Design analysis of composite laminate structures for light-weight armored vehicle by homogenization method
Introduction
Composite materials have been extensively used in various engineering fields in recent years. Especially, aerospace industry and land vehicle manufacturers are developing many products made with fiber reinforced composite materials. The major advantage of using composite materials is their high strength to weight ratios. Their highly oriented properties can be combined to produce a laminate tailored for a high performance specific structure.
In order to increase reliability of the use of various newly developed composites, their mechanical properties must be carefully studied because of a quite different type of failure mechanism from metals applied in most structures which have been studied for a long time. Extensive theoretical development of mechanics of composite materials made in the 1950s, and the most widely used classical lamination theory was developed in the 1960s and 1970s mostly for aerospace-type applications. Such a classical mechanics theory of composite and composite structures can be found in Jones (1970) [1] as well as their rich references, and we shall not describe their details here. However, we shall pay more attention to the mathematical theory of composite mechanics based on the homogenization theory that was mainly developed in the middle of the 1970s [2], [3] by applied mathematicians in France, Italy, and Russia, and we shall apply the basic methodology of the homogenization to a laminate to develop a new, but closely linked theory of a laminate with the existing theories introduced so far including the classical lamination theory, the first-order shear deformation theory, and others, in order to emphasize the effect of geometric and mechanical heterogeneity in the microstructure of composite materials. The main attraction of the homogenization method is its systematic capability of development of the homogenized macroscopic (phenomenological global) constitutive relation of composite materials and its straightforward methodology to compute the stress even in the microstructure of composites. In other words, both macro- and micro-mechanics can be treated in the same context by using the homogenization method, and they are naturally related, that is, the global and local stress analysis can be performed within the same context. The mathematical homogenization is, as well-known, based on the idea of asymptotic expansion of the displacement fieldwith respect to the size of the microstructure, and it examines the limiting case that goes to zero, while two scale coordinates x and y are introduced to represent the global macroscopic behavior and “small” variations (local microscopic behavior) due to microscopic heterogeneity of composite materials.
In this work, we shall apply the homogenization method to a laminate, and we shall examine its relation to the existing other theories for a laminate in order to emphasize the nature of the theory of homogenization methods, which are extensively applied to solids of composites. It is noted that there are few works on the application of the homogenization method to a laminate consisting of many laminae, although it has been applied to study mechanics of a heterogeneous lamina, more precisely, a plate. After making a brief review of lamination theories, we shall develop a homogenization theory of a laminate, and we shall apply the new theory to make global–local stress analysis of a laminate in order to show its strength and validity.
Many plate theories have been developed for composite laminates. The most famous one is the classical lamination theory (CLT). But it has been shown to produce poor results for moderately thick plates. To improve the analysis, many other theories have been developed. These include the first-order shear deformation theory (FSDT), higher-order shear deformation theory (HSDT), and discretized lamination theory [4], [5], [6], [7], [8].
The CLT is based on the well-known Kirchhoff assumption. It neglects effect of the transverse shear deformations. This assumption gives the simplest analysis model but also limits the usage of this theory. So CLT is adequate only when the thickness (to the side or radius ratio) is small. For a moderately thick composite laminate, CLT is used to underestimate the deflections and overestimate the natural frequencies and buckling loads. In order to remedy the flaw in CLT, it is appropriate to develop a higher-order theory that can be applied to moderately thick plates. Based on this idea, the FSDT and HSDT were developed. The significant difference between CLT and SDT is that SDT includes effects of transverse shear deformation and rotatory inertia.
But there are also some common points among all three models. These models are based on an assumed global linear (CLT, FSDT) or non-linear (HSDT) distribution of the in-plane displacements in the thickness direction. This assumption generally results in incompatible shearing stresses between adjacent layers. But we also know from the theory of elasticity that the displacements and stresses at the interface between two bonded layers must satisfy the continuity conditions. So these theories fail to satisfy the continuity conditions at the interface between laminations. Therefore, CLT, FSDT, and HSDT are adequate to describe global behavior but inadequate to describe the local behavior such as stress distributions through the layers.
In order to analyze the local behavior, other discretized lamination theories were developed. These theories can predict both global and local behaviors of laminated composites very well. The major drawback is that the formulas are usually too complex to do efficiency analysis.
Based on the trade-off between the accuracy and the efficiency, many commercial finite element codes (for example, ABAQUS1) use FSDT instead of other higher-order theories. It can save time by using FSDT in plate analysis, but it will lose the ability to describe the structures local behavior.
