Mixed Mode Fracture Behaviour of Piezoelectric Materials

Piezoelectrics present an interactive electromechanical behaviour that, especially in recent years, has generated much interest since it renders these materials adapt for use in a variety of electronic and industrial applications like sensors, actuators, transducers, smart structures. Both mechanical and electric loads are generally applied on these devices and can cause high concentrations of stress, particularly in proximity of defects or inhomogeneities, such as flaws, cavities or included particles. A thorough understanding of their fracture behaviour is crucial in order to improve their performances and avoid unexpected failures. Therefore, a considerable number of research works have addressed this topic in the last decades. Most of the theoretical studies on this subject find their analytical background in the complex variable formulation of plane anisotropic elasticity. This theoretical approach bases its main origins in the pioneering works of Muskelishvili and Lekhnitskii who obtained the solution of the elastic problem in terms of independent analytic functions of complex variables. In the present work, the expressions of stresses and elastic and electric displacements are obtained as functions of complex potentials through an analytical formulation which is the application to the piezoelectric static case of an approach introduced for orthotropic materials to solve elastodynamics problems. This method can be considered an alternative to other formalisms currently used, like the Stroh’s formalism. The equilibrium equations are reduced to a first order system involving a six-dimensional vector field. After that, a similarity transformation is induced to reach three independent Cauchy-Riemann systems, so justifying the introduction of the complex variable notation. Closed form expressions of near tip stress and displacement fields are therefore obtained. In the theoretical study of cracked piezoelectric bodies, the issue of assigning consistent electric boundary conditions on the crack faces is of central importance and has been addressed by many researchers. Three different boundary conditions are commonly accepted in literature: the permeable, the impermeable and the semipermeable (“exact”) crack model. This thesis takes into considerations all the three models, comparing the results obtained and analysing the effects of the boundary condition choice on the solution. The influence of load biaxiality and of the application of a remote electric field has been studied, pointing out that both can affect to a various extent the stress fields and the angle of initial crack extension, especially when non-singular terms are retained in the expressions of the electro-elastic solution. Furthermore, two different fracture criteria are applied to the piezoelectric case, and their outcomes are compared and discussed. The work is organized as follows: Chapter 1 briefly introduces the fundamental concepts of Fracture Mechanics. Chapter 2 describes plane elasticity formalisms for an anisotropic continuum (Eshelby-Read-Shockley and Stroh) and introduces for the simplified orthotropic case the alternative formalism we want to propose. Chapter 3 outlines the Linear Theory of Piezoelectricity, its basic relations and electro-elastic equations. Chapter 4 introduces the proposed method for obtaining the expressions of stresses and elastic and electric displacements, given as functions of complex potentials. The solution is obtained in close form and non-singular terms are retained as well. Chapter 5 presents several numerical applications aimed at estimating the effect of load biaxiality, electric field, considered permittivity of the crack. Through the application of fracture criteria the influence of the above listed conditions on the response of the system and in particular on the direction of crack branching is thoroughly discussed.

Effect of Laser Shock Peening on Fatigue Crack Propagation of Aeronautical Structures

Laser Shock Peening (LSP) is a surface enhancement treatment which induces a significant layer of beneficial compressive residual stresses up to several mm underneath the surface of metal components in order to improve the detrimental effects of crack growth behavior rate in it. The aim of this thesis is to predict the crack growth behavior of thin Aluminum specimens with one or more LSP stripes defining a compressive residual stress area. The LSP treatment has been applied as crack retardation stripes perpendicular to the crack growing direction, with the objective of slowing down the crack when approaching the LSP patterns. Different finite element approaches have been implemented to predict the residual stress field left by the laser treatment, mostly by means of the commercial software Abaqus/Explicit. The Afgrow software has been used to predict the crack growth behavior of the component following the laser peening treatment and to detect the improvement in fatigue life comparing to the specimen baseline. Furthermore, an analytical model has been implemented on the Matlab software to make more accurate predictions on fatigue life of the treated components. An educational internship at the Research and Technologies Germany- Hamburg department of Airbus helped to achieve knowledge and experience to write this thesis. The main tasks of the thesis are the following: -To up to date Literature Survey related to laser shock peening in metallic structures -To validate the FE models developed against experimental measurements at coupon level -To develop design of crack growth slow down in centered and edge cracked tension specimens based on residual stress engineering approach using laser peened patterns transversal to the crack path -To predict crack growth behavior of thin aluminum panels -To validate numerical and analytical results by means of experimental tests.

