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Consequently, any attempt to solve a problem related to a smart composite must successfully decouple the two scales and treat the two problems macroscopic and microscopic independently. One such technique that has enjoyed significant success for many years is that of asymptotic homogenization. The pertinent mathematical details. Many problems in elasticity, thermoelasticity, and piezo-magneto-elasticity have been solved via asymptotic homogenization.

We want to mention particularly the works of Kalamkarov [45], who analyzed a wide variety of problems, such as composite and reinforced plates and shells, network-reinforced shells, plates with corrugated surfaces and other structures, Kalamkarov and Kolpakov [46] who used these models to design and optimize various composite structures on account of strength and stiffness requirements, the pioneering work of Guedes and Kikuchi [47] on computational aspects of homogenization, the modification of asymptotic homogenization for problems related to elasticity and thermal conductivity of thin plates appearing in the works of Duvaut [48,49], Andrianov et al.

Recent years have witnessed the emergence of smart composite plates and shells as the preeminent structural members for many practical applications. Enhanced strength, reduced weight, materials savings and ease of fabrication are among the reasons that make these structures attractive. More recently, advancements in the field of nanotechnology and the increasing popularity of nanocomposite thin films, plates and shells [57] have further enhanced the application potential of such structures. The periodic or nearly periodic nature of smart composite and nanocomposite plates and shells renders asymptotic homogenization a valuable tool in their analysis, design and optimization.

The "classical" asymptotic homogenization approach however cannot be applied directly to a thin plate or shell if the scale of the spatial inhomogeneity is comparable to the thickness of the structure. In that case, a refined approach developed by Caillerie [52, 53] in his heat conduction studies is needed. In particular, a two-scale formalism is applied, whereby a set of microscopic variables is used for the tangential directions in which periodicity exists and another microscopic variable is used for the transverse direction in which periodicity considerations do not apply.

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Kohn and Vogelius [] adopted this approach in their study of the pure bending of a thin, linearly elastic homogeneous plate. Kalamkarov [45] and Kalamkarov and Kolpakov [46] applied this modified two-scale methodology to determine the effective elastic, thermal expansion and thermal conductivity coefficients of thin curvilinear composite layers.

Challagulla et al. Kalamkarov and Geor-giades [60] and, Georgiades and Kalamkarov [61] developed comprehensive micromechanical models for smart composite wafer- and rib-reinforced plates. Saha et al. In that work only an in-plane temperature variation was considered and therefore any out-of-plane thermal effects were ignored.

Thus, unlike in the present work, the out-of-plane thermal expansion, pyroelectric and py-romagnetic coefficients were not captured in [63]. More important, however, is the fact that the micromechani-cal model in [63] is only applied to the case of simple laminated plates.

In contrast, the model developed in the present work explicitly allows for different periodicity in the lateral directions. As such, it is readily amenable to the design and analysis of magnetoelectric reinforced plates such as the wafer-reinforced structures shown in Section 4. To the authors' best knowledge, this is the first time completely coupled piezo-magneto-thermo-elastic effective coefficients for reinforced plates are presented and analyzed.

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Also relevant to the present papers are the works of Kalamkarov and Georgiades [60], Georgiades and Kalamkarov [61] and Hadjiloizi et al. In [60] and [61], Kalamkarov and Georgiades developed and illustrated the use of an asymptotic homogenization model for the analysis of reinforced piezoelectric plates. In their work, [60], [61] the authors adopted only a semi-coupled analysis, which results in expressions for the effective coefficients that do not reflect the influence of such parameters as the electric permittivity, magnetic permeability, primary magnetoelectricity etc.

In the present work and its companion paper [64], however, a fully coupled analysis is performed, and as a consequence the expressions for the effective coefficients involve all pertinent material parameters. As an example, the effective extensional elastic coefficients are dependent on not only the elastic properties of the constituent materials, but also on the piezoelectric, piezomagnetic, magnetic permeability, dielectric permittivity and magnetoelectric coefficients.

The same holds true for the remaining effective coefficients. This feature is captured in the present papers, but not in previously published works, such as [60] and [61]. Thus, the results presented here represent an important refinement of previously established results. To the authors' best knowledge completely coupled piezo-magneto-thermo-elastic effective coefficients for reinforced plates have not been presented and analyzed before.

In [25], [26], Hadjiloizi et al. However, these models employed the "classical" homogenization approach, see [43] for example, and consequently could not capture the mechanical, thermal, piezoelectric and piezomagnetic behavior that is related to bending, twisting and general out-of-plane deformation and electric and magnetic field generation.

The model developed in the current work and its companion paper [64], however, accomplishes precisely this; it employs the modified asymptotic homogenization technique discussed earlier which makes use of two sets of microscopic variables that permit the decoupling of inplane and out-of-plane behavior of the structure under consideration.

