On characterising fracture resistance in mode-I delamination

In this work we focus on the mode-I quasi-static crack propagation in adhesive joints or composite laminates. For this problems a number of di ff erent standards have been approved. The most widely used are based on the double cantilever beam (DCB) test and on linear elastic fracture mechanics (LEFM) but di ff er in some aspects of the testing procedure and the recommended data-reduction schemes. The applicability of these methods is still a matter of debate in the scientific community, particularly in the case of ductile interfaces. We revisit the accuracy of the most used standards and compare it with other methods based on either LEFM or J-integral theory. All the methods analysed in our work are based on either Euler-Bernoulli or Timoshenko beam theories. We present a number of numerical examples where we compare di ff erent expressions for fracture resistance obtained with di ff erent methods. The input for the analysis, which includes applied load, cross-head displacement and rotation, crack length and cohesive zone length, is obtained from the numerical model which simulates real experiments. In these simulations, we use a Timoshenko beam model with a bi-linear CZM, which allows us accurate comparison with analytical formulas for fracture resistance based on Euler-Bernoulli and Timoshenko beam theory

Enhanced simple beam theory for characterising mode-I fracture resistance via a double cantilever beam test

We study a double-cantilever beam (DCB), in which either the crack-mouth opening displacement or the end rotations are prescribed, in the linear-elastic-fracture-mechanics (LEFM) limit of an infinitely stiff and brittle interface.Wepresentanovel,yetextremelysimple,derivationoftheclosed-formsolutionofthisproblemwhen thearmsaremodelledwithTimoshenkobeamtheory.Weremovetheassumptionthatthecrosssectionsofthe DCB arms are assumed not to rotate (i.e. that they are clamped) at the crack tip, which is made in so-called ‘simple beam theory’(SBT).Therefore, with our‘ enhanced simple beam theory’(ESBT),in front of the crack tip, cross section sareallowedtorotate,althoughthebeamaxisstaysundeformed.Thus,wecandeterminethecracktiprotationcausedbythedeformationofthebeaminfrontofthecracktipalsointheLEFMlimit.Asaresult, mostoftheinaccuraciesoftheSBTareeliminated,withouttheneedforacrack-length correction, use dinthe ‘corrected beam theory’(CBT).Inthisway,wecanderiveaveryaccuratedatareductionformulaforthecritical energy release rate, Gc, which does not require the measurement of the crack length, unlike CBT. In our numerical results we show that, compared to the most effective data reduction methods currently available (including CBT), our formula is either as accurate or more accurate for the case of brittle delamination of thick compositeplates,inwhichsheardeformabilitycanplayasignificantrole

A computational framework for polyconvex large strain elasticity for geometrically exact beam theory

In this paper, a new computational framework is presented for the analysis of nonlinear beam finite elements subjected to large strains. Specifically, the methodology recently introduced in Bonet et al. (Comput Methods Appl Mech Eng 283:1061–1094, 2015) in the context of three dimensional polyconvex elasticity is extended to the geometrically exact beam model of Simo (Comput Methods Appl Mech Eng 49:55–70, 1985), the starting point of so many other finite element beam type formulations. This new variational framework can be viewed as a continuum degenerate formulation which, moreover, is enhanced by three key novelties. First, in order to facilitate the implementation of the sophisticated polyconvex constitutive laws particularly associated with beams undergoing large strains, a novel tensor cross product algebra by Bonet et al. (Comput Methods Appl Mech Eng 283:1061–1094, 2015) is adopted, leading to an elegant and physically meaningful representation of an otherwise complex computational framework. Second, the paper shows how the novel algebra facilitates the re-expression of any invariant of the deformation gradient, its cofactor and its determinant in terms of the classical beam strain measures. The latter being very useful whenever a classical beam implementation is preferred. This is particularised for the case of a Mooney–Rivlin model although the technique can be straightforwardly generalised to other more complex isotropic and anisotropic polyconvex models. Third, the connection between the two most accepted restrictions for the definition of constitutive models in three dimensional elasticity and beams is shown, bridging the gap between the continuum and its degenerate beam description. This is carried out via a novel insightful representation of the tangent operator

A comparison of Finite Elements for Nonlinear Beams: The absolute nodal coordinate and geometrically exact formulations

Two of the most popular finite element formulations for solving nonlinear beams are the absolute nodal coordinate and the geometrically exact approaches. Both can be applied to problems with very large deformations and strains, but they differ substantially at the continuous and the discrete levels. In addition, implementation and run-time computational costs also vary significantly. In the current work, we summarize the main features of the two formulations, highlighting their differences and similarities, and perform numerical benchmarks to assess their accuracy and robustness. The article concludes with recommendations for the choice of one formulation over the other

Complete analytical solutions for double cantilever beam specimens with bi-linear quasi-brittle and brittle interfaces

