Composite Material

Right. So, you want to understand composites. Fascinating. A material born from the deliberate marriage of unlike substances, each retaining its own identity, yet contributing to a whole that is, ideally, greater than the sum of its parts. It's a concept that resonates, wouldn't you agree?

Composite Material

A material that is meticulously crafted from at least two distinct constituent materials. These components, with their inherently dissimilar chemical or physical properties, are integrated to forge a material possessing characteristics that transcend those of the individual elements. Crucially, within the final structure, these constituent materials remain separate and discernible, setting composites apart from mere mixtures or uniform solid solutions. When a composite material exhibits multiple distinct layers, it is then classified as a composite laminate.

Typically, engineered composite materials are composed of a binding agent, which forms the matrix, and a filler material, be it particulates or fibres, that provides substance. Consider these examples, if you must:

• Concrete, the ubiquitous building block, is a composite. Reinforced concrete, with its embedded steel, is a more sophisticated iteration. Even masonry, with its mortar binder – itself a composite – fits the bill.
• Composite wood products, like the robust glulam or the layered plywood, rely on wood glue to bind their constituent wood elements.
• Reinforced plastics, such as the familiar fiberglass or the high-strength fibre-reinforced polymer, utilize resin or thermoplastics as their binding matrix.
• Ceramic matrix composites, which can include composite ceramic and metal matrices for demanding applications.
• Metal matrix composites, where metals are reinforced with other materials.
• And then there are the advanced composite materials, often born from the relentless pursuit of performance in spacecraft and aircraft design.

The allure of composite materials lies in their potential to be less expensive, lighter, stronger, or more durable than conventional materials. Some are even elegant imitations of nature's own designs, found in the intricate structures of plants and animals. [3]

Even more complex are robotic materials, which are composites that have been imbued with sensing, actuation, computation, and communication capabilities. [4] [5]

The applications for composite materials are far-reaching, extending to construction and intricate technical structures such as the hulls of boat hulls, the panels of swimming pool enclosures, the sleek bodies of racing cars, the enclosures for shower stalls, the smooth surfaces of bathtubs, the robust walls of storage tanks, and the sophisticated imitations of granite and cultured marble found in sinks and countertops. [6] [7] Their adoption in general automotive applications is also on a steep upward trajectory. [8]

History

The genesis of composite materials can be traced back to the rudimentary yet effective combination of straw and [mud] to form bricks for building construction. The ancient Egyptians, with their meticulous documentation, even captured the art of early brick-making in their tomb paintings. [9]

One might argue that wattle and daub, with its impressive lineage of over 6000 years, represents the most ancient form of composite material. [10]

• The natural world itself provides a wealth of composite materials. Woody plants, from the true wood of trees to the fibrous structures of palms and bamboo, yield materials that have been utilized by humankind since prehistory, and continue to be essential in construction and scaffolding.

• Plywood, with its origins dating back to 3400 BC, [11] was ingeniously crafted by the Ancient Mesopotamians. By strategically gluing wood veneers at different angles, they achieved properties superior to those of natural wood.

• Cartonnage, a material composed of layers of linen or papyrus saturated with plaster, emerged during Egypt's First Intermediate Period, around 2181–2055 BC. [11] It served a solemn purpose, notably in the creation of death masks.

• Cob, a building material consisting of mud mixed with straw or gravel as a binder, has been employed for millennia in the form of mud bricks or entire mud walls. [12]

• Concrete itself was meticulously described by Vitruvius in his work "Ten Books on Architecture," written around 25 BC. He differentiated between various aggregates suitable for preparing lime mortars. For structural applications, he specifically recommended pozzolana, volcanic sands found near Pozzuoli and in both brownish-yellow-gray and reddish-brown hues near Naples and Rome, respectively. Vitruvius's specified ratios of 1 part lime to 3 parts pozzolana for building mortars, and 1:2 for underwater work, are remarkably close to modern concrete formulations used in marine environments. [13] Furthermore, natural cement-stones, when calcined, produced cements that were used in concrete from post-Roman times well into the 20th century, some exhibiting superior properties to manufactured Portland cement.

• Papier-mâché, a composite of paper and glue, has graced decorative and functional objects for centuries. [14]

• The pioneering work in artificial fibre reinforced plastic occurred in 1935, a collaboration between Al Simison and Arthur D Little at Owens Corning Company, resulting in a composite of fiberglass and bakelite. [15]

• Perhaps one of the most widely recognized composites is fibreglass. This material consists of small glass fibres embedded within a polymeric matrix, typically epoxy or polyester. While the glass fibres offer considerable strength and stiffness, they are also brittle. The polymer, on the other hand, is ductile but relatively weak and flexible. The resulting fibreglass strikes a balance, exhibiting commendable stiffness, strength, flexibility, and ductility. [16]

• The sophisticated composite bow leverages the properties of layered materials.

• Historical weaponry also featured composite designs, such as leather cannon and wooden cannon.

Examples

Composite Materials

Concrete stands as the most ubiquitous artificial composite material globally. In 2009 alone, an estimated 7.5 billion cubic meters of concrete were produced annually. [17]

Concrete is typically formulated from loose stones, acting as construction aggregate, bound together by a matrix of cement. This composition renders concrete an economical material, exceptionally resistant to compressive forces, [18] but regrettably susceptible to tensile loading. [19] To counteract this weakness and imbue concrete with the capacity to resist stretching, steel bars, which possess high tensile strength, are frequently incorporated, thereby creating reinforced concrete. [20]

Observe the stark contrast: a black carbon fibre, a critical reinforcement component, juxtaposed against the delicate fragility of a human hair.

