Polymers, Elastomers and Composite Materials

Introduction

Polymers and polymer-based composites are central to modern mechanical engineering. Their low density, corrosion resistance, manufacturability, vibration damping and design flexibility make them important in automotive, aerospace, biomedical, electrical and consumer products. At the same time, their behavior differs fundamentally from that of metals and ceramics. Polymer properties depend not only on chemical composition, but also on molecular weight, chain architecture, crystallinity, temperature, time, processing history and environmental exposure.

Composite materials extend this design flexibility by combining a matrix with a reinforcing phase. Properly designed composites can provide high specific stiffness and strength, but they are anisotropic and may fail through several interacting mechanisms, including fiber fracture, matrix cracking, debonding, delamination and buckling.

This chapter relates molecular architecture, crystallinity and glass transition to polymer behavior; distinguishes thermoplastics, thermosets and elastomers; and develops the basic micromechanics and failure concepts needed to evaluate fiber-, particle- and laminate-reinforced composites.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Explain how polymer molecular structure influences stiffness, strength, toughness and thermal behavior.
2. Distinguish glass transition, melting and degradation temperatures.
3. Compare thermoplastics, thermosets and elastomers in terms of structure, processing and applications.
4. Use basic stress–strain and micromechanics relationships for composite materials.
5. Explain anisotropy, lamina behavior and laminate construction.
6. Identify common composite failure mechanisms and appropriate design practices.
7. Select polymer and composite materials while considering service conditions, manufacturing, sustainability and safety.

1. Molecular Architecture and Polymer Behavior

A polymer is composed of long molecules formed by repeating structural units called monomers. Polymer chains may contain hundreds or millions of covalently bonded atoms. The properties of a polymer are therefore governed by both the chemistry of each repeat unit and the organization of many chains.

1.1 Chain structure

Important architectural features include:

• Molecular weight: Higher molecular weight generally increases entanglement and toughness, although it can increase melt viscosity and processing difficulty.
• Chain branching: Branches can reduce packing efficiency and crystallinity. Long-chain branching may improve melt strength.
• Cross-linking: Covalent links between chains restrict molecular motion and improve dimensional stability and heat resistance.
• Tacticity: The arrangement of side groups along a chain affects regularity and crystallizability. Isotactic chains are usually more regular than atactic chains.
• Polarity: Polar groups increase intermolecular attraction and may raise stiffness, strength and glass-transition temperature.
• Copolymerization: Combining different monomers can tailor flexibility, impact resistance, chemical resistance and processing behavior.

The distinction between primary and secondary bonding is especially important. Covalent bonds along a polymer backbone are strong, whereas interactions between chains may be van der Waals forces, dipole interactions or hydrogen bonds. Deformation often involves rotation and sliding of chains rather than rupture of every covalent bond.

1.2 Molecular weight and entanglement

A polymer sample contains chains of different lengths. The number-average molecular weight is

where N_i is the number of molecules having molecular weight M_i. The weight-average molecular weight is

The ratio M_w/M_n is the dispersity, a measure of molecular-weight distribution. These relationships apply to a population of polymer molecules and assume that the molecular weights and number of molecules are appropriately characterized. Greater chain length and entanglement commonly improve tensile toughness and resistance to creep, but increase viscosity during processing.

1.3 Amorphous and semicrystalline polymers

In an amorphous polymer, chains are arranged without long-range order. Such materials generally soften progressively above the glass-transition temperature. Examples include polystyrene, polycarbonate and polymethyl methacrylate.

A semicrystalline polymer contains ordered crystallites dispersed in an amorphous phase. Perfect crystallinity is uncommon because chain irregularities, branching and processing constraints prevent complete packing. Crystallites act as physical cross-links and often increase stiffness, strength, chemical resistance and dimensional stability. Examples include polyethylene, polypropylene, nylon and polyether ether ketone.

Crystallinity depends on cooling rate, nucleation, chain regularity, molecular weight, pressure and thermal history. Slow cooling usually permits greater crystal growth than rapid cooling. Crystalline regions also produce directional shrinkage during molding, which can create residual stress and warpage.

1.4 Glass transition and melting

The glass-transition temperature, T_g, is the approximate temperature range in which cooperative segmental motion becomes possible in the amorphous phase. Below T_g, an amorphous polymer is relatively rigid and glassy; above T_g, it is softer and more viscoelastic. The glass transition is not a first-order phase change and occurs over a range rather than at one exact temperature.

