The need to improve the performance of materials used in the form of plates has led to the development of new types of materials, composite materials. Among the most important composite materials are fiber-reinforced ones and those derived from wood, and they typically have orthotropic behavior. The characterization of these materials requires methods that are different from those used for more traditional materials. Among these methods, there is the Virtual Fields Method (VFM), which bases its definition on the use of the Principle of Virtual Works (PVW). A fundamental basis is the Kirchhoff-Love theory (KLT), valid for thin plates, which provides a simple relationship between deformations (measured) and curvatures (input of the VFM). The focus moved on the definition of the VFM equations. Starting from the PVW, it was derived the basic equation containing the components of the bending stiffness matrix (which for an orthotropic material are four) which represent the unknowns of the problem. It was therefore necessary to define four functions of the virtual fields, and two approaches were followed. In the first, the equations were defined manually, choosing them in compliance with the boundary conditions. In the second case, the formulation of piecewise optimised functions was exploited, which would have allowed to obtain better results. From these functions the relative four equations derived from the basic one of the PVW have been obtained. Having defined the equations, it was necessary to obtain the curvatures. In particular, a bending test of a plate (material: OSB) constrained in three points was simulated, to which a displacement in the center was then imposed. Two models were built: 2D and 3D. The curvatures of the individual nodes were then given as input into the VFM. In the 2D case, the results have been very good. Given the goodness of the results, a test was made by adding simulated noise to the curvature data, to simulate the experimental conditions. Two levels of noise have been used, and two differenty "smoothing" methods, necessary to smooth the noise present. With less noise, there is a slight deterioration of the results, but the smoothing functions even worsen the values obtained, operating in an inadequate way, but with the highest noise, the smoothing methods managed to slightly improve the results. For the 3D case the plate was constrained with flat surfaces. The data obtained were bad, so some changes were made. Firstly, focusing on the piecewise optimised functions defined , editing the virtual constraints imposed on the nodes of the virtual mesh. The result was a small improvement in the components of the matrix D obtained. Subsequently, the geometry of the plate and the imposed deflection were investigated, both factors influencing the internal stress state. Several plate displacements with different thicknesses have been imposed. The best values were obtained with thicker plates, in which the influence of the membrane components of tension is negligible. Finally, a last improvement of the model was made using the reduced integration in the simulation of the bending test, since the "solid" elements, subjected to full integration, suffer from "shear locking", which leads these elements to manifest a stiffness greater than real. Thus there is an improvement in the value of the component associated with the Poisson effect (D12). The residual percentage error is attributable to the approximation assumed with the KLT: in fact, it assumes that the curvatures depend only on the displacement z, while in a 3D field they also depend on the displacements along x and y. However, most of the error is attributable to D12, typically more difficult to be detected.
La necessità di migliorare le performance dei materiali impiegati sotto forma di piastre ha portato allo sviluppo di nuove tipologie di materiali, i materiali compositi. Tra i più importanti materiali compositi ci sono quelli fibrorinforzati e quelli derivati dal legno, ed hanno tipicamente comportamento ortotropo. La caratterizzazione di questi materiali richiede metodi diversi da quelli usati per i materiali più tradizionali. Tra questi metodi figura il Virtual Fields Method (VFM), che basa la sua definizione sull’utilizzo del Principio dei Lavori Virtuali. Base fondamentale è stata la teoria di Kirchhoff-Love (KLT), valida per piastre sottili, che fornisce una semplice relazione tra deformazioni (misurate) e curvature (input del VFM). Si è passati quindi alla definizione delle equazioni del VFM. Partendo dal Principio dei Lavori Virtuali, è stata derivata l’equazione base contenente le componenti della matrice di rigidezza a flessione (che per un materiale ortotropo sono quattro) che rappresentano le incognite del problema. È stato quindi necessario definire quattro funzioni dei campi virtuali, e sono stati seguiti due approcci. Nel primo le equazioni sono state definite manualmente, scegliendole nel rispetto delle condizioni al contorno del modello. Nel secondo caso, è stata sfruttata la formulazione di funzioni ottimizzate definite a tratti, che avrebbero permesso di ottenere risultati migliori. Da queste funzioni sono state ricavate le relative quattro equazioni derivate da quella base del PLV. Definite le equazioni, è stato necessario ottenere le curvature. In particolare, è stata simulata una prova a flessione di una piastra (materiale: OSB) vincolata in tre punti, a cui poi è stato imposto uno spostamento al centro. Due i modelli costruiti: uno 2D e uno 3D. Le curvature dei singoli nodi sono state quindi date in input nel VFM. Nel caso 2D, i risultati sono stati molto buoni. Vista la bontà dei risultati, è stata fatta una prova aggiungendo del rumore simulato ai dati delle curvature, così da simulare le condizioni sperimentali. Due livelli di rumore sono stati utilizzati, e altrettanti