Fibre-reinforced polymers encompass a broad category, including carbon-fiber-reinforced polymers and glass-reinforced plastic. When categorized by their matrix, we find thermoplastic composites, short fibre thermoplastics, long fibre thermoplastics, and long-fiber-reinforced thermoplastics. The realm of thermoset composites is equally vast, featuring materials like paper composite panels. Many advanced thermoset polymer matrix systems are designed with aramid fibre and carbon fibre embedded within an epoxy resin matrix. [21] [22]

Shape-memory polymer composites represent a class of high-performance materials. They are meticulously formulated using fibre or fabric reinforcements and a shape-memory polymer resin as the matrix. This unique matrix grants these composites the remarkable ability to be easily molded into various configurations upon heating above their activation temperatures, while maintaining high strength and stiffness at lower temperatures. Furthermore, they can be repeatedly reshaped by reheating without degradation of their material properties. These attributes make them ideal for applications demanding lightweight, rigid, deployable structures, rapid manufacturing processes, and dynamic reinforcement capabilities. [23] [24]

High strain composites are another category of high-performance composites, engineered to withstand and perform under conditions of significant deformation. They find frequent use in deployable systems where structural flexibility is a distinct advantage. [ citation needed ] While exhibiting many similarities to shape-memory polymers, the performance of high strain composites is primarily dictated by the fibre layout rather than the resin content of the matrix. [25]

Composites can also incorporate metal fibres to reinforce other metals, giving rise to metal matrix composites (MMC), or ceramic fibres in ceramic matrix composites (CMC). [26] [27] Biological structures like bone, with its hydroxyapatite (hydroxyapatite) reinforced by collagen fibres, are natural examples. The category also includes cermet (a composite of ceramic and metal) and, as mentioned, concrete. The primary design goal for ceramic matrix composites is fracture toughness, rather than sheer strength. Another distinct class of composite materials comprises woven fabric composites, characterized by longitudinal and transverse laced yarns, offering inherent flexibility due to their fabric form.

Organic matrix/ceramic aggregate composites include materials such as asphalt concrete, polymer concrete, mastic asphalt, mastic roller hybrids, dental composite, syntactic foam, and the iridescent mother of pearl. [28] A specialized composite armour, known as Chobham armour, finds application in military contexts. [ citation needed ]

Furthermore, thermoplastic composite materials can be engineered to incorporate specific metal powders, resulting in materials with densities ranging from 2 g/cm³ to 11 g/cm³ – a density comparable to that of lead. These materials are often referred to as "high gravity compounds" (HGC), though "lead replacement" is also a common designation. Their utility extends to applications where they can substitute traditional materials like aluminum, stainless steel, brass, bronze, copper, lead, and even tungsten, particularly in weighting, balancing (for instance, adjusting the center of gravity in a tennis racquet), vibration damping, and radiation shielding. High-density composites offer an economically viable alternative when certain materials are deemed hazardous and are consequently banned (such as lead), or when the cost of secondary operations like machining, finishing, or coating becomes a significant factor. [29]

Research has consistently shown that interleaving stiff and brittle epoxy-based carbon-fiber-reinforced polymer laminates with flexible thermoplastic laminates can yield highly toughened composites with enhanced impact resistance. [30] An intriguing characteristic of these interleaved composites is their capacity for shape memory behavior, achievable without the necessity of shape-memory polymers or shape-memory alloys. Examples include balsa plies interleaved with hot glue, [31] aluminum plies interleaved with acrylic polymers or PVC, [32] and carbon-fiber-reinforced polymer laminates interleaved with polystyrene. [33]

A specialized class of composite, the sandwich-structured composite, is constructed by bonding two thin yet rigid skins to a lightweight but thick core. While the core material typically possesses low strength, its substantial thickness imparts high bending stiffness to the sandwich composite, all while maintaining a low overall density. [34] [35]

Plywood, a testament to engineered wood, finds extensive application in construction.

Wood itself is a naturally occurring composite, characterized by cellulose fibres embedded within a matrix of lignin and hemicellulose. [36] Engineered wood encompasses a diverse array of products, including wood fibre board, plywood, oriented strand board, wood plastic composite (a matrix of recycled wood fibre in polyethylene), Pykrete (a curious blend of sawdust and ice), plastic-impregnated or laminated paper or textiles, Arborite, Formica (plastic), and Micarta. Other engineered laminate composites, such as Mallite, utilize a central core of end-grain balsa wood bonded to surface skins of lightweight alloy or GRP, yielding materials of exceptional rigidity and low weight. [37]

Particulate composites feature particles dispersed within a matrix, which can be non-metallic, such as glass or epoxy. The ubiquitous automobile tire is a prime example of a particulate composite. [38]

Advanced diamond-like carbon (DLC) coated polymer composites have been reported, [39] where the coating significantly enhances surface hydrophobicity, hardness, and wear resistance.

Ferromagnetic composites, including those with a polymer matrix and nanocrystalline Fe-based powder fillers, are also being developed. The use of amorphous and nanocrystalline powders, often derived from metallic glasses, allows for the creation of controllable magnetic properties in these ferromagnetic nanocomposites. [40]

Products

Due to their desirable combination of lightweight and strength, capable of withstanding harsh loading conditions, fibre-reinforced composite materials have become increasingly popular, despite their often higher cost. Their applications span across aerospace components (such as tails, wings, fuselages, and propellers), the hulls of boats and sculls, bicycle frames, and the bodies of racing cars. Other uses include fishing rods, storage tanks, swimming pool panels, and composite baseball bats. The Boeing 787 and Airbus A350 aircraft, for instance, extensively utilize composites in their structures, including their wings and fuselages. [41] Composite materials are also finding increasing integration in orthopedic surgery, [42] and they dominate the market for hockey sticks.

Carbon composite materials are fundamental to modern launch vehicles and are critical for the heat shields used during the re-entry phase of spacecraft missions. They are widely employed in solar panel substrates, antenna reflectors, and spacecraft yokes. Their utility extends to payload adapters, inter-stage structures, and the heat shields of launch vehicles. Moreover, the disk brake systems of airplanes and racing cars frequently incorporate carbon/carbon material. Notably, a composite material featuring carbon fibres and a silicon carbide matrix has been introduced into the design of luxury vehicles and sports cars.

In 2006, a fibre-reinforced composite pool panel was introduced as a non-corrosive alternative to galvanized steel for both residential and commercial in-ground swimming pools.

The year 2007 saw the unveiling of an all-composite military Humvee by TPI Composites Inc and Armor Holdings Inc, marking the first all-composite military vehicle. The use of composites in this vehicle resulted in a lighter platform, thereby allowing for increased payloads. [43] In 2008, ECS Composites developed military transit cases utilizing a combination of carbon fibre and DuPont Kevlar (five times stronger than steel) with enhanced thermoset resins, achieving cases that were 30 percent lighter yet possessed exceptional strength.

Pipes and fittings designed for a variety of purposes, including the transportation of potable water, fire-fighting systems, irrigation, seawater, desalinated water, chemical and industrial waste, and sewage, are now commonly manufactured from glass reinforced plastics.

