2 Setting-up the mathematical stage
In this chapter, we define the problem mathematically. Symbols and notation are introduced in Section 2.1, as well as the fundamental assumptions that we make. Then, the kernel of the hessian of the energy is introduced in Section 2.2 as a subspace of the space of admissible states.
2.1 Mathematical setting
The space of admissible states of the system under consideration is denoted \(U\). It has the structure of a real vector space. The energy of the system is \(\E(u, \lambda)\), where \(\lambda\) denotes a loading parameter. It is assumed that the fundamental branch of the equilibrium diagram, \(u^\star(\lambda)\) is known. Then the energy is stationary with respect to the state \(u\) along the whole branch. In other words, for all \(\hat{u} \in U\) \[ \E_{,u}[u^{\star}(\lambda), \lambda; \hat{u}]=0, \tag{2.1}\] where \(\E_{,u}(u, \lambda; \hat{u})\) denotes the (real) value of the differential of the energy \(\E\) with respect to the state \(u\), evaluated at \((u, \lambda)\), for the test function \(\hat{u}\). Similarly, evaluation of the second-, third-, etc., order differential of the energy will be denoted \(\E_{,uu}(u, \lambda; \hat{u}, \hat{v})\), \(\E_{,uuu}(u, \lambda; \hat{u}, \hat{v}, \hat{w})\), etc. It is assumed that a finite value \(\lambda_0 > 0\) of \(\lambda\) can be found (critical load), such that
- \(\E_{,uu}[u(\lambda), \lambda] > 0\) for all \(0 < \lambda < \lambda_0\),
- \(\E_{,uu}(u_0, \lambda_0) \geq 0\) but \(\E_{,uu}(u_0, \lambda_0) \ngtr 0\).
The load \(\lambda_0\) will be referred to as the critical load; similarly, the state \(u_0\) of the system at the critical load will be referred to as the critical state; finally, the pair \((u_0, \lambda_0)\) is the critical point of the system. Assumption 1 implies that equilibria along the fundamental branch are stable below the critical load. Conversely, it results from assumption 3 that equilibrium points on the fundamental branch are unstable above the critical load. Stability at the critical load is yet undetermined.
The goal of these notes is to analyze all equilibrium paths that pass through the critical point \((u_0, \lambda_0)\).
We introduce the following notations \[ u_0 = u^\star(\lambda_0), \quad \dot{u}_0 = \frac{\D u^\star}{\D \lambda} \biggr \rvert_{\lambda=\lambda_0}, \quad \ddot{u}_0 = \frac{\D^2 u^\star}{\D \lambda^2} \biggr \rvert_{\lambda=\lambda_0}, \quad \dddot{u}_0 = \ldots, \quad \ddddot{u}_0 = \ldots \] and \[ \E_2 = \E_{,uu}(u_0, \lambda_0), \quad \E_3 = \E_{,uuu}(u_0, \lambda_0), \quad \E_4 = \E_{,uuuu}(u_0, \lambda_0). \]
Note that \(\E_2\), \(\E_3\) and \(\E_4\) thus defined are bi-, tri- and quadrilinear forms, respectively. The following derivatives are also introduced \[ \dot{\E}_2(\hat{u}, \hat{v}) = \frac{\D}{\D\lambda} \E_{,uu}[ u^\star(\lambda), \lambda; \hat{u}, \hat{v}] \biggr \rvert_{\lambda = \lambda_0} = \E_{,uuu}(u_0, \lambda_0; \dot{u}_0, \hat{u}, \hat{v}) + \E_{,uu\lambda}(u_0, \lambda_0; \hat{u}, \hat{v}) \tag{2.2}\] \[ \begin{aligned}[b] \ddot{\E}_2(\hat{u}, \hat{v}) = \frac{\D^2}{\D\lambda^2} \E_{,uu}[ u^\star(\lambda), \lambda; \hat{u}, \hat{v}] \biggr \rvert_{\lambda = \lambda_0} ={} & \E_{,uuuu}(u_0, \lambda_0; \dot{u}_0, \dot{u}_0, \hat{u}, \hat{v}) + 2\E_{,uuu\lambda}(u_0, \lambda_0; \dot{u}_0, \hat{u}, \hat{v})\\ & + \E_{,uu\lambda\lambda}( u_0, \lambda_0; \hat{u}, \hat{v}) + \E_{,uuu}(u_0, \lambda_0, \ddot{u}_0), \end{aligned} \] and, similarly, \(\dot{\E}_3\), \(\ddot{\E}_3\), etc.