The melting temperature, T_m, is associated with loss of ordered crystalline regions. Only crystalline or semicrystalline polymers exhibit a true melting transition. An amorphous polymer has no T_m in the same sense, although it may soften and eventually degrade.

The service temperature must be judged relative to T_g and T_m, not from a single tabulated value. Plasticizers generally lower T_g, while rigid backbones, polar groups and cross-links tend to raise it. Moisture can lower the effective T_g of hygroscopic polymers such as nylon.

1.5 Viscoelasticity and time dependence

Polymers exhibit both elastic and viscous behavior. An ideal elastic solid stores recoverable energy, while an ideal viscous liquid dissipates energy continuously. Real polymers show creep, stress relaxation and hysteresis.

For a constant applied stress, creep is the increase in strain with time. For a constant imposed strain, stress relaxation is the decrease in stress with time. A simple linear viscoelastic representation is the Maxwell model, consisting of a spring and dashpot in series. Its stress-relaxation equation is

where σ₀ is the initial stress, t is time and τ = η/E is the relaxation time. Here η is dashpot viscosity and E is spring modulus. This idealized equation applies to a linear Maxwell material under a constant total strain; real polymers commonly require several relaxation times.

Temperature and loading rate strongly affect response. Increasing temperature or decreasing strain rate generally allows more molecular rearrangement, producing lower apparent stiffness and strength but greater ductility. Time–temperature superposition can sometimes shift data obtained at different temperatures along a logarithmic time scale, but it requires thermorheologically simple behavior and should not be assumed for all polymers.

2. Polymer Classes and Processing

2.1 Thermoplastics

Thermoplastics consist of linear or branched chains that are not permanently cross-linked. They soften when heated and solidify when cooled, allowing repeated processing if degradation is avoided. They may be amorphous or semicrystalline.

Typical advantages include weldability, recyclability, impact resistance and rapid processing. Limitations include creep, sensitivity to temperature and, for some grades, solvent or moisture sensitivity. Common processes are injection molding, extrusion, blow molding, thermoforming and additive manufacturing.

Examples include:

2.2 Thermosets

Thermosets begin as low-viscosity resins and undergo curing reactions that create a three-dimensional network. Once cured, they do not melt for reshaping; excessive heating causes degradation or charring. Epoxy, unsaturated polyester, vinyl ester and phenolic resins are common examples.

Cross-linking improves heat resistance, creep resistance and dimensional stability, but usually reduces melt-processability and repairability. Thermosets are widely used as matrices in fiber-reinforced composites because they wet fibers effectively before curing and provide strong interfacial bonding.

Cure schedules must control temperature, time, resin flow, volatile removal and exothermic heat. Inadequate cure produces low strength and poor environmental resistance; excessive cure temperature can generate voids, residual stress or thermal damage.

2.3 Elastomers

Elastomers are lightly cross-linked polymers capable of large, largely recoverable strains. Their operating temperature is normally above T_g, where chain segments can move, while cross-links prevent permanent flow. Natural rubber, silicone rubber, styrene–butadiene rubber and nitrile rubber are important examples.

Elasticity in an ideal rubber is largely entropic. Stretching reduces the number of possible chain conformations; release allows chains to return toward a more probable state. Real elastomers also show hysteresis, strain-rate dependence, temperature sensitivity, swelling in solvents and fatigue crack growth.

Vulcanization introduces sulfur or other cross-links into rubber. Increasing cross-link density generally raises modulus and dimensional stability but can reduce extensibility. Fillers such as carbon black improve strength, wear resistance and electrical properties.

3. Composite Materials

A composite contains at least two distinct phases whose combined behavior is intended to exceed that of the individual constituents. The matrix transfers load, protects reinforcement and defines shape. The reinforcement supplies stiffness, strength, wear resistance or other targeted properties. The interface controls load transfer and can dominate failure.

Major reinforcement forms are:

• continuous fibers;
• short or discontinuous fibers;
• particles;
• flakes, whiskers or natural fibers; and
• layered sheets or laminates.

Matrices may be polymeric, metallic or ceramic. This chapter emphasizes polymer-matrix composites, including glass-fiber-reinforced polymer (GFRP), carbon-fiber-reinforced polymer (CFRP) and natural-fiber composites.

