1. Introduction
The development of modern technology is inseparable from composite materials. Composite material structures have been extensively used in various engineering fields in the past few decades. However, the heterogeneity of microstructures brings high computational costs to the modeling and simulation of composite structures. Since composite structures have various scale characteristics in nature, multiscale homogenization methods are an important tool for the analysis of composite structures from the point of view of accuracy and efficiency.
In recent decades, many multiscale homogenization methods have been proposed to deal with composite structures, such as the two-scale asymptotic homogenization method (AHM) [
1,
2,
3], the multiscale eigenelement method (MEM) [
4,
5,
6], the heterogeneous multiscale method (HMM) [
7,
8], the variational asymptotic method (VAM) [
9,
10], and for many other multiscale homogenization methods referred to [
11,
12] and the references cited therein. Among the above numerical homogenization methods [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10], the AHM [
1,
2,
3] is one of the most representative ones with a rigorous mathematical foundation and has been widely used in the homogenization analysis of periodic composite structures for statics [
13,
14,
15,
16,
17,
18] and dynamics [
19,
20,
21,
22]. However, the AHM [
1,
2,
3,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22] was mainly confined to coping with structural problems with omnidirectional periodicity, such as two-dimensional (2D) plane or three-dimensional (3D) solid problems, where the governing equations are second-order elliptic partial differential equations (PDEs) with periodically oscillating coefficients. Whereas, studies about the transverse homogenization of composite plate structures with only in-plane periodicity are still inadequate. A unique property of this kind of problem is that the plates deform in the thickness direction, which is perpendicular to the direction in which the periodicity exists. To realize the transverse homogenization of periodic plates, we have to choose one of the plate theories, which are briefly reviewed below, before reviewing the works related to the homogenization.
The Kirchhoff plate theory [
23] is the simplest plate theory in which only deflection is the independent variable. Brunelle et al. [
24] pointed out that the Kirchhoff plate theory is not suitable for moderately thick plates or those with a large ratio of elastic modulus to shear modulus, since the Kirchhoff plate theory ignores the transverse shear deformation. To improve the accuracy of the Kirchhoff plate theory, Reissner and Mindlin et al. [
25,
26] constructed the first-order shear plate theory by introducing two additional independent angles to describe the transverse shear deformations, but it cannot capture warping deformation, so the shear correction coefficient was introduced to update shear stiffness. Furthermore, several higher-order shear plate theories [
27,
28,
29,
30,
31,
32,
33,
34] were proposed by expanding in-plane displacements into higher-order power functions of thickness coordinates, one representative of which is the third-order shear deformation theory [
32]. Additionally, layer-wise theories [
35,
36,
37,
38] provide a powerful tool to simulate the stress distribution of laminates, achieving higher accuracy compared with single-layer plate theories [
39,
40].
However, the existing plate theories [
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40] were developed for homogeneous plates or laminated composite plates. For composite plate structures with periodic microstructures, these plate theories cannot be used directly, and they should be used together with the solutions of the unit cell problem to effectively fulfill the static and dynamic analyses of the periodic plates.
To deal with periodic thin plate structures, some work focused on the simplification of 3D periodic thin plate problems to 2D homogenized Kirchhoff plate models in analytical manners. Libove et al. [
41] were one of the earliest researchers to investigate the homogenized properties of sandwich plates and state that the sandwich panel can be transformed into an equivalent homogeneous panel with elastic constants. Briassoulis et al. [
42] held that corrugated thin plates composed of anisotropic materials could be equivalent to homogenized orthotropic thin plates, and derived the analytical expressions of the effective bending stiffnesses in accordance with force-distortion relationships. Richard et al. [
43], Samanta et al. [
44], and Kress et al. [
45] achieved the analytical results of effective stiffnesses of sinusoidal, trapezoidal, and circular corrugated plates. Based on the principle of strain energy equivalence, Xia et al. [
46] established a homogenization-based analytical model to obtain the explicit expressions of effective in-plane tensile stiffness and out-of-plane bending stiffness for various sandwich plates. As for periodic moderately thick plate structures, transverse shear effects cannot be neglected, so the Reissner–Mindlin plate model should be considered in the process of stiffness predictions. According to the relation between force and deformation, Libove et al. [
41] derived the effective bending stiffness and transverse shear stiffness of corrugated sandwich plates. Following the work of Libove et al. [
41], Lok et al. [
47,
48], Fung et al. [
49,
50,
51], and Leekitwattana et al. [
52] derived the effective properties of some periodic, moderately thick plates in accordance with force-distortion relationships.
