Introduction
This article explains how crack driving forces extend the J-integral and allow for analysis of the fracture response of viscoplastic materials under any load history. Before reading this article you may want to review my article on Linear Elastic Fracture Mechanics (LEFM), and my article on Elastic Plastic Fracture Mechanics (EPFM). The topics that I will describe here are in many ways an extension of the J-integral!
Quick Summary of the J-Integral
- The J-integral is equal to the energy that is released during crack growth.
- The J-integral is path independent.
- The J-integral is suitable for non-linear elastic, time-independent materials.
John Eshelby (1916-1981)
Crack Driving Forces
- Can be used with arbitrary material models.
- Can be used to study crack growth during cyclic loading.
- Configurational forces are thermodynamic forces responsible for motion of defects in a materials.
- A configurational force is generated if the total energy of a body varies for different positions of the defect(s).
- Crack driving forces can be calculated and plotted in Ansys Mechanical.
The foundation of configurational mechanics and crack driving forces is the Eshelby energy momentum transfer tensor, which in the reference configuration can be written: \( \mathbf{C} = W \mathbf{I} – \mathbf{F}^\top \mathbf{S}^{PK1} = W \mathbf{I} -J \mathbf{F}^\top \boldsymbol{\sigma} \mathbf{F}^{-\top}. \)
The crack driving force (in the reference configuration) is given by the material divergence of the C tensor: \( \mathbf{f} = -\nabla_X \cdot \mathbf{C}. \)
Note that the crack driving force can be calculated for any material model and load history! The total crack driving force on a finite element is given by:
\(\displaystyle F_i = -\sum_{p=1}^{GP} \sum_{k=1}^4 \left[ \frac{\partial N_k}{\partial X_1} C_{i1}^k + \frac{\partial N_k}{\partial X_2} C_{i2}^k + \frac{\partial N_k}{\partial X_3} C_{i3}^k \right] w_p.\)
The force on a node is obtained by assembling the element forces.
The J-integral at the crack tip is given by: \(J_{tip} = -\mathbf{f}_{tip} \cdot \mathbf{e}_c\), which by using the divergence theorem can be written:
\( J_{tip} = \displaystyle\mathbf{e}_c \cdot \lim_{r \rightarrow 0} \int_{\Gamma_r} \mathbf{C} \mathbf{m} \,dl. \)
Note the J-integral based on the crack driving forces is path dependent.
Crack Driving Forces in Ansys
Ansys Mechanical supports the calculation of the crack driving forces for arbitrary geometries. The following example (from the Ansys documentation) shows on one example. The blue arrows in the image show the nodal crack driving forces.
One limitation of the Ansys implementation is that it is restricted to the material models listed below.
Summary
- Crack Driving Forces is an extension of the J-integral.
- Can be used with any material model.
- Can be used with any loading history.
- Is an interesting tool for investigating crack growth and fracture of polymers
