Hybrid equilibrium formulation with adaptive element side orientation for cohesive crack prediction

The present article proposes an hybrid equilibrium element (HEE) formulation for the prediction of cohesive fracture formation and propagation with the crack modelled by extrinsic interface embedded at element sides. The hybrid equilibrium element formulation can model high order (quadratic, cubic and quartic) stress fields which strongly satisfy homogeneous equilibrium equations, inter-element and boundary equilibrium equations. The HEE can implicitly model both the initially rigid behaviour of an extrinsic interface and its debonding condition with separation displacement and softening. The extrinsic interface is embedded at the element sides and its behaviour is governed by means of the same degrees of freedom of HEE (generalized stresses), without any additional degree of freedom. The proposed extrinsic cohesive model is developed in the thermodynamic framework of damage mechanics. The proposed crack propagation criterion states that crack grows when the maximum principal stress reaches the tensile strength value, in a direction orthogonal to the principal stress direction. The crack is embedded at an element side and the mesh around crack tip is adapted, by rotation of the element sides, in order to have the interface aligned to the crack growth direction. Three classic two-dimensional problems of fracture propagation are numerically reproduced and the results compared to the experimental data or to the other numerical results

Integration of finite displacement interface element in reference and current configurations

In the present paper the non-linear behaviour of a solid body with embedded cohesive interfaces is examined in a finite displacements context. The principal target is the formulation of a two dimensional interface finite element which is referred to a local reference frame, defined by normal and tangential unit vectors to the interface middle surface. All the geometric operators, such as the interface elongation and the reference frame, are computed as function of the actual nodal displacements. The constitutive cohesive law is defined in terms of Helmholtz free energy for unit undeformed interface surface and, in order to obtain the same nodal force vector and stiffness matrix by the two integration schemes, the cohesive law in the deformed configuration is defined in terms of Cauchy traction, as a function of separation displacement and of interface elongation. Explicit expression of the nodal force vector is integrated either over the reference configuration or over the current configuration, which is shown to produce the same analytical finite element operators. No differences between the integration carried out in the reference and in the current configuration are shown, provided that elongation of the interface is taken in to account

Elimination of spurious kinematic modes in hybrid equilibrium elements

Hybrid stress elements are proposed as alternative to standard finite elements for linear and non linear analysis. Hybrid stress formulation is developed in a rigorous mathematical setting and an original approach for elimination of spurious kinematic modes is presented. Hybrid equilibrium method is compared to classical displacement based method by linear elastic analysis of some well known structural examples

an extrinsic interface developed in an equilibrium based finite element formulation

Abstract The phenomenon of delamination in composite material is studied in the framework of hybrid equilibrium based formulation with extrinsic cohesive zone model. The hybrid equilibrium formulation is a stress based approaches defined in the class of statically admissible solutions. The formulation is based on the nine-node triangular element with quadratic stress field which implicitly satisfy the homogeneous equilibrium equations. The inter-element equilibrium condition and the boundary equilibrium condition are imposed by considering independent side displacement fields as interfacial Lagrangian variable, in a classical hybrid formulation. The hybrid equilibrium element formulation is coupled with an extrinsic interface, for which the interfacial separation is zero for a sound interface. The extrinsic interface is defined as a rigid-damage cohesive zone model (CZM) in the rigorous thermodynamic framework of damage mechanics and is defined as embedded interface at the hybrid equilibrium element sides

multiple surface cracking and debonding failure for thin thermal coatings

Abstract A mechanical analysis of thin films of quasi-brittle materials used as thermal coatings for superalloy substrate is proposed. The study considers a bi-material element subjected to uniform tension formed by a thin layer of quasi-brittle material (typically a ceramic) bonded on an elastic substrate. The bounding between the coating film and the substrate is realized by a very thin primer which mechanically modeled as a zero thickness cohesive frictional interface. The analysis is developed by a non-linear finite element simulation in which, in order to consider damage size effects, a non-local isotropic damage model is adopted for the quasi-brittle coating. The results of the analysis shows the formation of multiple cracks on the coating surface which propagate up to the interface. At the same time, due to the mismatch between the elastic moduli between the coating and the substrate and the development of the transverse cracks, a competing debonding mechanism along the interface develops. The numerical results show also, for thick coating layers, the development of skew crack bands, which forecast coating spalling