The application of composite materials in tensile structures for facade designs offers the distinct advantage of translucency. The woven base cloth, coupled with an appropriate coating, facilitates superior light transmission, creating a comfortable level of illumination that contrasts with the harsh brightness of direct sunlight. [44]

For several years, wind turbine blades, which are continually increasing in size, reaching lengths of approximately 50 meters, have been fabricated using composites. [45] Composites are also utilized in marine energy structures, such as tidal turbine blades. [46]

Amputees can achieve running speeds comparable to non-amputees, thanks to prosthetic lower legs constructed from carbon-fiber composite. [47]

High-pressure gas cylinders, typically in the 7–9 liter volume range and designed for 300 bar pressure, are now commonly made from carbon composite. Type-4 cylinders incorporate metal solely for the boss that carries the thread for valve attachment.

On September 5, 2019, HMD Global introduced the Nokia 6.2 and Nokia 7.2 smartphones, which were claimed to feature frames made from polymer composite. [48]

Overview

The creation of composite materials involves combining individual materials, known as constituent materials, into two primary categories: the matrix (or binder) and the reinforcement. [49] A certain proportion of each is essential. The matrix serves to support the reinforcement, enveloping it and maintaining its relative position. Simultaneously, the reinforcements imbue the matrix with their exceptional physical and mechanical properties, thereby enhancing the matrix's inherent characteristics. Through this synergistic interaction, the composite material achieves mechanical properties unattainable by its individual constituent materials. This offers designers a broad spectrum of options for selecting optimal combinations of matrix and reinforcement materials.

The fabrication of engineered composites requires a molding process. The reinforcement is positioned within the mold surface or cavity, and the matrix is subsequently introduced, either before or after this placement. A critical step involves a melding event, where the matrix solidifies, thereby setting the part's shape. This melding event can manifest in various ways, depending on the nature of the matrix: solidification from a molten state for thermoplastic polymer matrix composites, or chemical polymerization for thermoset polymer matrix composites.

The choice of molding method is dictated by the specific requirements of the end-item design and is fundamentally influenced by the chosen matrix and reinforcement materials. The overall production volume also plays a significant role. For large quantities, substantial capital investment in rapid and automated manufacturing technologies is justifiable. Conversely, smaller production runs necessitate lower capital investment but incur higher labor and tooling expenses, with a correspondingly slower production rate.

Many commercially produced composites utilize a polymer matrix material, often referred to as a resin solution. A wide array of polymers is available, derived from various starting raw ingredients. These can be broadly categorized, with numerous variations within each category. The most common include polyester, vinyl ester, epoxy, phenolic, polyimide, polyamide, polypropylene, PEEK, among others. The reinforcement materials are frequently fibres, but ground minerals are also common. The various fabrication methods described below have been developed to optimize the resin content in the final product, or conversely, to increase the fibre content. As a general guideline, hand lay-up processes typically yield a product containing 60% resin and 40% fibre, whereas vacuum infusion results in a final product with 40% resin and 60% fibre content. The product's strength is significantly influenced by this ratio.

Martin Hubbe and Lucian A Lucia propose that wood can be considered a natural composite, with cellulose fibres forming the reinforcement within a matrix of lignin. [50] [51]

Cores in Composites

Certain composite layup designs involve the co-curing or post-curing of prepreg materials with various other media, such as foam or honeycomb structures. This approach is generally known as a sandwich structure. This represents a more generalized layup configuration, suitable for the production of cowlings, doors, radomes, or non-structural components.

Open- and closed-cell foams – including polyvinyl chloride (polyvinyl chloride), polyurethane (polyurethane), polyethylene (polyethylene), or polystyrene (polystyrene) foams – along with balsa wood, syntactic foams, and honeycombs, are commonly employed as core materials. Both open- and closed-cell metal foam can also serve this purpose. More recently, three-dimensional graphene structures, often referred to as graphene foam, have also been utilized as core structures. A comprehensive review by Khurram and Xu et al. provides a summary of the state-of-the-art fabrication techniques for these 3D graphene structures and illustrates their application as cores for polymer composites. [52]

Semi-crystalline Polymers

While the two phases within semi-crystalline polymers are chemically identical, they can be quantitatively and qualitatively described as composite materials. The crystalline portion exhibits a higher elastic modulus, thereby reinforcing the less stiff, amorphous phase. The degree of crystallinity, or volume fraction, in polymeric materials can range from 0% to 100%, [53] depending on molecular structure and thermal history. Various processing techniques can be employed to manipulate the percentage of crystallinity and, consequently, the mechanical properties of these materials, as detailed in the physical properties section. This phenomenon is observable in diverse applications, from industrial plastics like polyethylene shopping bags to the remarkably varied mechanical properties of spider silks. [54] In many instances, these materials behave akin to particle composites, with randomly dispersed crystals known as spherulites. However, they can also be engineered to exhibit anisotropy, behaving more like fiber-reinforced composites. [55] In the case of spider silk, the crystal size, independent of volume fraction, can influence the material's properties. [56] Ironically, single-component polymeric materials represent some of the most easily tunable composite materials known.

Methods of Fabrication

The fabrication of composites typically involves the process of wetting, mixing, or saturating the reinforcement material with the matrix. Subsequently, the matrix is induced to bind, either through heat or a chemical reaction, forming a rigid structure. This operation is usually carried out within an open or closed forming mold. However, the sequence and methods of introducing the constituents can vary considerably. Composite fabrication employs a wide array of methods, including advanced fibre placement (automated fiber placement), [57] the fibreglass spray lay-up process, [58] filament winding, [59] the lanxide process, [60] tailored fibre placement, [61] tufting, [62] and z-pinning. [63]

Overview of Mould

The reinforcing and matrix materials are combined, compacted, and cured (processed) within a mold to undergo a melding event. The final shape of the part is fundamentally determined after this melding event. However, under specific process conditions, deformation can occur. For thermoset polymer matrix materials, the melding event is a curing reaction initiated by the application of heat or chemical reactivity, such as an organic peroxide. For thermoplastic polymer matrix materials, the melding event involves solidification from a molten state. In the case of metal matrix materials, such as titanium foil, the melding event is a fusion process occurring at high pressure and a temperature close to the melting point.

