The rank-two laminate is constructed by properly embedding a reinforcement phase a in a softer matrix b. In particular, such a heterogeneous transducer can be designed in two different ways [2], according to the layouts sketched in Fig. 1 that are independent of direction \(x_3\). In the first case, referred to as ‘Tree a’, the device is obtained by layering a core rank-one composite (whose relevant variables are henceforth labelled with ‘R1’) with layers of the soft material b acting as a shell. In our terminology, the shell is a purely homogeneous material and its quantities are labelled with ‘sh’. In the second case, referred to as ‘Tree b’, the rank-one core is sandwiched between layers of the stiffer material a, here playing the role of the shell.
We assume separation of length-scales such that each rank can be homogenised independently. More specifically, the local fields within the rank-one composite are microscopic fields, whereas, at a much larger scale, the mesoscopic fields are the local fields for the rank-two composite, in which the rank-one core is modelled as a homogeneous phase.
Within this picture, at the mesoscopic level, \(c^{\mathrm{R1}}\) and \(c^{\mathrm{sh}}=1-c^{\mathrm{R1}}\) denote the volume fractions of core and shell, respectively, the former reading \(c^{\mathrm{R1}} = c_a^{\mathrm{R1}}+c_b^{\mathrm{R1}}\). At the microscopic level, the homogenisation of the rank-one core requires the use of the volume fractions \(c_a^{\mathrm{R1}}/c^{\mathrm{R1}}\) and \(c_b^{\mathrm{R1}}/c^{\mathrm{R1}}\) for the two phases a and b, respectively. Finally, \(c^a\) and \(c^b\) are the overall volume fractions of the two homogeneous materials entering the whole composite, such that \(c^a+c^b=1\). In particular, it results:
$$\begin{aligned} c^a=c_a^{\mathrm{R1}} \ \ {\hbox {and}} \ \ c^b=c_b^{\mathrm{R1}}+c^{\mathrm{sh}} \ \ \ {\hbox {in `Tree a' configuration,}} \end{aligned}$$
while
$$\begin{aligned} c^a=c_a^{\mathrm{R1}}+c^{\mathrm{sh}} \ \ {\hbox {and}} \ \ c^b=c_b^{\mathrm{R1}} \ \ \ {\hbox {in `Tree b' configuration.}} \end{aligned}$$
The interfaces between phases a and b in the core are henceforth denoted as microscopic, whereas the mesoscopic interfaces are those separating shell and rank-one phases in the rank-two composite. The normal and tangential unit vectors defining the microscopic and mesoscopic interfaces belong to planes \(x_3=\mathrm {const}\) and are indicated as \(({{\mathbf{n }}}^1, {{\mathbf{m }}}^1)\) and \(({{\mathbf{n }}}^0, {{\mathbf{m }}}^0)\), respectively (see Fig. 1); we express their Cartesian components in terms of the interfaces’ angles with respect to the axis \(x_1\), denoted as \(\theta _{\mathrm{R1}}\) and \(\theta _{\mathrm{sh}}\), respectively. These angles are positive if anti-clockwise, as illustrated in Fig. 1. In both ‘Tree a’ and ‘Tree b’ configurations, the normal vector \({{\mathbf{n }}}^1\) always points from the stiffer phase a towards the softer phase b, while the normal vector \({{\mathbf{n }}}^0\) points from the rank-one core towards the shell.
In this work, the relevant mesoscopic fields, assumed to be spatially uniform, are the deformation gradient \({{\mathbf{F }}}_k\)\((k={\mathrm sh},{\mathrm R1})\) and the nominal electric field \({{\mathbf{E }}}_k\), along with their work-conjugate quantities, that are the first Piola–Kirchhoff stress \({{\mathbf{S }}}_k\) and the nominal electric displacement \({{\mathbf{D }}}_k\). The analogous macroscopic electro-mechanical quantities governing the overall response of the actuator are indicated as \({{\mathbf{F }}}\), \({{\mathbf{E }}}\), \({{\mathbf{S }}}\), \({{\mathbf{D }}}\).