metodi di "smoothing", necessari per attutire il rumore presente. Con il rumore minore, si ha un piccolo peggioramento dei risultati, ma le funzioni di smoothing peggiorano addirittura i valori ottenuti, operando in maniera non adeguata, andando evidentemente ad alterare troppo i dati su cui agiscono, ma con il rumore più elevato, i metodi di smoothing riescono a migliorare leggermente i risultati. Per il caso 3D la piastra è stata vincolata con superfici piane. I dati ottenuti sono risultati insoddisfacenti, così si è cercato di migliorare le funzioni dei campi virtuali ottimizzate definite a tratti, andando a modificare i vincoli virtuali imposti sui nodi della mesh virtuale. Il risultato è stato un piccolo miglioramento delle componenti della matrice D ricavate. Successivamente è stata investigata la geometria della piastra e la flessione imposta, entrambi fattori che influenzano lo stato tensionale interno. Sono stati imposti diversi spostamenti a piastre con diverso spessore. I migliori valori si ottengono con piastre più spesse, in cui l’influenza delle componenti membranali di tensione è minore. Infine, un ultimo miglioramento del modello è stato fatto utilizzando l’integrazione ridotta nella fase di simulazione del test a flessione, in quanto gli elementini "solid", sottoposti a integrazione completa soffrono di "shear locking", che porta questi elementini a manifestare una rigidezza maggiore di quella reale. Si ha così un miglioramento del valore della componente associata all’effetto Poisson D12. L’errore percentuale residuo è attribuibile all’approssimazione assunta con la KLT: essa, infatti, presuppone che le curvature dipendano solo dallo spostamento z, mentre in un campo 3D dipendono anche dagli spostamenti lungo x e y.
Identificazione dei componenti di rigidezza a flessione della piastra mediante il Virtual Fields Method: validazione numerica
SULPIZIO, LUCA
2019/2020
Abstract
The need to improve the performance of materials used in the form of plates has led to the development of new types of materials, composite materials. Among the most important composite materials are fiber-reinforced ones and those derived from wood, and they typically have orthotropic behavior. The characterization of these materials requires methods that are different from those used for more traditional materials. Among these methods, there is the Virtual Fields Method (VFM), which bases its definition on the use of the Principle of Virtual Works (PVW). A fundamental basis is the Kirchhoff-Love theory (KLT), valid for thin plates, which provides a simple relationship between deformations (measured) and curvatures (input of the VFM). The focus moved on the definition of the VFM equations. Starting from the PVW, it was derived the basic equation containing the components of the bending stiffness matrix (which for an orthotropic material are four) which represent the unknowns of the problem. It was therefore necessary to define four functions of the virtual fields, and two approaches were followed. In the first, the equations were defined manually, choosing them in compliance with the boundary conditions. In the second case, the formulation of piecewise optimised functions was exploited, which would have allowed to obtain better results. From these functions the relative four equations derived from the basic one of the PVW have been obtained. Having defined the equations, it was necessary to obtain the curvatures. In particular, a bending test of a plate (material: OSB) constrained in three points was simulated, to which a displacement in the center was then imposed. Two models were built: 2D and 3D. The curvatures of the individual nodes were then given as input into the VFM. In the 2D case, the results have been very good. Given the goodness of the results, a test was made by adding simulated noise to the curvature data, to simulate the experimental conditions. Two levels of noise have been used, and two differenty "smoothing" methods, necessary to smooth the noise present. With less noise, there is a slight deterioration of the results, but the smoothing functions even worsen the values obtained, operating in an inadequate way, but with the highest noise, the smoothing methods managed to slightly improve the results. For the 3D case the plate was constrained with flat surfaces. The data obtained were bad, so some changes were made. Firstly, focusing on the piecewise optimised functions defined , editing the virtual constraints imposed on the nodes of the virtual mesh. The result was a small improvement in the components of the matrix D obtained. Subsequently, the geometry of the plate and the imposed deflection were investigated, both factors influencing the internal stress state. Several plate displacements with different thicknesses have been imposed. The best values were obtained with thicker plates, in which the influence of the membrane components of tension is negligible. Finally, a last improvement of the model was made using the reduced integration in the simulation of the bending test, since the "solid" elements, subjected to full integration, suffer from "shear locking", which leads these elements to manifest a stiffness greater than real. Thus there is an improvement in the value of the component associated with the Poisson effect (D12). The residual percentage error is attributable to the approximation assumed with the KLT: in fact, it assumes that the curvatures depend only on the displacement z, while in a 3D field they also depend on the displacements along x and y. However, most of the error is attributable to D12, typically more difficult to be detected.| File | Dimensione | Formato | |
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Descrizione: Tesi di Luca Sulpizio
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https://hdl.handle.net/20.500.12075/3385