It is conventional to refer to one piece of the mold as the "lower" mold and another as the "upper" mold for many molding methods. These terms do not denote the mold's spatial orientation but rather the different faces of the molded panel. Following this convention, there is always a lower mold, and sometimes an upper mold. Part construction begins by applying materials to the lower mold. "Lower mold" and "upper mold" serve as more generalized descriptors than common and specific terms like male side, female side, a-side, b-side, tool side, bowl, hat, mandrel, etc. Continuous manufacturing processes utilize a different nomenclature.

Typically, the molded product is referred to as a panel. For certain geometries and material combinations, it may be termed a casting. In some continuous processes, it is referred to as a profile. Some of the common processes include autoclave moulding, [64] vacuum bag moulding, [65] pressure bag moulding, [66] resin transfer moulding, [67] and light resin transfer moulding. [68]

Other Fabrication Methods

Additional fabrication techniques include casting, [69] centrifugal casting, [70] braiding (onto a former), continuous casting, [71] filament winding, [72] press molding, [73] transfer moulding, pultrusion molding, [74] and slip forming. [75] Forming capabilities also extend to CNC filament winding, vacuum infusion, wet lay-up, compression moulding, and thermoplastic molding, among others. The implementation of curing ovens and paint booths is also a requisite for certain projects.

Finishing Methods

The finishing of composite parts is a critical aspect of the final design. Many of these finishes involve specialized coatings, such as rain-erosion coatings or polyurethane coatings.

Tooling

The molds and mold inserts are collectively referred to as "tooling." The tooling can be constructed from various materials, including aluminium, carbon fibre, invar, nickel, reinforced silicone rubber, and steel. The selection of tooling material is typically based on, but not limited to, factors such as the coefficient of thermal expansion, the anticipated number of cycles, the required tolerances of the end item, the desired surface finish, the cure method, the glass transition temperature of the material being molded, the molding method, the matrix type, cost, and a range of other considerations.

Physical Properties

Main article: Rule of mixtures

The plot illustrates the overall strength of a composite material as a function of fiber volume fraction, bounded by the upper limit (rule of mixtures) and the lower limit (inverse rule of mixtures).

Generally, the physical properties of a composite material are direction-dependent, rendering them anisotropic. This characteristic applies to numerous properties, including elastic modulus, [76] ultimate tensile strength, thermal conductivity, and electrical conductivity. [77] The rule of mixtures and the inverse rule of mixtures provide upper and lower bounds for these properties, respectively. The actual value will reside somewhere between these bounds and can be influenced by a multitude of factors, such as:

• The orientation under consideration.
• The length of the fibres.
• The precision of fibre alignment.
• The intrinsic properties of the matrix and fibres.
• The degree of delamination between fibres and matrix.
• The presence of any impurities.

Figure (a) depicts the isostress condition, where composite layers are oriented perpendicular to the applied force, while figure (b) illustrates the isostrain condition, where the layers are parallel to the force. [78]

For a given material property, denoted by $E$, the rule of mixtures posits that the overall property in the direction parallel to the fibers can be expressed as:

$E_{\parallel }=fE_{f}+\left(1-f\right)E_{m}$

The inverse rule of mixtures suggests that in the direction perpendicular to the fibers, the elastic modulus of a composite could be as low as:

$E_{\perp }=\left({\frac {f}{E_{f}}}+{\frac {1-f}{E_{m}}}\right)^{-1}$

where:

• $f = \frac{V_{f}}{V_{f}+V_{m}}$ represents the volume fraction of the fibers.
• $E_{\parallel }$ is the material property of the composite parallel to the fibers.
• $E_{\perp }$ is the material property of the composite perpendicular to the fibers.
• $E_{f}$ is the material property of the fibers.
• $E_{m}$ is the material property of the matrix.

The majority of commercially produced composites are manufactured with randomly dispersed and oriented strengthening fibers. In such cases, the composite's Young's modulus will fall between the isostrain and isostress bounds. However, in applications demanding the highest possible strength-to-weight ratio, such as in the aerospace industry, fibre alignment may be meticulously controlled.

In stark contrast to composites, isotropic materials (like aluminum or steel in their standard wrought forms) typically exhibit uniform stiffness regardless of the direction of applied forces and/or moments. The relationship between forces/moments and strains/curvatures for an isotropic material is described by relatively straightforward mathematical relationships involving Young's Modulus, the shear modulus, and the Poisson's ratio. For anisotropic materials, however, the mathematics becomes more complex, requiring a second-order tensor and up to 21 material property constants. In the specific case of orthogonal isotropy, three distinct material property constants are needed for each of Young's Modulus, Shear Modulus, and Poisson's ratio, totaling nine constants to fully express the relationship between forces/moments and strains/curvatures.

Techniques that leverage the anisotropic properties of materials include mortise and tenon joints, commonly found in natural composites like wood, and pi joints used in synthetic composites.

Mechanical Properties of Composites

Particle Reinforcement

Generally, particle reinforcement provides less strengthening to composites compared to fiber reinforcement. Its primary purpose is to enhance the stiffness of the composites while simultaneously increasing their strength and toughness. Due to their favorable mechanical properties, they are employed in applications requiring wear resistance. For instance, reinforcing cement with gravel particles can dramatically increase its hardness. Particle reinforcement stands out as a highly advantageous method for tuning material mechanical properties due to its ease of implementation and low cost. [79] [80] [81] [82]

The elastic modulus of particle-reinforced composites can be expressed by the following equation:

$E_{c}=V_{m}E_{m}+K_{c}V_{p}E_{p}$

where $E$ represents the elastic modulus, $V$ denotes the volume fraction. The subscripts $c$, $p$, and $m$ indicate the composite, particle, and matrix, respectively. $K_{c}$ is an empirically determined constant.

Similarly, the tensile strength of particle-reinforced composites can be expressed as:

$(T.S.)_{c}=V_{m}(T.S.)_{m}+K_{s}V_{p}(T.S.)_{p}$

where T.S. is the tensile strength, and $K_{s}$ is a constant (distinct from $K_{c}$) that can be found empirically.

Short Fiber Reinforcement (Shear Lag Theory)

• See also: Short_fiber_thermoplastics § Mechanical_properties

Short fibers often present a more cost-effective or convenient manufacturing option compared to longer, continuous fibers, while still offering superior properties over particle reinforcement. A common illustration of this is found in carbon fiber-reinforced 3D printing filaments, which incorporate chopped short carbon fibers within a matrix, typically PLA or PETG.

