The principle of virtual power (PVP) provides a systematic way of deriving field equations and boundary conditions for arbitrary mechanical and coupled problems (cf. [
22‐
24]). In the present work it is used in the following form: The virtual power of the inertia forces
\(\mathscr {P}^{*}_{\text {a}}\) balances the virtual power
\(\mathscr {P}^{*}_{\text {int}}\) of the internal and
\(\mathscr {P}^{*}_{\text {ext}}\) of the external forces acting on any sub-domain
\(\mathscr {S}\) of the material body
\(\mathscr {B}\) for any admissible virtual velocity field
\(\mathbf {v}^{*}\) and virtual rate of order parameter field
\(\dot{ \phi }^*\), i.e.,
$$\begin{aligned} \mathscr {P}^{*}_{\text {a}} = \mathscr {P}^{*}_{\text {int}} + \mathscr {P}^{*}_{\text {ext}}\,. \end{aligned}$$
(23.13)
For the sake of simplicity we disregard any higher order inertia terms [
25] as well as inertial forces acting on the order parameter, resulting in
$$\begin{aligned} \mathscr {P}^{*}_{\text {a}} = \int \limits _{\mathscr {S}} \rho \dot{\mathbf {v}} \cdot \mathbf {v}^{*} \,\text {d}V\,. \end{aligned}$$
(23.14)
The power of internal forces is given by
$$\begin{aligned} \mathscr {P}^{*}_{\text {int}} = - \int \limits _{\mathscr {S}} \left( \mathbf {S}^{\top }:\mathbf {L}^{*} + \varvec{\mathcal {T}}\,\vdots \,\mathrm{grad}\,{\mathbf {L}^{*}} - \pi \,\dot{ \phi }^*+ \mathbf {\xi }\cdot \mathrm{grad}\,{\dot{ \phi }^*} \right) \,\text {d}V, \end{aligned}$$
(23.15)
with
\(\mathbf {L}^{*}:=\mathrm{grad}\,{\mathbf {v}^{*}}\). Here
\(\mathbf {S}\) and
\(\varvec{\mathcal {T}}\) are the
Cauchy and higher order stresses, respectively, while
\(\pi \) and
\(\mathbf {\xi }\) are thermodynamic forces that directly correspond to the internal microforce and microstress introduced by Gurtin [
26]. We note that the invariance requirement of
\(\mathscr {P}^{*}_{\text {int}}\) with respect to superimposed rigid body motions results in
\(\mathbf {S}= \mathbf {S}^{\top }\). For the power of external forces we consider the very simple case of no body or contact forces acting on
\(\mathbf {L}^{*}\) and
\(\mathrm{grad}\,{\dot{ \phi }^*}\), and only a contact (micro)force
\(\zeta \) acting
\(\dot{ \phi }^*\)$$\begin{aligned} \mathscr {P}^{*}_{\text {ext}} = \int \limits _{\mathscr {S}} \mathbf {f}\cdot \mathbf {v}^{*} \rho \,\text {d}V+ \int \limits _{\partial \mathscr {S}}\left( \mathbf {t}\cdot \mathbf {v}^{*} + \zeta \,\dot{ \phi }^*\right) \,\text {d}a\,. \end{aligned}$$
(23.16)
In order to obtain the consequences of the PVP, the integrals in (
23.15) are transformed using the following identities
$$\begin{aligned}&\mathrm{div} {(\mathbf {S}\cdot \mathbf {v}^{*})} = (\mathrm{div} \,{\mathbf {S}})\cdot \mathbf {v}^{*} + \mathbf {S}:\mathbf {L}^{*},\end{aligned}$$
(23.17)
$$\begin{aligned}&\mathrm{div} {(\varvec{\mathcal {T}}:\mathbf {L}^{*})} = (\mathrm{div} \,{\varvec{\mathcal {T}}}):\mathbf {L}^{*} + \varvec{\mathcal {T}}\,\vdots \,\mathrm{grad}\,{\mathbf {L}^{*}}, \end{aligned}$$
(23.18)
$$\begin{aligned}&\mathrm{div} {\bigl ((\mathrm{div} \,{\varvec{\mathcal {T}}})\cdot \mathbf {v}^{*}\bigr )} = (\mathrm{div} {\,\mathrm{div} \,{\varvec{\mathcal {T}}}})\cdot \mathbf {v}^{*} + (\mathrm{div} \,{\varvec{\mathcal {T}}}):\mathbf {L}^{*}, \end{aligned}$$
(23.19)
$$\begin{aligned}&\mathrm{div} {(\mathbf {\xi }\,\dot{ \phi }^*)} = (\mathrm{div} \,{\mathbf {\xi }})\,\dot{ \phi }^*+ \mathbf {\xi }\cdot \mathrm{grad}\,{\dot{ \phi }^*}, \end{aligned}$$
