Editing Composite material (section)

== Mechanical properties of composites ==

=== Particle reinforcement ===
In general, particle reinforcement is [[strengthening mechanisms of materials|strengthening]] the composites less than [[fiber]] reinforcement. It is used to enhance the [[stiffness]] of the composites while increasing the [[yield (engineering)|strength]] and the [[toughness]]. Because of their [[mechanical properties]], they are used in applications in which [[wear]] resistance is required. For example, hardness of [[engineered cementitious composite|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.<ref>{{cite journal |last1=Wu |first1=Xiangguo |last2=Yang |first2=Jing |last3=Mpalla |first3=Issa B. |title=Preliminary design and structural responses of typical hybrid wind tower made of ultra high performance cementitious composites |journal=Structural Engineering and Mechanics |date=25 December 2013 |volume=48 |issue=6 |pages=791–807 |doi=10.12989/sem.2013.48.6.791 }}</ref><ref>{{cite journal |last1=Li |first1=Mo |last2=Li |first2=Victor C. |title=Rheology, fiber dispersion, and robust properties of Engineered Cementitious Composites |journal=Materials and Structures |date=March 2013 |volume=46 |issue=3 |pages=405–420 |doi=10.1617/s11527-012-9909-z |hdl=2027.42/94214 |hdl-access=free }}</ref><ref>{{cite journal |date=2008 |title=Large-Scale Processing of Engineered Cementitious Composites |journal=ACI Materials Journal |volume=105 |issue=4 |doi=10.14359/19897 }}</ref><ref>{{cite journal |last1=Zeidi |first1=Mahdi |last2=Kim |first2=Chun IL |last3=Park |first3=Chul B. |date=2021 |title=The role of interface on the toughening and failure mechanisms of thermoplastic nanocomposites reinforced with nanofibrillated rubbers |journal=Nanoscale |volume=13 |issue=47 |pages=20248–20280 |doi=10.1039/D1NR07363J |pmid=34851346 }}</ref>

The [[elastic modulus]] of particle-reinforced composites can be expressed as,
:<math>E_c = V_m E_m + K_c V_p E_p</math>

where E is the [[elastic modulus]], V is the [[volume fraction]]. The subscripts c, p and m are indicating composite, particle and matrix, respectively. <math>K_c</math> is a constant can be found empirically.

Similarly, tensile strength of particle-reinforced composites can be expressed as,
:<math>(T.S.)_c = V_m (T.S.)_m + K_s V_p (T.S.)_p</math>

where T.S. is the [[Ultimate tensile strength|tensile strength]], and <math>K_s</math> is a constant (not equal to <math>K_c</math>) that can be found empirically.

=== Short fiber reinforcement (shear lag theory) ===
{{ see also | Short_fiber_thermoplastics#Mechanical_properties}}

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 [[Polylactic acid|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 <math>\tau_i</math> 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 <math>r_0</math>:

:<math> \frac{d\sigma_f}{dx} = -\frac{2\tau_i}{r_0} </math>

This leads to the average fiber stress over the full length of the fibre being given by:

:<math> \sigma_f = E_f\varepsilon_1\left(1-\frac{\tanh(ns)}{ns}\right) </math>

where
* <math>\varepsilon_1</math> is the macroscopic strain in the composite
* <math>s</math> is the ''fiber aspect ratio'' (length over diameter)
* <math> n = \left( \frac{2E_m}{E_f(1+\nu_m)\ln(1/f)} \right)^{1/2}</math> is a dimensionless constant<ref>{{cite journal | title=On the Use of Shear-Lag Methods for Analysis of Stress Transfer in Unidirectional Composites | author=John A. Nairn | journal=Mechanics of Materials | year = 1997 | doi=10.1016/S0167-6636(97)00023-9}}</ref>
* <math> \nu_m </math> is the [[Poisson's ratio]] of the matrix

By assuming a uniform tensile strain, this results in:<ref>{{cite journal | author=P.J. WITHERS | title=4.02 - Elastic and Thermoelastic Properties of Brittle Matrix Composites | journal=Comprehensive Composite Materials | year=2000 | doi=10.1016/B0-08-042993-9/00087-5}}</ref>