The shear lag theory, employing a shear lag model, is utilized to predict properties such as the Young's modulus for short fiber composites. This model operates under the assumption that load is transferred from the matrix to the fibers exclusively through interfacial shear stresses, denoted by $\tau_{i}$, acting along the cylindrical interface. According to shear lag theory, the rate of change of axial stress within the fiber, as one moves along its length, is directly proportional to the ratio of the interfacial shear stresses to the fiber's radius, $r_{0}$:

$\frac{d\sigma _{f}}{dx}=-{\frac {2\tau _{i}}{r_{0}}}$

This relationship leads to the average fiber stress over the entire fiber length being given by:

$\sigma _{f}=E_{f}\varepsilon _{1}\left(1-{\frac {\tanh(ns)}{ns}}\right)$

where:

• $\varepsilon _{1}$ represents the macroscopic strain within the composite.
• $s$ is the fiber aspect ratio (the ratio of length to diameter).
• $n=\left({\frac {2E_{m}}{E_{f}(1+\nu _{m})\ln(1/f)}}\right)^{1/2}$ is a dimensionless constant [83]
• $\nu _{m}$ is the Poisson's ratio of the matrix.

By assuming a uniform tensile strain, the following expression for the composite's Young's modulus is derived: [84]

$E_{1}={\frac {\sigma _{1}}{\varepsilon _{1}}}=fE_{f}\left(1-{\frac {\tanh(ns)}{ns}}\right)+(1-f)E_{m}$

As the aspect ratio $s$ increases, this equation converges towards the rule of mixtures, which accurately represents the Young's modulus parallel to continuous fibers.

Continuous Fiber Reinforcement

In general, continuous fiber reinforcement is achieved by incorporating a fiber as the stronger phase within a weaker phase, the matrix. The widespread use of fibers stems from the fact that materials can exhibit extraordinary strength in their fiber form. Non-metallic fibers, due to the covalent nature of their bonds, typically display a very high strength-to-density ratio compared to metallic fibers. The most prominent example of this is carbon fibers, which find extensive applications ranging from sports gear to protective equipment and the space industries. [85] [86]

The stress experienced by the composite can be expressed in terms of the volume fraction of the fiber and the matrix:

$\sigma _{c}=V_{f}\sigma _{f}+V_{m}\sigma _{m}$

where $\sigma$ represents stress, and $V$ denotes volume fraction. The subscripts $c$, $f$, and $m$ refer to the composite, fiber, and matrix, respectively.

While the stress–strain behavior of fiber composites can only be definitively determined through testing, a predictable trend emerges, characterized by three distinct stages in the stress–strain curve. The first stage encompasses the region where both the fiber and the matrix undergo elastic deformation. This linearly elastic region can be mathematically described as follows: [85]

$\sigma _{c}-E_{c}\epsilon _{c}=\epsilon _{c}(V_{f}E_{f}+V_{m}E_{m})$

where $\sigma$ is stress, $\epsilon$ is strain, $E$ is the elastic modulus, and $V$ is the volume fraction. The subscripts $c$, $f$, and $m$ denote the composite, fiber, and matrix, respectively.

Following the elastic region for both fiber and matrix, the stress–strain curve enters its second stage. In this phase, the fiber remains elastically deformed, while the matrix begins to deform plastically, as it is the weaker constituent. The instantaneous modulus during this stage can be determined from the slope of the stress–strain curve. The relationship between stress and strain can be expressed as:

$\sigma _{c}=V_{f}E_{f}\epsilon _{c}+V_{m}\sigma _{m}(\epsilon _{c})$

where $\sigma$ is stress, $\epsilon$ is strain, $E$ is the elastic modulus, and $V$ is the volume fraction. The subscripts $c$, $f$, and $m$ indicate the composite, fiber, and matrix, respectively. To ascertain the modulus in the second region, the derivative of this equation can be employed, as the slope of the curve directly corresponds to the modulus.

$E_{c}'={\frac {d\sigma _{c}}{d\epsilon _{c}}}=V_{f}E_{f}+V_{m}\left({\frac {d\sigma _{c}}{d\epsilon _{c}}}\right)$

In most practical scenarios, it can be reasonably assumed that $E_{c}'=V_{f}E_{f}$ since the second term is significantly smaller than the first. [85]

In reality, the derivative of stress with respect to strain does not always directly yield the modulus due to the binding interaction between the fiber and the matrix. The strength of this interaction can significantly alter the mechanical properties of the composite. The compatibility between the fiber and matrix serves as a measure of internal stress. [85]

High-strength fibers with covalent bonds, such as carbon fibers, primarily undergo elastic deformation before fracture, as plastic deformation typically occurs through dislocation motion. In contrast, metallic fibers offer greater scope for plastic deformation, resulting in composites that exhibit a third stage where both the fiber and the matrix undergo plastic deformation. Metallic fibers possess numerous applications in cryogenic hardening and operation at cryogenic temperatures, presenting a distinct advantage of metal fiber composites over their non-metallic counterparts. The stress in this region of the stress–strain curve can be described as:

$\sigma _{c}(\epsilon _{c})=V_{f}\sigma _{f}\epsilon _{c}+V_{m}\sigma _{m}(\epsilon _{c})$

where $\sigma$ is stress, $\epsilon$ is strain, $E$ is the elastic modulus, and $V$ is the volume fraction. The subscripts $c$, $f$, and $m$ denote the composite, fiber, and matrix, respectively. $\sigma _{f}(\epsilon _{c})$ and $\sigma _{m}(\epsilon _{c})$ represent the fiber and matrix flow stresses, respectively. Shortly after the third region, the composite exhibits necking. The necking strain of the composite tends to fall between the necking strains of the fiber and the matrix, mirroring the behavior of other mechanical properties. The necking strain of the weaker phase is delayed by the presence of the stronger phase. The extent of this delay is contingent upon the volume fraction of the stronger phase. [85]

Consequently, the tensile strength of the composite can be expressed in terms of the volume fraction: [85]

$(T.S.)_{c}=V_{f}(T.S.)_{f}+V_{m}\sigma _{m}(\epsilon _{m})$

where T.S. is the tensile strength, $\sigma$ is stress, $\epsilon$ is strain, $E$ is the elastic modulus, and $V$ is the volume fraction. The subscripts $c$, $f$, and $m$ denote the composite, fiber, and matrix, respectively. The composite tensile strength can be expressed as:

$(T.S.)_{c}=V_{m}(T.S.)_{m}$

for $V_{f}$ less than or equal to $V_{c}$ (an arbitrary critical volume fraction).

