Particle reinforcement In general, particle reinforcement is
strengthening the composites less than
fiber reinforcement. It is used to enhance the
stiffness of the composites while increasing the
strength and the
toughness. Because of their
mechanical properties, they are used in applications in which
wear resistance is required. For example, hardness of
cement can be increased by reinforcing gravel particles, drastically. Particle reinforcement a highly advantageous method of tuning mechanical properties of materials since it is very easy implement while being low cost. The
elastic modulus of particle-reinforced composites can be expressed as, :E_c = V_m E_m + K_c V_p E_p where E is the
elastic modulus, V is the
volume fraction. The subscripts c, p and m are indicating composite, particle and matrix, respectively. K_c is a constant can be found empirically. Similarly, 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 (not equal to K_c) that can be found empirically.
Short fiber reinforcement (shear lag theory) Short fibers are often cheaper or more convenient to manufacture than longer continuous fibers, but still provide better properties than particle reinforcement. A common example is carbon fiber reinforced
3D printing filaments, which use chopped short
carbon fibers mixed into a matrix, typically
PLA or
PETG. Shear lag theory uses the shear lag model to predict properties such as the Young's modulus for short fiber composites. The model assumes that load is transferred from the matrix to the fibers solely through the interfacial shear stresses \tau_i acting on the cylindrical interface. Shear lag theory says then that the rate of change of the axial stress in the fiber as you move along the fiber is proportional to the ratio of the interfacial shear stresses over the radius of the fibre r_0: : \frac{d\sigma_f}{dx} = -\frac{2\tau_i}{r_0} This leads to the average fiber stress over the full length of the fibre being given by: : \sigma_f = E_f\varepsilon_1\left(1-\frac{\tanh(ns)}{ns}\right) where • \varepsilon_1 is the macroscopic strain in the composite • s is the
fiber aspect ratio (length over diameter) • n = \left( \frac{2E_m}{E_f(1+\nu_m)\ln(1/f)} \right)^{1/2} is a dimensionless constant • \nu_m is the
Poisson's ratio of the matrix By assuming a uniform tensile strain, this results in: : E_1 = \frac{\sigma_1}{\varepsilon_1} = fE_f \left( 1 - \frac{\tanh(ns)}{ns}\right) + (1-f) E_m As
s becomes larger, this tends towards the rule of mixtures, which represents the Young's modulus parallel to continuous fibers.
Continuous fiber reinforcement In general, continuous
fiber reinforcement is implemented by incorporating a
fiber as the strong phase into a weak phase, matrix. The reason for the popularity of fiber usage is materials with extraordinary strength can be obtained in their fiber form. Non-metallic fibers are usually showing a very high strength to density ratio compared to metal fibers because of the
covalent nature of their
bonds. The most famous example of this is
carbon fibers that have many applications extending from
sports gear to
protective equipment to
space industries. The stress on 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 is the stress, V is the
volume fraction. The subscripts c, f and m are indicating composite, fiber and matrix, respectively. Although the
stress–strain behavior of fiber composites can only be determined by testing, there is an expected trend, three stages of the
stress–strain curve. The first stage is the region of the stress–strain curve where both fiber and the matrix are
elastically deformed. This linearly elastic region can be expressed in the following form. The
Tsai-Hill criterion provides a more complete description of fiber composite tensile strength as a function of orientation angle by coupling the contributing yield stresses: \sigma^{*}_\parallel, \sigma^{*}_\perp, and \tau_m. \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 assumed that they can be treated as thi lamina and so out–of–plane stresses and strains are negligible. That is \sigma_3 = \sigma_4 = \sigma_5 = 0 and \varepsilon_4 = \varepsilon_5 = 0. This allows the stiffness and compliance matrices to be reduced to 3x3 matrices as follows: 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} For fiber-reinforced composite, the fiber orientation in material affect anisotropic properties of the structure. From characterizing technique i.e. tensile testing, the material properties were measured based on sample (1-2) coordinate system. The tensors above express stress-strain relationship in (1-2) coordinate system. While the known material properties is in the principal coordinate system (x-y) of material. Transforming the tensor between two coordinate system help identify the material properties of the tested sample. The
transformation matrix with \theta degree rotation is The angle of fiber orientation is very important because of the anisotropy of fiber composites (please see the section "
Physical properties" for a more detailed explanation). The mechanical properties of the composites can be tested using standard
mechanical testing methods by positioning the samples at various angles (the standard angles are 0°, 45°, and 90°) with respect to the orientation of fibers within the composites. In general, 0° axial alignment makes composites resistant to longitudinal bending and axial tension/compression, 90° hoop alignment is used to obtain resistance to internal/external pressure, and ± 45° is the ideal choice to obtain resistance against pure torsion.