2.2 Kernel of the hessian of the energy
The kernel of the hessian of the energy, \(\E_2\), is defined as follows \[ V = \bigl\{ u \in U, \E_2(u, u) = 0 \bigr\}. \]
Since \(\E_2\) is a bilinear, symmetric and positive (but not poitive definite!) form, \(V\) is a vector subspace of \(U\).
To prove this result, we must show that, for all \(u, v \in V\) and \(\alpha \in \reals\), \(w = u + \alpha \, v \in V\). From the bilinearity and symmetry of \(\E_2\) \[ \E_2(w, w) = \E_2(u + \alpha \, v, u + \alpha \, v) = \E_2(u, u) + 2\alpha \, \E_2(u, v) + \alpha^2 \, \E_2(v, v), \]
Since \(u, v \in \ker\E_2\), the first and the last term vanish, and the above identity reduces to \[ \E_2(w, w) = 2\alpha \, \E_2(u, v) \]
The bilinear form \(\E_2\) is positive, therefore the left-hand side is positive, for all values of \(\alpha \in \reals\). The quantity \(\E_2(u, v) = 0\) is necessarily null, and \(\E_2(w, w) = 0\), which proves that \(w \in V\) and that \(V\) is a vector subspace of \(U\). The following characterization of \(V\) holds \[ v \in V \quad \iff \quad \E_2(v, \hat{u}) = 0 \quad \text{for all} \quad \hat{u} \in U. \tag{2.3}\]
Indeed, if for all \(\hat{u} \in U\), \(\E_2(v, \hat{u}) = 0\), then in particular \(\E_2(v, v) = 0\) and \(v \in V\). Conversely, let \(v \in V\), \(\hat{u} \in U\) and \(\alpha \in \reals\). Similarly to the previous proof, we write that \(\E_2(w, w) \geq 0\), with \(w = \hat{u} + \alpha \, v\) \[ \E_2(w, w) = \E_2(\hat{u}, \hat{u}) + 2\alpha \, \E_2(\hat{u}, v) + \alpha^2 \, \E_2(v, v) = 2\alpha \, \E_2(\hat{u}, v) + \E_2(\hat{u}, \hat{u}) \geq 0, \] (\(\E_2(v, v) = 0\) since \(v \in V\)). The above expression is of degree 1 in \(\alpha\), with a constant sign. Therefore the linear term in \(\alpha\) must vanish: \(\E_2(\hat{u}, v) = 0\), which proves the characterization (2.3) of \(V\).
It will be assumed in the remainder of these notes the dimension of \(V\) is finite: \(m = \dim V < +\infty\); \(m\) is the multiplicity of the critical point. A (finite) basis \((v_1, \ldots, v_m)\) of \(V\) can therefore be introduced, that is orthonormal in the sense of \(\langle \bullet, \bullet \rangle\) \[ \langle v_i, v_j \rangle = \delta_{ij} \quad \text{for all} \quad i, j = 1, \ldots, m. \]
To close this section, we define the complementary subspace \(W\), orthogonal to \(V\) for the scalar product \(\langle \bullet, \bullet \rangle\) \[ U = V \overset{\perp}{\otimes} W \quad \text{and} \quad \langle \hat{v}, \hat{w} \rangle = 0 \quad \text{for all} \quad \hat{v} \in V \quad \text{and} \quad \hat{w} \in W. \]
It results from definition (2.2) that, for all \(\hat{v} \in V\) \[ \begin{aligned}[b] \E_{,uu}[ u^\star(\lambda), \lambda; \hat{v}, \hat{v}] &= \E_2(\hat{v}, \hat{v}) + \bigl( \lambda - \lambda_0 \bigr) \, \dot{\E}_2(\hat{u}, \hat{u}) + o(\lambda - \lambda_0)\\ &= \bigl( \lambda - \lambda_0 \bigr) \, \dot{\E}_2(\hat{v}, \hat{v}) + o(\lambda - \lambda_0) \end{aligned} \]
For \(\lambda < \lambda_0\), it has been assumed that the hessian is positive definite along the fundamental branch. From the above equation, it therefore results that \(\dot{\E}_2(\hat{v}, \hat{v}) < 0\). In other words, \(\dot{\E}_2\) is a negative definite form over \(V\).
