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Micromechanics of Composite Materials
A Generalized Multiscale Analysis Approach
By Jacob Aboudi, Steven M. Arnold, Brett A. Bednarcyk
ElsevierCopyright © 2013 Elsevier Inc.
All rights reserved.
1.1 Fundamentals of Composite Materials and Structures 2
1.2 Modeling of Composites 9
1.3 Description of the Book Layout 15
1.4 Suggestions on How to Use the Book 17
Micromechanics of Composite Materials: A Generalized Multiscale Analysis Approach is the culmination of nearly 30 years of work by the first author and his co-workers on the development, implementation, and application of micromechanics theories for composites. The intent of the book is to place these theories in context, provide their theoretical underpinnings in a clear and concise manner, and illustrate their utility for the design and analysis of advanced composites, particularly in the nonlinear regime. The power of these theories becomes particularly clear with their application in multiscale modeling of composites. Because they provide an effective anisotropic constitutive equation for composite materials, these theories can be used to represent the macroscopic (global) nonlinear, inelastic, viscoelastic, or finite strain behavior at a point in a composite structure that is being analyzed using a higher scale model such as finite element analysis. In this context, nonlinearity in the composite constituent materials due to inelasticity and/or damage will affect the composite behavior, and this change will impact the higher scale structural response. Thus, the physics of damage and deformation in composites can be captured at a more fundamental scale by conducting multiscale analyses. However, for multiscale problems to remain tractable, the micromechanics methods must be very efficient—and efficiency is a hallmark of the micromechanics theories presented herein as they are closed form or semi-closed form.
Throughout this book, a basic knowledge of solid mechanics is assumed. Consequently, there is no chapter on the basics of solid and structural mechanics (e.g., introducing the concepts of stress and strain). Rather, Chapter 2 presents the constitutive models associated with deformation and damage that will be used throughout the book to describe the behavior of the constituent materials of composites. For the advanced topics covered in Chapters 8 to 12, it is further assumed that the reader has a general knowledge of each topic.
This introductory chapter provides some fundamental information about composites and then focuses on introducing modeling of composites, particularly micromechanics and multiscale modeling. There are many excellent texts, however, that go into much greater detail regarding the how and the why of composite materials and structures. Rather than repeat this information, the reader is referred to Jones (1975), Christensen (1979), Carlsson and Gillespie (1990), Herakovich (1998), Hyer (1998), Zweben and Kelly (2000), Miracle and Donaldson (2001), and Barbero (2011). It should also be noted that this book follows the foundation laid by the Aboudi (1991) book on micromechanics, which summarizes a great deal of his early work on the subject.
1.1 Fundamentals of Composite Materials and Structures
In the fields of Structural Engineering and Materials Science and Engineering, the difference between a structure and a material comes down to the presence of a boundary. A material is the substance of which a body is composed. The material itself has no boundaries, but rather may be thought of as what is present at a point in the body. Scientists and engineers have developed ways to represent materials through properties that describe how the material behaves at a point in a body, such as Young's modulus, thermal conductivity, density, yield stress, Poisson's ratio, and coefficient of thermal expansion. The body itself, on the other hand, is a structure. It has boundaries and its behavior is dependent on the conditions at these boundaries. For example, given a steel beam, the beam itself is a structure, while the material is steel. This distinction between materials and structures is natural and extremely convenient for structural engineers and materials scientists. Imagine attempting to combine properties of materials and structures in the case of the aforementioned beam. A beam's bending characteristic is dictated by its flexural rigidity (the Young's modulus E times the cross-section moment of inertia I, or EI, and not just its Young's modulus). If this were not separated into a material property (E) and a structural property (I) but rather kept as a combined property, one would need to look up a value for every combination of beam shape and material.
The above discussion implies that a material is a continuum, meaning it is continuous and completely fills the region of space it occupies. The material can thus be modeled using continuum mechanics, which considers the material to be amorphous and does not explicitly account for any internal details within the material, such as the presence of inclusions, grain orientation, or molecular arrangement. To account for such internal details, some additional theory beyond standard continuum mechanics is needed.
In its broadest context, a composite is anything comprised of two or more entities. A composite structure would then be any body made up of two or more parts or two or more materials. Likewise, a composite material is a material composed of two or more materials with a recognizable interface between them. Because it is a material, it has no external boundaries; once an external boundary is introduced, it becomes a structure composed of composite materials, which is a particular type of composite structure. Clearly, however, a composite material does have distinct internal boundaries. If these internal boundaries are ignored, continuum mechanics can be used to model composite materials as pseudo-homogeneous anisotropic materials with directionally dependent 'effective,' 'homogenized,' or 'smeared' material properties. Micromechanics, on the other hand, attempts to account for the internal boundaries within a composite material and capture the effects of the composite's internal arrangement. In micromechanics, the individual materials (typically referred to as constituents or phases) that make up a composite are each treated as continua via continuum mechanics, with their individual representative properties and arrangement dictating the overall behavior of the composite material.
