User:Oh Isaac/A Novel Continuum-Decohesive Finite Element for Modeling In-plane Fracture in Fiber Reinforced Composites

Title
A Novel Continuum-Decohesive Finite Element for Modeling In-plane Fracture in Fiber Reinforced Composites

Principle of Virtual Work (PVW)
The principle of virtual work, also known as the principle of virtual displacements, can be stated in the form below:

For continuum body,any quasi-static and admissible virtual displacement $$\delta w$$ from an equilibrium configuration, the increament of strain energy stored is equal to the  increment of work done by body forces $$f$$ in volume $$V$$ and surface tractions $$\Phi$$ on surface $$S$$.

For a body with discontinuity, the traction $$T$$ over the discontinuity also contributes the increment of strain energy.

Cohesive Laws
For Mode I:

A linear soften cohesive zone model is often used to describe the progressive failure in some quasi-brittle materials. The relationship between cohesive traction and separation displacement is shown below. When traction is removed gradually before the displacement reaching the critical separating point, the separation displacement will recover through another path, or to the opposite direction on the same path if traction is added again.

{{NumBlk|:|$$ F=\left\{\begin{matrix} \sigma_{IC}\left [ 1-\frac{\delta_I}{\delta_{IC}} \right ]& \mbox{if } \delta_I>0,\Delta\sigma_I>0\\ k_I^*\sigma_I& \mbox{if } \delta_I>0,\Delta\sigma_I<0,\mbox{where }k_I^*=\frac{\sigma_I^*}{\delta_I^*} \end{matrix}\right. $$|$$}}

The cohesive energy density is define as the area under the line described above, and the total energy needed for the critical separation is the whole area:

Criterion to decide whether the traction separate the body by a mixed model energy criterion is defined as:

Build Cohesive Element
Consider a element is cut alone the crack path(known already),describe the elements with virtual work principle in form of matrix.

where,

using static condensation to write the matrix function in terms of three triangle nodes to get the displacement $$\{d\}$$