A comprehensive study of the 26th December 2004 Sumatra earthquake: possible implications on Earth rotation and investigations on the coseismic and postseismic stress diffusion associated with the seismic rupture

In this work a multidisciplinary study of the December 26th, 2004 Sumatra earthquake has been carried out. We have investigated both the effect of the earthquake on the Earth rotation and the stress field variations associated with the seismic event. In the first part of the work we have quantified the effects of a water mass redistribution associated with the propagation of a tsunami wave on the Earth’s pole path and on the length-of-day (LOD) and applied our modeling results to the tsunami following the 2004 giant Sumatra earthquake. We compared the result of our simulations on the instantaneous rotational axis variations with some preliminary instrumental evidences on the pole path perturbation (which has not been confirmed yet) registered just after the occurrence of the earthquake, which showed a step-like discontinuity that cannot be attributed to the effect of a seismic dislocation. Our results show that the perturbation induced by the tsunami on the instantaneous rotational pole is characterized by a step-like discontinuity, which is compatible with the observations but its magnitude turns out to be almost one hundred times smaller than the detected one. The LOD variation induced by the water mass redistribution turns out to be not significant because the total effect is smaller than current measurements uncertainties. In the second part of this work of thesis we modeled the coseismic and postseismic stress evolution following the Sumatra earthquake. By means of a semi-analytical, viscoelastic, spherical model of global postseismic deformation and a numerical finite-element approach, we performed an analysis of the stress diffusion following the earthquake in the near and far field of the mainshock source. We evaluated the stress changes due to the Sumatra earthquake by projecting the Coulomb stress over the sequence of aftershocks taken from various catalogues in a time window spanning about two years and finally analyzed the spatio-temporal pattern. The analysis performed with the semi-analytical and the finite-element modeling gives a complex picture of the stress diffusion, in the area under study, after the Sumatra earthquake. We believe that the results obtained with the analytical method suffer heavily for the restrictions imposed, on the hypocentral depths of the aftershocks, in order to obtain the convergence of the harmonic series of the stress components. On the contrary we imposed no constraints on the numerical method so we expect that the results obtained give a more realistic description of the stress variations pattern.

Fracture Mechanics Investigation of Structures with Defects

Fracture mechanics plays an important role in the material science, structure design and industrial production due to the failure of materials and structures are paid high attention in human activities. This dissertation, concentrates on some of the fractural aspects of shaft and composite which have being increasingly used in modern structures, consists four chapters within two parts. Chapters 1 to 4 are included in part 1. In the first chapter, the basic knowledge about the stress and displacement fields in the vicinity of a crack tip is introduced. A review involves the general methods of calculating stress intensity factors are presented. In Chapter 2, two simple engineering methods for a fast and close approximation of stress intensity factors of cracked or notched beams under tension, bending moment, shear force, as well as torque are presented. New formulae for calculating the stress intensity factors are proposed. One of the methods named Section Method is improved and applied to the three dimensional analysis of cracked circular section for calculating stress intensity factors. The comparisons between the present results and the solutions calculated by ABAQUS for single mode and mixed mode are studied. In chapter 3, fracture criteria for a crack subjected to mixed mode loading of two-dimension and three-dimension are reviewed. The crack extension angle for single mode and mixed mode, and the critical loading domain obtained by SEDF and MTS are compared. The effects of the crack depth and the applied force ratio on the crack propagation angle and the critical loading are investigated. Three different methods calculating the crack initiation angle for three-dimension analysis of various crack depth and crack position are compared. It should be noted that the stress intensity factors used in the criteria are calculated in section 2.1.

