In this research, a plane rigid body spring model (RBSM) was developed to deal with damage and crack propagation problems in isotropic and orthotropic brittle materials. The RBSMs belong to the family of the discrete computational methods. They are based on the idea of discretizing a solid in particles of a certain shape and interacting among them according to specific rules. Thanks to an algebraic formulation they have been widely applied to the study of failure problems in brittle materials, that are characterized by the rapid spread of cracks as soon as the elastic limit is reached. The most common downsides are an intrinsic failure anisotropy and a limitation on the Poisson's ratio, that are related to the intrinsic micro-structure of the interactions of the discrete model. The RBSM proposed here has been developed from the one proposed by Casolo in 2004. This method is characterized by quadrilateral rigid elements connected to the beside ones by normal and shear springs. This RBSM is characterized by a simple mechanistic structure, that has allowed, its application also to the study of the compression response of a completely fractured tempered glass bonded to an ionoplastic interlayer. The fundamental idea of the here proposed RBSM is the addiction of an interaction also between the elements that share only one node, that are the elements on the diagonals for quadrilateral rigid elements. Adding a diagonal spring that connects the element centres of gravity, the RBSM is able to model isotropic materials with a Poisson's ratio between -1 and 1/3 differently from the original model that shows an anisotropic response for materials with a Poisson's ratio different from 0. Furthermore, in the new model the mechanistic model structure is preserved, being the diagonal spring modulus proportional to the Poisson's ratio. The addition of the diagonal springs, introducing also a new direction in which the failure criterion is checked, reduces significantly the failure anisotropy. This is further reduced defining a coupled failure criterion that couples the responses of the normal and shear springs. The capability of the model is proven by reproducing the experimental results for crack propagation problems on concrete notched beams. Both pure mode I and mixed mode crack propagation problems are successfully reproduced, getting also the concrete size effect. The analysis have been conducted comparing different solving algorithms looking for an efficient solving algorithm that avoids convergence issues. The Casolo's RBSM was widely applied to the study of orhtotropic materials, like masonry, taking also into account its texture at the macro-scale, defining the model parameters in relation with an equivalent Cosserat continuum. Hence, in the end of this research, the capability of the new model of dealing with orthotropic Cauchy or Cosserat materials was investigated. Dealing with orthotropic Cauchy materials, the biggest challenge arises when the principal orthotropy directions are not aligned with the rigid element grid, because, in this case, there is a coupling between the axial and the shear responses, like in a generic anisotropic material. The study has showed that for square rigid elements the continuum equivalent to the RBSM has always the coupling terms equal for the two orthogonal directions. Similarly, comparing the RBSM with an orthotropic Cosserat continuum, the model topology and the kinematic of the springs impose that the coupling term between the shear responses has to be equal to the coupling term between the axial responses. Despite these limits, some possible solutions, to reasonably control the committed approximations, have been proposed. In conclusion, this study has allowed to extend the range of applicability of this quadrilateral RBSM, characterized by a simple mechanistic topology. The good results obtained, both for the elastic and the crack propagation problems, despite a limited number of springs, in comparison to other discrete approaches, prove the efficiency of the proposed model to study failure problems with a limited computational effort.
In questa ricerca si è sviluppato un modello piano a masse rigide e molle (RBSM), al fine di riprodurre problemi di propagazione della frattura in materiali fragili, sia isotropi che ortotropi. I modelli a masse rigide e molle appartengono alla famiglia dei metodi computazionali discreti. Questi metodi sono basati sull’idea di discretizzare un solido in particelle con una certa forma e che interagiscono fra di loro secondo regole specifiche. Grazie ad una formulazione algebrica, questi approcci sono stati ampliamente applicati allo studio della rottura di materiali fragili, che sono caratterizzati dalla rapida diffusione di fratture non appena il limite elastico del materiale è stato raggiunto. Tuttavia, i più comuni svantaggi di questi approcci sono: una intrinseca anisotropia a rottura e dei limiti sul coefficiente di Poisson del continuo isotropo equivalente. Questi limiti sono dettati della micro-struttura interna, legata alle interazioni fra particelle dello specifico modello discreto. Il modello discreto qui descritto è stato sviluppato dal modello a masse e molle proposto da Casolo nel 2004. Questo metodo è caratterizzato da elementi rigidi quadrangolari collegati agli elementi accanto con molle normali ed a taglio. Grazie a questa topologia, questo modello è caratterizzato da una struttura meccanicistica semplice, che ha consentito la sua applicazione allo studio della risposta a compressione di un vetro temprato completamente fratturato vincolato ad un interlayer ionoplastico. L’idea fondamentale del modello a masse e molle qui proposto e l’aggiunta di una interazione anche fra gli elementi che condividono solo un nodo, ovvero gli elementi sulle diagonali. Aggiungendo una molla diagonale che connette i centri di gravità degli elementi, il modello proposto è capace di modellare materiale isotropi con un coefficiente di Poisson compreso fra -1 e 1/3, differentemente dal modello originale che mostrava una risposta anisotropa per materiali con un coefficiente di Poisson diverso da zero. Inoltre si è preservata una semplice struttura meccanicistica, essendo il ruolo della molla diagonale direttamente relazionato al coefficiente di Poisson. L’aggiunta di una molla diagonale, introducendo una nuova direzione in cui verificare il criterio di rottura, riduce significativamente l’anisotropia a rottura. Questa è stata ulteriormente ridotta definendo un criterio di rottura che accoppia la risposta delle molle normali ed a taglio. Le capacità del