(23.20)
and the divergence theorem, resulting in
$$\begin{aligned} \mathscr {P}^{*}_{\text {int}}&= \int \limits _{\mathscr {S}} \bigl ( \mathrm{div} \,{\mathbf {S}} - \mathrm{div} {\,\mathrm{div} \,{\varvec{\mathcal {T}}}}\bigr )\cdot \mathbf {v}^{*}\,\text {d}V- \int \limits _{\partial \mathscr {S}}\mathbf {n}\cdot \bigl (\mathbf {S}^{\top } - \mathrm{div} \,{\varvec{\mathcal {T}}}\bigr )\cdot \mathbf {v}^{*}\,\text {d}a\nonumber \\&\,\, - \int \limits _{\partial \mathscr {S}}\mathbf {n}\cdot \varvec{\mathcal {T}}:\mathbf {L}^{*}\,\text {d}a+ \int \limits _{\mathscr {S}} \bigl ( \pi + \mathrm{div} \,{\mathbf {\xi }} \bigr )\,\dot{ \phi }^*\,\text {d}V- \int _{\partial \mathscr {S}}\mathbf {n}\cdot \mathbf {\xi }\,\dot{ \phi }^*\,\text {d}a\,. \end{aligned}$$
(23.21)
Introducing the surface gradient operator
$$\begin{aligned} \mathrm{grad_{S}}(\cdot ) = \mathrm{grad}\,(\cdot ) - \partial _{\mathbf {n}}(\cdot )\otimes \mathbf {n}, \end{aligned}$$
(23.22)
where
\(\partial _{\mathbf {n}}\) is the directional derivative in the direction of the outward normal
\(\mathbf {n}\), the third integral in expression (
23.21) can be rewritten as
$$\begin{aligned} \int \limits _{\partial \mathscr {S}}\mathbf {n}\cdot \varvec{\mathcal {T}}:\mathbf {L}^{*}\,\text {d}a&=\int \limits _{\partial \mathscr {S}}\mathbf {n}\cdot \varvec{\mathcal {T}}:\mathrm{grad_{S}}{\mathbf {v}^{*}}\,\text {d}a+ \int \limits _{\partial \mathscr {S}}\mathbf {n}\cdot \varvec{\mathcal {T}}:\partial _{\mathbf {n}}\mathbf {v}^{*}\otimes \mathbf {n}\,\text {d}a\end{aligned}$$
(23.23)
$$\begin{aligned} \nonumber&= \int \limits _{\partial \mathscr {S}}{\text {div}}_{S}{\bigl (\mathbf {n}\cdot \varvec{\mathcal {T}}\cdot \mathbf {v}^{*}\bigr )}\,\text {d}a- \int \limits _{\partial \mathscr {S}}{\text {div}}_{S}{\bigl (\mathbf {n}\cdot \varvec{\mathcal {T}}\bigr )}\cdot \mathbf {v}\,\text {d}a\,+\\&\qquad + \int \limits _{\partial \mathscr {S}}\mathbf {n}\cdot \varvec{\mathcal {T}}:\partial _{\mathbf {n}}\mathbf {v}^{*}\otimes \mathbf {n}\,\text {d}a. \end{aligned}$$
(23.24)
Finally, applying the surface divergence theorem and, for the sake of simplicity, neglecting any wedge line and corner contributions, we find
$$\begin{aligned} \int \limits _{\partial \mathscr {S}}{\text {div}}_{S}{\bigl (\mathbf {n}\cdot \varvec{\mathcal {T}}\cdot \mathbf {v}^{*}\bigr )}\,\text {d}a= \int \limits _{\partial \mathscr {S}}\bigl ( {\text {div}}_{S}{\mathbf {n}} \bigr )\mathbf {n}\otimes \mathbf {n}:\varvec{\mathcal {T}}\cdot \mathbf {v}^{*}\,\text {d}a. \end{aligned}$$
(23.25)
Enforcing (
23.13) we arrive after a number of straightforward algebraic manipulations at the following field equations on
\(\mathscr {B}\)$$\begin{aligned}&\rho \dot{\mathbf {v}} = \mathrm{div} {\left( \mathbf {S}- \mathrm{div} \,\varvec{\mathcal {T}}\right) } + \rho \mathbf {f},\end{aligned}$$
(23.26a)
$$\begin{aligned}&0 = \mathrm{div} \,{\mathbf {\xi }} + \pi , \end{aligned}$$
(23.26b)
and boundary conditions on
\(\partial \mathscr {B}\)$$\begin{aligned}&\mathbf {t}= \left( \mathbf {S}- \mathrm{div} \,\varvec{\mathcal {T}}\right) \cdot \mathbf {n}- {\text {div}}_{S}{\left( \mathbf {n}\cdot \varvec{\mathcal {T}}\right) },\end{aligned}$$
(23.26c)
$$\begin{aligned}&\zeta = \mathbf {\xi }\cdot \mathbf {n}\,. \end{aligned}$$
(23.26d)