3.1 Rule of mixtures

For a unidirectional composite loaded parallel to continuous fibers, the simple longitudinal rule of mixtures is

where E_L is longitudinal elastic modulus, E_f and E_m are fiber and matrix moduli, and V_f and V_m are volume fractions. For a two-phase composite, V_f + V_m = 1. The equation assumes aligned, continuous fibers, uniform strain, perfect load transfer, linear elasticity and negligible voids.

The corresponding longitudinal strength is not generally obtained by simply averaging strengths. A useful idealized expression at fiber failure is

where σ_f is fiber stress at composite failure and σ'_m is matrix stress at that same strain. This relation is conditional on simultaneous strain compatibility and does not account fully for defects, statistical fiber strength or local stress concentrations.

For loading transverse to aligned fibers, the inverse rule of mixtures is often used as a first estimate:

It is less reliable for real composites because transverse behavior is strongly affected by interface quality, fiber packing and matrix yielding.

Worked Example 1: Longitudinal stiffness

A unidirectional carbon/epoxy composite contains 60% fibers by volume. Let E_f = 230 GPa and E_m = 3.5 GPa. Assuming perfect alignment and elastic uniform strain,

The estimate is much closer to the fiber modulus because the stiff fibers carry most of the longitudinal load. Actual values may be lower because of waviness, voids, imperfect bonding and nonuniform fiber distribution.

3.2 Specific properties

Density is important in transportation design. Specific modulus and specific strength are

and

where E is elastic modulus, σ is strength and ρ is density. Consistent units are required; for example, modulus in Pa and density in kg/m³ produce specific modulus in m²/s². Composites are attractive when stiffness or strength per unit mass is more important than absolute stiffness or strength.

3.3 Short-fiber composites

Short fibers are easier to process than continuous fibers but do not develop full fiber stress unless their length is sufficient. The critical fiber length is approximately

where l_c is critical length, σ_f is the fiber tensile strength, d is fiber diameter and τ_i is the interfacial shear stress. The expression assumes a cylindrical fiber, uniform interfacial shear stress and simplified shear-lag load transfer.

Fibers much longer than l_c are more efficient. Fiber orientation, length distribution, dispersion and processing-induced alignment strongly affect properties. Randomly oriented short fibers produce more nearly isotropic behavior but lower directional stiffness than aligned fibers.

3.4 Particle-reinforced composites

Particles can increase hardness, wear resistance, dimensional stability, thermal conductivity or stiffness. Examples include mineral-filled polymers, rubber-toughened plastics and cemented carbides. Particles may be reinforcing or weakening depending on size, shape, adhesion, concentration and dispersion.

Fine, well-bonded particles can impede deformation. Poorly bonded particles or agglomerates act as void-like defects and initiate cracks. Rubber particles in a brittle thermoplastic can improve impact toughness by promoting shear yielding and localized energy absorption, even though they reduce modulus.

4. Anisotropy, Laminae and Laminates

An isotropic material has properties independent of direction. A unidirectional composite is strongly anisotropic: longitudinal stiffness may be dominated by fibers, whereas transverse stiffness and shear strength depend heavily on the matrix and interface.

A single composite ply is called a lamina. Its material axes are usually designated 1 along the fibers, 2 transverse to the fibers and 3 through the thickness. A laminate is a bonded stack of laminae with selected fiber orientations such as 0°, ±45° and 90°.

For a unidirectional lamina under plane stress, the reduced constitutive relation is

where stresses and strains are expressed in material axes and [Q] is the reduced stiffness matrix. For an orthotropic lamina,

with reciprocity condition

Here E₁ and E₂ are principal-direction moduli, G₁2 is in-plane shear modulus, ν₁2 and ν₂1 are Poisson ratios, and σ₁, σ₂, τ₁2 and corresponding strains are material-axis components. These equations assume linear elastic orthotropic behavior and plane stress.

When a ply is placed at an angle θ to the global loading direction, its stiffness must be transformed. Off-axis plies couple normal and shear deformation. Classical laminate theory assumes thin, perfectly bonded layers; plane sections remain approximately plane; each ply is linearly elastic; and transverse stresses are small. It is useful for preliminary design but does not fully capture free-edge stresses, delamination, nonlinear matrix behavior or damage evolution.

