Equivalently, an appropriate restriction for a constitutive model is polyconvexity with respect to the beam strain measures
\(\{\varvec{\Gamma },\varvec{K}\}\) [
35]. It is well known that this condition ensures ellipticity of the static problem and, equivalently, hyperbolicity of its dynamic counterpart [
35]. Essentially, the strain energy
\(\Psi _b\)
7 per unit undeformed length of the beam must be a function of the deformation gradient via a convex multi-valued function
\(W_b\) as
$$\begin{aligned} \Psi _b\left( \varvec{\nabla }_0\varvec{x}\right) =W_b\left( \varvec{\Gamma },\varvec{K}\right) , \end{aligned}$$
(76)
where
\(W_b\) is convex with respect to its 6 variables, namely, the
\(3\times 1\) components of
\(\varvec{\Gamma }\) and
\(\varvec{K}\). Moreover, invariance with respect to rotations is automatically satisfied since
\(\varvec{\Gamma }\) and
\(\varvec{K}\) are defined in the reference configuration. Following a similar approach to that in Sect.
2.2, it is possible to define work conjugates
\(\varvec{\Sigma _{\Gamma }}\) and
\(\varvec{\Sigma _{K}}\) to the beam strain measures
\(\varvec{\Gamma }\) and
\(\varvec{K}\), respectively, as
$$\begin{aligned} \varvec{\Sigma _{\Gamma }}\left( \varvec{\Gamma },\varvec{K}\right) =\frac{\partial W_b}{\partial \varvec{\Gamma }};\quad \varvec{\Sigma _K}\left( \varvec{\Gamma },\varvec{K}\right) =\frac{\partial W_b}{\partial \varvec{K}}. \end{aligned}$$
(77)
It is easy to realise that
\(\varvec{\Sigma }_{\varvec{\Gamma }}\) represents the axial-shear force vector rotated back to the reference undeformed configuration, whilst
\(\varvec{\Sigma }_{\varvec{K}}\) is the torsional–bending moment rotated back to the reference undeformed configuration, work conjugates of the axial-shear strain vector
\(\varvec{\Gamma }\) and the torsional–bending strain vector
\(\varvec{K}\), respectively. This set of conjugate measures enables the directional derivative of the strain energy
\(W_b\) to be expressed as
$$\begin{aligned} DW_b\left[ D\varvec{\Gamma }[\delta \varvec{u}_0,\delta {\varvec{\theta }}],D\varvec{K}[\delta \varvec{u}_0,\delta {\varvec{\theta }}]\right]= & {} \varvec{\Sigma _{\Gamma }}\cdot D\varvec{\Gamma }[\delta \varvec{u}_0,\delta {\varvec{\theta }}]\nonumber \\&+ \varvec{\Sigma _K}\cdot D\varvec{K}[\delta \varvec{u}_0,\delta {\varvec{\theta }}].\nonumber \\ \end{aligned}$$
(78)
Recalling the relation between the strain energies
\(W_b\) and
\(\bar{W}\) in (
68) and hence, between
\(\Psi _b\) and
\({\Psi }\) and Eq. (
71), it is possible to obtain the following expression for the directional derivative of
\(\Psi _b\)
$$\begin{aligned} D\Psi _b[\delta \varvec{u}_0,\delta {\varvec{\theta }}]= & {} \int _{A(s)}D\Psi [\delta \varvec{u}_0,\delta {\varvec{\theta }}]\;dA\nonumber \\= & {} \int _{A(s)} \varvec{\Sigma _B}\cdot D\varvec{B}[\delta \varvec{u}_0,\delta {\varvec{\theta }}]\; dA. \end{aligned}$$
(79)
The directional derivative of
\(\varvec{B}\) can be expressed in terms of the directional derivatives of