$(T.S.)_{c}=V_{f}(T.S.)_{f}+V_{m}(\sigma _{m})$

for $V_{f}$ greater than or equal to $V_{c}$.

The critical value of volume fraction can be determined using the following equation:

$V_{c}={\frac {[(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}{[(T.S.)_{f}+(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}}$

Evidently, the composite tensile strength can surpass that of the matrix if $(T.S.)_{c}$ is greater than $(T.S.)_{m}$.

Thus, the minimum fiber volume fraction required can be expressed as:

$V_{c}={\frac {[(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}{[(T.S.)_{f}-\sigma _{m}(\epsilon _{f})]}}$

Although this minimum value is often quite low in practice, it is crucial to recognize its significance. The primary objective of incorporating continuous fibers is to enhance the mechanical properties of the materials/composites, and this critical volume fraction represents the threshold at which such improvement becomes demonstrable. [85]

The Effect of Fiber Orientation

Aligned Fibers

Variations in the angle between the applied stress and the fiber orientation will inevitably impact the mechanical properties of fiber-reinforced composites, particularly their tensile strength. This angle, denoted by $\theta$, can be used to predict the dominant tensile fracture mechanism.

At small angles, $\theta \approx 0^{\circ}$, the primary fracture mechanism mirrors that observed when the load is aligned with the fibers: tensile fracture. The resolved force acting along the fiber length is reduced by a factor of $\cos \theta$ due to rotation.

$F_{\mbox{res}}=F\cos \theta$

The resolved area upon which the fiber experiences this force is increased by a factor of $\cos \theta$ due to rotation.

$A_{\mbox{res}}=A_{0}/\cos \theta$

Taking the effective tensile strength to be $(T.S.)_{c} = F_{\mbox{res}}/A_{\mbox{res}}$ and the aligned tensile strength as $\sigma _{\parallel }^{*}=F/A$, the composite tensile strength for longitudinal fracture is given by: [85]

$({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{longitudinal fracture}})={\frac {\sigma _{\parallel }^{*}}{\cos ^{2}\theta }}$

At moderate angles, $\theta \approx 45^{\circ}$, the material is prone to shear failure. The effective force direction is reduced relative to the aligned direction.

$F_{\mbox{res}}=F\cos \theta$

The resolved area upon which this force acts is:

$A_{\mbox{res}}=A_{m}/\sin \theta$

The resulting tensile strength is then dependent on the shear strength of the matrix, $\tau_{m}$. [85]

$({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{shear failure}})={\frac {\tau _{m}}{\sin {\theta }\cos {\theta }}}$

At extreme angles, $\theta \approx 90^{\circ}$, the dominant failure mode becomes tensile fracture within the matrix in the perpendicular direction. Analogous to the isostress case in layered composite materials, the strength in this direction is diminished compared to the aligned direction. The effective forces and areas act perpendicular to the aligned direction, both scaling by $\sin \theta$. The resolved tensile strength is proportional to the transverse strength, $\sigma _{\perp }^{*}$. [85]

$({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{transverse fracture}})={\frac {\sigma _{\perp }^{*}}{\sin ^{2}\theta }}$

The critical angles at which the dominant fracture mechanism transitions can be calculated as follows:

$\theta _{c_{1}}=\tan ^{-1}\left({\frac {\tau _{m}}{\sigma _{\parallel }^{*}}}\right)$

$\theta _{c_{2}}=\tan ^{-1}\left({\frac {\sigma _{\perp }^{*}}{\tau _{m}}}\right)$

where $\theta _{c_{1}}$ represents the critical angle between longitudinal fracture and shear failure, and $\theta _{c_{2}}$ denotes the critical angle between shear failure and transverse fracture. [85]

This model, by neglecting length effects, is most accurate for continuous fibers and does not fully capture the strength-orientation relationship for short fiber reinforced composites. Furthermore, most real-world systems do not exhibit the local maxima predicted at the critical angles. [87] [88] [89] [90] The Tsai-Hill criterion offers a more comprehensive description of composite tensile strength as a function of orientation angle by integrating the contributing yield stresses: $\sigma _{\parallel }^{*}$, $\sigma _{\perp }^{*}$, and $\tau_{m}$. [91] [85]

({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{Tsai-Hill}})={\bigg [}{\frac {\cos ^{4}\theta }{({\sigma _{\parallel }^{*}})^{2}}}+\cos ^{2}\theta \sin ^{2}\theta \left({\frac {1}{({\tau _{m}})^{2}}}-{\frac {1}{({\sigma _{\parallel }^{*}})^{2}}}\right)+{\frac {\sin ^{4}\theta }{({\sigma _{\perp }^{*}})^{2}}}{\bigg ]}^{-1/2}}

Randomly Oriented Fibers

The anisotropy inherent in the tensile strength of fiber-reinforced composites can be mitigated by randomly orienting the fiber directions within the material. This approach sacrifices the ultimate strength achieved in the aligned direction for an overall, isotropically strengthened material.