Under voltage-controlled actuation in plane-strain conditions, we assign the through-the-thickness macroscopic nominal electric field component
$$\begin{aligned} E_2={\Delta \phi \over h^0}, \end{aligned}$$
in which \(\Delta \phi\) is the electric potential jump applied across the electrodes and \(h^0\) is the initial laminate thickness. Additionally, we impose
$$\begin{aligned} E_1=0, \end{aligned}$$
which is consistent with disregarding edge effects. This is coupled with macroscopic traction-free boundary conditions:
$$\begin{aligned} S_{22} &= 0, \end{aligned}$$
(1a)
$$\begin{aligned} S_{12} &= 0, \end{aligned}$$
(1b)
$$\begin{aligned} S_{11} &= 0, \end{aligned}$$
(1c)
the latter allowing free stretch \(\lambda\) along the \(x_1\) direction (see Fig. 1). Under these conditions, the deformation gradient assumes the form
$$\begin{aligned} {{\mathbf{F }}}= \left[ {\begin{array}{ccc} \lambda &{}\quad \xi /\lambda &{}\quad 0 \\ 0 &{}\quad 1/\lambda &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \\ \end{array} } \right] , \end{aligned}$$
in which the term \(\xi\) represents the amount of shear associated with the actual shear angle \(\gamma =\arctan {\xi }\).
We remark that the methodology developed in this investigation can be equally applied when the deformation along \(x_3\) is unconstrained. Hence, in Appendix, we indicate how to solve this dual boundary-value problem by utilising exactly the same equations as those obtained in the following. More importantly, on the basis of numerical investigations, in Appendix, we draw conclusions on the appropriateness of the plane-strain assumption.
Within the proposed two-scale framework, we obtain the overall actuation response through homogenisation of the mesoscopic level by following the same technique proposed in [4] for the rank-one composite. The main novelty in our study of the rank-two composite consists of using for the rank-one core phase the free energy density obtained by Spinelli and Lopez-Pamies [12], which is different from that characterising the shell. Conversely, in the computational and analytical investigations of Gei et al. [4] and Spinelli and Lopez-Pamies [12], the two phases constituting the rank-one composite therein studied have the same form of free energy density. In other words, the effective energy density of Spinelli and Lopez-Pamies [12] is in our context the analytical result of the microscopic rank-one homogenisation into a homogeneous mesoscopic phase. We remark that at both levels the homogenisation takes advantage of the condition that the interface normal (either \({{\mathbf{n }}}^1\) or \({{\mathbf{n }}}^0\)) is spatially uniform.
The homogenisation technique consists of coupling of information on the continuity of electro-mechanical variables at the mesoscopic interface with the definition of macroscopic quantities as weighed averages of mesoscopic fields [16,17,18].
On the one hand, continuity at the mesoscopic interface is expressed by [19]
$$\begin{aligned}&({{\mathbf{F }}}_{\mathrm{sh}}-{{\mathbf{F }}}_{\mathrm{R1}}){{\mathbf{m }}}^0={{\mathbf{0 }}}, \end{aligned}$$
(2a)
$$\begin{aligned}&({{\mathbf{S }}}_{\mathrm{sh}}-{{\mathbf{S }}}_{\mathrm{R1}}){{\mathbf{n }}}^0={{\mathbf{0 }}}, \end{aligned}$$
(2b)
$$\begin{aligned}&({{\mathbf{E }}}_{\mathrm{sh}}-{{\mathbf{E }}}_{\mathrm{R1}})\cdot {\mathbf{m }}^0=0, \end{aligned}$$
(2c)
$$\begin{aligned}&({\mathbf{D }}_{\mathrm{sh}}-{\mathbf{D }}_{\mathrm{R1}})\cdot {\mathbf{n }}^0=0. \end{aligned}$$
(2d)
where Eq. (2c) is obtained from the general relation \({\mathbf{n }}^0\times ({\mathbf{E }}_{\mathrm{sh}}-{\mathbf{E }}_{\mathrm{R1}})={\mathbf{0 }}\) particularised to the case here of interest, in which both \({\mathbf{n }}^0\) and the electric field vector have vanishing component along the \(x_3\) direction (see Fig. 1). Moreover, here and henceforth, \(\times\) and \(\cdot\) denote, respectively, the vector and inner products.