2.3 Two canonical variational problems
The bilinear form \(\E_2\) is elliptic over \(W\). Therefore, variational problems of the type: find \(w \in W\) such that \[ \E_2(w, \hat{w})+\ell(\hat{w}) = 0 \quad \text{for all} \quad \hat{w} \in W \] are well-posed for any linear form \(\ell\) over \(W\). In particular, for \(\ell=0\), the unique solution to the variational problem \[ \E_2(w, \hat{w}) = 0 \quad \text{for all} \quad \hat{w} \in W \] is \(w = 0\). For \(\ell(\bullet) = \dot{\E}_2(v_i, \bullet)\) and \(\ell(\bullet) = \E_3(v_i, v_j, \bullet)\), \(w_i \in W\) and \(w_{ij} \in W\) are defined as the unique solutions in \(W\) of the following variational problems \[ \E_2(w_{i}, \hat{w}) + \dot{\E}_2(v_i, \hat{w}) = 0 \tag{2.4}\] and \[ \E_2(w_{ij}, \hat{w}) + \E_3(v_i, v_j, \hat{w}) = 0, \tag{2.5}\] for all \(\hat{w} \in W\). These variational problems (and their solutions) will pop-up recurrently in what follows.
2.4 Additional symbols
We will make use of the following symbols \[ \dot{E}_{ij} = \dot{\E}_2(v_i, v_j), \tag{2.6}\] \[ \ddot{E}_{ij} = \ddot{\E}_2(v_i, v_j) + \dot{\E}_2(v_i, w_{j}) + \dot{\E}_2(v_j, w_{i}), \tag{2.7}\] \[ E_{ijk} = \E_3(v_i, v_j, v_k), \tag{2.8}\] \[ \dot{E}_{ijk} = \dot{\E}_3(v_i, v_j, v_k) + \dot{\E}_2(v_i, w_{jk}) + \dot{\E}_2(v_j, w_{ik}) + \dot{\E}_2(v_k, w_{ij}), \tag{2.9}\] \[ \begin{aligned}[b] E_{ijkl} ={}& \E_4(v_i, v_j, v_k, v_l) + \E_3(v_i, v_j, w_{kl})\\ &+ \E_3(v_i, v_k, w_{jl}) + \E_3(v_i, v_l, w_{jk}). \end{aligned} \tag{2.10}\]
These symbols define second, third and fourth-order, fully symmetric, tensors over \(V\).
Since \(\E_2(v, \bullet) = 0\) for all \(v \in V\), definitions (2.6) and (2.8) of \(\dot{E}_{ij}\) and \(E_{ijk}\) can also be written \[ \dot{E}_{ij} = \dot{\E}_2(v_i, v_j) + \E_2(v_i, w_{j}) + \E_2(v_j, w_{i}) \] and \[ E_{ijk} = \E_3(v_i, v_j, v_k) + \E_2(v_i, w_{jk}) + \E_2(v_j, w_{ik}) + \E_2(v_k, w_{ij}), \] wich shows the consistency with definitions (2.7) and (2.9) of \(\ddot{E}_{ij}\) and \(\dot{E}_{ijk}\).