:<math> E_1 = \frac{\sigma_1}{\varepsilon_1} = fE_f \left( 1 - \frac{\tanh(ns)}{ns}\right) + (1-f) E_m </math>

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 bond|covalent]] nature of their [[chemical bond|bonds]]. The most famous example of this is [[carbon fibers]] that have many applications extending from [[sports gear]] to [[Protective gear in sports|protective equipment]] to [[SpaceX|space industries]].<ref name=":1">{{cite book |last1=Courtney |first1=Thomas H. |title=Mechanical Behavior of Materials |date=2005 |publisher=Waveland Press |isbn=978-1-4786-0838-7 }}{{pn|date=January 2025}}</ref><ref>{{cite book |doi=10.1007/978-981-13-0538-2 |title=Carbon Fibers |series=Springer Series in Materials Science |date=2018 |volume=210 |isbn=978-981-13-0537-5 |first1=Soo-Jin |last1=Park }}{{pn|date=January 2025}}</ref>

The stress on the composite can be expressed in terms of the [[volume fraction]] of the fiber and the matrix.
:<math>\sigma_c = V_f \sigma_f + V_m \sigma_m</math>

where <math>\sigma</math> 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 analysis|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 [[elastic deformation|elastically deformed]]. This linearly elastic region can be expressed in the following form.<ref name=":1"/>
:<math>\sigma_c - E_c \epsilon_c = \epsilon_c (V_f E_f + V_m E_m)</math>

where <math>\sigma</math> is the stress, <math>\epsilon</math> is the strain, E is the [[elastic modulus]], and V is the [[volume fraction]]. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively.

After passing the elastic region for both fiber and the matrix, the second region of the stress–strain curve can be observed. In the second region, the fiber is still elastically deformed while the matrix is plastically deformed since the matrix is the weak phase. The instantaneous [[elastic modulus|modulus]] can be determined using the slope of the stress–strain curve in the second region. The relationship between [[stress (mechanics)|stress]] and strain can be expressed as,
:<math>\sigma_c = V_f E_f \epsilon_c + V_m \sigma_m (\epsilon_c)</math>

where <math>\sigma</math> is the stress, <math>\epsilon</math> is the strain, E is the [[elastic modulus]], and V is the [[volume fraction]]. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively. To find the modulus in the second region derivative of this equation can be used since the [[slope|slope of the curve]] is equal to the modulus.
:<math>E_c' = \frac{d \sigma_c}{d \epsilon_c} = V_f E_f + V_m \left(\frac{d \sigma_c}{d \epsilon_c}\right)</math>

In most cases it can be assumed <math>E_c'= V_f E_f</math> since the second term is much less than the first one.<ref name=":1"/>

In reality, the [[derivative]] of stress with respect to strain is not always returning the modulus because of the [[chemical bond|binding interaction]] between the fiber and matrix. The strength of the interaction between these two phases can result in changes in the [[list of materials properties|mechanical properties]] of the composite. The compatibility of the fiber and matrix is a measure of [[stress (mechanics)|internal stress]].<ref name=":1"/>

The [[covalent bond|covalently bonded]] high strength fibers (e.g. [[carbon fibers]]) experience mostly [[deformation (engineering)|elastic deformation]] before the fracture since the [[deformation (engineering)|plastic deformation]] can happen due to [[dislocation|dislocation motion]]. Whereas, [[metallic fiber]]s have more space to plastically deform, so their composites exhibit a third stage where both fiber and the matrix are plastically deforming. [[Metallic fiber]]s have [[Cryogenic hardening|many applications]] to work at [[cryogenics|cryogenic temperatures]] that is one of the advantages of composites with [[steel fibre-reinforced shotcrete|metal fibers]] over nonmetallic. The stress in this region of the [[stress–strain curve]] can be expressed as,
:<math>\sigma_c (\epsilon_c) = V_f \sigma_f \epsilon_c + V_m \sigma_m (\epsilon_c)</math>

where <math>\sigma</math> is the stress, <math>\epsilon</math> is the strain, E is the [[elastic modulus]], and V is the [[volume fraction]]. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively. <math>\sigma_f (\epsilon_c)</math> and <math>\sigma_m (\epsilon_c)</math> are for fiber and matrix flow stresses respectively. Just after the third region the composite exhibit [[necking (engineering)|necking]]. The necking strain of composite is happened to be between the necking strain of the fiber and the matrix just like other mechanical properties of the composites. The necking strain of the weak phase is delayed by the strong phase. The amount of the delay depends upon the volume fraction of the strong phase.<ref name=":1"/>