Detailed study of the seismotectonic setting in the Lucanian Apennines and surrounding areas (Southern Italy)

In this research work I analyzed the instrumental seismicity of Southern Italy in the area including the Lucanian Apennines and Bradano foredeep, making use of the most recent seismological database available so far. I examined the seismicity occurred during the period between 2001 and 2006, considering 514 events with magnitudes M ≥ 2.0. In the first part of the work, P- and S-wave arrival times, recorded by the Italian National Seismic Network (RSNC) operated by the Istituto Nazionale di Geofisica e Vulcanologia (INGV), were re-picked along with those of the SAPTEX temporary array (2001–2004). For some events located in the Upper Val d'Agri, I also used data from the Eni-Agip oil company seismic network. I computed the VP/VS ratio obtaining a value of 1.83 and I carried out an analysis for the one-dimensional (1D) velocity model that approximates the seismic structure of the study area. After this preliminary analysis, making use of the records obtained in the SeSCAL experiment, I incremented the database by handpicking new arrival times. My final dataset consists of 15,666 P- and 9228 S-arrival times associated to 1047 earthquakes with magnitude ML ≥ 1.5. I computed 162 fault-plane solutions and composite focal mechanisms for closely located events. I investigated stress field orientation inverting focal mechanism belonging to the Lucanian Apennine and the Pollino Range, both areas characterized by more concentrated background seismicity. Moreover, I applied the double difference technique (DD) to improve the earthquake locations. Considering these results and different datasets available in the literature, I carried out a detailed analysis of single sub-areas and of a swarm (November 2008) recorded by SeSCAL array. The relocated seismicity appears more concentrated within the upper crust and it is mostly clustered along the Lucanian Apennine chain. In particular, two well-defined clusters were located in the Potentino and in the Abriola-Pietrapertosa sector (central Lucanian region). Their hypocentral depths are slightly deeper than those observed beneath the chain. I suggest that these two seismic features are representative of the transition from the inner portion of the chain with NE-SW extension to the external margin characterized by dextral strike-slip kinematics. In the easternmost part of the study area, below the Bradano foredeep and the Apulia foreland, the seismicity is generally deeper and more scattered and is associated to the Murge uplift and to the small structures present in the area. I also observed a small structure NE-SW oriented in the Abriola-Pietrapertosa area (activated with a swarm in November 2008) that could be considered to act as a barrier to the propagation of a potential rupture of an active NW-SE striking faults system. Focal mechanisms computed in this study are in large part normal and strike-slip solutions and their tensional axes (T-axes) have a generalized NE-SW orientation. Thanks to denser coverage of seismic stations and the detailed analysis, this study is a further contribution to the comprehension of the seismogenesis and state of stress of the Southern Apennines region, giving important contributions to seismotectonic zoning and seismic hazard assessment.

Magma injections and destabilization of basaltic volcanoes: a numerical study. Application to La Reunion (Indian ocean, France) and Stromboli (Tyrrhenian sea, Italy)

Most basaltic volcanoes are affected by recurrent lateral instabilities during their evolution. Numerous factors have been shown to be involved in the process of flank destabilization occurring over long periods of time or by instantaneous failures. However, the role of these factors on the mechanical behaviour and stability of volcanic edifices is poorly-constrained as lateral failure usually results from the combined effects of several parameters. Our study focuses on the morphological and structural comparison of two end-member basaltic systems, La Reunion (Indian ocean, France) and Stromboli (southern Tyrrhenian sea, Italy). We showed that despite major differences on their volumes and geodynamic settings, both systems present some similarities as they are characterized by an intense intrusive activity along well-developed rift zones and recurrent phenomena of flank collapse during their evolution. Among the factors of instability, the examples of la Reunion and Stromboli evidence the major contribution of intrusive complexes to volcano growth and destruction as attested by field observations and the monitoring of these active volcanoes. Classical models consider the relationship between vertical intrusions of magma and flank movements along a preexisting sliding surface. A set of published and new field data from Piton des Neiges volcano (La Reunion) allowed us to recognize the role of subhorizontal intrusions in the process of flank instability and to characterize the geometry of both subvertical and subhorizontal intrusions within basaltic edifices. This study compares the results of numerical modelling of the displacements associated with high-angle and low-angle intrusions within basaltic volcanoes. We use a Mixed Boundary Element Method to investigate the mechanical response of an edifice to the injection of magmatic intrusions in different stress fields. Our results indicate that the anisotropy of the stress field favours the slip along the intrusions due to cointrusive shear stress, generating flank-scale displacements of the edifice, especially in the case of subhorizontal intrusions, capable of triggering large-scale flank collapses on basaltic volcanoes. Applications of our theoretical results to real cases of flank displacements on basaltic volcanoes (such as the 2007 eruptive crisis at La Reunion and Stromboli) revealed that the previous model of subvertical intrusions-related collapse is a likely mechanism affecting small-scale steeply-sloping basaltic volcanoes like Stromboli. Furthermore, our field study combined to modelling results confirms the importance of shallow-dipping intrusions in the morpho-structural evolution of large gently-sloping basaltic volcanoes like Piton de la Fournaise, Etna and Kilauea, with particular regards to flank instability, which can cause catastrophic tsunamis.

