An approximate solution for the inverted four-point bending test in symmetric specimens

An approximate solution is derived for the interfacial energy release rate of the inverted four-point bending test. The analysis builds on a previous model developed by one of the authors for an orthotropic edge-cracked layer subject to arbitrary generalized end forces. Contact forces exerted from the upper on the lower layer of the edge-cracked portion behind the delamination tip are introduced. Their value is chosen such that the two layers undergo the same deflection. The effects of both shear deformations along the layers and friction at the point of contact are taken into account within beam theories and approximate Coulomb model with a prescribed friction coefficient. The delamination energy release rate is derived for homogeneous and symmetric specimens. A parametric analysis is performed on varying the mechanical parameters of the model to analize the influence of shear deformations and friction. The results show that both shear deformations and friction affect the value of the energy release rate for short/intermediate interfacial cracks. For very long interfacial cracks the delamination energy release rate tends to a constant limit value which corresponds to that obtained within classical Euler-Bernoulli beam theory in absence of friction

Local zigzag effects and brittle delamination fracture of n-layered beams using a structural theory with three displacement variables

Equivalent single layer theories for layered beams effectively and accurately predict global displacements and internal force and moment resultants using a limited number of displacement variables. However, they cannot reproduce local effects due to material architecture or weak/imperfect bonding of the layers, such as zigzag  displacement fields, displacement jumps at the layer interfaces and complex transverse stress fields, nor can they simulate delamination damage growth. In this work we will present some applications and discuss advantages and limitations of a recently formulated zigzag model. The model, through a modification of the equilibrium equations of an equivalent single layer theory, which maintains the same number of variables, reproduces local effects and delamination fracture under mode II dominant conditions. The approach is based on a local enrichment of the displacement field of first order shear deformation theory, the introduction of cohesive interfaces and homogenization

An analytical beam model for the evaluation of crack tip root rotations and displacements in orthotropic specimens

Explicit and simple expressions for root compliance coefficients, which can be used to define root rotations and root displacements at the crack tip cross section of orthotropic cracked beams, are derived under general self-equilibrated loading conditions at the crack tip. The effects of both shear deformations and transverse elasticity are taken into account in order to accurately define displacement fields and energy release rate. The derivation builds on and extends one-dimensional formulations in the literature. The employment of the novel analytical expressions requires the determination of one a priori unknown parameter which describes the effects of the transverse elasticity and is determined through matching of well established 2D results in the literature. The one-dimensional model accurately reproduces crack tip effects in symmetric isotropic and orthotropic specimens; shear deformations are included in the formulation for an accurate derivation of the root displacement coefficients; the accuracy reduces in asymmetric specimens where the matching parameter becomes load dependent

Modeling imperfect interfaces in layered beams through multi- and single-variable zigzag kinematics

Multiscale structural models based on the coupling of a zigzag kinematics and a cohesive crack approach have been recently formulated to analyze the response of shear deformable layered structures with imperfect interfaces and describe progressive delamination fracture (Massabò, in Handbook of Damage Mechanics, Springer, 2022, pp.665-698). The zigzag kinematics accounts for zigzag effects associated to the elastic mismatch of the layers and displacement jumps due to interfacial imperfections, using a limited number of variables, which is independent of the number of layers. The effects of imperfect interfaces on the response of structures subjected to thermo-mechanical loading and on wave propagation and dispersion have been studied analytically and the advantages of this approach over discrete layer models and layerwise theories have been highlighted and discussed. In the presentation we review and discuss these models and present preliminary results on novel single-variable formulations, which have been inspired by a technique developed for homogeneous Timoshenko beams in (Kiendl et al

Response of two-layer composite beams with interlayer slip and damaging interfaces