Thus, the [[ultimate tensile strength|tensile strength]] of the composite can be expressed in terms of the [[volume fraction]].<ref name=":1"/>
:<math>(T.S.)_c=V_f(T.S.)_f+V_m \sigma_m(\epsilon_m)</math>

where T.S. is the [[ultimate tensile strength|tensile strength]], <math>\sigma</math> is the stress, <math>\epsilon</math> is the strain, E is the [[elastic modulus]], and V is the [[volume fraction]]. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively. The composite tensile strength can be expressed as
:<math>(T.S.)_c=V_m(T.S.)_m</math> for <math>V_f</math> is less than or equal to <math>V_c</math> (arbitrary critical value of volume fraction)
:<math>(T.S.)_c= V_f(T.S.)_f + V_m(\sigma_m)</math> for <math>V_f</math> is greater than or equal to <math>V_c</math>

The critical value of [[volume fraction]] can be expressed as,
:<math>V_c= \frac{[(T.S.)_m - \sigma_m(\epsilon_f)]}{[(T.S.)_f + (T.S.)_m - \sigma_m(\epsilon_f)]}</math>

Evidently, the composite [[ultimate tensile strength|tensile strength]] can be higher than the matrix if <math>(T.S.)_c</math> is greater than <math>(T.S.)_m
</math>.

Thus, the minimum volume fraction of the fiber can be expressed as,
:<math>V_c= \frac{[(T.S.)_m - \sigma_m(\epsilon_f)]}{[(T.S.)_f - \sigma_m(\epsilon_f)]}</math>

Although this minimum value is very low in practice, it is very important to know since the reason for the incorporation of continuous fibers is to improve the mechanical properties of the materials/composites, and this value of volume fraction is the threshold of this improvement.<ref name=":1"/>

===The effect of fiber orientation===

====Aligned fibers====
A change in the angle between the applied stress and fiber orientation will affect the mechanical properties of fiber-reinforced composites, especially the tensile strength. This angle, <math>\theta</math>, can be used predict the dominant tensile fracture mechanism.

At small angles, <math>\theta \approx 0^{\circ}</math>, the dominant fracture mechanism is the same as with load-fiber alignment, tensile fracture. The resolved force acting upon the length of the fibers is reduced by a factor of <math>\cos \theta</math> from rotation. <math>F_{\mbox{res}}=F\cos\theta</math>. The resolved area on which the fiber experiences the force is increased by a factor of <math>\cos \theta</math> from rotation. <math>A_{\mbox{res}}=A_{0}/\cos\theta</math>. Taking the effective [[ultimate tensile strength|tensile strength]] to be <math>(\mbox{T.S.})_{\mbox{c}}=F_{\mbox{res}}/A_{\mbox{res}}</math> and the aligned [[Ultimate tensile strength|tensile strength]] <math>\sigma^*_\parallel=F/A</math>.<ref name=":1"/>
:<math>(\mbox{T.S.})_{\mbox{c}}\;(\mbox{longitudinal fracture})=\frac{\sigma^*_\parallel}{\cos^2\theta}</math>

At moderate angles, <math>\theta \approx 45^{\circ}</math>, the material experiences shear failure. The effective force direction is reduced with respect to the aligned direction. <math>F_{\mbox{res}}=F\cos\theta</math>. The resolved area on which the force acts is <math>A_{\mbox{res}}=A_m/\sin\theta</math>. The resulting [[ultimate tensile strength|tensile strength]] depends on the [[shear strength]] of the matrix, <math>\tau_m</math>.<ref name=":1"/>
:<math>(\mbox{T.S.})_{\mbox{c}}\;(\mbox{shear failure})=\frac{\tau_m}{\sin{\theta}\cos{\theta}}</math>

