2.3 Variational principle for elastodynamics

We modify the HPR principle [14] for the case of Cosserat elastodynamics in the following way: for any set Av of all admissible states s ¼ ½ � u;ϕ; γ;χ; σ; μ that satisfy the strain-displacement and torsion-rotation relations Eq. (5), the zero variation

$$
\delta\Theta(\mathfrak{s}) = \mathbf{0}
$$

of the functional

$$\begin{split} \Theta(\mathfrak{s}) &= \int\_{t\_0}^{t\_k} \left[ U\_K + T\_C - \int\_{B\_0} \left( \sigma \cdot \mathfrak{y} + \mathfrak{a} \cdot \mathfrak{x} + \mathfrak{p} \frac{\partial \mathfrak{u}}{\partial t} + \mathfrak{q} \frac{\partial \phi}{\partial t} \right) dv \right] dt \\ &+ \int\_{t\_0}^t \int\_{\mathcal{G}\_1} [\sigma\_\mathbf{n} \cdot (\mathfrak{u} - \mathfrak{u}\_0) + \mu\_\mathbf{n} (\phi - \phi\_\mathbf{0})] da dt + \int\_{t\_0}^t \int\_{\mathcal{G}\_2} [\sigma\_\mathbf{0} \cdot \mathfrak{u} + \mu\_\mathbf{0} \cdot \phi] da dt \end{split} \tag{29}$$

at s ∈ A is equivalent of s to be a solution of the system of equilibrium Eqs. (1)–(2), constitutive relations Eqs. (6)–(7), which satisfies the mixed boundary conditions Eqs. (8)–(9).

Proof of the variational principle for elastodynamics Let us consider the variation of the functional Θð Þs :

$$\begin{aligned} \delta\Theta(\mathbf{s}) &= \int\_{t\_0}^{t\_0} [\delta U\_K + \delta T\_C] dt \\\\ &- \int\_{t\_0}^{t\_0} \int\_{B\_0} \left[ \delta \mathbf{\sigma} \cdot \mathbf{y} + \boldsymbol{\sigma} \cdot \delta \mathbf{\hat{y}} + \delta \boldsymbol{\mu} \cdot \boldsymbol{\chi} + \boldsymbol{\mu} \cdot \delta \boldsymbol{\chi} + \frac{\partial \mathbf{u}}{\partial t} \delta \mathbf{p} + \mathbf{p} \cdot \delta \left( \frac{\partial \mathbf{u}}{\partial t} \right) + \delta \mathbf{\hat{q}} \cdot \frac{\partial \boldsymbol{\Phi}}{\partial t} + \mathbf{q} \cdot \delta \left( \frac{\partial \boldsymbol{\Phi}}{\partial t} \right) \right] dv dt \\\\ &+ \int\_{t\_0}^{t} \int\_{\mathcal{J}\_1} [\delta \boldsymbol{\sigma}\_{\mathbf{t}} \cdot (\mathbf{u} - \mathbf{u}\_0) + \boldsymbol{\sigma}\_{\mathbf{n}} \delta \mathbf{u} + \delta \boldsymbol{\mu}\_{\mathbf{n}} \cdot (\boldsymbol{\Phi} - \boldsymbol{\Phi}\_0) + \boldsymbol{\mu}\_{\mathbf{n}} \delta \boldsymbol{\Phi}] da dt \\\\ &+ \int\_{t\_0}^{t} \int\_{\mathcal{J}\_2} [\sigma\_{\mathbf{0}} \cdot \delta \mathbf{u} + \boldsymbol{\mu}\_{\mathbf{0}} \cdot \delta \boldsymbol{\Phi}] da dt \end{aligned}$$

Taking into account Eq. (5) we can perform the integration by parts

$$\begin{aligned} \int\_{B\_0} \boldsymbol{\sigma} \cdot \delta \boldsymbol{\eta} \boldsymbol{dv} &= \int\_{\partial B\_0} \boldsymbol{\sigma}\_{\mathfrak{n}} \cdot \delta \mathbf{u} da - \int\_{B\_0} \delta \mathbf{u} \cdot \mathbf{div} \boldsymbol{\sigma} d\boldsymbol{v} + \int\_{B\_0} \boldsymbol{\varepsilon} \boldsymbol{\sigma} \cdot \delta \boldsymbol{\phi} d\boldsymbol{v} \\\\ \int\_{B\_0} \boldsymbol{\mu} \cdot \delta \boldsymbol{\chi} d\boldsymbol{v} &= \int\_{\partial B\_0} \boldsymbol{\mu}\_{\mathfrak{n}} \cdot \delta \boldsymbol{\phi} da - \int\_{B\_0} \delta \boldsymbol{\phi} \cdot \mathbf{div} \boldsymbol{\mu} d\boldsymbol{v} \end{aligned}$$

and based on Eqs. (17)–(23)

$$
\delta\Phi = \frac{\partial\Phi}{\partial\sigma} \cdot \delta\sigma + \frac{\partial\Phi}{\partial\mu} \cdot \delta\mu, \quad \delta\Upsilon\_C = \frac{\partial\Upsilon\_C}{\partial\left(\frac{\partial\mathbf{u}}{\partial t}\right)} \cdot \delta\left(\frac{\partial\mathbf{u}}{\partial t}\right) + \frac{\partial\Upsilon\_C}{\partial\left(\frac{\partial\phi}{\partial t}\right)} \cdot \delta\left(\frac{\partial\phi}{\partial t}\right).
$$

Then keeping in mind that δTK ¼ �δT and Eq. (28) we can rewrite the expression for the variation of the functional δΘð Þs in the following form

Dynamical Systems Theory

δΘð Þ¼ s ðt t0 ð B0 ∂Φ <sup>∂</sup><sup>σ</sup> � <sup>γ</sup> � � � δσ � �dvdt <sup>þ</sup> ðt t0 ð B0 ∂Φ <sup>∂</sup><sup>μ</sup> � <sup>χ</sup> � � � δμ � �dvdt þ ðt t0 ð B0 ρ ∂u ∂t � p � � � δ ∂u ∂t � � � � dvdt <sup>þ</sup> ðt t0 ð B0 J ∂φ ∂t � q � � � δ ∂φ ∂t � � � � dvdt þ ðt t0 ð B0 div<sup>σ</sup> � <sup>∂</sup><sup>p</sup> ∂t � � � δu � �dvdt <sup>þ</sup> ðt t0 ð B0 div<sup>μ</sup> <sup>þ</sup> <sup>ε</sup> � <sup>σ</sup> � <sup>∂</sup><sup>q</sup> ∂t � � � δu � �dvdt þ ðt t0 ð G1 ½ � ð Þ� u � u0 δσ<sup>n</sup> dadt þ ðt t0 ð G1 ϕ � ϕ<sup>0</sup> ð Þ� δμ<sup>n</sup> ½ �dadt þ ðt t0 ð G2 ½ � ð Þ� σ<sup>n</sup> � σ<sup>0</sup> δu dadt þ ðt t0 ð G2 μ<sup>n</sup> � μ<sup>0</sup> ½ � ð Þ� δϕ dadt