Modelling and analysis of thin-walled beams in the context of the Generalized Beam Theory

In this work, the Generalized Beam Theory (GBT) is used as the main tool to analyze the mechanics of thin-walled beams. After an introduction to the subject and a quick review of some of the most well-known approaches to describe the behaviour of thin-walled beams, a novel formulation of the GBT is presented. This formulation contains the classic shear-deformable GBT available in the literature and contributes an additional description of cross-section warping that is variable along the wall thickness besides along the wall midline. Shear deformation is introduced in such a way that the classical shear strain components of the Timoshenko beam theory are recovered exactly. According to the new kinematics proposed, a reviewed form of the cross-section analysis procedure is devised, based on a unique modal decomposition. Later, a procedure for a posteriori reconstruction of all the three-dimensional stress components in the finite element analysis of thin-walled beams using the GBT is presented. The reconstruction is simple and based on the use of three-dimensional equilibrium equations and of the RCP procedure. Finally, once the stress reconstruction procedure is presented, a study of several existing issues on the constitutive relations in the GBT is carried out. Specifically, a constitutive law based on mirroring the kinematic constraints of the GBT model into a specific stress field assumption is proposed. It is shown that this method is equally valid for isotropic and orthotropic beams and coincides with the conventional GBT approach available in the literature. Later on, an analogous procedure is presented for the case of laminated beams. Lastly, as a way to improve an inherently poor description of shear deformability in the GBT, the introduction of shear correction factors is proposed. Throughout this work, numerous examples are provided to determine the validity of all the proposed contributions to the field.

Salt stress responses in pear and quince: physiological and molecular aspects

The productivity of agricultural crops is seriously limited by salinity. This problem is rapidly increasing, particularly in irrigated lands. Like almost all the fruit tree species, Pyrus communis is generally considered a salt sensitive species, but only little information is available on its behavior under saline conditions. Previous studies, carried out in the Department of Fruit Tree and Woody Plant Science (University of Bologna), focused their attention on pear and quince salt stress responses to understand which rootstock would be the most suitable for pear in order to tolerate a salt stress condition. It has been reported that pear and quince have different ability in the uptake, translocation and accumulation of chloride (Cl-) and sodium (Na+) ions, when plants were irrigated for one season with saline water (5 dS/m). The aim of the present work was to deepen these aspects and investigate salt stress responses in pear and quince. Two different experiments have been performed: a “short-term” trial in a growth chamber and a “long-term” experiment in the open field. In the short-term experiment, three different genotypes usually adopted as pear rootstocks (MC, BA29 and Farold®40) and the pear variety Abbé Fétel own rooted have been compared under salt stress conditions. The trial was performed in a hydroponic culture system, applying a 90 mM NaCl stress to half of the plants, after five weeks of normal growth in Hoagland’s solution. During the three-weeks of salt stress treatment, physiological, mineral and molecular analyses were performed in order to monitor, for each genotype, the development of the salt stress responses in comparison with the corresponding “unstressed” plants. Farold®40 and Abbé Fétel own rooted showed the onset of leaf necrosis, due to salt toxicity, one week before quinces. Moreover, quinces displayed a significant delay in premature senescence of old leaves, while pears emerged for their ability to regenerate new leaves from apparently dead foliage with the salt stress still running. Physiological measurements, such as shoots length, chlorophyll (Chl) content, and photosynthesis, have been carried out and revealed that pears exhibited a significant reduction in water content and a wilting aspect, while for quinces a decrease in Chl content and a growth slowdown were observed. At the end of the trial, all plants were collected and organs separated for dry weight estimation and mineral analyses (Cu, Fe, Mn, Zn Mg, Ca, K, Na and Cl). Mineral contents have been affected by salinity; same macro/micro nutrients were altered in some organs or relocated within the plant. This plant response could have partially contributed to face the salt stress. Leaves and roots have been harvested for molecular analyses at four different times during stress conditions. Molecular analyses consisted of the gene expression study of three main ion transporters, well known in Arabidopsis thaliana as salt-tolerance determinants in the “SOS” pathway: NHX1 (tonoplast Na+/H+ antiporter), SOS1 (plasmalemma Na+/H+ antiporter) and HKT1 (K+ high-affinity and Na+ low-affinity transporter). These studies showed that two quince rootstocks adopted different responsive mechanisms to NaCl stress. BA29 increased its Na+ sequestration activity into leaf vacuoles, while MC enhanced temporarily the same ability, but in roots. Farold®40, instead, exhibited increases in SOS1 and HKT1 expression mainly at leaf level in the attempt to retrieve Na+ from xylem, while Abbé Fétel differently altered the expression of these genes in roots. Finally, each genotype showed a peculiar response to salt stress that was the sum of its ability in Na+ exclusion, osmotic tolerance and tissue tolerance. In the long-term experiment, potted trees of the pear variety Abbé Fétel grafted on different rootstocks (MC, BA29 and Farold®40), or own rooted and also rootstocks only were subjected to a salt stress through saline water irrigation with an electrical conductivity of 5 dS/m for two years. The purposes of this study were to evaluate salinity effects on physiological (shoot length, number of buds, photosynthesis, etc.) and yield parameters of cultivar Abbé Fétel in the different combinations and to determine the salt amount that pear is able to tolerate over the years. With this work, we confirmed the previous hypothesis that pear, despite being classified as a salt-sensitive fruit tree, can be cultivated for two years under saline water irrigation, without showing any salt toxicity symptoms or severe drawbacks on plant development and production. Among different combinations, Abbé Fétel grafted on MC resulted interesting for its peculiar behaviors under salt stress conditions. In the near future, further investigations on physiological and molecular aspects will be necessary to enrich and broaden the knowledge of salt stress responses in pear.