At extreme angles, <math>\theta \approx 90^{\circ}</math>, the dominant mode of failure is tensile fracture in the matrix in the perpendicular direction. As in the [[#Isostress rule of mixtures|isostress case]] of layered composite materials, the strength in this direction is lower than in the aligned direction. The effective areas and forces act perpendicular to the aligned direction so they both scale by <math>\sin\theta</math>. The resolved tensile strength is proportional to the transverse strength, <math>\sigma^{*}_{\perp}</math>.<ref name=":1"/>
:<math>(\mbox{T.S.})_{\mbox{c}}\;(\mbox{transverse fracture})=\frac{\sigma^*_{\perp}}{\sin^2\theta}</math>

The critical angles from which the dominant fracture mechanism changes can be calculated as,
:<math>\theta_{c_1}=\tan^{-1}\left({\frac{\tau_m}{\sigma^*_\parallel}}\right)</math>
:<math>\theta_{c_2}=\tan^{-1}\left({\frac{\sigma^*_\perp}{\tau_m}}\right)</math>

where <math>\theta_{c_1}</math> is the critical angle between longitudinal fracture and shear failure, and <math>\theta_{c_2}</math> is the critical angle between shear failure and transverse fracture.<ref name=":1"/>

By ignoring length effects, this model is most accurate for continuous fibers and does not effectively capture the strength-orientation relationship for short fiber reinforced composites. Furthermore, most realistic systems do not experience the [[maxima and minima|local maxima]] predicted at the critical angles.<ref>{{cite journal |last1=Lasikun |last2=Ariawan |first2=Dody |last3=Surojo |first3=Eko |last4=Triyono |first4=Joko |date=2018 |title=Effect of fiber orientation on tensile and impact properties of Zalacca Midrib fiber-HDPE composites by compression molding |journal=The 3rd International Conference on Industrial |series=AIP Conference Proceedings |volume=1927 |issue=1 |location=Jatinangor, Indonesia |pages=030060 |doi=10.1063/1.5024119 |bibcode=2018AIPC.1931c0060L |doi-access=free}}</ref><ref>{{cite journal |last1=Mortazavian |first1=Seyyedvahid |last2=Fatemi |first2=Ali |title=Effects of fiber orientation and anisotropy on tensile strength and elastic modulus of short fiber reinforced polymer composites |journal=Composites Part B: Engineering |date=April 2015 |volume=72 |pages=116–129 |doi=10.1016/j.compositesb.2014.11.041 }}</ref><ref>{{cite journal |id={{ProQuest|1030964421}} |last1=Banakar |first1=Prashanth |last2=Shivananda |first2=H K |last3=Niranjan |first3=H B |title=Influence of Fiber Orientation and Thickness on Tensile Properties of Laminated Polymer Composites |journal=International Journal of Pure and Applied Sciences and Technology |volume=9 |issue=1 |date=March 2012 |pages=61–68 }}</ref><ref>{{cite journal |last1=Brahim |first1=Sami Ben |last2=Cheikh |first2=Ridha Ben |title=Influence of fibre orientation and volume fraction on the tensile properties of unidirectional Alfa-polyester composite |journal=Composites Science and Technology |date=January 2007 |volume=67 |issue=1 |pages=140–147 |doi=10.1016/j.compscitech.2005.10.006 }}</ref> The [[Tsai-Hill failure criterion|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: <math>\sigma^{*}_\parallel</math>, <math>\sigma^{*}_\perp</math>, and <math>\tau_m</math>.<ref>{{cite journal |last1=Azzi |first1=V. D. |last2=Tsai |first2=S.W. |date=1965 |title=Anisotropic Strength of Composites |journal=Experimental Mechanics |volume=5 |issue=9 |pages=283–288 |doi=10.1007/BF02326292 }}</ref><ref name=":1"/>
:<math>(\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}</math>

====Randomly oriented fibers====
Anisotropy in the tensile strength of fiber reinforced composites can be removed by randomly orienting the fiber directions within the material. It sacrifices the ultimate strength in the aligned direction for an overall, isotropically strengthened material.
:<math>E_c=KV_{f}E_{f}+V_{m}E_{m}</math>

Where K is an empirically determined reinforcement factor; similar to the [[#Particle Reinforcement|particle reinforcement]] equation. For fibers with randomly distributed orientations in a plane, <math>K \approx 0.38</math>, and for a random distribution in 3D, <math>K \approx 0.20</math>.<ref name=":1"/>

