8 Asymmetric bifurcation
In this chapter, we consider a bifurcated branch for which \(\order[1]{\lambda} \neq 0\). The bifurcation equation (5.4) shows that necessarily, \(E_{ijk}\) is not identically nul. This equation has at most \((2^m - 1)\) pairs of real solutions \((\order[1]{\lambda}, \order[1]{u})\) et \((-\order[1]{\lambda}, -\order[1]{u})\); furthermore, multiplication by \(\order[1]{\xi_i}\) shows that
\[ \order[1]\lambda = -\frac{E_{ijk} \, \order[1]{\xi}_i \, \order[1]{\xi}_j \, \order[1]{\xi}_k}{2 \dot{E}_{ij} \, \order[1]{\xi}_i \, \order[1]{\xi}_j}. \tag{8.1}\]
I can’t prove that the bifurcation equation (5.4) has at most \((2^m - 1)\) pairs of real solutions.
Along the bifurcated branch, we have \(\lambda = \lambda_0 + \eta \, \order[1]\lambda + o(\eta)\), and \(\eta\) can be eliminated. In other words, \(\eta=\lambda\) (\(\order[1]\lambda = 1\) and \(\order[2]\lambda = \order[3]\lambda = \cdots = 0\)) can be selected as a parameter. It is therefore possible to express the bifurcated branch as a function of \(\lambda\): \(u(\lambda)\). For example, combining Eqs. (5.7) and (6.2), we find that
\[ \begin{aligned}[b] \E_{, uu}[u(\eta), \lambda(\eta); \order[1]u, \order[1]u] &= \eta \, \bigl[\E_3(\order[1]u, \order[1]u, \order[1]u) + \order[1]\lambda \, \dot{\E}_2(\order[1]u, \order[1]u)\bigr] + o(\eta)\\ &= - \eta \, \order[1]\lambda \, \dot{\E}_2(\order[1]u, \order[1]u) + o(\eta), \end{aligned} \] or \[ \E_{, uu}[u(\lambda), \lambda; \order[1]u, \order[1]u] = -\bigl( \lambda - \lambda_0 \bigr) \, \dot{\E}_2(\order[1]u, \order[1]u) + o(\lambda - \lambda_0). \tag{8.2}\]
For \(\lambda < \lambda_0\), the above quantity is negative (since \(\dot{\E}_2\) is negative definite). In other words: for asymmetric bifurcations, below the critical load, the bifurcated branch is unstable.
To investigate the stability above the critical load, we need to analyse the sign of the eigenvalues \(\alpha\) of the Hessian. At first order, \(\alpha = \eta \, \order[1]\alpha + o(\eta)\), where \(\order[1]\alpha\) is an eigenvalue of \((E_{ijk} \, \order[1]{\xi}_k + \order[1]{\lambda} \, \dot{E}_{ij})\) (see Chapter 7). Let \(\alpha_{\min}\) and \(\alpha_{\max}\) be the minimum and maximum eigenvalues of this second-order tensor. Three cases must be discussed
- If \(\alpha_{\min} \, \alpha_{\max} > 0\), then \((E_{ijk} \, \order[1]{\xi}_k + \order[1]{\lambda} \, \dot{E}_{ij})\) is positive or negative definite: all eigenvalues have the same sign, \(\epsilon \in \{-1, +1\}\). Then the sign of the eigenvalues \(\alpha\) of the Hessian is \(\epsilon \, \eta\) and there is a stability switch at the critical load. Since the bifurcated branch is unstable below the critical load, this means that it is stable above the critical load.
- If \(\alpha_{\min} \, \alpha_{\max} < 0\), then the extremal eigenvalues of the Hessian are \(\eta \, \alpha_{\min}\) and \(\eta \, \alpha_{\max}\), the product of which is \(\eta^2 \, \alpha_{\min} \, \alpha_{\max} < 0\). The bifurcated branch is unstable for all values of \(\lambda\).
- If \(\alpha_{\min} \, \alpha_{\max} = 0\), the analysis is inconclusive.
To close this section, it is observed that the dominant term of the expansion (6.1) of the potential energy along the bifurcated branch is of the third order in \(\eta\) \[ \E[u(\eta), \lambda(\eta)] = \E[u^\star \circ \lambda(\eta), \lambda(\eta)\} + \tfrac{1}{6} \order[1]{\lambda} \, \eta^3 \, \dot{E}_{ij} \, \order[1]{\xi}_i \, \order[1]{\xi}_j + o(\eta^3). \]
Eliminating \(\lambda\) and plugging expression (8.1) of \(\order[1]\lambda\) delivers the expression of the potential energy, where \(\lambda\) is the parameter \[ \begin{aligned}[b] \E[u(\lambda), \lambda] &= \E[u^{\star}(\lambda), \lambda] + \frac{\bigl( \lambda - \lambda_0 \bigr)^3}{6\bigl( \order[1]{\lambda} \bigr)^2} \, \dot{E}_{ij} \, \order[1]{\xi}_i \, \order[1]{\xi}_j + o(\lambda^3)\\ &= \E[u^{\star}(\lambda), \lambda] + \frac{2 \bigl( \dot{E}_{ij} \, \order[1]{\xi}_i \, \order[1]{\xi}_j \bigr)^3}{3 \bigl( E_{ijk} \, \order[1]{\xi}_i \, \order[1]{\xi}_j \order[1]{\xi}_k \bigr)^2} \, \bigl(\lambda - \lambda_0\bigr)^3 + o(\lambda^3). \end{aligned} \]
Recalling that \(\dot{E}_{ij} \, \order[1]{\xi}_i \, \order[1]{\xi}_j < 0\), it is found that, above the critical load, the potential energy is smaller along the bifurcated branch than along the fundamental branch.
As expected, the above expression does not depend on the scaling of \(\order[1]u\) (of the \(\order[1]{\xi_i}\)).
It has been shown in Chapter 3 that, when \(E_{ijk}\) is not identically null, the critical point is unstable.