In thiswork we develop a complete analytical solution for a double cantilever beam (DCB) where the arms are modelled as Timoshenko beams, and a bi-linear cohesive-zone model (CZM) is embedded at the interface. The solution is given for two types of DCB; one with prescribed rotations (with steady-state crack propagation) and one with prescribed displacement (where the crack propagation is not steady state). Because the CZM is bi-linear, the analytical solutions are given separately in three phases, namely (i) linearelastic behaviour before crack propagation, (ii) damage growth before crack propagation and (iii) crack propagation. These solutions are then used to derive the solutions for the casewhen the interface is linear-elastic with brittle failure (i.e. no damage growth before crack propagation) and the case with infinitely stiff interface with brittle failure (corresponding to linear-elastic fracture mechanics (LEFM) solutions). If the DCB arms are shear-deformable, our solution correctly captures the fact that they will rotate at the crack tip and in front of it even if the interface is infinitely stiff. Expressions defining the distribution of contact tractions at the interface, as well as shear forces, bending moments and cross-sectional rotations of the arms, at and in front of the crack tip, are derived for a linear-elastic interface with brittle failure and in the LEFM limit. For a DCB with prescribed displacement in the LEFM limit we also derive a closed-form expression for the critical energy release rate, Gc. This formula, compared to the so-called ‘standard beam theory’ formula based on the assumptions that the DCB arms are clamped at the crack tip (and also used in standards for determining fracture toughness in mode-I delamination), has an additional term which takes into account the rotation at the crack tip. Additionally, we provide all the mentioned analytical solutions for the case when the shear stiffness of the arms is infinitely high, which corresponds to Euler–Bernoulli beam theory. In the numerical examples we compare results for Euler–Beronulli and Timoshenko beam theory and analyse the influence of the CZM parameters

Dynamic sensitivity of multi-block stacks subjected to pulse base excitation - experimental evidence and non smooth contact dynamics simulations

Experimental and computational dynamic sensitivity study of Multi-Block Stacks subjected to Pulse Base Excitation is considered. Advanced non contact optical measuring technique based on the GOM Aramis and Pontos systems, as well as the corresponding processing software (displacement history of control sensor points, with a high resolution high speed cameras) have been applied to replace conventional displacement measuring systems and accelerometers. The Non Smooth Contact Dynamics (NSCD) time integration simulation framework SOLFEC http://code.google.com/p/solfec/ is adopted here for comparative NSCD analyses, including a sensitivity study on interface characteristics, as a validation process. Series of test experiments were conducted and recorded on a bespoke platform with and without lateral constraints in the Oxford Impact Engineering Laboratory and Rijeka University Structural Dynamics Laboratory for an extensive series of controlled pulse base excitation tests of multi block stacks configurations. Impact is generated by a pin-ball mechanism with spring and a wooden projectile, attached to an optical bench. For the NSCD simulations the base was subjected to a constant acceleration over a finite time, thereby facilitating the characterisation of multi block stacks tumbling modes of failures (global or partial), as a function of stop gap distance. Creation of well documented benchmarks for the validation of simulation paradigms for discontinuous media will be extremely valuable for researchers and code developers (non smooth contact dynamics, discrete elements, discontinuous deformation analysis), as well as for safety case engineers and industry regulators

Dynamic sensitivity of multi-block stacks subjected to pulse base excitation - experimental evidence and non smooth contact dynamics simulations

Experimental and computational dynamic sensitivity study of Multi-Block Stacks subjected to Pulse Base Excitation is considered. Advanced non contact optical measuring technique based on the GOM Aramis and Pontos systems, as well as the corresponding processing software (displacement history of control sensor points, with a high resolution high speed cameras) have been applied to replace conventional displacement measuring systems and accelerometers. The Non Smooth Contact Dynamics (NSCD) time integration simulation framework SOLFEC http://code.google.com/p/solfec/ is adopted here for comparative NSCD analyses, including a sensitivity study on interface characteristics, as a validation process. Series of test experiments were conducted and recorded on a bespoke platform with and without lateral constraints in the Oxford Impact Engineering Laboratory and Rijeka University Structural Dynamics Laboratory for an extensive series of controlled pulse base excitation tests of multi block stacks configurations. Impact is generated by a pin-ball mechanism with spring and a wooden projectile, attached to an optical bench. For the NSCD simulations the base was subjected to a constant acceleration over a finite time, thereby facilitating the characterisation of multi block stacks tumbling modes of failures (global or partial), as a function of stop gap distance. Creation of well documented benchmarks for the validation of simulation paradigms for discontinuous media will be extremely valuable for researchers and code developers (non smooth contact dynamics, discrete elements, discontinuous deformation analysis), as well as for safety case engineers and industry regulators