=== Stiffness and Compliance Elasticity ===
Composite materials are generally [[anisotropic]], and in many cases are [[Orthotropic material|orthotropic]]. [[Voigt notation]] can be used to reduce the rank of the stress and strain tensors such that the [[stiffness]] <math>C</math> (often also referred to by <math>Q</math>) and compliance <math>S</math> can be written as a [[Matrix (mathematics)|matrix]]:<ref>{{cite book |last1=Lekhnit͡skiĭ |first1=Sergeĭ Georgievich |title=Theory of Elasticity of an Anisotropic Elastic Body |date=1963 |publisher=Holden-Day |oclc=652279972 }}{{pn|date=January 2025}}</ref>

<math>\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}</math> and <math>\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}</math>

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 <math>\sigma_3 = \sigma_4 = \sigma_5 = 0</math> and <math>\varepsilon_4 = \varepsilon_5 = 0</math>.<ref name=":0">{{cite book |doi=10.1007/978-94-011-4489-6 |title=Mechanics of Composite Materials and Structures |date=1999 |isbn=978-0-7923-5871-8 |editor-last1=Soares |editor-last2=Soares |editor-last3=Freitas |editor-first1=Carlos A. Mota |editor-first2=Cristóvão M. Mota |editor-first3=Manuel J. M. }}</ref> This allows the stiffness and compliance matrices to be reduced to 3x3 matrices as follows:

<math>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 </math> and <math> \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} </math>
[[File:Transform coordinate system.png|thumb|331x331px|Two different coordinate systems of material. The structure has a (1-2) coordinate system. The material has a (x-y) principal coordinate system.]]
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 <math>\theta      </math> degree rotation is <ref name=":0" />

<math>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}      </math> for <math>\begin{bmatrix} \acute{\epsilon} \end{bmatrix} = T(\theta)_\epsilon \begin{bmatrix} \epsilon \end{bmatrix}

      </math><math>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}      </math> for <math>\begin{bmatrix} \acute{\sigma} \end{bmatrix} = T(\theta)_\sigma \begin{bmatrix} \sigma \end{bmatrix}      </math>

===Types of fibers and mechanical properties===
The most common types of fibers used in industry are [[glass fiber]]s, [[carbon fibers]], and [[kevlar]] due to their ease of production and availability. Their mechanical properties are very important to know, therefore the table of their mechanical properties is given below to compare them with S97 [[steel]].<ref>{{cite web |title=Carbon Fibre, Tubes, Profiles – Filament Winding and Composite Engineering |url=http://www.performance-composites.com/carbonfibre/carbonfibre.asp |website=www.performance-composites.com |access-date=2020-05-22 |archive-date=2020-05-05 |archive-url=https://web.archive.org/web/20200505133007/http://www.performance-composites.com/carbonfibre/carbonfibre.asp |url-status=live}}</ref><ref>{{cite web |title=Composite Manufacturing {{!}} Performance Composites|url=https://www.performancecomposites.com/|website=www.performancecomposites.com|access-date=2020-05-22|archive-date=2020-05-03|archive-url=https://web.archive.org/web/20200503120735/http://www.performancecomposites.com/|url-status=live}}</ref><ref>{{cite web |title=Composite Materials • Innovative Composite Engineering |url=http://www.innovativecomposite.com/materials/ |website=Innovative Composite Engineering |language=en-US |access-date=2020-05-22 |archive-date=2020-05-05 |archive-url=https://web.archive.org/web/20200505134923/http://www.innovativecomposite.com/materials/ |url-status=live}}</ref><ref>{{cite web |title=Reinforcement Fabrics – In Stock for Same Day Shipping {{!}} Fibre Glast|url=https://www.fibreglast.com/category/Composite-Fabrics|website=www.fibreglast.com|access-date=2020-05-22|archive-date=2020-07-16|archive-url=https://web.archive.org/web/20200716204826/https://www.fibreglast.com/category/Composite-Fabrics|url-status=live}}</ref> The angle of fiber orientation is very important because of the anisotropy of fiber composites (please see the section "[[#Physical properties|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.<ref>{{cite web |title=Filament Winding, Carbon Fibre Angles in Composite Tubes |url=http://www.performance-composites.com/carbonfibre/fibreangles.asp |website=www.performance-composites.com |access-date=2020-05-22 |archive-date=2020-05-05 |archive-url=https://web.archive.org/web/20200505132959/http://www.performance-composites.com/carbonfibre/fibreangles.asp |url-status=live}}</ref>