Studio di caratteri correlati a stress idrico in una collezione di cultivar di frumento duro (Triticum durum Desf.)

L’agricoltura si trova ad affrontare una diminuzione della disponibilità d’acqua ed una crescente domanda della produzione di cereali per scopi alimentari. Sono perciò necessarie strategie di coltivazione innovative per migliorare la produttività e nuovi genotipi migliorati nell'efficienza dell’uso delle risorse in condizioni di siccità. Questi rappresentano gli obietti principali del progetto “DROPS” (Drought tolerant yielding Plants) all’interno del quale ha avuto luogo il mio progetto di Dottorato. La mia attività di ricerca è stata svolta come segue: 1. Caratterizzazione molecolare di un panel di188 accessioni di frumento duro con marcatori SSR e DaRT; 2. Esperimenti in serra su 100 accessioni del panel per valutare la Water-Use Efficiency (WUE) in sei repliche secondo un Alpha Lattice design; 3. Prove sul campo, effettuate secondo un Alpha Lattice design, in due stagioni di crescita: a. 2010/11, valutazione di 100 accessioni presso l’Azienda sperimentale dell'Università di Cadriano (BO); b. 2011/12, valutazione del panel completo in 3 ambienti, con due diversi regimi irrigui In entrambi gli anni, abbiamo valutato caratteri agronomici correlati con il ciclo di sviluppo, la resa di granella e sue componenti, nonché diversi fattori ambientali e del suolo. Per quanto riguarda WUE, abbiamo trovato differenze altamente significative tra accessioni; inoltre, cinque accessioni hanno mostrato elevati valori di WUE e cinque accessioni valori molto bassi di WUE in tutte e sei le repliche. Gli esperimenti di campo nelle stagioni 2011 e 2012 hanno evidenziato differenze altamente significative tra le accessioni del panel per la maggior parte dei caratteri analizzati, confermando inoltre che il panel di fiorisce entro una settimana. L'esperimento del secondo anno ci ha permesso osservare un significativa interazione Genotipo X Ambiente. Questi risultati saranno integrati con ulteriori analisi QTL, per identificare regioni cromosomiche coinvolte nel controllo genetico dei caratteri di interesse e verificare la stabilità dei QTL in diversi ambienti.