Mechanical properties of fiber composite materials Carbon fiber & fiberglass composites vs. aluminum alloy and steel Although strength and stiffness of
steel and
aluminum alloys are comparable to fiber composites,
specific strength and
stiffness of composites (i.e. in relation to their weight) are significantly higher.
Failure Shock, impact of varying speed, or repeated cyclic stresses can provoke the laminate to separate at the interface between two layers, a condition known as
delamination. Individual fibres can separate from the matrix, for example,
fibre pull-out. Composites can fail on the
macroscopic or
microscopic scale. Compression failures can happen at both the macro scale or at each individual reinforcing fibre in compression buckling. Tension failures can be net section failures of the part or degradation of the composite at a microscopic scale where one or more of the layers in the composite fail in tension of the matrix or failure of the bond between the matrix and fibres. Some composites are brittle and possess little reserve strength beyond the initial onset of failure while others may have large deformations and have reserve energy absorbing capacity past the onset of damage. The distinctions in fibres and matrices that are available and the
mixtures that can be made with blends leave a very broad range of properties that can be designed into a composite structure. The most famous failure of a brittle ceramic matrix composite occurred when the carbon-carbon composite tile on the leading edge of the wing of the
Space Shuttle Columbia fractured when impacted during take-off. It directed to the catastrophic break-up of the vehicle when it re-entered the Earth's atmosphere on 1 February 2003. Composites have relatively poor bearing strength compared to metals. Another failure mode is fiber tensile fracture, which becomes more likely when fibers are aligned with the loading direction, so is the possibility of fiber tensile fracture, assuming the tensile strength exceeds that of the matrix. When a fiber has some angle of misorientation θ, several fracture modes are possible. For small values of θ the stress required to initiate fracture is increased by a factor of (cos θ)−2 due to the increased cross-sectional area (
A cos θ) of the fibre and reduced force (
F/cos θ) experienced by the fiber, leading to a composite tensile strength of
σparallel /cos2 θ where
σparallel is the tensile strength of the composite with fibers aligned parallel with the applied force. Intermediate angles of misorientation θ lead to matrix shear failure. Again the cross sectional area is modified but since
shear stress is now the driving force for failure the area of the matrix parallel to the fibers is of interest, increasing by a factor of 1/sin θ. Similarly, the force parallel to this area again decreases (
F/cos θ) leading to a total tensile strength of
τmy /sin θ cos θ where
τmy is the matrix shear strength. Finally, for large values of θ (near π/2) transverse matrix failure is the most likely to occur, since the fibers no longer carry the majority of the load. Still, the tensile strength will be greater than for the purely perpendicular orientation, since the force perpendicular to the fibers will decrease by a factor of 1/sin θ and the area decreases by a factor of 1/sin θ producing a composite tensile strength of
σperp /sin2θ where
σperp is the tensile strength of the composite with fibers align perpendicular to the applied force.
Testing Composites are tested before and after construction to assist in predicting and preventing failures. Pre-construction testing may adopt finite element analysis (FEA) for ply-by-ply analysis of curved surfaces and predicting wrinkling, crimping and dimpling of composites. Materials may be tested during manufacturing and after construction by various non-destructive methods including ultrasonic, thermography, shearography and X-ray radiography, and laser bond inspection for NDT of relative bond strength integrity in a localized area. ==See also==