Anisotropy and Fibre Composites

Materials selection

• Conditions to be fulfilled when selecting materials:

  • Stiffness

  • Strength

  • Fracture toughness

  • Thermal conductivity

  • Corrosion resistance

  • Mass

  • Cost

Composite materials

• Material that is a combination of two or more materials (constituents), and have material properties derived form the individual constituents.

• The properties of the composite materials are influenced by the properties of each constituent material, as well as the distribution, orientation, and interaction of these components within the composite structure.

    Four commonly accepted types of composite materials:

• Fibre reinforced - fibres in a matrix

• Particle reinforced - particles in a matrix

• Laminated - layers of varios materials

• Combination of some or all of the first three types

    Homogeneos:         uniform properties throughout the body, i.e. independent of                                      position in the body.

    Inhomogeneous:    nonuniform properties over the body, i.e. depend on position in                                      the body

Isotropy vs. Anisotropy; Fundamental elastic constants

Composite materials can be isotropic or anisotropic

• most fibre-reinforced materials are anisotropic

Isotropic materials (macroscopic isotropy in a statistical sense):

• Four elastic constants, with only two being independent:

• Modulus of elasticity: E

• Shear modulus: G

• Bulk modulus: K

• Poisson’s ratio: \nu

• Relationships:

  • E = 2G(1 + ν)

  • G=\frac{E}{2\left(1+\nu\right)}

  • K=\frac{E}{3\left(1-2\nu\right)}

Anisotropic materials

• material properties depend on direction

• in its most general form, 21 elastic constants exist (e.g., wood, bone, and most fibre-reinforced materials)

Lamina and laminate concepts

• Lamina/ply: the basic reinforcing layer in a composite; fibres are typically unidirectional within a lamina

• Matrix is soft relative to fibres

• The main roles of the matrix:

  • Provides shape to the structure

  • Transfers loads between fibres

  • Maintains fibre spacing and protects fibres

• Fibre types: Glass, Carbon, Aramid (Kevlar)

• Matrices: Thermosets (Polyester, Epoxy, Vinyl Ester) and Thermoplastics (Polyethylene, Polystyrene)

• Laminate: a bonded stack of laminae/plies; each layer can have different orientation and can be made of different materials

Mechanical behavior and properties of composites

• Mechanical behavior often differs from conventional materials due to inhomogeneity and anisotropy

• Key concepts:

  • Density and tensile strength values commonly differ significantly between fibre and matrix

  • Typical fibre/matrix stiffness ratio ranges from about 20/1 to 100/1

  • Fibres are generally linear-elastic; the matrix often exhibits nonlinear behavior

• Material comparisons often focus on: Young’s modulus, density, and tensile strength (e.g., carbon vs glass fibres; steel vs carbon fibres in composites)

Fibre volume fraction and fibre packing in composites

• Fibre volume fraction vf (or f) depends on fibre arrangement in the ply

• For common packing geometries:

  • Hexagonal packing (close-packed hexagonal arrangement):

  • Maximum vf ≈ 0.907

  • vfhex = vf{hex} = rac{ ext{area of fibre} }{ ext{area per fiber in hex lattice} } = rac{
    }{ 2\sqrt{3} } = \frac{\pi}{2\sqrt{3}} \approx 0.9069

  • Square packing: vf_max ≈ 0.785

  • vfsquare = vf{sq} = rac{ ext{area of fibre} }{ ext{area per fiber in square lattice} } = rac{ \pi r^2 }{ 4 r^2 } = \frac{\pi}{4} \approx 0.7854

• The fibre volume fraction depends on fibre radius and arrangement; maximum packing fractions are given above

• References for fibre/matrix geometry: D. Hull and T. W. Clyne, An Introduction to Composite Materials, 2nd ed., Cambridge University Press, 1996

Design and performance considerations for composites

• Typical performance metrics include:

  • Specific stiffness: E/ρ

  • Specific strength: σ_t/ρ

  • Cost and life-cycle cost considerations

• Example context: aircraft structures (e.g., wing spars), wind turbine blades, and other aerospace/automotive applications

• Example: doubly tapered wing spar discussed in literature (R. M. Jones, Mechanics of Composite Materials, 2nd ed., 1999)

• Advantages of composites:

  • High mass-specific strength and stiffness

  • Tailored materials and structures

  • Ability to adapt to load paths via anisotropic properties

  • Design flexibility and complex shapes

  • Good damping and fatigue, corrosion resistance, low thermal expansion, non-magnetic, potential for low energy in production

