) + \tfrac{1}{6} \xi^3 \, \E_3(v, v, v)\\ & + \tfrac{1}{2} \xi^2 \, \eta \, \E_3(v, v, w) + \tfrac{1}{2} \xi \, \eta^2 \, \E_3(v, w, w)\\ & + \tfrac{1}{6} \eta^3 \, \E_3(w, w, w) + \tfrac{1}{24} \xi^4 \, \E_4(v, v, v, v)\\ & + \tfrac{1}{6} \xi^3 \, \eta \, \E_4(v, v, v, w) + \tfrac{1}{4} \xi^2 \, \eta^2 \, \E_4(v, v, w, w)\\ & + \tfrac{1}{6} \xi \, \eta^3 \, \E_4(v, w, w, w) + \tfrac{1}{24} \eta^4 \, \E_4(w, w, w, w)\\ & + o\bigl[\bigl(\xi^2 + \eta^2\bigr)^2\bigr]. \end{aligned} \]
For the equilibrium to be stable, the above expression must be \(\geq 0\) for all \(\xi\) et \(\eta\) small enough. Taking first \(\eta = 0\), we get the following necessary conditions \[ \E_3(v, v, v) = 0 \quad \text{and} \quad \E_4(v, v, v, v) \geq 0 \quad \text{for all} \quad v \in V. \tag{3.1}\]
Note that the first of these two conditions is equivalent to \(E_{ijk}=0\), for all \(i, j, k = 1, \ldots m\).
In other words, if there exists \(v \in V\) such that \(\E_3(v, v, v) \neq 0\) or \(\E_4(v, v, v, v) < 0\), then the equilibrium is unstable at the critical point. The above conditions are not sufficient. Indeed, assuming conditions (3.1) to hold, we now take \(\eta = \xi^2\) \[ \E(u_0 + u, \lambda_0) - \E(u_0, \lambda_0) = \tfrac{1}{2} \xi^4 \bigl[ \E_2(w, w) + \E_3(v, v, w) + \tfrac{1}{12} \E_4(v, v, v, v) \bigr] + o(\xi^4) \] and we get the further necessary condition \[ \E_2(w, w) + \E_3(v, v, w) + \tfrac{1}{12} \E_4(v, v, v, v) \geq 0 \quad \text{for all} \quad v \in V \quad \text{and} \quad w \in W. \tag{3.2}\]
The direction \(v \in V\) being fixed, the above expression is minimal when \(w\) satisfies the following variational problem \[ 2\E_2(w, \hat{w}) + \E_3(v, v, \hat{w}) = 0 \quad \text{for all} \quad \hat{w} \in W. \tag{3.3}\]
Expanding \(v \in V\) in the \((v_i)\) basis as follows: \(v = \xi_i \, v_i\), it is observed that the solution to the above variational problem is \(w = \tfrac{1}{2} \xi_i \, \xi_j \, w_{ij}\), where \(w_{ij}\) is the solution to the elementary variational problem (2.5) . For this value of \(w\), condition (3.2) reads \[ \bigl[\E_4(v_i, v_j, v_k, v_l) - 3\E_2(w_{ij}, w_{kl})\bigr] \, \xi_i \, \xi_j \, \xi_k \, \xi_l \geq 0 \quad \text{for all} \quad \xi_1, \ldots, \xi_m \in \reals, \] which, in view of definition (2.10) of \(E_{ijkl}\), is equivalent to \[ E_{ijkl} \, \xi_i \, \xi_j \, \xi_k \, \xi_l \geq 0 \quad \text{for all} \quad \xi_m, \ldots, \xi_m \in \reals. \]
Note that Eq. (3.2) implies \(\E_4(\lambda_0; v, v, v, v) \geq 0\), which becomes a redundant necessary condition. Indeed, plugging \(w= \xi_i \xi_j w_{ij}\) into Eq. (3.2) cancels the first two terms. To sum up, we have the following necessary conditions for stability \[ E_{ijk} \, \xi_i \, \xi_j \, \xi_k = 0 \quad \text{and} \quad E_{ijkl} \, \xi_i \, \xi_j \, \xi_k \, \xi_l \geq 0 \quad \text{for all} \quad \xi_m, \ldots, \xi_m \in \reals. \tag{3.4}\]
It is claimed in Koiter and Heijden (2009) that the above condition is also sufficient to ensure stability at the critical point. I can’t prove this result, though.