However, the abovementioned analytical approaches [
41,
42,
43,
44,
45,
46,
47,
48,
49,
50,
51,
52] to effective constant predictions may encounter difficulties when addressing periodic plates with complex microstructures. Hence, numerical homogenization methods may be a necessary choice in such situations. For composite plates with only in-plane periodic microstructures, Nasution et al. [
53] used the AHM to achieve the in-plane effective properties by relieving the periodicity in the thickness direction. Nevertheless, the traditional AHM [
1,
2,
3,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22] cannot well capture out-of-plane deformations produced by transverse loads due to few or no repeated unit cells in the thickness direction. To achieve the out-of-plane effective stiffnesses of periodic thin plates, some work [
54,
55,
56,
57] simplified 3D plate problems to 2D homogenized Kirchhoff plate problems by assuming the behaviors in the thickness direction. However, these works [
54,
55,
56,
57] required the solution of complex 3D unit cell problems and were difficult to achieve the micro deformations under real loads.
Different from previous work, the main purpose of this work is to develop a homogenization method for composite thin plates with in-plane periodicity by combining the Kirchhoff plate theory with the two-scale AHM to predict the effective bending stiffness and achieve two-scale displacements. To the author’s knowledge, few studies have been conducted on the homogenization of the 2D periodic Kirchhoff plate. Kolpakov [
58] applied the AHM to deal with the 2D inhomogeneous periodic plate with initial stresses. Recently, Faraci et al. [
59] held that the two-scale AHM could be used to analyze sufficiently thin periodic plates based on the Kirchhoff plate theory, and the first two-order asymptotic solutions were presented. However, the above references [
58,
59] did not involve the solutions of higher perturbed terms or their physical interpretations.
In this work, we focus on the two-scale asymptotic solutions of 2D periodic Kirchhoff plates. The main novel contributions to this work are listed as follows:
- (1)
The asymptotic expansion solutions of the fourth-order PDE with rapidly periodic oscillating coefficients were presented, and the accuracy of high-order perturbed terms was investigated quantitatively.
- (2)
Constraint conditions for the unit cell problem were given and elaborated physically.
- (3)
The influence functions were interpreted physically for a better understanding of the two-scale AHM.
- (4)
The explicit analytically homogenized stiffness for periodic plates with layered structures was presented.
The rest of this paper is organized as follows: In
Section 2, a 3D periodic plate problem is simplified as a 2D Kirchhoff plate model, and then the process of asymptotic homogenization for the 2D Kirchhoff plate model is presented;
Section 3 provides the solutions for unit cell problems; in
Section 4, the physical interpretations of influence functions are presented; in
Section 5, numerical experiments are conducted to verify the proposed method; and conclusions are drawn in
Section 6.
3. Solutions for Unit Cell Problems
In this section, constraint conditions are first proposed to determine influence functions which are used to calculate the homogenized stiffness in Equation (37) and the two-scale solution in Equation (15); then, the analytical homogenized stiffnesses are presented for periodic layered structures; finally, finite element formulations are provided for unit cell problems.
3.1. Constraint Conditions for Unit Cell Problems
A key factor for obtaining the influence functions
(
n = 2, 3) is to determine the constraint conditions for unit cell problems (32) and (34). Since the perturbed term
(
n = 1, 2, ⋯) is periodic in
Y, implying that
(
n = 2, 3) has the same periodicity in
y as follows:
where
and
respectively represent two pairs of periodic boundaries, as shown in
Figure 3:
Equation (38) implies the continuity conditions of the interface displacements. Likewise, to satisfy the continuity conditions of normal rotation angles, normal bending moments, as well as equivalent shear forces, another three periodic conditions concerning
(
n = 2, 3) is required as follows:
It is worth noting that periodic boundary conditions (38) eliminate the rigid rotation modes with respect to the
and
axes. In order to remove the remaining rigid translation mode along the transverse direction,
(
n = 2, 3) is required to satisfy the following normalization constraint condition:
According to the Kirchhoff plate theory,
C1 continuity is supposed to be satisfied at the interface of unit cells for the periodic plate, and thus, another two periodic conditions are used to ensure the continuity of tangential rotation angles as follows:
With the proposed constraint conditions (38)–(43), one can solve the unit cell problems (32) and (34) by analytical methods for simple structures or by the finite element method (FEM) for general structures. It is worth noting that the perturbed displacements (n = 2, 3) are defined as the products of the influence function and the corresponding n-th derivative of . Here, reflecting the inhomogeneity of unit cells, is the solution of the unit cell problems (32) and (34), while is the homogenized (macroscopic) solution to the global problem (36) depending on the obtained homogenized stiffness and the external load q. As a result, only meets the boundary conditions of the plate, whereas cannot in general due to the use of periodic boundary conditions.