Beams and columns made of different materials or reinforced with steel or composite elements are only some examples of two-layer structural systems. The combination of different elements improves the performances of the system but introduces weak elements such as interfaces. A perfect connection that retains relative displacements between the layers would allow a complete transmission of the stresses and ensure optimal performance in terms of global stiffness and strength. However, in practice, this is difficult to obtain, so that relative displacements between the layers can occur and only a partial composite action follows. As a consequence, the mechanical behavior of multi-layer composite beams depends not only on the geometrical and mechanical properties of the single layers but also on the nature of the bond between them. The optimal design of such composite systems needs to account for the response of interfaces and the progressive interface damaging as further global failure mechanism. The problem of the equilibrium of multi-layer beams consisting of elastic layers elastically bonded has been the object of a large number of studies. Only a little attention has been focused on the nonlinear interfacial behavior. On the other hand, under loading conditions that generate interlaminar stresses flaws and defects due to manifacturing errors or impacts may propagate or, alternatively, mechanical shear devices such as nails and steel studs may yield. This leads to a further reduction of the degree of the connection between the layers and, as a consequence, of the global stiffness and strength of the system and even to its premature collapse when the layers are still elastic. This work, trying to fill this gap, develops basic understanding of the essential features of such particular failure mechanism that affects the mechanical response and strength of structural composite elements, in order to optimize their design and performance in practical applications. In the framework of a multi-scale treatment of the problem, composite beams are modelled as beams having a higher number of degrees of freedom than classical homogeneous beams and governed by additional compatibility and constitutive equations accounting for relative displacements between the layers. The analysis builds on previous works by the authors in which fundamental analytical solutions for two-layer beams with interlayer slip and non-proportional linear interface constitutive laws are obtained. The formulation is restricted to the analysis of bonds realized, for instance, by the use of nails for which the stiffness in the transverse direction can be assumed to be infinite, so that uplifts between the layers are not allowed and only slips between the layers in their longitudinal direction can occur. According to classical elastic beam theory, each layer is modelled as an elastic Euler-Bernoulli beam. The connection in the longitudinal direction is modelled as a continuous distribution of shear tractions related to displacement discontinuities between the layers through a multi-linear law. Such a multi-linear law represents the evolution of the interfacial behavior during a loading process and allows simple analytical solutions for each its linear branch representing a regime the interface can undergo. With reference to beams subjected to simple loading and constraining conditions, a complete simulation from damage initiation to ultimate failure of the damage process at the bond is conducted in order to investigate the mechanical response and the collapse of the system and to understand which parameters characterizing the interface law have the most influence on the global response of the composite system. Future developments deal with the interaction of this failure mechanism with failure mechanisms involving layer materials and the influence of such interaction on the global mechanical response

Analisi di lastre contenenti fessure interagenti

L'evoluzione della frattura di lastre finite multi-fessurate viene simulata mediante un processo di carico incrementale in cui \ue8 assunto quale parametro di controllo la lunghezza delle fessure; possono cos\uec essere rilevate anche eventuali instabilit\ue0 di tipo snap-back. Il problema \ue8 affrontato nell'ambito della Meccanica della Frattura Elastica Lineare, con riferimento a lastre di dimensioni finite contenenti distribuzioni ordinate o casuali di fessure interagenti in condizioni di carico, rispettivamente, di modo I e di modo misto. I relativi diagrammi carico-spostamento, compresi eventuali tratti instabili, sono tracciati utilizzando un codice di calcolo automatico basato sul Metodo degli Elementi di Contorno a Discontinuit\ue0 di Spostamento

Descrizione anisotropa del danno in interfacce imperfette

Viene proposto un modello d\u2019interfaccia imperfetta espresso in termini di trazioni e discontinuit\ue0 di spostamento e basato su una descrizione al continuo del danno conseguente a stati di microfessurazione diffusa nell\u2019interfaccia. Si considerano due micromeccanismi responsabili del comportamento macroscopico, le cui modalit\ue0 di attuazione dipendono dalla condizione attuale dell\u2019interfaccia: distacco e/o scorrimento con attrito tra le superfici opposte e propagazione del danno. Il modello dipende da cinque costanti e da tre variabili interne delle quali una, di tipo tensoriale, necessaria per descrivere il carattere anisotropo del danno ed altre due atte a descrivere il contatto monolatero attritivo. Con riferimento al caso particolare di interfacce non dilatanti, l\u2019equazione costitutiva viene scritta in forma incrementale, finalizzata alla definizione di un elemento finito d\u2019interfaccia da utilizzarsi in applicazioni numeriche