$E_{c}=KV_{f}E_{f}+V_{m}E_{m}$

Here, $K$ is an empirically determined reinforcement factor. For fibers with randomly distributed orientations within a plane, $K \approx 0.38$, and for a random distribution in three dimensions, $K \approx 0.20$. [85]

Stiffness and Compliance Elasticity

Composite materials are generally anisotropic, and frequently orthotropic. Voigt notation can be employed to reduce the rank of the stress and strain tensors, allowing the stiffness $C$ (often also referred to as $Q$) and compliance $S$ matrices to be represented as: [92]

$\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}$

and

$\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}={\begin{bmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}$

When considering each ply individually, it is generally assumed that they can be treated as a lamina, with out-of-plane stresses and strains being negligible. That is, $\sigma _{3}=\sigma _{4}=\sigma _{5}=0$ and $\varepsilon _{4}=\varepsilon _{5}=0$. [93] This simplification allows the stiffness and compliance matrices to be reduced to 3x3 matrices:

C={\begin{bmatrix}{\tfrac {E_{\rm {1}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&{\tfrac {E_{\rm {2}}{\nu _{\rm {12}}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&0\\{\tfrac {E_{\rm {2}}{\nu _{\rm {12}}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&{\tfrac {E_{\rm {2}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&0\\0&0&G_{\rm {12}}\\\end{bmatrix}}\quad }

and

$\quad S={\begin{bmatrix}{\tfrac {1}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {21}}}{E_{\rm {2}}}}&0\\-{\tfrac {\nu _{\rm {12}}}{E_{\rm {1}}}}&{\tfrac {1}{E_{\rm {2}}}}&0\\0&0&{\tfrac {1}{G_{\rm {12}}}}\\\end{bmatrix}}$

Two distinct coordinate systems are considered for the material: the structure's (1-2) coordinate system and the material's (x-y) principal coordinate system.

For fiber-reinforced composites, the orientation of the fibers within the material significantly influences the anisotropic properties of the structure. Through characterization techniques such as tensile testing, material properties are measured based on samples oriented in the (1-2) coordinate system. The tensors presented above describe the stress-strain relationship within this (1-2) coordinate system. However, the known material properties are typically provided in the principal coordinate system (x-y) of the material. Transforming the tensor between these two coordinate systems is essential for accurately identifying the material properties of the tested sample. The transformation matrix, with a rotation of $\theta$ degrees, is given by:

$T(\theta )_{\epsilon }={\begin{bmatrix}\cos ^{2}\theta &\sin ^{2}\theta &\cos \theta \sin \theta \\\sin^{2}\theta &\cos ^{2}\theta &-\cos \theta \sin \theta \\-2\cos \theta \sin \theta &2\cos \theta \sin \theta &\cos ^{2}\theta -\sin ^{2}\theta \end{bmatrix}}$

for $\begin{bmatrix}{\acute {\epsilon }}\end{bmatrix}=T(\theta )_{\epsilon }\begin{bmatrix}\epsilon \end{bmatrix}$, and

$T(\theta )_{\sigma }={\begin{bmatrix}\cos ^{2}\theta &\sin ^{2}\theta &2\cos \theta \sin \theta \\\sin^{2}\theta &\cos ^{2}\theta &-2\cos \theta \sin \theta \\-\cos \theta \sin \theta &\cos \theta \sin \theta &\cos ^{2}\theta -\sin ^{2}\theta \end{bmatrix}}$

for $\begin{bmatrix}{\acute {\sigma }}\end{bmatrix}=T(\theta )_{\sigma }\begin{bmatrix}\sigma \end{bmatrix}$.

Types of Fibers and Mechanical Properties

The most prevalent types of fibers utilized in industry are glass fibers, carbon fibers, and Kevlar, owing to their ease of production and widespread availability. Their mechanical properties are of paramount importance for comparative analysis. The following table presents a comparison of their mechanical properties against S97 steel. [94] [95] [96] [97] The angle of fiber orientation is a critical factor due to the inherent anisotropy of fiber composites (refer to the "Physical Properties" section for a more detailed explanation). The mechanical properties of composites can be rigorously tested using standard mechanical testing methods by orienting samples at various angles (typically 0°, 45°, and 90°) relative to the fiber orientation within the composites. Generally, 0° axial alignment provides resistance to longitudinal bending and axial tension/compression; 90° hoop alignment is employed to withstand internal/external pressure; and ±45° orientation is optimal for resistance against pure torsion. [98]

Mechanical Properties of Fiber Composite Materials

Fibres @ 0° (UD), 0/90° (fabric) to loading axis, Dry, Room Temperature, V f = 60% (UD), 50% (fabric) Fibre / Epoxy Resin (cured at 120 °C) [99]

Symbol	Units	Standard	Carbon Fiber Fabric	High Modulus Carbon Fiber Fabric	E-Glass Fibre Glass Fabric	Kevlar Fabric	Standard Unidirectional Carbon Fiber Fabric	High Modulus Unidirectional Carbon Fiber Fabric	E-Glass Unidirectional Fiber Glass Fabric	Kevlar Unidirectional Fabric	Steel S97
Young's Modulus 0°	E1	GPa	70	85	25	30	135	175	40	75	207
Young's Modulus 90°	E2	GPa	70	85	25	30	10	8	8	6	207
In-plane Shear Modulus	G12	GPa	5	5	4	5	5	5	4	2	80
Major Poisson's Ratio	v12		0.10	0.10	0.20	0.20	0.30	0.30	0.25	0.34	–
Ult. Tensile Strength 0°	Xt	MPa	600	350	440	480	1500	1000	1000	1300	990
Ult. Comp. Strength 0°	Xc	MPa	570	150	425	190	1200	850	600	280	–
Ult. Tensile Strength 90°	Yt	MPa	600	350	440	480	50	40	30	30	–
Ult. Comp. Strength 90°	Yc	MPa	570	150	425	190	250	200	110	140	–
Ult. In-plane Shear Stren.	S	MPa	90	35	40	50	70	60	40	60	–
Ult. Tensile Strain 0°	ext	%	0.85	0.40	1.75	1.60	1.05	0.55	2.50	1.70	–
Ult. Comp. Strain 0°	exc	%	0.80	0.15	1.70	0.60	0.85	0.45	1.50	0.35	–
Ult. Tensile Strain 90°	eyt	%	0.85	0.40	1.75	1.60	0.50	0.50	0.35	0.50	–
Ult. Comp. Strain 90°	eyc	%	0.80	0.15	1.70	0.60	2.50	2.50	1.35	2.30	–
Ult. In-plane shear strain	es	%	1.80	0.70	1.00	1.00	1.40	1.20	1.00	3.00	–
Density		g/cc	1.60	1.60	1.90	1.40	1.60	1.60	1.90	1.40	–

Fibres @ ±45 Deg. to loading axis, Dry, Room Temperature, V f = 60% (UD), 50% (fabric) [99]

Symbol	Units	Standard Carbon Fiber	High Modulus Carbon Fiber	E-Glass Fiber Glass	Standard Carbon Fibers Fabric	E-Glass Fiber Glass Fabric	Steel Al
Longitudinal Modulus	E1	GPa	17	17	12.3	19.1	12.2
Transverse Modulus	E2	GPa	17	17	12.3	19.1	12.2
In Plane Shear Modulus	G12	GPa	33	47	11	30	8
Poisson's Ratio	v12		.77	.83	.53	.74	.53
Tensile Strength	Xt	MPa	110	110	90	120	120
Compressive Strength	Xc	MPa	110	110	90	120	120
In Plane Shear Strength	S	MPa	260	210	100	310	150
Thermal Expansion Co-ef	Alpha1	Strain/K	2.15 E-6	0.9 E-6	12 E-6	4.9 E-6	10 E-6
Moisture Co-ef	Beta1	Strain/K	3.22 E-4	2.49 E-4	6.9 E-4

Note: Steel Al properties are presented for comparison.