Balanced laminates containing +θ and −θ plies reduce extension–shear coupling. Symmetric laminates reduce bending–extension coupling. Quasi-isotropic layups approximate isotropic in-plane response but do not eliminate all three-dimensional effects.

5. Composite Failure and Durability

Composite failure is progressive rather than necessarily catastrophic. Common mechanisms include:

1. Fiber fracture: dominant in the fiber direction and often sudden.
2. Matrix cracking: caused by transverse tension, compression, shear or thermal stress.
3. Fiber–matrix debonding: reduces load transfer and may produce local stress concentrations.
4. Delamination: separation between plies, often initiated by impact or free-edge stresses.
5. Fiber kinking and microbuckling: important under longitudinal compression.
6. Pull-out: occurs when fibers debond and slide from the matrix.
7. Buckling: structural instability may occur before material strength is reached.

A simple maximum-stress criterion declares failure when any stress component exceeds its allowable value. For example, in a lamina,

where X, Y and S are allowable longitudinal, transverse and shear stresses. This criterion is easy to use but does not represent interaction between stresses.

An interactive quadratic criterion such as Tsai–Hill is more representative for some applications, but its coefficients and sign conventions must be selected consistently with the material data. No single failure criterion is universally valid. Design should consider mode-specific allowables, environmental knock-down factors, damage tolerance and testing.

Fatigue behavior depends on stress ratio, frequency, temperature, moisture, fiber orientation and defects. Polymer matrices may soften through self-heating under cyclic loading. Impact can cause barely visible internal damage, especially delamination, so visual inspection alone may be inadequate. Ultrasonic inspection, radiography, thermography or other nondestructive methods may be required.

Moisture absorption can plasticize a matrix, lower T_g, reduce interface strength and promote swelling. Elevated temperature accelerates creep and oxidation. Ultraviolet radiation can embrittle exposed polymers. Fire behavior is also critical: combustion may produce heat, smoke and toxic gases. Flame retardants alter performance and must be evaluated for environmental and health effects.

6. Materials Selection and Manufacturing Practice

Selection should begin with the required function rather than with a familiar material. Important questions include:

• What stiffness, strength, toughness and fatigue life are required?
• What are the temperature, humidity, chemical and radiation conditions?
• Is the load static, cyclic, impact, compressive or multiaxial?
• Is low mass more important than low cost?
• Can the chosen process control fiber orientation, void content and cure?
• Can the component be inspected, repaired, recycled or safely disposed of?

Manufacturing defects commonly include voids, dry spots, resin-rich regions, fiber waviness, poor cure, porosity, inclusions and dimensional distortion. Process monitoring should control resin viscosity, fiber volume fraction, temperature history, pressure, consolidation and cure degree. Design allowables should be based on representative manufacturing processes rather than ideal laboratory coupons alone.

Thermoplastic composites offer welding, rapid consolidation and potential recyclability. Thermoset composites often provide excellent performance but are difficult to separate into reusable constituents. Mechanical recycling may reduce fiber length and performance; chemical recycling and pyrolysis are developing alternatives. A life-cycle assessment should include raw-material production, manufacturing energy, service life, maintenance and end-of-life treatment.

Standards and regulations vary by sector. Aerospace, automotive, medical and pressure-containing applications require traceability, qualification testing and documented quality systems. Engineers have an ethical duty to communicate uncertainty, avoid unsupported allowables and design against foreseeable misuse and environmental degradation.

7. Applied Case: Illustrative Lightweight Composite Leaf Spring

Background

Consider an illustrative engineering case in which a manufacturer seeks to replace a steel vehicle leaf spring with a glass-fiber/epoxy composite spring. The objective is to reduce mass while retaining vertical stiffness, fatigue resistance and safe failure behavior.

Problem

A unidirectional composite candidate has fiber modulus E_f = 70 GPa, matrix modulus E_m = 3 GPa, fiber volume fraction V_f = 0.55, and density approximately ρ_c = 2.0 Mg/m³. The spring experiences bending, so the material is not loaded purely longitudinally. The design team must avoid treating the longitudinal rule of mixtures as a complete design solution.

Analysis

A first stiffness estimate is

This suggests a large stiffness-to-mass advantage over many steels, but bending introduces tensile and compressive stresses, transverse stresses near attachments, and interlaminar stresses. A constant-thickness unidirectional plate would be vulnerable to delamination and local crushing at bolt holes.