3.2. Analytical Homogenized Stiffnesses for Periodic Layered Plates
This section presents the analytical solution of homogenized stiffness for periodic layered plates, where the material properties change only in the
direction, refer to
Figure 4b as an example. Hence,
is the function of
, that is
Taking into account the constraint conditions (38)–(43) and the distribution of
as Equation (44), the solution of the cell problem (32) is assumed as a function of
, and has the following form:
By substituting Equation (45) into (32), the unit cell problem (32) becomes a fourth-order ordinary differential equation as follows:
Inserting Equation (45) into Equation (37) results in:
Thus, the homogenized stiffness can be achieved if
is determined. With the periodic conditions (39) and (40),
can be obtained from Equation (46), as
where
.
By substituting Equation (48) into Equation (47), one can obtain the explicit expressions of as:
It can be seen from Equation (49) that depends on the proportions of materials of layered structures in a highly nonlinear way.
3.3. Finite Element Formulations for Solving Unit Cell Problems
This section aims to provide the finite element formulations of the boundary value problems for
and
. The Lagrange multiplier method [
64] was used to consider the normalization constraint condition (42), and the master-slave method [
65] was employed to deal with the periodic boundary conditions (38), (39) and (43).
It follows from the governing equations of
and
in Equations (32) and (34) that the right-side terms are actually self-balanced quasi-loads depending on the material constants. According to the principle of virtual work and the Lagrange multiplier method, the unit cell problems (32) and (34) with the normalization constraint condition (42) can be respectively written as:
where
and
denote the Lagrange multipliers.
Considering the symmetric property of
and the arbitrariness of the virtual displacements
and
, Equations (50) and (51) can be respectively discretized in matrix form as:
where
where
K denotes the global stiffness matrix of the microscopic unit cell model;
is the domain of the element
e in the unit cell model;
N represents the shape function row vector;
and
stand for the second-order and third-order quasi-load matrices, respectively.
Since the node parameters of the plate elements contain deflection and two rotation angles, the periodic conditions (38), (39) and (43) can be implemented by the master-slave method to make the node parameters
for
n = 2, 3 on slave boundaries be identical to those on the corresponding master boundaries. Then, the nodal degrees of freedom on the slave boundaries are eliminated by applying the multifreedom constraints such that
where
T represents the transformation matrix. By inserting Equations (65) and (66) into Equations (52) and (53), one has
By solving Equations (67) and (68) in conjunction with Equations (65) and (66),
and
are finally achieved. With the obtained
and Equation (37), the numerical result of the homogenized stiffness matrix of a periodic Kirchhoff plate is achieved as follows:
Then for a static problem (14) of the periodic plate, one can easily figure out the homogenized solution by using the finite element methods. For clarity, the finite element formulations for this problem are given as follows:
where
where
and
denote the global stiffness matrix and global load vector of the macroscopic plate model, respectively;
represents the element nodal displacement vector;
is the domain of the plate element
e.
After solving
from Equation (70), the asymptotic solution perturbed to the third order in the plate element
e can be determined in a matrix form as follows:
where
It follows from Equation (74) that the accuracy of the asymptotic solutions depends on the accuracy of the homogenized displacements and their derivatives if the unit cell problems are solved accurately.
Here, a new method is introduced for calculating higher-order derivatives of the homogenized displacements whose order is beyond the order of the shape functions.
Without loss of generality, take an example of the four-node rectangular element with nodal parameters
, denoted as ACM12. The form of
containing twelve nodal parameters is as follows:
According to the property of the shape function
N, one has
where
represents
, and the same interpretations are also true for other similar expressions. Substituting Equations (77)–(79) into Equation (76) results in:
Then, the first-order derivative of
can be computed by:
or
where
Note that above is not the direct derivative of the node displacement vector, but the column vector of the nodal displacement partial derivative concerning . The same interpretations are also true for the following higher-order terms in Equations (84)–(86).