====Mechanical properties of fiber composite materials====
{| class="wikitable"
|+Fibres @ 0° (UD), 0/90° (fabric) to loading axis, Dry, Room Temperature, V<sub>f</sub> = 60% (UD), 50% (fabric) Fibre / Epoxy Resin (cured at 120&nbsp;°C)<ref name=":2">{{cite web |title=Mechanical Properties of Carbon Fibre Composite Materials |url=http://www.performance-composites.com/carbonfibre/mechanicalproperties_2.asp |website=www.performance-composites.com |access-date=2020-05-22 |archive-date=2020-06-03 |archive-url=https://web.archive.org/web/20200603174526/http://www.performance-composites.com/carbonfibre/mechanicalproperties_2.asp |url-status=live}}</ref>
|
!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
| –
|}
<br/>
{| class="wikitable"
|+Fibres @ ±45 Deg. to loading axis, Dry, Room Temperature, Vf = 60% (UD), 50% (fabric)<ref name=":2"/>
!
!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
|207
|72
|-
!Transverse Modulus
|E2
|GPa
|17
|17
|12.3
|19.1
|12.2
|207
|72
|-
!In Plane Shear Modulus
|G12
|GPa
|33
|47
|11
|30
|8
|80
|25
|-
!Poisson's Ratio
|v12
|
|.77
|.83
|.53
|.74
|.53
|
|
|-
!Tensile Strength
|Xt
|MPa
|110
|110
|90
|120
|120
|990
|460
|-
!Compressive Strength
|Xc
|MPa
|110
|110
|90
|120
|120
|990
|460
|-
!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
|11 E-6
|23 E-6
|-
!Moisture Co-ef
|Beta1
|Strain/K
|3.22 E-4
|2.49 E-4
|6.9 E-4
|
|
|
|
|}

==== Carbon fiber & fiberglass composites vs. aluminum alloy and steel ====

Although strength and stiffness of [[steel]] and [[Aluminium alloy|aluminum alloy]]s are comparable to fiber composites, [[specific strength]] and [[Specific modulus|stiffness]] of composites (i.e. in relation to their weight)  are significantly higher.

{| class="wikitable"
|+Comparison of Cost, Specific Strength, and Specific Stiffness<ref>{{cite web |title=Carbon Fiber Composite Design Guide |url=https://www.performancecomposites.com/about-composites-technical-info/124-designing-with-carbon-fiber.pdf |website=www.performancecomposites.com |access-date=2020-05-22 |archive-date=2020-10-30 |archive-url=https://web.archive.org/web/20201030130724/https://www.performancecomposites.com/about-composites-technical-info/124-designing-with-carbon-fiber.pdf |url-status=live}}</ref>
|
|'''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<sup>6</sup>– 50 x 10<sup>6</sup>
|8 x 10<sup>6</sup> – 10 x 10<sup>6</sup>
|1 x 10<sup>6</sup> – 1.5 x 10<sup>6</sup>
|10 x 10<sup>6</sup>
|30 x 10<sup>6</sup>
|-
|'''Density (lb/in3)'''
|0.050
|0.050
|0.055
|0.10
|0.30
|-
|'''<u>Specific Strength</u>'''
|<u>1.8 x 10<sup>6</sup> – 4 x 10<sup>6</sup></u>
|<u>1 x 10<sup>6</sup> – 1.8 x 10<sup>6</sup></u>
|<u>363,640–636,360</u>
|<u>350,000</u>
|<u>200,000</u>
|-
|'''<u>Specific Stiffness</u>'''
|<u>200 x 10<sup>6</sup> – 1,000 x 10<sup>6</sup></u>
|<u>160 x 10<sup>6</sup> – 200 x 10<sup>6</sup></u>
|<u>18 x 10<sup>6</sup> – 27 x 10<sup>6</sup></u>
|<u>100 x 10<sup>6</sup></u>
|<u>100 x 10<sup>6</sup></u>
|}

