Effective reference and current integration for large displacement interface

The most common interface formulations proposed in literature are generally based on the restrictivehypothesis of small strains and small displacements and, even though their application togeometrically nonlinear problems is of paramount interest, only few contributions are available in literature.Motivations are probably due to the difficulties encountered on such formulation, as alreadymentioned by several authors.A pioneering formulation is the finite displacement three-dimensional interface developed by Ortizand Pandolfi in [1], where normal and tangential traction components are evaluated with respectto the middle surface in the current configuration, producing a non-symmetric geometric stiffnessmatrix.More recently, an interface element formulation for geometrical non-linearity and material nonlinearity,which is developed in the reference configuration, has been proposed by Reinoso andPaggi in [2]. The constitutive model is formulated on the local reference, defined by normal axisand tangential axis with respect to the middle surface in the current configuration. The interfaceformulation generates a non symmetric geometric stiffness matrix, which is simplified by neglectingthe non symmetric contribution, in order to reduces computational cost by the use of symmetricsolver.The state of the art of cohesive models for the material separation is presented by Mosler andScheider in [3], focusing the attention on the thermodynamics and variational consistency. In [3]the authors state that many proposed models do not verify fundamental requirements such as thermodynamicprinciples, frame invariance, equilibrium conditions. Such problems are magnified foranisotropic models in geometrically nonlinear context. Attention is also focused on the unphysicaldissipation produced in elastic paths due to unsymmetrical stiffness matrix.Some existing cohesive-zone models are analyzed under conditions of large displacement andlarge strain by Ottosen et al in [4], and CZMs are also evaluated with respect to thermodynamicconsistency and the fundamental laws such as balance of angular momentum and frame invariance.It is shown that in elastic regime only isotropic models, with traction vector aligned to separationdisplacement vector, fulfill the physical principles, as already shown in [5].In [6] some cohesive-zone models are compared at finite strain condition, by a wedge test andby a peel test. The paper [6] shows that some models available in literature, or implemented incommercial finite element codes, which integrate the weak form equilibrium condition over thecurrent configuration, produce significant error in terms of fracture energy. On the contrary, modelsintegrated over the reference configuration produce negligible numerical error.The present paper investigates reasons of the different results between current and reference integrationschemes. It is shown that interface formulations integrated over current configuration violateenergy conservation principle, due to the elastic energy generated by the finite interface elongationwith constant elastic stiffness parameters. Moreover, an original mechanical interpretation of theelastic stiffness parameters, defined as a density of elastic springs between the two interface edges,can be considered an effective solution for interface integrated over the current configuration. In fact,the interface elongation modify the density of springs, as well as volume change modifies the massdensity, and integration over current configuration and integration over the reference one producetwo identical solutions.In t