User:Eml4500.f08.ateam.carr/HW5

Composite Materials
Composite materials consist of a strengthening fiber and matrix substrate. The addition of these fibers to a substrate is termed 'doping'. Due to the mixed material properties, composite materials often contain a varying elastic modulus. In other words, the Young's modulus is a function of position in the material.

Some examples of composite materials with this characteristic:


 * Carbon fiber
 * Fiber Glass

The Elastic Modulus is much higher in these materials in the fiber direction (longitudinal direction). However, perpendicular to the fibers (transverse direction) the Elastic Modulus is smaller as the fibers lose their strength in shear.

Free body diagram of a rigid body cross section


$$ \sum{F_x}=0=-N(x,t) + N(x+dx,t)+f(x,t)dx-m(x)\ddot{u}dx $$

$$ \sum{F_x}=0=\frac{\delta N}{\delta x}(x,t)dx+H.O.T.+f(x,t)dx-m(x)\ddot{u}dx $$

Cancel dx and neglect the higher order terms (H.O.T.)

(Eqn. 1): $$ \sum{F_x}=0=\frac{\delta N}{\delta x}(x,t)+f(x,t)-m(x)\ddot{u} $$

Recall Taylor Series Expansion:

$$f(x+dx)=f(x)+\frac{df(x)}{dx}dx+[\frac{d^2f(x)}{2dx^2}+...] $$

Where the content inside the brackets represent higher order terms.

(Eqn. 2):  $$\frac{\delta N}{\delta x}+ f =m\ddot{u} $$

Where Equation 2 represents the equation of motion(EOM) of the elastic bar.

$$ N(x,t)=A(x)\sigma(x,t) $$ where: $$ \sigma(x,t)=E(x)\epsilon (x,t)$$ and: $$ \epsilon (x,t)=\frac{\delta u}{\delta x }(x,t)$$ (Eqn. 3)

Substituting Eqn. 3 into Eqn. 2:

$$ \frac{\delta }{\delta x}(A(x)E(x)\frac{\delta u}{\delta x})+f(x,t)=m(x)\ddot{u} $$, where: $$ \ddot{u}=\frac{\delta ^2u}{\delta t^2}$$

Partial Differential Equation of Motion
In order to integrate differential equations of motion, boundary conditions are needed to evaluate the constants of integration. For the problem above, this translates to two initial conditions (due to the 2nd order nature of the equations).

Boundary Conditions:



$$u(0,t)=0=u(L,t)$$



$$u(0,t)=0$$

$$N(L,t)=F(t)$$