• Disadvantages of composites:

  • Higher design and production complexity

  • Material properties are largely generated during production; QC/QA can be challenging

  • Limited high-temperature resistance for some fibres/matrices

  • Relatively higher material and automation costs; less experience with lifecycle behavior

Applications and real-world relevance

• Polymer composite applications – overview:

  • Aeronautics: sailplanes, light aircraft, propellers, military and civil aircraft, helicopter structures

  • Space: antennas, satellite structures, pressure vessels, measurement platforms

  • Energy: wind turbine rotor blades, housings, wind tunnel components

  • Shipbuilding: military and civilian vessels, yachts, offshore structures

  • Vehicle technology: cars, trucks, racing cars, trains

  • Civil and mechanical engineering, medicine technology, sporting equipment, electronics

• Composite usage in aircraft structures:

  • Historical examples: Akaflieg Stuttgart FS-24 Phoenix – first glider to use fibreglass; evolution from wood to fibre-reinforced sandwich designs

  • Helicopter blades among early adopters of composites

• Aircraft and wind turbine statistics:

  • By weight, composites account for a substantial and growing share of aircraft and blade components

  • Wind turbine blades: composites account for more than 90% of blade weight (typical WT blade composition)

• End-of-life and sustainability concerns:

  • Early-stage recycling routes for composite blades are not yet established globally

  • Projections indicate significant blade waste by 2050 under various scenarios; recycling and reuse strategies are actively explored

• End-of-life reuse in architecture and design:

  • Reuse of blade materials in urban furniture, benches, charging stations, etc., as part of circular economy initiatives

  • Examples and initiatives include projects by Superuse Studios and related literature

Design project and course logistics (summary)

• Design project:

  • Group work (2 students per group)

  • Two reports: an individualized group report and an individual report; combined for final grade

• Course logistics:

  • Lectures and exercises with weekly topics and reading (Chap. references as per plan)

  • Reading materials and updates provided through DTU Learn

  • Project work and mid-term evaluation scheduled during weeks 6–7

Key references and sources mentioned

• Foundations of Fibre Composites, 2nd edition (Zenkert & Battley) – core reference for fundamentals

• Engineering Materials 1: An Introduction to Properties, Applications, and Design (Ashby & Jones) – material selection and property indices

• An Introduction to Composite Materials (Hull & Clyne) – fibre-matrix concepts and properties

• Mechanics of Composite Materials (R. M. Jones) – structural application context

• Cambridge Materials Property Site (specific stiffness/strength vs density) for comparisons

• DTU Learn – course materials, updates, and design project details

Summary of key formulas and numbers to remember

• Isotropic elastic constants relationships:

  • G = \frac{E}{2(1+\nu)}

  • K = \frac{E}{3(1-2\nu)}

  • E = 2G(1+\nu)

• Fibre volume fraction (packing geometries):

  • Hexagonal packing: vf_hex = \frac{\pi}{2\sqrt{3}} \approx 0.9069

  • Square packing: vf_sq = \frac{\pi}{4} \approx 0.7854

• Material index for lightweight design:

  • M1 = \frac{\rho}{E}

• Specific performance concepts:

  • Specific stiffness: \dfrac{E}{\rho}

  • Specific strength: \dfrac{\sigma_t}{\rho}

Ethical, philosophical, and practical implications

• End-of-life considerations highlight the ethical and environmental responsibility of materials choice and design:

  • Need for recycling pathways and sustainable design for composites

  • Balance between performance, cost, and lifecycle impact

• Practical implications include: design for manufacturability, quality control challenges, and life-cycle analysis integration early in the design process

Connections to foundational principles

• Connects classical elasticity with anisotropic and composite-specific theories

• Bridges material science (constituent properties, interfaces) with structural analysis (laminate theory, A/B/D matrices)

• Illustrates how optimization (material index, stacking sequence) blends physics with engineering design

Notable terminology recap

• Lamina/ply: single reinforcing layer with a specific fibre orientation

• Laminate: stack of laminae with different orientations/materials

• A-matrix: extensional stiffness of the laminate

• B-matrix: bending–extensional coupling (mid-plane shift effects)

• D-matrix: bending stiffness

• FPF: Failure Prediction of Failure Criteria for laminates

• LPF: Laminate Progressive Failure

• Vf or f: fibre volume fraction

• ρ: density; E: Young’s modulus; G: shear modulus; K: bulk modulus; ν: Poisson’s ratio