modello sono state dimostrate, riproducendo alcuni risultati sperimentali di problemi di propagazione della frattura in travi in calcestruzzo pre-fessurate. Sono stati considerati sia problemi di propagazione della frattura per modo I che per modo misto, tenendo conto anche dell’effetto scala tipico del calcestruzzo. Per le analisi sono stati confrontati diversi algoritmi risolutivi al fine di eliminare problemi di convergenza. Il modello a masse e molle proposto da Casolo è stato ampliamente applicato allo studio di materiali ortotropi, come la muratura, considerando anche alla macro-scala gli effetti della tessitura muraria, definendo i parametri del modello in relazione con un continuo di Cosserat. Quindi, al termine di questa ricerca, sono state studiate le capacità del nuovo modello sviluppato di modellare materiali ortotropi sia secondo la teoria dei solidi di Cauchy che quella di Cosserat. Modellando materiali ortotropi alla Cauchy, la più grande sfida è modellare il comportamento del materiale quando gli assi principali di ortotropia non sono allineati con la griglia degli elementi rigidi. In questo caso insorge un accoppiamento fra la risposta a taglio e quella assiale, come in un materiale genericamente anisotropo. Lo studio condotto ha mostrato che per elementi rigidi quadrati, il continuo equivalente al modello discreto ha sempre i termini di accoppiamento uguali per le due direzioni ortogonali. Similmente confrontando il modello con un continuo ortotropo alla Cosserat, la topologia del modello e la cinematica delle molle impongono che il termine di accoppiamento fra le risposte a taglio sia uguale al termine di accoppiamento fra le risposte assiali. Nonostante questi limiti, alcune soluzioni per contenere gli errori sono state proposte. In conclusione, questo studio ha consentito di estendere il campo di applicabilità di un modello a masse rigide e molle, con elementi quadrangolari e caratterizzato da una semplice struttura meccanicistica. I buoni risultati ottenuti, sia dalle analisi elastiche che dai problemi di propagazione della frattura, nonostante un numero limitato di molle, se confrontato con altri approcci, provano l’efficacia del modello proposto per studiare problemi di frattura con uno sforzo computazione contenuto.
Discrete models and solid mechanics of brittle materials
Tateo, Vito
2020/2021
Abstract
In this research, a plane rigid body spring model (RBSM) was developed to deal with damage and crack propagation problems in isotropic and orthotropic brittle materials. The RBSMs belong to the family of the discrete computational methods. They are based on the idea of discretizing a solid in particles of a certain shape and interacting among them according to specific rules. Thanks to an algebraic formulation they have been widely applied to the study of failure problems in brittle materials, that are characterized by the rapid spread of cracks as soon as the elastic limit is reached. The most common downsides are an intrinsic failure anisotropy and a limitation on the Poisson's ratio, that are related to the intrinsic micro-structure of the interactions of the discrete model. The RBSM proposed here has been developed from the one proposed by Casolo in 2004. This method is characterized by quadrilateral rigid elements connected to the beside ones by normal and shear springs. This RBSM is characterized by a simple mechanistic structure, that has allowed, its application also to the study of the compression response of a completely fractured tempered glass bonded to an ionoplastic interlayer. The fundamental idea of the here proposed RBSM is the addiction of an interaction also between the elements that share only one node, that are the elements on the diagonals for quadrilateral rigid elements. Adding a diagonal spring that connects the element centres of gravity, the RBSM is able to model isotropic materials with a Poisson's ratio between -1 and 1/3 differently from the original model that shows an anisotropic response for materials with a Poisson's ratio different from 0. Furthermore, in the new model the mechanistic model structure is preserved, being the diagonal spring modulus proportional to the Poisson's ratio. The addition of the diagonal springs, introducing also a new direction in which the failure criterion is checked, reduces significantly the failure anisotropy. This is further reduced defining a coupled failure criterion that couples the responses of the normal and shear springs. The capability of the model is proven by reproducing the experimental results for crack propagation problems on concrete notched beams. Both pure mode I and mixed mode crack propagation problems are successfully reproduced, getting also the concrete size effect. The analysis have been conducted comparing different solving algorithms looking for an efficient solving algorithm that avoids convergence issues. The Casolo's RBSM was widely applied to the study of orhtotropic materials, like masonry, taking also into account its texture at the macro-scale, defining the model parameters in relation with an equivalent Cosserat continuum. Hence, in the end of this research, the capability of the new model of dealing with orthotropic Cauchy or Cosserat materials was investigated. Dealing with orthotropic Cauchy materials, the biggest challenge arises when the principal orthotropy directions are not aligned with the rigid element grid, because, in this case, there is a coupling between the axial and the shear responses, like in a generic anisotropic material. The study has showed that for square rigid elements the continuum equivalent to the RBSM has always the coupling terms equal for the two orthogonal directions. Similarly, comparing the RBSM with an orthotropic Cosserat continuum, the model topology and the kinematic of the springs impose that the coupling term between the shear responses has to be equal to the coupling term between the axial responses. Despite these limits, some possible solutions, to reasonably control the committed approximations, have been proposed. In conclusion, this study has allowed to extend the range of applicability of this quadrilateral RBSM, characterized by a simple mechanistic topology. The good results obtained, both for the elastic and the crack propagation problems, despite a limited number of springs, in comparison to other discrete approaches, prove the efficiency of the proposed model to study failure problems with a limited computational effort.| File | Dimensione | Formato | |
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https://hdl.handle.net/10589/177081