The team therefore evaluates a tapered laminate with 0° plies carrying the principal bending load, ±45° plies carrying shear and stabilizing the structure, and limited 90° plies controlling transverse splitting. The laminate is made symmetric to reduce bending–extension coupling. Rounded transitions and bonded or molded-in metallic inserts replace sharp bolt-hole stress concentrations.

Approach and solution

Finite-element analysis is used to determine ply stresses, but the model is calibrated against coupon and subcomponent testing. Tests include tensile and compression strength, interlaminar shear, fatigue, moisture conditioning, temperature exposure and impact followed by residual-strength testing. Manufacturing controls specify fiber volume fraction, maximum void content, cure temperature and nondestructive inspection.

Outcome

The illustrative design achieves the mass target only after the attachment regions are locally reinforced. The initial uniform laminate fails by delamination near the mounting hole in a simulated overload. The revised tapered design, with improved load introduction and a tougher resin system, survives the qualification sequence with acceptable residual stiffness.

Learning points

1. A high longitudinal modulus does not guarantee safe structural performance.
2. Layup, geometry and load introduction are as important as constituent properties.
3. Compression, fatigue, impact and environmental tests are essential for polymer composites.
4. Defect tolerance and inspection must be designed into the component.
5. Material selection and manufacturing process cannot be separated.

8. Contemporary Developments

Current research and industrial development include recyclable thermoplastic composites, bio-based polymers, natural-fiber reinforcement, automated fiber placement, additive manufacturing of reinforced polymers, nanocomposites and embedded sensing. Carbon nanotubes, graphene and nanoclays can modify electrical, barrier, thermal or mechanical properties, but dispersion, cost, health assessment and scale-up remain challenges.

Artificial intelligence can assist formulation optimization, process monitoring and defect detection from image or sensor data. However, machine-learning predictions require representative training data and physical validation. AI should support—not replace—mechanics-based modeling, testing, engineering judgment and safety certification.

Chapter Summary

Polymer behavior is controlled by molecular architecture, intermolecular bonding, molecular weight, crystallinity, cross-linking and thermal history. The glass transition marks increased segmental mobility in an amorphous phase, while melting describes loss of crystalline order. Thermoplastics can be repeatedly softened and reshaped, thermosets form permanent networks during curing, and elastomers are lightly cross-linked materials capable of large recoverable strains.

Composite performance depends on the matrix, reinforcement, interface, geometry, orientation, defects and loading environment. The longitudinal rule of mixtures provides a useful first estimate for aligned continuous fibers, while transverse, short-fiber and strength predictions require greater caution. Unidirectional composites are anisotropic, and laminates are designed by combining plies at selected orientations. Failure may involve fibers, matrix, interface, delamination or structural instability.

Reliable composite design requires constituent selection, micromechanics, laminate analysis, manufacturing control, environmental evaluation, damage tolerance and testing. Sustainability, inspection, regulatory qualification and ethical communication of uncertainty are part of professional engineering practice.

Key Terms

Polymer: A material composed of long chains of repeating molecular units.

Molecular weight: The mass of a polymer molecule or molecular population measure.

Dispersity: The ratio M_w/M_n, describing molecular-weight distribution.

Crystallinity: The fraction or degree of ordered molecular structure in a polymer.

Glass-transition temperature: The temperature range associated with increased segmental mobility in an amorphous polymer phase.

Melting temperature: The temperature at which crystalline polymer regions lose their ordered structure.

Viscoelasticity: Time- and rate-dependent behavior combining elastic and viscous responses.

Creep: Time-dependent increase in strain under sustained stress.

Stress relaxation: Time-dependent decrease in stress under sustained strain.

Thermoplastic: A polymer that softens on heating and can generally be reshaped repeatedly.

Thermoset: A permanently cross-linked polymer network formed by curing.

Elastomer: A lightly cross-linked polymer capable of large recoverable deformation.

Matrix: The continuous phase that surrounds and transfers load to reinforcement.

Reinforcement: The phase added to improve stiffness, strength or other properties.

Lamina: A single composite ply with defined material directions.

Laminate: A bonded stack of composite laminae.

Anisotropy: Directional dependence of material properties.