Replacing in Equation (81) by yields:
Here, if
, we define
By using Equations (82)–(85), one can calculate any high-order derivatives of
by:
4. Physical Interpretation of AHM
For a better understanding of the present work, this section aims to interpret the AHM for plates from a physical point of view. In the AHM, the actual (or two-scale) displacements are the superposition of homogenized displacement
and the perturbed displacements
, which are the products of
and the
n-th derivatives of
. Therefore, the physical interpretation [
17] of the AHM is essentially equivalent to that of the influence function
, or more specifically, the quasi-load
for solving
.
It can be seen from the unit cell problems (32) and (34) as well as their finite element formulations (59) and (60) that (n = 2, 3) completely depend on the material constants and also the microstructure of the unit cell. In other words, , independent of external forces, are caused by the inhomogeneity of the unit cells. Hence, is called quasi-load in the present work, and its values are zero for homogenized structures. Note that is self-balanced in a unit cell, implying the zero mean value of in Y.
For clarity, a unit cell with a single inclusion, as shown in
Figure 5, is used to demonstrate the physical interpretation of the AHM. The unit cell is
in height, and its in-plane size is
with a square inclusion at its center. Both the matrix and the inclusion are isotropic, with the same Poisson’s ratio of 0.3, and Young’s moduli
for inclusion and
for the matrix, respectively. The unit cell is modeled by
four-node cubic Hermite rectangular elements here.
Figure 6 and
Figure 7, respectively, present the diagrams of each column of
and
, where
Q3 denotes the nodal shear forces along the thickness direction,
M22 and
M11 represent the nodal bending moments around the
y1 and
y2 axes, respectively. Some conclusions can be drawn from
Figure 6 and
Figure 7, as follows:
- (1)
The quasi-shear forces are zero for and , and the quasi-bending moments for is equal to zero. Compared with , has nonzero quasi-bending moments and nonzero quasi-shear forces, and their distributions are more complex than those of .
- (2)
is self-balanced in the unit cell, and has nonzero values only at the interfaces of the matrix and inclusion as well as the boundaries, implying that is caused by the discontinuities of materials. Additionally, the simple distributions of shows that is the most fundamental quasi-load reflecting the inhomogeneity of unit cells, implying that is the primary perturbed term. Since at all nodes is not zero, captures more microscopic information compared with . In general, the AHM perturbed to the third order is accurate enough.
- (3)
The unit of is , demonstrating that behaves as a moment. In fact, the first two columns of are the quasi-bending moments caused by the unit bending curvatures around the and axes, and the third column of denotes the quasi-torsional moment attributable to the unit torsional strain. Accordingly, represent three fundamental deformations caused by . Since is the product of and the second derivative of , see Equation (31), thus, the second derivative of acts as the modal coordinates in the superposition method.
- (4)
The unit of is . Since the six columns of are independent, thus, denotes six independent microscopic deformations accordingly, and the third derivative of acts as the modal coordinates.
6. Conclusions
This paper proposes a two-scale method by combining the Kirchhoff plate theory with the two-scale asymptotic homogenization method to deal with the static and dynamic problems of 3D periodic thin plates. In this work, the solutions of the fourth-order elliptic PDE with periodically oscillating coefficients were given in an asymptotic expansion form, where perturbed terms were the multiplications of influence functions and the derivatives of homogenized displacements. To determine the influence functions from the unit cell problems, periodic boundary and normalization constraint conditions were given and elaborated physically.
It was found that the first-order perturbed term of the asymptotic expansion solution of the fourth-order PDE with periodically oscillating coefficients should be zero. The reason for this phenomenon is that the first-order perturbed terms cannot reflect the microdeformations. In addition, with the physical interpretation of the AHM, it was shown that the second-order influence functions are the fundamental terms for capturing the microscopic information, since the second-order quasi-loads for solving the second-order influence functions are simple line loads, with nonzero values only along the interfaces of matrices and inclusions as well as boundaries.
Finally, the free vibrations and static problems of several periodic composite plates with different boundary conditions were investigated to validate the effectiveness of the proposed method, showing that the present AHM’s solutions are meaningful and physically acceptable. Additionally, it has been shown that homogenized displacements play a significant part in the prediction of microscale solutions and that second-order or even higher-order perturbed displacements are necessary for achieving accurate rotation angles for periodic plates. This work lays the foundation for the study of moderately thick periodic plates.