On the other hand, the macroscopic fields read
$$\begin{aligned} {\mathbf{F }} &= c^{\mathrm{sh}}{\mathbf{F }}_{\mathrm{sh}}+c^{\mathrm{R1}}{\mathbf{F }}_{\mathrm{R1}}, \end{aligned}$$
(3a)
$$\begin{aligned} {\mathbf{E }} &= c^{\mathrm{sh}}{\mathbf{E }}_{\mathrm{sh}}+c^{\mathrm{R1}}{\mathbf{E }}_{\mathrm{R1}}, \end{aligned}$$
(3b)
$$\begin{aligned} {\mathbf{D }} &= c^{\mathrm{sh}}{\mathbf{D }}_{\mathrm{sh}}+c^{\mathrm{R1}}{\mathbf{D }}_{\mathrm{R1}}, \end{aligned}$$
(3c)
such that fulfillment of (2a), (2c), (2d) requires the following forms of the mesoscopic fields in terms of the scalar coefficients \(\alpha\), \(\beta\), and \({\bar{\beta }}\) [11]
$$\begin{aligned} {\mathbf{F }}_{\mathrm{sh}} &= {\mathbf{F }}({\mathbf{I }}+\alpha \,c^{\mathrm{R1}}{\mathbf{m }}^0\otimes {\mathbf{n }}^0), \end{aligned}$$
(4a)
$$\begin{aligned} {\mathbf{F }}_{\mathrm{R1}} &= {\mathbf{F }}({\mathbf{I }}-\alpha \,c^{\mathrm{sh}}{\mathbf{m }}^0\otimes {\mathbf{n }}^0), \end{aligned}$$
(4b)
where \(\otimes\) denotes the outer product,
$$\begin{aligned} {\mathbf{E }}_{\mathrm{sh}} &= {\mathbf{E }}+c^{\mathrm{R1}}\beta \,{\mathbf{n }}^0, \end{aligned}$$
(5a)
$$\begin{aligned} {\mathbf{E }}_{\mathrm{R1}} &= {\mathbf{E }}-c^{\mathrm{sh}}\beta \,{\mathbf{n }}^0, \end{aligned}$$
(5b)
$$\begin{aligned} {\mathbf{D }}_{\mathrm{sh}} &= {\mathbf{D }}+c^{\mathrm{R1}}{\bar{\beta }}\,{\mathbf{m }}^0, \end{aligned}$$
(6a)
$$\begin{aligned} {\mathbf{D }}_{\mathrm{R1}} &= {\mathbf{D }}-c^{\mathrm{sh}}{\bar{\beta }}\,{\mathbf{m }}^0. \end{aligned}$$
(6b)
The coefficient \(\alpha\) is a dimensionless parameter, whereas \(\beta\) and \({\bar{\beta }}\) have the dimensions of an electric field and an electric displacement, respectively. Parameters \(\alpha\), \(\beta\), and \({\bar{\beta }}\) are determined by imposing further electro-mechanical conditions, involving the mesoscopic constitutive laws, that can be expressed in terms of the free energy densities \(W_k({\mathbf{F }}_k,{\mathbf{E }}_k)\) [20]
$$\begin{aligned} {\mathbf{S }}_k &= \frac{\partial W_k}{\partial {\mathbf{F }}_k}-p_k{\mathbf{F }}_k^{-T}, \end{aligned}$$
(7a)
$$\begin{aligned} {\mathbf{D }}_k &= -\frac{\partial W_k}{\partial {\mathbf{E }}_k} . \end{aligned}$$
(7b)
As an important issue to address in this investigation, in Eq. (7a) we have assumed materials constrained to undergo isochoric deformation, such that the stress depends on the Lagrangian multiplier \(p_k\), to be determined on each phase.