Carbon Fiber & Fiberglass Composites vs. Aluminum Alloy and Steel

While the strength and stiffness of steel and aluminum alloys are comparable to those of fiber composites, the specific strength and stiffness (strength and stiffness relative to weight) of composites are significantly higher.

Comparison of Cost, Specific Strength, and Specific Stiffness [100]

	Carbon Fiber Composite (aerospace grade)	Carbon Fiber Composite (commercial grade)	Fiberglass Composite	Aluminum 6061 T-6	Steel, Mild
Cost $/LB	$20 –$250+	$5 –$20	$1.50 –$3.00	$3	$0.30
Strength (psi)	90,000 – 200,000	50,000 – 90,000	20,000 – 35,000	35,000	60,000
Stiffness (psi)	10 x 10 6 – 50 x 10 6	8 x 10 6 – 10 x 10 6	1 x 10 6 – 1.5 x 10 6	10 x 10 6	30 x 10 6
Density (lb/in³)	0.050	0.050	0.055	0.10	0.30
Specific Strength	1.8 x 10 6 – 4 x 10 6	1 x 10 6 – 1.8 x 10 6	363,640–636,360	350,000	200,000
Specific Stiffness	200 x 10 6 – 1,000 x 10 6	160 x 10 6 – 200 x 10 6	18 x 10 6 – 27 x 10 6	100 x 10 6	100 x 10 6

Failure

Failures in composite materials can manifest in various ways. Shock, impact at varying speeds, or repeated cyclic stresses can induce the laminate to separate at the interface between layers, a phenomenon known as delamination. [101] [102] Individual fibers may also detach from the matrix, a process referred to as fibre pull-out.

Composites can fail on both the macroscopic and microscopic scales. Compression failures can occur at the macro level or at the individual reinforcing fiber level through compression buckling. Tensile failures can involve net section failures of the entire part or degradation of the composite at a microscopic level, where one or more layers within the composite fail due to tensile stress on the matrix or a breakdown in the bond between the matrix and fibers.

Some composites exhibit brittle behavior with limited reserve strength beyond the initial onset of failure, while others can undergo significant deformations and possess considerable energy absorption capacity even after damage initiation. The diverse range of available fibers and matrices, along with the flexibility in their blending and mixing, allows for the design of composite structures with an exceptionally broad spectrum of properties. A notable failure involving a brittle ceramic matrix composite occurred when a carbon-carbon composite tile on the leading edge of the Space Shuttle Columbia wing fractured upon impact during takeoff. This incident tragically led to the catastrophic disintegration of the vehicle upon its re-entry into Earth's atmosphere on February 1, 2003.

Compared to metals, composites generally exhibit relatively poor bearing strength.

The provided graph illustrates the three primary fracture modes that a composite material may experience, contingent upon the angle of misorientation relative to fibers aligned parallel to the applied stress.

Another failure mode is fiber tensile fracture, which becomes more probable when fibers are aligned with the loading direction. This is assuming the tensile strength of the fibers exceeds that of the matrix. When a fiber is oriented at an angle of misorientation $\theta$, several fracture modes become possible. For small values of $\theta$, the stress required to initiate fracture increases by a factor of $(\cos \theta)^{-2}$. This is attributed to the increased cross-sectional area $(A \cos \theta)$ of the fiber and the reduced force $(F/\cos \theta)$ experienced by the fiber, leading to a composite tensile strength of $\sigma _{\parallel }^{*} / \cos ^{2}\theta$, where $\sigma _{\parallel }^{*}$ is the tensile strength of the composite with fibers aligned parallel to the applied force.

Intermediate angles of misorientation $\theta$ tend to induce matrix shear failure. Again, the cross-sectional area is modified, but since shear stress is now the primary driver of failure, the area of the matrix parallel to the fibers becomes relevant. This area increases by a factor of $1/\sin \theta$. Similarly, the force acting parallel to this area decreases $(F/\cos \theta)$, resulting in a composite tensile strength of $\tau _{m} / \sin \theta \cos \theta$, where $\tau _{m}$ is the matrix shear strength.

Finally, for large values of $\theta$ (approaching $\pi/2$), transverse matrix failure is the most likely outcome, as the fibers no longer bear the majority of the load. Nevertheless, the tensile strength in this scenario will still be greater than that for a purely perpendicular orientation. This is because the force perpendicular to the fibers decreases by a factor of $1/\sin \theta$, and the area also decreases by a factor of $1/\sin \theta$, yielding a composite tensile strength of $\sigma _{\perp }^{*} / \sin ^{2}\theta$, where $\sigma _{\perp }^{*}$ is the tensile strength of the composite with fibers aligned perpendicular to the applied force. [103]

Testing

Composites undergo rigorous testing both before and after construction to aid in predicting and preventing potential failures. Pre-construction testing often employs finite element analysis (FEA) for detailed ply-by-ply analysis of curved surfaces, predicting phenomena such as wrinkling, crimping, and dimpling in composites. [104] [105] [106] [107] [108] During manufacturing and after construction, materials can be inspected using various non-destructive methods, including ultrasonic testing, thermography, shearography, and X-ray radiography, [109] along with laser bond inspection for assessing the integrity of relative bond strength in localized areas.

See also

• 3D composites
• Aluminium composite panel
• American Composites Manufacturers Association
• Chemical vapour infiltration
• Composite laminate
• Discontinuous aligned composite
• Epoxy granite
• Hybrid material
• Lay-up process
• Nanocomposite
• Pykrete
• Rule of mixtures
• Scaled Composites
• Smart material
• Smart Materials and Structures
• Void (composites)