The magnitude of the driving force on a defect decides whether this defect can move or extend, or not. Fracture mechanics deals with the behavior of cracks in materials. A classical crack driving force parameter is the J-integral J, and it has been found that a crack in a certain type of material can grow, if J exceeds a certain critical value Jc. However, the conventional J-integral is, in general, an accurate measure of the crack driving force only for elastic materials, but not for elastic-plastic (or creeping) materials.
The concept of configurational forces has allowed us also to shed new light on this problem. Configurational forces are thermodynamic forces that act on all types of defects in materials, e.g. dislocations, interfaces, cracks, voids. The configurational force vectors originating from a defect indicate, in which direction a defect would like to move in order to minimize the total energy of the system. The higher the gain in total energy is, the higher the driving forces on the defect become. The configurational forces are evaluated by post-processing after a conventional finite element stress and strain analysis.
Elastic plate containing a hole, subjected to uniform tension in vertical direction. Configurational forces originate at the outer boundaries and the hole surface. The small bulk configurational forces appear due to numerical inaccuracy.
If several defects are present in a material, they interact with each other. A good example for such an interaction is that a material inhomogeneity influences the driving force on a nearby crack. This has been called the material inhomogeneity effect. The configurational force concept has allowed us to quantify the crack driving force and the crack growth direction in many types of inhomogeneous materials. The influence of eigenstrains and thermal residual stresses on the crack driving force can be assessed, too.
The application of the conventional J-integral for elastic-plastic materials relies on “deformation theory of plasticity”, i.e. elastic-plastic materials are treated as if they were nonlinear elastic. This theory is not applicable in cases of non-proportional loading, i.e. if unloading processes appear in elastic-plastic materials. However, such unloading processes are inevitable during crack extension or during cyclic loading of a structure.
Crack extension and variation of crack tip plastic zone (a) in material with deformation plasticity, (b) in material with incremental plasticity.
Based on the configurational force concept, Simha et al., J.Mech.Phys.Solids (2008) derived a J-integral for elastic-plastic materials and incremental theory of plasticity, Jep, which overcomes these limitations. Since configurational forces are induced in the plastically deformed regions of the material, Jep becomes path dependent. In order to decide whether a given, stationary crack can start to grow, the crack driving force should be evaluated by calculating Jep for a contour ΓPZ drawn around the crack-tip plastic zone. For growing cracks, the crack driving force is given by Jep evaluated for a contour ΓactPZ drawn around the active plastic zone at the current crack tip position, in contrast to the plastic wake. The configurational force concept has enabled us also to study the driving force for stationary and growing cracks under cyclic loading conditions.
Fracture mechanics specimen made of elastic-plastic material with a stationary crack (left). Specimen with a growing crack after crack extension Δa with active plastic zone (right).
Distribution of bulk configurational forces in an elastic-plastic fracture mechanics specimen (left); the large configurational force originating from the crack tip is not drawn. Map of the equivalent plastic strain (right).
Currently, we apply the configurational force concept to investigate the driving force of cracks in elastic-plastic, creeping materials. Several parameters have been worked out in order to estimate the behavior of cracks under these conditions, e.g. the C*-integral and the Ct-parameter, but it is unclear whether these parameters do reflect the crack driving force. In order to answer this question, a J-integral for elastic-plastic, creeping materials Jepc has been recently defined, analogously to Jep.
Parts of these investigations have been funded by the COMET program within the K2 Center “Integrated Computational Material, Process and Product Engineering, IC-MPPE” (strategic projects A4.20, P1.3 and P1.12).