A literary review on micro-lattice structures for the design of orthopedic implants has been performed and two parameters, i.e. Young modulus (E) and pore size (p) (defined as the diameter of the biggest inscribed sphere inside the lattice domain) have been found crucial to avoid stress shielding the former and to promote bone ingrowth and vascularization the latter. Design intervals of those variables has been defined to ensure the above-mentioned properties (300-4000 MPa for ‘E’ and 500-1000 micro-meter for ‘p’) and 3D printed via SLM using Ti6Al4V powder. Among seven initial unit cell geometries, two of them (Diamond and Tetrahedron) have been selected on the basis of the overmentioned constraints on ‘E’ and ‘p’. For the numerical estimation of the elastic modulus, a homogenized approach has been used and proper PBC (Periodic Boundary Conditions) have been imposed to the boundary of a periodic micro-lattice unit volume representative of the infinite homogenized mean. Different ways to apply PBC has been studied for symmetric and non-symmetric structures. A Matlab code has been developed to calculate pore size of ideal and real geometry reconstructed from CT scan. Static compression and tension tests have been conducted and the homogenized stress vs strain curves have been obtained with the aid of DIC (Digital Image Correlation). CT scans have been performed in order to evaluate the differences between real and ideal geometry. Numerical elasto-plastic simulations under compression loading condition have been realized either on the ideal and real structures and the values of Young modulus, yield stress and pore size have been estimated and compared with experimental data. An attempt to substitute solid with beam elements thus saving computational time and enabling the modelling of big structures has been performed; even if it resulted promising, several open questions have been left for future developments. A strong relationship has been observed between relative density and printing quality on one side and mechanical properties on the other side. With reference to Tetrahedron topology, an important deviation between as-designed and as-manufactured Young modulus and yield stress (nearly 50 %) is present because of poor geometry other than lower relative density (16.6 % ideal vs 14.5 % real). Nevertheless, real values of ‘E’ and ‘p’ of both cells shape still fall within the interval defined by the constraints relevant for the biomedical application. With reference to the numerical compressive simulations, a difference, caused by boundary effects, between the elastic modulus of the periodic unit with PBC and the structures composed by a finite number of cells (like the models of the specimens) has been assessed and quantified for both cell topologies as a function of the cross-sectional area. These boundary effects are mainly due to the presence of free surfaces at the boundary of the specimens. Regarding the non-linear regime, Buckling has been observed on Tetrahedron in compression tests; the same behavior has been predicted by numerical analysis. The observation, already present in literature, that one periodic unit cell with PBC is not enough to represent the infinite mean in non-linear regime in case of buckling has been confirmed. No buckling phenomena has been observed for Diamond cell.
Una ricerca legata all’applicazione di strutture micro-lattice finalizzata alla progettazione di protesi ossee è stata realizzata in letteratura e due parametri, ovvero modulo elastico (E) e dimensione di poro (p) sono risultati cruciali per evitare l’intorpidimento osseo il primo e per promuovere vascolarizzazione e ricrescita ossea il secondo. Gli intervalli di progettazione per garantire tali proprietà sono stati definiti (300-4000 MPa per ‘E’ e 500-1000 micro-metri for ‘p’). Partendo da sette celle unitarie candidate, due di loro (Tetraedro e Diamante) sono state selezionate sulla base della migliore compatibilità con ‘E’ e ‘p’ e stampate in 3D con la tecnica SLM usando polvere di Titanio Ti6Al4V. Per la stima numerica del modulo elastico, un approccio di omogeneizzazione è stato applicato e opportune condizioni al contorno periodiche sono state imposte all’interfaccia di un volume unitario di micro-lattice in modo tale che esso sia rappresentativo del mezzo omogeneo infinito. Differenti modalità di applicazione delle PBC sono state studiate su strutture simmetriche e non. Un codice Matlab è stato sviluppato per calcolare la dimesione di poro sia della geometria ideale sia di quella reale ricostruita a partire dalla tomografia. Test sperimentali a trazione e compressione sono stati condotti e le curve sforzi-deformazioni omogeneizzate sono state misurate tramite la tecnica DIC (Digital Image Correlation). Inoltre, le immagini tomografate sono state acquisite per valutare le differenze tra geometria reale ed ideale. Simulazioni numeriche con modello elasto-plastico sono state realizzate sia sulle strutture reali sia su quelle ricostruite a partire dalla tomografia e i valori di modulo elastico, dimesione di poro e carico di snervamento sono stati stimati e confrontati con i dati sperimemtali. Un approccio per sostituire gli elementi solidi con i beam nei modelli ad elementi finiti è