For example, the elastic coefficients can be distinguished into the familiar extensional, bending and coupling coefficients, which is not possible to achieve with the 3D models in [25] and [26]. What this amounts to is the fact that the two modeling approaches are essentially applicable to entirely different structures and geometries. The 3D models in [25], [26] can be used to analyze structures of comparable dimensions in the x, y, z directions such as thick laminates but cannot be used for thin structures such as wafer- and rib-reinforced plates. The micromechanical models developed in the present work, however, are applicable to structures with a much smaller dimension in the transverse direction than in the other two directions.

Thus, they can be used in the design and analysis of an impressive range of composite and reinforced plates such as the aforementioned wafer-and rib-reinforced structures see Section 4 , three-layered honeycomb-cored magnetoelectric plates, thin laminates Section 3 etc. To summarize, the present paper deals with the development and applications of appropriate plane stress mi-cromechanical models for thin magnetoelectric composite and reinforced plates.

The work is implemented in two parts. In part I [64] the pertinent micromechanical models are derived, and the unit cell problems, from which the effective coefficients including the product properties can be extracted, are obtained. The applications of the de-. Following this introduction the basic mathematical model and the pertinent unit cell problems are reviewed in Section 2.

Sections 3 and 4 present, respectively, the solution of the unit cell problems for magnetoelectric laminates of constant thickness and for wafer-reinforced magnetoelectric plates. Section 4 also compares the results of the developed model with previously reported results, and, finally, Section 5 concludes the work. In view of the applications mentioned earlier in this section, the most important aspect of this publication is the development of closed-form design-oriented equations that can be used in the analysis and design of magnetoelectric composite and reinforced plates.

It is shown that thermoelasticity, piezoelectricity and piezomagnetism are entirely coupled and the solution of one affects the solutions of the others. The boundary value problem characterizing the thin smart composite plate of rapidly varying thickness, Fig. Here aij is the mechanical stress, D;- and B;- are, respectively, the electric displacement and magnetic induction, Ei and Hi are the electric and magnetic fields, Pi represents a generic body force, pi represents the surface tractions, and ui is the mechanical displacement. Finally, T is the change in temperature with respect to a suitable reference.

We note that because the composite layer is periodic only in the tangential directions, see Fig.

We finally note that the overall thickness of the structure must be small compared to the other two dimensions. For the analysis of a thick piezo-magneto-thermo-elastic laminate, one should consider an appropriate 3D model, e. The in-plane force and moment resultants pertaining to the homogenized plate are given in Hadjiloizi et al. Finally, the expressions for the mechanical displacement and the electric and magnetic potentials can be written.

Likewise, the averaged electric displacement and magnetic induction, see Hadjiloizi et al. Note that in the previous sentence and from this point onwards, for the sake of convenience, all equations that are referenced from Hadjiloizi et al. For example, Eq. We will illustrate our work by means of several examples. The first examples are for laminates of constant thickness, as shown in Fig. As shown in the unit cell of Fig. SM where M is the total number of layers.

The real thickness of the mth layer as measured in the original x1,x2,x3 coordinate system is S Sm-Sm-1 , where S is the thickness of the laminate again with respect to the original coordinate system. Furthermore, Lijm, My, Nj etc. It is apparent that all material parameters are independent of y1 and y2, and consequently, all partial derivatives in Eqs.

We will consider laminates made up of perfectly bonded laminae of piezoelectric and piezomagnetic materials with the poling and magnetization directions along the z-axis. The "perfect bonding" assumption is akin to neglecting the interphase layers between adjacent plies. Because the interphase regions might be important in the case of nano-laminates, application of the derived models to such structures might need to take the interphase lay-. Furthermore, the overall thickness of the laminate is considered to be small compared to the in-plane dimensions.

For the sake of generality, we will also assume that the constituent materials are made of orthotropic materials, with the principal material coordinate axes not necessarily coinciding with the yi, y2, z system but with a system that has been rotated by an arbitrary angle with respect to the z axis.

As such, the pertinent coefficient matrices tensors are the same as those of a monoclinic material, as far as the number and location of the non-zero coefficients is concerned, see Reddy [68]. In Eq. We will now proceed with the solution of the unit cell problems and the determination of general expressions for the effective coefficients. To obtain the effective properties, the following procedure will be adhered to: The unit cell problems 2.

The pertinent boundary conditions will also be simplified, because the normal vector N becomes 0,0,1. Subsequently, the reduced unit cell problems will be solved in a straight-forward manner, giving the coefficient functions bj, nfv, Ska etc. Because each of these functions is in turn a function of three local functions for example Eq.