In many cases, especially with composite materials used in structural engineering, the geometric arrangement of one phase is continuous and serves to hold the other constituent(s) together. This constituent is referred to as the matrix material. Whereas the other constituent(s), often referred to as inclusion(s) and/or reinforcement(s), are materials that can be either continuous or discontinuous and are held together by the matrix. There may also be interface materials, or interphases, present between the matrix and inclusion. A fundamental descriptor of composites that should always be indicated when denoting a given system (since it strongly influences the effective behavior) is the volume fraction of phases present. Typically, only the reinforcement phase is indicated unless multiple phases are present since the sum of all phases must equal 100%; for example, in a two-phase fiber-reinforced composite vf is the volume fraction of fibers and vm = 1 - vf is that of the matrix. Composites are typically classified at two distinct levels. The first level of designation is usually made with respect to the matrix constituent. This divides composites into three main categories: polymer matrix composites (PMCs), metal matrix composites (MMCs), and ceramic matrix composites (CMCs). The second level of classification refers to the form of the reinforcement: discontinuous (particulate or whisker), continuous fiber, or woven (textile) (braided or knitted fiber architectures are included in this classification). In the case of woven and braided composites, the weave or braiding pattern (e.g., plane weave, triaxially braided) is also often indicated. Examples of some of these types of composites are shown in Figure 1.1. Note that particulate composites are typically isotropic whereas most other composite forms have some level of anisotropy (e.g., a unidirectional continuous fiber composite is usually transversely isotropic).
Composites, particularly PMCs, are often manufactured as an assembly of thin layers joined together to form a laminate (see Figure 1.1(b)). Each layer is referred to as a lamina or ply. By orienting the reinforcement direction of each ply, the properties and behavior of the resulting laminate can be controlled. A quasi-isotropic laminate can be formed by balancing the orientations of the plies such that the extensional stiffness of the laminate is constant in all in-plane directions. Quasi-isotropic laminates have thus been very popular as—under in-plane elastic extension—they behave like isotropic metals, with which most engineers are familiar. However, this has also led to engineers attempting to simply replace metals with quasi-isotropic laminates in structures that were designed based on isotropic metallic properties. This is the origin of the expression 'black aluminum,' which refers to a black quasi-isotropic carbon/epoxy laminate, whose in-plane effective elastic properties are often very close to those of aerospace aluminum alloys. The tremendous pitfall of this approach, which has in many ways slowed the realization of the full potential of composites, is that quasi-isotropic carbon/epoxy laminates are not even close to isotropic in terms of their out-of-plane behavior. They are highly prone to delamination and interlaminar failure, failure modes which do not afflict isotropic metals. Care must therefore be taken to minimize out-of-plane loads and quantify out-of-plane margins of safety when designing structures with this type of composite laminate. The 'black aluminum' design approach, while simple, is typically very inefficient.
A key distinction among PMCs, MMCs, and CMCs is their maximum service temperature. As shown in Figure 1.2, most PMCs are limited to an operating temperature under 450 °F. Metal matrix composites extend this range to approximately 1200 °F, depending upon the capability of the chosen matrix, and typical CMCs can remain functional to over 2000 °F. Obviously, the temperature limitations are dependent on the limitations of the composite constituent materials. Indeed, the groups of small ovals in Figure 1.2 representing CMCs with maximum service temperatures of approximately 1100 °F are tungsten carbide ceramic matrix materials with particulate metallic cobalt inclusions. Thus, the lower operating temperature is due to the metallic reinforcement; most CMCs are composed of ceramic matrices and ceramic reinforcements.
The vertical axes in Figure 1.2 represent the composite's (a) effective Young's modulus and (b) specific strength (strength divided by density). The wide spread in properties, especially in the case of PMCs, is indicative of the anisotropy present in continuously reinforced composites. For the carbon/epoxy composite labeled in Figure 1.2, there is a factor of 20 between the Young's moduli in the longitudinal direction (along the continuous carbon fibers, 0°) and the transverse direction (perpendicular to the fiber direction, 90°). For the specific strength, the corresponding factor is close to 40. The composite labeled 'HS C/epoxy, QI lam' represents the effective in-plane properties of a quasi-isotropic laminate composed of the previously discussed high strength carbon/epoxy material. As would be expected, this laminate's effective properties are intermediate to those of its plies in each direction. It is also noteworthy that this quasi-isotropic laminate, which is actually a structure with external boundaries, is compared here to unidirectional composite materials. Such a laminate would only behave like a material if appropriate extensional in-plane boundary conditions were applied. If it were subjected to bending it would behave like an anisotropic plate, and its properties would be dependent on its thickness and ply stacking sequence, which are structural rather than material properties.
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