Editing Composite material (section)

=== 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.