Hence, we will look for the three unit cell problems that use. For example, local functions Nm, Aka, and Aka appear in unit cell problems 2. Finally, after obtaining these local functions, we will back-substitute them into the appropriate expressions for the coefficient functions which in turn yield the effective coefficients after applying the homogenization procedure of Eq.

Entire solutions of nonlocal elasticity models for composite materials

Solving 3. From Eq.


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Hence, we will need two more equations which will come from unit-cell problems 2. Following the same procedure as above, and keeping Eq. The second feature that is evident in Eqs. Had we chosen polarization and magnetization directions along y1 or y2 the results would be drastically different. Herein lies a significant advantage of our. Using these solutions, the in-plane elastic, piezoelectric and piezomagnetic functions from which the effective coefficients will be computed in the sequel may be calculated as follows:.

From the second expression in Eq. Two features are worth mentioning here. First of all, it can be seen that the elastic functions and as a consequence the effective extensional elastic coefficients depend not only on the elastic parameters of the constituent phases, but also on the piezoelectric, piezomagnetic, and magne-toelectric coefficients, as well as the dielectric permittivities and magnetic permeabilities. This is in marked contrast with previous simpler models, see Kalamkarov and Georgiades [60], which predicted that the extensional elastic coefficients depend only on the elastic parameters of the constituents.

Therefore, the present work constitutes an important refinement over previously established results. Also, for the case of simple laminates such as the ones considered in this section, the work presented here represents an extension of the classical composite laminate theory see e. Of course, if piezoelectric and piezo-magnetic effects are completely ignored, then Eq. As we need two more equations, we resort to unit cell problems 2.

Following the same procedure as above we end up with:. Using Eqs. The reason why the in-plane piezoelectric functions vanish was explained above. Also, we note the appearance of the magnetoelectric functions first product properties. Similarly to the previous three unit cell problems, Eqs.

Using these results, the in-plane piezomagnetic, magnetic permeability and magnetoelectric functions are derived as follows:. With these results we can calculate the thermal expansion, pyroelectric and pyromagnetic functions related to the mid-plane temperature variation.

We recall from Eq. The set of coefficients stemming from Eqs. Hence, the solution of the secondary pyroelectric and py-romagnetic as well as in-plane thermal expansion functions all related to the through-the-thickness temperature variation is given simply by:. The last set of functions that we need to consider are the elastic bending functions shown in Eq.

It is evident that:. The effective coefficients are obtained in a straightforward fashion by directly applying the homogenization procedure in Eq. For example, referring to Fig. We will illustrate our work by considering a simple 4-ply laminate consisting of alternating barium titanate top layer and cobalt ferrite laminae. The overall thickness of the laminate is 1 mm. The pertinent material parameters are given in Table 1. For the sake of discussion we will further assume that the Barium Titanate is doped with Fe so that it exhibits primary magnetoelectric-ity.

Thus, in the parlance of our present work we define the magnetoelectric coefficients slightly differently, see Section 2 and for a dielectric permittivity value of around We will also assume that the Fe doping does not affect the remaining properties of BaTiO3 as shown in Table 1. Likewise, we will assume that the cobalt ferrite is doped with the rare earth element Dy, see Dascalu et al. Thus, for a dielectric permittivity value of around 0.

It can clearly be seen that the effective elastic coefficients change. Since cobalt ferrite is generally stiffer than its barium titanate counterpart, increasing the overall thickness of the latter causes a corresponding reduction in the value of the effective bending coefficients. As expected, the absence of barium titanate renders the laminate symmetric and nullifies the effective coupling coefficients. As the thickness of this constituent increases,. Figure 4: Plot of effective magnetic permeability coefficient vs.

Finally, as the thickness of barium titanate approaches 0. As expected, reducing the volume fraction of Dy-doped CoFe2O4 results in a corresponding reduction of the effective magnetic permeability coefficients and an increase in the effective magnetoelectric coefficients. In essence, our derived model is comprehensive enough, in that it affords complete flexibility to the designer to customize the effective properties of the smart composite structure to conform to the requirements of a particular engineering application.

This is also evident in the next example considered in this paper. The following examples will be concerned with a different type of structure, namely a wafer-reinforced magnetoelectric plate, shown in Fig. For generality we will assume that the material of the base-plate is different than that of the ribs.

For example, the base-plate may be elastic or piezomagnetic and the ribs may be piezoelectric. Each constituent material maybe assumed to be orthotropic. We are interested in calculating the effective elastic, piezoelectric, thermal expansion, dielectric permittivity, magneto-electric, pyroelectric etc coefficients for this structure.

