Editing Composite material (section)

=== 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"/>