Configurational force in fracture mechanics

The crack driving force in a non-linear elastic material is described by the J−integral. However, this description of crack driving force is not valid under non-proportional loading, which usually occurs with crack extension. A recent application of concept of configurational forces has made it possible to describe the J−integral, which is independent of the constitutive response of the material. Configurational forces are thermodynamic driving forces on defects in materials. Examples are voids, cracks, interfaces, dislocations, etc. In a homogeneous material the bulk configurational forces are described as,

f = −∇ · (φI − FTS)                         (1)

where, φ is the strain energy density, FT is the transpose of the deformation gradient and S is the first Piola-Kirchhoff Stress tensor. The divergence is represented by ∇ and identity tensor by I, respectively.

The J−integral is calculated by adding the configurational forces in a contour around the crack tip, i.e.

 J = e ·∫fdl                                   (2)

where e denotes the unit vector in the direction of crack extension and dl is a line element. In an elastic-plastic material, the strain energy density includes both elastic strain energy as well as plastic strain energy. However, only the elastic strain energy density is recoverable and can describe the available energy for crack extension. Therefore, the true crack driving force can be described by a near tip J−integral as a sum of configurational forces appearing by considering only the elastic strain energy density in Eq. (1), denoted as fep, i.e. 

Jep = e ·∫fepdl                              (3)

• As configurational forces are independent of constitutive behaviour of material, the J integral calculated using Eq.(1) and (2) are also independent of constitutive response. 

• The difference in Eq.(1) and (2), is the non-linear elastic behaviour and elastic plastic behaviour. The conventional J integral which assumed non-linear elastic behavior equals Eq.(1) and the true crack driving force cannot be described by this for an elastic plastic material deformation.

• For an elastic plastic material, as only elastic energy is available for crack growth, the formulation of Eq.(2) can only be applied.