Inter-Element Crack Propagation with High-Order Stress Equilibrium Element

The present contribution proposes a formulation based on the use of hybrid equilibrium elements (HEEs), for the analysis of inter-element delamination and fracture propagation problems. HEEs are defined in terms of quadratic stress fields, which strongly verify both the homogeneous and inter-element equilibrium equations and they are employed with interfaces, initially exhibiting rigid behavior, embedded at the elements’ sides. The interface model is formulated in terms of the same degrees of freedom of the HEE, without any additional burden. The cohesive zone model (CZM) of the extrinsic interface is rigorously developed in the damage mechanics framework, with perfect adhesion at the pre-failure condition and with linear softening at the post-failure regime. After a brief review, the formulation is computationally tested by simulating the behavior of a double-cantilever-beam with diagonal loads; the obtained numerical results confirm the accuracy and potential of the method

Effective reference and current integration for large displacement interface

The most common interface formulations proposed in literature are generally based on the restrictive hypothesis of small strains and small displacements and, even though their application to geometrically 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 already mentioned by several authors. A pioneering formulation is the finite displacement three-dimensional interface developed by Ortiz and Pandolfi in [1], where normal and tangential traction components are evaluated with respect to the middle surface in the current configuration, producing a non-symmetric geometric stiffness matrix. 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 and Paggi in [2]. The constitutive model is formulated on the local reference, defined by normal axis and tangential axis with respect to the middle surface in the current configuration. The interface formulation generates a non symmetric geometric stiffness matrix, which is simplified by neglecting the non symmetric contribution, in order to reduces computational cost by the use of symmetric solver. The state of the art of cohesive models for the material separation is presented by Mosler and Scheider 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 thermodynamic principles, frame invariance, equilibrium conditions. Such problems are magnified for anisotropic models in geometrically nonlinear context. Attention is also focused on the unphysical dissipation produced in elastic paths due to unsymmetrical stiffness matrix. Some existing cohesive-zone models are analyzed under conditions of large displacement and large strain by Ottosen et al in [4], and CZMs are also evaluated with respect to thermodynamic consistency 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 separation displacement 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 and by a peel test. The paper [6] shows that some models available in literature, or implemented in commercial finite element codes, which integrate the weak form equilibrium condition over the current configuration, produce significant error in terms of fracture energy. On the contrary, models integrated over the reference configuration produce negligible numerical error. The present paper investigates reasons of the different results between current and reference integration schemes. It is shown that interface formulations integrated over current configuration violate energy conservation principle, due to the elastic energy generated by the finite interface elongation with constant elastic stiffness parameters. Moreover, an original mechanical interpretation of the elastic 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 mass density, and integration over current configuration and integration over the reference one produce two identical solutions. In the present paper the interface formulation is rigorously developed under large displacement conditions, assuming as local reference for the constitutive model, normal and tangential axes to the middle surface, as already proposed in [1]. The geometric operators in the current configuration, such as the normal and tangential axes to the middle surface and elongation of the middle surface, are defined as functions of nodal displacements, and first order and second order derivatives, with respect to nodal displacements, are developed. Finally, nodal force vector and consistent stiffness matrix are developed for a two-dimensional interface element, showing the symmetry condition of the geometric stiffness matrix, if the second order derivative are not neglected. The proposed interface formulation is implemented in the FEAP finite element code [7] and the cohesive formulation proposed in [8] is considered as constitutive model. Results of numerical some simulations are proposed with times of convergence obtained with a symmetric solver

Mechanistic insight into ligand binding to G-quadruplex DNA

Specific guanine-rich regions in human genome can form higher-order DNA structures called G-quadruplexes, which regulate many relevant biological processes. For instance, the formation of G-quadruplex at telomeres can alter cellular functions, inducing apoptosis. Thus, developing small molecules that are able to bind and stabilize the telomeric G-quadruplexes represents an attractive strategy for antitumor therapy. An example is 3-(benzo[d]thiazol-2-yl)-7-hydroxy-8-((4-(2-hydroxyethyl)piperazin-1-yl)methyl)-2H-chromen-2-one (compound 1), recently identified as potent ligand of the G-quadruplex [d(TGGGGT)]4 with promising in vitro antitumor activity. The experimental observations are suggestive of a complex binding mechanism that, despite efforts, has defied full characterization. Here, we provide through metadynamics simulations a comprehensive understanding of the binding mechanism of 1 to the G-quadruplex [d(TGGGGT)]4. In our calculations, the ligand explores all the available binding sites on the DNA structure and the free-energy landscape of the whole binding process is computed. We have thus disclosed a peculiar hopping binding mechanism whereas 1 is able to bind both to the groove and to the 3' end of the G-quadruplex. Our results fully explain the available experimental data, rendering our approach of great value for further ligand/DNA studie