At the macroscopic level, the effective electro-elastic free energy density is the sum of the weighed mesoscopic energies, namely
$$\begin{aligned} W({\mathbf{F }},{\mathbf{E }})=c^{\mathrm{sh}}W_{\mathrm{sh}}({\mathbf{F }}_{\mathrm{sh}},{\mathbf{E }}_{\mathrm{sh}})+ c^{\mathrm{R1}}W_{\mathrm{R1}}({\mathbf{F }}_{\mathrm{R1}},{\mathbf{E }}_{\mathrm{R1}}) =c^{\mathrm{sh}}\tilde{W}_{\mathrm{sh}}(\alpha ,\beta )+c^{\mathrm{R1}}\tilde{W}_{\mathrm{R1}}(\alpha ,\beta ), \end{aligned}$$
(8)
where, by resorting to Eqs. (4) and (5), we have expressed the mesoscopic energies as functions of \(\alpha\) and \(\beta\). More precisely, the function W in Eq. (8) is the effective energy only for \(\alpha\) and \(\beta\) fulfilling the required conditions of the set boundary-value problem. We note that, in the voltage-controlled problem here of concern, the effective response of the rank-two composite is completely determined by the parameters \(\alpha\) and \(\beta\), while \({\bar{\beta }}\) would directly enter the effective response in the dual charge-controlled problem.
Analogously to (7), the macroscopic constitutive equations read [6, 10]
$$\begin{aligned} {\mathbf{S }} &= \frac{\partial W}{\partial {\mathbf{F }}}-p{\mathbf{F }}^{-T}, \end{aligned}$$
(9a)
$$\begin{aligned} {\mathbf{D }} &= -\frac{\partial W}{\partial {\mathbf{E }}}, \end{aligned}$$
(9b)
where
$$\begin{aligned} p=c^{\mathrm{sh}}p_{\mathrm{sh}}+c^{\mathrm{R1}}p_{\mathrm{R1}}. \end{aligned}$$
Now, we focus on some general results, inherent to our homogenisation framework, which can be conveniently employed in the computations, as explained in detail in Sects. 4 and 5.
First, by combining Eqs. (9b) and (8), we obtain
$$\begin{aligned} {\mathbf{D }}=-\left[ c^{\mathrm{sh}}\left( \frac{\partial W_{\mathrm{sh}}}{\partial {\mathbf{F }}_{\mathrm{sh}}}\frac{{\mathrm{d}}{\mathbf{F }}_{\mathrm{sh}}}{{\mathrm{d}}{\mathbf{E }}} +\frac{\partial W_{\mathrm{sh}}}{\partial {\mathbf{E }}_{\mathrm{sh}}}\frac{{\mathrm{d}}{\mathbf{E }}_{\mathrm{sh}}}{{\mathrm{d}}{\mathbf{E }}}\right) +c^{\mathrm{R1}}\left( \frac{\partial W_{\mathrm{R1}}}{\partial {\mathbf{F }}_{\mathrm{R1}}}\frac{{\mathrm{d}}{\mathbf{F }}_{\mathrm{R1}}}{{\mathrm{d}}{\mathbf{E }}} +\frac{\partial W_{\mathrm{R1}}}{\partial {\mathbf{E }}_{\mathrm{R1}}}\frac{{\mathrm{d}}{\mathbf{E }}_{\mathrm{R1}}}{{\mathrm{d}}{\mathbf{E }}}\right) \right] , \end{aligned}$$
(10)
in which our notation for the chian rule implies \([(\partial W/\partial {\mathbf{F }})({\mathrm{d}}{\mathbf{F }}/{\mathrm{d}}{\mathbf{E }})]_k \equiv (\partial W/\partial F_{ij})({\mathrm{d}}F_{ij}/{\mathrm{d}}E_k)\) with i, j, k indices with respect to a Cartesian system.