\(\{\varvec{\Gamma },\varvec{K}\}\) through Eq. (
62), leading to the following modified version of Eq. (
79)
$$\begin{aligned} \begin{aligned} D\Psi _b[\delta \varvec{u}_0,\delta {\varvec{\theta }}]&=\int _{A(s)} \varvec{\Sigma _B}\,dA\cdot D\varvec{\Gamma }[\delta \varvec{u}_0,\delta {\varvec{\theta }}] \\&\quad \,- \int _{A(s)} \varvec{\Sigma _B}\,dA\cdot (\varvec{I}\varvec{\times }\,\bar{\varvec{X}})D\varvec{K}[\delta \varvec{u}_0,\delta {\varvec{\theta }}]\\&=\int _{A(s)} \varvec{\Sigma _B}\,dA\cdot D\varvec{\Gamma }[\delta \varvec{u}_0,\delta {\varvec{\theta }}]\\&\quad \,+ \int _{A(s)} (\bar{\varvec{X}}\times \varvec{\Sigma _B})\;dA\cdot D\varvec{K}[\delta \varvec{u}_0,\delta {\varvec{\theta }}] \end{aligned} \end{aligned}$$
(80)
where use of the tensor cross product formula (
130) has been made. Comparison of Eq. (
80) with (
78) leads to the relation between the conjugate beam measures
\(\varvec{\Sigma _{\Gamma }}\) and
\(\varvec{\Sigma _{K}}\) (
77) with
\(\varvec{\Sigma _B}\) (
70) as
$$\begin{aligned} \varvec{\Sigma _{\Gamma }}=\int _{A(s)}\varvec{\Sigma _B}\,dA;\quad \varvec{\Sigma _{K}}=\int _{A(s)}\bar{\varvec{X}}\times \varvec{\Sigma _B}\,dA. \end{aligned}$$
(81)
Note that the tangent operator of the strain energy
\(W_b\) naturally emerges as
$$\begin{aligned} \begin{aligned}&D^2W_b[\delta \varvec{u}_0,\delta {\varvec{\theta }};\varvec{u}_0,\Delta {\varvec{\theta }}]\\&\quad = \int _{A(s)}D^2\Psi [\delta \varvec{u}_0,\delta {\varvec{\theta }};\varvec{u}_0,\Delta {\varvec{\theta }}]\,dA\\&\quad = \begin{bmatrix} \mathbb {S}_{\delta } \end{bmatrix}_{W_b}^T\begin{bmatrix} \mathbb {H}_{W_b} \end{bmatrix}\begin{bmatrix} \mathbb {S}_{\Delta } \end{bmatrix}_{W_b} \\&\qquad + \varvec{\Sigma _{\Gamma }}\cdot D^2\varvec{\Gamma }[\delta \varvec{u}_0,\delta {\varvec{\theta }};\varvec{u}_0,\Delta {\varvec{\theta }}]\\&\qquad + \varvec{\Sigma _{K}}\cdot D^2\varvec{K}[\delta \varvec{u}_0,\delta {\varvec{\theta }};\varvec{u}_0,\Delta {\varvec{\theta }}], \end{aligned} \end{aligned}$$
(82)
with
$$\begin{aligned}&\begin{bmatrix} \mathbb {S}_{\delta } \end{bmatrix}_{W_b}^T = \begin{bmatrix} D\varvec{\Gamma }[\delta \varvec{u}_0,\delta {\varvec{\theta }}]\cdot&D\varvec{K}[\delta \varvec{u}_0,\delta {\varvec{\theta }}]\cdot \end{bmatrix};\nonumber \\&\begin{bmatrix} \mathbb {S}_{\Delta } \end{bmatrix}_{W_b} = \begin{bmatrix} D\varvec{\Gamma }[\varvec{u}_0,\Delta {\varvec{\theta }}] \\ D\varvec{K}[\varvec{u}_0,\Delta {\varvec{\theta }}] \end{bmatrix}. \end{aligned}$$
(83)
and the Hessian operator
\([\mathbb {H}_{W_b}]\) is
$$\begin{aligned} \begin{aligned} \left[ \mathbb {H}_{W_b}\right] =\begin{bmatrix} \frac{\partial ^2 W_b}{\partial \varvec{\Gamma }\partial \varvec{\Gamma }}&\frac{\partial ^2 W_b}{\partial \varvec{\Gamma }\partial \varvec{K}}\\ \frac{\partial ^2 W_b}{\partial \varvec{K}\partial \varvec{\Gamma }}&\frac{\partial ^2 W_b}{\partial \varvec{K}\partial \varvec{K}} \end{bmatrix} \end{aligned} \end{aligned}$$
(84)