A solution of the local problems relevant to this kind of geometry may be found assuming that the thickness of each of the three elements of the unit cell is small in comparison with the other two dimensions, i. The local problems can then be approximately solved for each of the unit cell elements assuming that the discontinuities at the joints are highly localized.

Consequently, the local problems can be solved independently for regions and as shown in Fig. The analytical procedure followed in this example is similar to its counterpart in the previous example. First of all, the unit cell problems are simplified in each of the three regions of the unit cell. In particular, periodicity conditions in yi and y2 reduce the pertinent partial differential equations in Region 3 to ordinary differential equations in z. Likewise, since Region 1 is thin and entirely oriented in the y2 direction it is characterized by independence in y2.

Hence, the corresponding unit cell problem is reduced from a partial differential equation in variables yi, y2, and z into one involving yi and z only. Similarly, the appropriate differential equation for Region 2 is reduced to one involving variables y2 and z only. The solution of the unit cell problem in each region involves coefficient functions, e.

Nm, Aka, and Aka. Since we need three equations to solve for the three unknown local functions, we need to simultaneously consider all three unit cell problems which involve the given local functions. For example, unit cell problems 2. Once the local functions are determined, they are back substituted into the expressions for the coefficient functions, Eqs.

The results from each region are then superimposed. However, our approximation will be quite accurate, since these regions of intersection are highly localized and do not contribute significantly to the integral over the entire unit cell domain. A complete mathematical justification for this argument in the form of the so-called princi-. We will first tackle unit cell problem 2. If we assume that the structure is piece-wise homogeneous, then the elastic coefficients in each region of Fig.

Now, recalling Eqs. The elastic bj functions from which the effective elastic coefficients may be readily determined are then given by:. This problem will be solved in each of the regions fl2, and separately and will yield the in-plane nka piezomagnetic functions which will later give the effective piezomagnetic coefficients according to:.

Again, this problem will be solved in each of the regions Q1, Q2, and Q3 separately and will yield the in-plane 8fa ii Region Q-. As was the case of the laminate of the previous example, we expect coupled solutions of the current unit cell problems. Hence, Eqs. We begin by setting up the boundary conditions in each region. Thus, from Eqs. Thus, using 4.

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We observe that these expressions are the same as those in Eqs. We are now ready to solve the unit cell problems in Eqs. We will begin with the T? Region Q3. Because of periodicity in y1 and y2, and considering differential equations 4. The latter expression in Eq. Region In this region we have independence of the y2 coordinate, since the element is oriented entirely in the y2 direction. Thus, from differential equations 4.

The latter two expressions in Eq. Realizing that we need one more equation, we turn our attention to unit cell problem 4. The solution may be given as:. From the appropriate definition in Eq. The solution of system 4. Region Q2. In this region we have independence of the y1 coordinate since the element is oriented entirely in the y1 direction.

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Thus, the solution of differential equations 4. The solution of ther! The expressions of the appropriate local functions are given as:. We need two more equations in order to be able to solve for the unknown functions. Therefore, we resort to unit cell problem 4. On account of the pertinent definition in Eq. The solution of the tJ2, t12, and nf2 problems proceeds in the same manner. In this case the algebraic systems involved are trivial and the expressions of the appropriate local functions are obtained in a straight-forward fashion as:.

Before explaining how the effective coefficients may be obtained from the aforementioned local coefficient functions, we will first solve the corresponding unit cell problems associated with the out-of-plane deformation and electric and magnetic field generation of the reinforced magnetoelectric plate. We now turn our attention to unit cell problems 2. For a piecewise homogeneous unit cell, problem 2. The coupling elastic bj1va functions from which the effective elastic coupling coefficients may be determined are then given by:.

Again, this problem will be solved in each of the regions Q1, Q2, and separately and will yield the out-of-plane piezoelectric functions which in turn give the effective out-of-plane piezoelectric coefficients according to:. Thus, using Eqs. Finally, the third unit cell problem that will be solved in conjunction with the aforementioned two problems,.

The same conclusion, namely that t ;. We will begin with the Ty 22, t-1 22, and nP22 functions. Following the same methodology as per the corresponding t? Here, functions n1-fl9 are given in Eqs.

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Likewise, functions Tj, t11 , n11 are obtained in a similar way:. Comparing this system with Eqs. Clearly, this is to be expected, because the three unit cell problems under discussion pertain to out-of-plane deformation of the magneto-electric composite. Hashin, Z. Google Scholar. Solids 10 , — Solids 11 , — Walpole, L. Solids 14 , — Solids 17 , — Beran, M. Willis, J. Solids 25 , — McCoy, J. Recent Advances in Engineering Sciences, vol. Silnutzer, N. Thesis, Univ. Milton, G. Solids 30 , — London Ser.

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