By accounting, in Eq. (10), for the dependence of the mesoscopic fields on the macroscopic quantities \({\mathbf{F }}({\mathbf{E }})\), \(\alpha ({\mathbf{E }})\), \(\beta ({\mathbf{E }})\) through Eqs. (4) and (5), we find out that the sums of the contributions multiplying \({\mathrm{d}}\alpha /{\mathrm{d}}{\mathbf{E }}\) and \({\mathrm{d}}\beta /{\mathrm{d}}{\mathbf{E }}\) turn out into the left-hand sides of continuity conditions (2b) and (2d), respectively. Similarly, two contributions involving \({\mathrm{d}}{\mathbf{F }}/{\mathrm{d}}{\mathbf{E }}\) multiply terms that allow one to single-out both the left-hand side of condition (2b) and the macroscopic stress \({\mathbf{S }}\). Finally, the product \(({\mathbf{F }}^{-T}{\mathrm{d}}{\mathbf{F }}/{\mathrm{d}}{\mathbf{E }})_k\equiv F^{-1}_{ji}{\mathrm{d}}F_{ij}/{\mathrm{d}}E_k\) represents the variation of \(\det {{\mathbf{F }}}\), to be neglected in the case of isochoric deformation. Hence, all these terms vanish. Because of this, it results that, in order to correctly evaluate Eq. (3c) through Eq. (10), we may completely disregard the dependence of \(\alpha\), \(\beta\), and \({\mathbf{F }}\) on \({\mathbf{E }}\).
Analogously, by combining Eqs. (9a) and (8), we obtain
$$\begin{aligned} {\mathbf{S }} &=c^{\mathrm{sh}}\left( \frac{\partial W_{\mathrm{sh}}}{\partial {\mathbf{F }}_{\mathrm{sh}}}\frac{{\mathrm{d}}{\mathbf{F }}_{\mathrm{sh}}}{{\mathrm{d}}{\mathbf{F }}} +\frac{\partial W_{\mathrm{sh}}}{\partial {\mathbf{E }}_{\mathrm{sh}}}\frac{{\mathrm{d}}{\mathbf{E }}_{\mathrm{sh}}}{{\mathrm{d}}{\mathbf{F }}}\right) \\&\quad +c^{\mathrm{R1}}\left( \frac{\partial W_{\mathrm{R1}}}{\partial {\mathbf{F }}_{\mathrm{R1}}}\frac{{\mathrm{d}}{\mathbf{F }}_{\mathrm{R1}}}{{\mathrm{d}}{\mathbf{F }}} +\frac{\partial W_{\mathrm{R1}}}{\partial {\mathbf{E }}_{\mathrm{R1}}}\frac{{\mathrm{d}}{\mathbf{E }}_{\mathrm{R1}}}{{\mathrm{d}}{\mathbf{F }}}\right) -p{\mathbf{F }}^{-T}. \end{aligned}$$
(11)
Calculations similar to those concerned with \({\mathbf{D }}\) lead to the result that the relation
$$\begin{aligned} c^{\mathrm{sh}}{\mathbf{S }}_{\mathrm{sh}}+c^{\mathrm{R1}}{\mathbf{S }}_{\mathrm{R1}} = {\mathbf{S }} \equiv {\mathbf{0 }} \end{aligned}$$
is correctly evaluated through Eq. (11) even by totally neglecting the derivatives \({\mathrm{d}}\alpha /{\mathrm{d}}{\mathbf{F }}\) and \({\mathrm{d}}\beta /{\mathrm{d}}{\mathbf{F }}\), as they multiply contributions that cancel out.
These observations are particularly useful in solving our problem as they allow us to end up with an algebraic system involving, among the unknowns, only a single Lagrangian multiplier p. Given the nonlinearity of the system providing the solution, the analytical development proposed in this work leads to a relevant computational advantage with respect to an approach directly implementing all the governing equations, where both mesoscopic pressures enter the system unknowns.
Moreover, within our homogenisation procedure, by combining Eqs. (11) and (1a), we can analytically obtain p as a function of \(\alpha\), \(\beta\), \(\lambda\), and \(\xi\), thus further reducing the dimension of the solving nonlinear system.
We finally note that in other analogous voltage-controlled problems, such as that with imposed vanishing shear deformation, \(F_{12}=0\), and \(S_{12}\) to be determined among the macroscopic unknowns, the foregoing observations about relations (10) and (11) still hold.
In the next section, we specify the mesoscopic free energy densities characterising the rank-two laminate.