stato implementato per favorire la modellazione di geometrie computazionalmente onerose; sebbene esso sia promettente, diversi sono i punti aperti lasciati a sviluppi futuri. Una forte dipendenza fra densità relativa e qualità di stampa da una parte e proprietà meccaniche dall’altra è stata osservata. Con rifermento alla geometria tetraedrica, un’importante differenza tra modulo elastico e snervamneto reali ed ideali è presente (50 % circa) a causa di una non perfetta riuscita di stampa oltre che alla differenza delle densità relative (16.6 % ideale contro 14.5 % ideale). Ciononostante, i valori reali di modulo elastico e dimensione di poro di entrambe le geometrie rispettano ancora i vincoli rilevanti in ambito biomedico. Con riferimento alle simulazioni numeriche, una differenza, causata da effetti di bordo, tra i moduli elatici della cella singola con PBC e di strutture composte da un numero finite di celle (come i modelli dei provini) è stata stimata e quantificata per entrambe le geometrie come funzione della dimensione di sezione. Tali effetti di bordo sono dovuti alla presenza di superfici libere alla periferia dei provini. Per quanto riguarda il regime non lineare, la deformazione di tipo buckling è stata osservata sulla struttura tetraedrica nei test di compressione; lo stesso comportamento è stato previsto anche dalle analisi numeriche. In linea con studi precendenti, si è confermato che una singola unità periodica non è sufficiente per rappresentare il mezzo omogeneo infinito in campo non lineare in presenza di buckling. Nessun cedimento di tipo buckling è stato osservarto sulla cella Diamante.
Topological design, experimental characterisation and elasto-plastic modelling of micro-lattice structures for orthopaedic implants
ZOIA, MAURIZIO
2018/2019
Abstract
A literary review on micro-lattice structures for the design of orthopedic implants has been performed and two parameters, i.e. Young modulus (E) and pore size (p) (defined as the diameter of the biggest inscribed sphere inside the lattice domain) have been found crucial to avoid stress shielding the former and to promote bone ingrowth and vascularization the latter. Design intervals of those variables has been defined to ensure the above-mentioned properties (300-4000 MPa for ‘E’ and 500-1000 micro-meter for ‘p’) and 3D printed via SLM using Ti6Al4V powder. Among seven initial unit cell geometries, two of them (Diamond and Tetrahedron) have been selected on the basis of the overmentioned constraints on ‘E’ and ‘p’. For the numerical estimation of the elastic modulus, a homogenized approach has been used and proper PBC (Periodic Boundary Conditions) have been imposed to the boundary of a periodic micro-lattice unit volume representative of the infinite homogenized mean. Different ways to apply PBC has been studied for symmetric and non-symmetric structures. A Matlab code has been developed to calculate pore size of ideal and real geometry reconstructed from CT scan. Static compression and tension tests have been conducted and the homogenized stress vs strain curves have been obtained with the aid of DIC (Digital Image Correlation). CT scans have been performed in order to evaluate the differences between real and ideal geometry. Numerical elasto-plastic simulations under compression loading condition have been realized either on the ideal and real structures and the values of Young modulus, yield stress and pore size have been estimated and compared with experimental data. An attempt to substitute solid with beam elements thus saving computational time and enabling the modelling of big structures has been performed; even if it resulted promising, several open questions have been left for future developments. A strong relationship has been observed between relative density and printing quality on one side and mechanical properties on the other side. With reference to Tetrahedron topology, an important deviation between as-designed and as-manufactured Young modulus and yield stress (nearly 50 %) is present because of poor geometry other than lower relative density (16.6 % ideal vs 14.5 % real). Nevertheless, real values of ‘E’ and ‘p’ of both cells shape still fall within the interval defined by the constraints relevant for the biomedical application. With reference to the numerical compressive simulations, a difference, caused by boundary effects, between the elastic modulus of the periodic unit with PBC and the structures composed by a finite number of cells (like the models of the specimens) has been assessed and quantified for both cell topologies as a function of the cross-sectional area. These boundary effects are mainly due to the presence of free surfaces at the boundary of the specimens. Regarding the non-linear regime, Buckling has been observed on Tetrahedron in compression tests; the same behavior has been predicted by numerical analysis. The observation, already present in literature, that one periodic unit cell with PBC is not enough to represent the infinite mean in non-linear regime in case of buckling has been confirmed. No buckling phenomena has been observed for Diamond cell.| File | Dimensione | Formato | |
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https://hdl.handle.net/10589/152358