Editing Composite material (section)

====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>