By checking the laminates global behavior, we can solve the deflection, natural frequency, and buckling load problems. But it is not enough to predict the failure based only on the global information. Usually, the composite laminate will fail in several types. They are
- 1.
Delamination between laminae.
- 2.
Debounded between fiber and matrix.
- 3.
Fiber fracture.
Then, there comes the homogenization theory. By using this theory, both the global and local behaviors can be achieved. Therefore, the problems can also be solved.
Section snippets
Homogenization methods for composite laminates
Application of the standard homogenization method to have an equivalent single lamina plate model from a laminate consisting of multiple laminae, yields a large amount of inaccurate representation of the bending and mixed stiffness of bending and membrane, while the membrane stiffness can be computed very accurately. This is the shortcoming of the homogenization method to develop a homogenized equivalent single lamina plate model for a long time.
Here we have examined the specific reason why the
Homogenization method for a composite beam
We shall apply the similar treatment to a composite beam including various sandwich-type beams made of composite materials. Again we shall assume the displacement field of a composite beam bywhere the x-axis coincides with the beam axis and (y, z) coordinate axes are mutually orthogonal to the x-axis. In this case, only the non-zero strain becomesand Hook’s law is written bywhere Eε is
An empirical weighting method
In the above homogenization method, we have to solve two unit cell problems for membrane and bending. More precisely, we have to obtain the characteristic function Xe and Xκ to identify the three stiffness matrices for laminate elasticity. Here we shall develop an approximation method to compute the homogenized bending stiffness matrix of a laminate that is a half-way homogenization method from the classical lamination theory in the sense that the elasticity matrices of laminae are modified by
Computational procedure
The following is a series of steps to determine the strains and stresses in the unit cell through the homogenization method and corresponding weighting factors:
- 1.
Build the unit cell model for computing the characteristic displacement
- 2.
Compute the homogenized elasticity matrix, EH, using E(i)
- 3.
Compute the weighting factor for each lamina, wf(i),
- 4.
Compute the weighted stiffness matrix for each lamina, Ê(i),
- 5.
Numerical example and results for a three-ply laminate
In order to verify the approach of the homogenization with a weighting factor, the cases studied by Pagano [9] are analyzed here by using the finite element method. Consider a three-ply ([0/90/0]) with layers of equal thickness under double sinusoidal transverse load q=q0 sin(πx/a)sin(πy/b) with simply supported boundary conditions on each side where a and b is lengths of the plate in the x- and y-directions, respectively. The laminate is constructed of a composite material with the following
Numerical example and results for a five-layer composite vehicle structure
After verifying the accuracy of the homogenization method with weighting factors (HMWF) in Section 6, it can be seen from Table 1, Table 2 that the method can predict reasonably approximate results for a composite laminate structure. In this section, the HMWF method will be used to analyze a three-point bending problem. The numerical results will be calculated and compared with other results from ABAQUS, FSDT, and the testing results from TACOM.
Optima design of tile configuration
After the homogenization method can successfully predict the laminate behavior, we will focus on the optima design of ceramic tile configuration inside the armor laminate. Two types of tile shape have been taken into consideration, they are square and hexagonal tiles. Other optimization parameters are volume ratio of tile (vt) for either shapes of the lamina and the offset ratio (or) for the square tiles only, Fig. 8, Fig. 9.
The objective function is to minimize the central deflection of
Discussion
In the above homogenization method, we have to solve two unit cell problems for membrane and bending to obtain the characteristic functions Xe and Xκ to identify the three stiffness matrices. Namely Dm, Dmb, and Db are computed without solving the unit cell problem for laminate elasticity. We have developed an approximation method to compute the homogenized bending stiffness matrix of a laminate that is a half-way homogenization method from the CLT in the sense that the elasticity matrices of
Commentary
The homogenization method is the most elegant theory to examine the micro-mechanics while it can be utilized to develop the homogenized equivalent constitutive model for macro-scale analysis of composites using an equivalent “homogeneous” anisotropic thermo-mechanical model. It can be regarded as an extension of the representative volume method.
While there are many researchers in the area of applied mechanics utilizing the so-called unit cell approach that is almost the same as the
Conclusion
A homogenization procedure has been improved by using the weighted material stiffness. The homogenization procedure is applied in the plate structure normal to the through-thickness coordinate in which the local structure is not so periodic. This procedure makes the global and local structure analysis possible through the use of the homogenization method.
Examples presented herein indicate that this procedure can give reasonable results compared with results from CLT, FSDT, or ABAQUS.
Acknowledgements
This project was funded by the in-house Laboratory Independent Research (ILIR) program at TACOM. Thanks are due to Dr. G. Gerhart for his encouragements and support of the project.
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