Delamination: Separation between adjacent composite layers.

Critical fiber length: The approximate length required for a short fiber to develop efficient load transfer.

Specific modulus: Elastic modulus divided by density.

Review Questions

Short Answer Questions

1. Why does increasing polymer molecular weight often improve toughness but complicate processing?
2. Distinguish glass-transition temperature from melting temperature.
3. State two structural differences between thermoplastics and thermosets.
4. What functions are performed by the matrix in a fiber-reinforced composite?
5. Why are unidirectional composites anisotropic?

Long Answer Questions

1. Explain how chain branching, tacticity, crystallinity and cross-linking influence polymer properties.
2. Compare thermoplastics, thermosets and elastomers in terms of molecular structure, processing, advantages and limitations.
3. Derive and discuss the assumptions behind the longitudinal rule of mixtures.
4. Explain the purpose of 0°, 90° and ±45° plies in a structural laminate.
5. Discuss composite failure mechanisms and explain why testing and manufacturing quality are essential.

Analytical / Application Questions

1. A composite contains 50% aligned fibers by volume, with E_f = 120 GPa and E_m = 4 GPa. Estimate the longitudinal modulus using the rule of mixtures and discuss two reasons the measured value may be lower.
2. A short-fiber composite uses fibers of diameter 20 μm, fiber strength 1.0 GPa and interfacial shear stress 25 MPa. Estimate the critical fiber length using l_c = σ_f d/(2τ_i), ensuring consistent units, and interpret the result.
3. Design a preliminary laminate for a panel carrying principal tension, in-plane shear and moderate transverse loading. Recommend a combination of ply orientations and justify the arrangement in terms of stiffness, coupling and failure resistance.

Multiple Choice Questions with Answers

Question 1

Which change generally increases the glass-transition temperature of an amorphous polymer?

A. Adding a plasticizer

B. Increasing backbone rigidity

C. Increasing free volume

D. Reducing intermolecular attraction

Correct Answer: B. Increasing backbone rigidity

Question 2

Which polymer class forms a permanent three-dimensional network during curing?

A. Thermoplastic

B. Elastomer only

C. Thermoset

D. Amorphous polymer only

Correct Answer: C. Thermoset

Question 3

For aligned continuous fibers loaded longitudinally, the basic rule of mixtures assumes:

A. Uniform stress in both phases

B. Uniform strain in both phases

C. No load transfer at the interface

D. Transverse isotropy is irrelevant to all loading

Correct Answer: B. Uniform strain in both phases

Question 4

A 60% fiber-volume composite has fiber modulus 100 GPa and matrix modulus 5 GPa. Its estimated longitudinal modulus is:

A. 42 GPa

B. 60 GPa

C. 62 GPa

D. 105 GPa

Correct Answer: C. 62 GPa

Question 5

What is the principal function of the interface in a fiber composite?

A. Preventing all thermal expansion

B. Transferring load between matrix and reinforcement

C. Melting the reinforcement during processing

D. Making the composite isotropic

Correct Answer: B. Transferring load between matrix and reinforcement

Question 6

Which failure mode involves separation between adjacent plies?

A. Fiber fracture

B. Matrix yielding

C. Delamination

D. Fiber crystallization

Correct Answer: C. Delamination

Question 7

For a polymer composite, increasing temperature well above its glass-transition temperature generally causes:

A. Lower molecular mobility

B. Increased stiffness in all cases

C. Greater viscoelastic deformation

D. Elimination of creep

Correct Answer: C. Greater viscoelastic deformation

Question 8

The critical fiber length increases when fiber diameter increases while all other variables remain constant because:

A. More length is required to transfer load through the interface

B. The matrix modulus becomes zero

C. Fiber density decreases automatically

D. Crystallinity becomes complete

Correct Answer: A. More length is required to transfer load through the interface

Question 9

A symmetric laminate is primarily used to reduce:

A. Molecular weight

B. Bending–extension coupling

C. Fiber strength

D. Matrix viscosity during service

Correct Answer: B. Bending–extension coupling

Question 10

Which statement about transverse composite properties is most accurate?

A. They are always equal to the fiber modulus

B. They depend strongly on matrix and interface behavior

C. They can always be calculated by averaging constituent strengths

D. They are independent of fiber packing

Correct Answer: B. They depend strongly on matrix and interface behavior