===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]].<ref>{{cite journal |last1=Ma |first1=Binlin |last2=Cao |first2=Xiaofei |last3=Feng |first3=Yu |last4=Song |first4=Yujian |last5=Yang |first5=Fei |last6=Li |first6=Ying |last7=Zhang |first7=Deyue |last8=Wang |first8=Yipeng |last9=He |first9=Yuting |title=A comparative study on the low velocity impact behavior of UD, woven, and hybrid UD/woven FRP composite laminates |journal=Composites Part B: Engineering |date=February 2024 |volume=271 |pages=111133 |doi=10.1016/j.compositesb.2023.111133 }}</ref><ref>{{cite journal |last1=Sanchez-Saez |first1=S. |last2=Barbero |first2=E. |last3=Zaera |first3=R. |last4=Navarro |first4=C. |title=Compression after impact of thin composite laminates |journal=Composites Science and Technology |date=October 2005 |volume=65 |issue=13 |pages=1911–1919 |doi=10.1016/j.compscitech.2005.04.009 |hdl=10016/7498 |hdl-access=free }}</ref> Individual fibres can separate from the matrix, for example, [[fiber pull-out|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 [[mixture]]s 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.

[[File:Composite Strength as a Function of Fiber Misalignment.png|thumb|The graph depicts the three fracture modes a composite material may experience depending on the angle of misorientation relative to aligning fibres parallel to the applied stress.]]

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)<sup>−2</sup> 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 ''σ<sub>parallel </sub>/''cos<sup>2</sup> θ where ''σ<sub>parallel </sub>'' 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 ''τ<sub>my</sub> /''sin&nbsp;θ cos&nbsp;θ where ''τ<sub>my</sub>'' 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 ''σ<sub>perp</sub> /''sin<sup>2</sup>θ where ''σ<sub>perp </sub>'' is the tensile strength of the composite with fibers align perpendicular to the applied force.<ref>
{{cite book |last=Courtney |first=Thomas H. |date=2000 |title=Mechanical Behavior of Materials |edition= 2nd |publisher=Waveland Press, Inc. |location=Long Grove, IL |pages=263–265 |isbn=978-1-57766-425-3}}
</ref>

===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.<ref name="Waterman">{{cite news |last1=Waterman |first1=Pamela |title=The Life of Composite Materials |url=https://www.digitalengineering247.com/article/the-life-of-composite-materials/ |work=Digital Engineering |date=1 May 2007 }}</ref><ref>{{cite journal |last1=Aghdam |first1=M.M. |last2=Morsali |first2=S.R. |title=Damage initiation and collapse behavior of unidirectional metal matrix composites at elevated temperatures |journal=Computational Materials Science |date=November 2013 |volume=79 |pages=402–407 |doi=10.1016/j.commatsci.2013.06.024}}</ref><ref>{{cite book |doi=10.1201/9781351228466 |title=Primary and Secondary Manufacturing of Polymer Matrix Composites |date=2017 |isbn=978-1-351-22846-6 |editor-last1=Debnath |editor-last2=Singh |editor-first1=Kishore |editor-first2=Inderdeep }}{{pn|date=January 2025}}</ref><ref>[https://coventivecomposites.com/explainers/what-is-finite-element-analysis/ What is Finite Element Analysis?]{{Dead link|date=August 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> Materials may be tested during manufacturing and after construction by various non-destructive methods including ultrasonic, thermography, shearography and X-ray radiography,<ref>{{cite journal
|last1=Matzkanin |first1=George A.
|last2=Yolken |first2=H. Thomas
|title=Techniques for the Nondestructive Evaluation of Polymer Matrix Composites
|journal=AMMTIAC Quarterly
|volume=2
|issue=4
|url=http://ammtiac.alionscience.com/pdf/AQV2N4.pdf
|url-status=dead
|archive-url=https://web.archive.org/web/20081217033116/http://ammtiac.alionscience.com/pdf/AQV2N4.pdf
|archive-date=2008-12-17
}}</ref> and laser bond inspection for NDT of relative bond strength integrity in a localized area.