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In this work, TWBs with open cross sections are considered. At first they are viewed as an assembly of plates each one of them being a wall of the TWB. Each plate is then modeled as a Koiter shell, initially flat. This allows for using the two fundamental forms of the middle surface of the shell as strain measures. On the middle surface of each plate two local coordinates are introduced: we will call them longitudinal and transversal, respectively. It is assumed that the components of the displacement field which characterize the middle surface kinematics can be expressed as a product of two functions: one defined along the longitudinal coordinate and one defined along the transversal coordinate. Given an explicit expression of the latter functions the 2D plate fields are reduced to 1D ones. The functions of the transversal coordinate are chosen so that the stretch along it together with the membrane shear, vanish. It is worth noting that the linearization of these constraints leads to the well known Vlasov’s assumptions. The walls are assembled by imposing suitable boundary conditions on their longitudinal edges. This procedure gives rise to an hyperelastic 1D beam model in which at least the warping is taken into account. The main features of the model are shown by means of some simple applications

We present and discuss a method to infer non invasively information on the fiber architecture in real LV walls. The method post–processes the echocardiographic data acquired by three–dimensional Speckle Tracking Echocardiography (3DSTE) through a MatLab–based protocol, already presented, discussed and val- idated in [5]. Our results reveals the difference between the role of endocardial and epicardial principal strain lines, at the systolic peak, and set the bases for possible future investigations aimed to analyze the onset of specific cardiac diseases through noninvasive analysis of LV fiber architecture

The aim of the paper is to solve and discuss the representation problem of a special class of integrable distortion fields in fibred cylindrical bodies and to analyze the correspond- ing induced shape changes. We find and discuss the compatibility conditions, i.e. the conditions to be satisfied to get a pair compatible distortion/shape change, when different fields of fibers are assigned on the cylindrical body, through the specification of the fields of fiber angles.

Stone-masonry constructions never disappeared from the horizons of architecture, even if the relationship between stone and architecture has been definitely loosened by Modern Movement. For the most part, it is survived in minor architectures, local or vernacular; however, some international architects have expressed their creativity within the classical stone language, also in very recent times. Moreover, the more recent technological innovation of the reinforced stone by post-tensioned steel cables has produced structures and structural elements of unusual slenderness, proving that stone can be fruitfully used in large architectures. Scientific research has never stopped within the field of masonry mechanics, for it has been deeply involved in conservation and restoration of the cultural heritage and so it is ready to help the renaissance of stone architecture.

A non-simple continuum model is adopted to grossly describe the behaviour of elastic microcracked bodies. The constitutive relations, obtained using a multiscale modelling based on the hypotheses of the classical molecular theory of elasticity, allow taking into account the microscopic features of the material. Referring to a one-dimensional microcracked bar, the possibility of the continuum to reveal the presence of internal heterogeneities is investigated. © (2010) Trans Tech Publications.

We present two applications of a direct one-dimensional beam model suitable for describing the buckling of thin-walled structures. The first application considers the buckling of a compressed beam with an intermediate stiffener under various warping constraints. The second describes the buckling of a two-bar frame, known as a Roorda frame, loaded by a dead force at the joint. Various warping constraints at the bar ends are considered and the relevant buckling modes and loads are numerically evaluated. Numerical results are presented for both cases; some of these appear to be new.

We present a reduced–order heart model aimed to introduce a novel point of view in the interpretation of the pressure–volume loops. The novelty of the approach is based on the definition of active contraction as opposed to that of active stress. The consequences of such assumption are discussed with reference to a specific pressure-volume loop characteristic of a normal human patient.

A beam model aimed at describing the flexural-torsional buckling of thin-walled members with non-symmetric cross-sections is presented. Two beam axes are introduced, and strain is defined with respect to both. The shearing strain between the cross-section and one of the two axes is assumed to vanish; the warping is supposed to be linear in the twist. Non-linear hyperelastic constitutive relations are introduced; by means of standard localization and static perturbation techniques, the field equations describing the flexural-torsional buckling are obtained. One benchmark example is given and some numerical values of the critical load for various warping constraints at the beam ends are provided. (c) 2008 Elsevier Ltd. All rights reserved.

Si consideri un sistema costituito da blocchi collegati da giunti di spessore sottile, disposti in modo da costituire un ‘arco’ o una ‘piattabanda’ di altezza e larghezza costanti. I solidi ‘arco’ e ‘piattabanda’ abbiano un asse costituto da una curva regolare piana, e da un segmento di retta, rispettivamente. I giunti siano disposti con regolarità in modo da determinare una variazione regolare della forma dei blocchi. In una prima fase di modellazione si assume che i blocchi siano corpi rigidi e i giunti interfacce elastiche.Successivamente si introduce un modello di trave monodimensionale con sezioni rigide per descrivere sommariamente il comportamento meccanico dei sistemi a blocchi. Si considera innanzitutto una configurazione iniziale in cui l’asse della trave coincide con l’asse del solido ‘arco’ o ‘piattabanda’ e le sezioni sono disposte lungo questo seguendo l’assetto dei giunti. Nel caso dell’arco, le sezioni sono ortogonali alla linea d’asse mentre nel caso della piattabanda formano un angolo variabile con continuità. Il caso più generale è quello in cui l’asse è una linea curva e le sezioni non sono ad esso ortogonali. Postulando una relazione di equivalenza in potenza tra i processi dinamici del modello discreto e quelli della trave, per atti di moto test corrispondenti, si esprimono le azioni di contatto della trave in funzione di quelle nel discreto ( vedi Masiani et al. 1995 ). Assegnata quindi una relazione costitutiva per le azioni di contatto del modello discreto, si ricava la relazione corrispondente per la azioni interne della trave.Tramite la relazione costitutiva il modello continuo, pur se sommario, si rivela capace di tenere in conto aspetti legati alla disomogeneità del materiale e alla geometria dei blocchi.

Using a direct one dimensional beam model with an internal structure including a kinematical descriptor which accounts for gross warping, the analysis of an L frame made up of thin walled beams is performed up to the initial postbuckling range. The beams have cross sections with only one axis of symmetry but the frame is symmetric and symmetrically loaded. Numerical results have been obtained by means of the finite element method using the COMSOL Multiphysics software. The results show that, despite the geometric and dynamic symmetry of the frame and depending on the transmission of warping between the beams, the frame can be imperfection sensitive.

A mechanics–based analysis of data from three-dimensional speckle tracking echocardiography can help to infer useful clinical data about the state of myocardium. Precisely, we recently conjectured that principal strain lines in the left ventricle change in presence of heart disease, and used data from healthy subjects and patients to support our conjecture. Here, we set the first steps of our analysis relative to the left atrium, describing our method of acquisition, and discussing the key mechanical characteristics delivered by the echocardiographic device for healthy subjects and patients affected by hypertrophic cardiomyopathy.

We study the morphing of three-dimensional objects within the framework of nonlinear elasticity with large distortions. A distortion field induces a target metric, and the configuration which is effectively realized by a material body is the one that minimizes the distance, measured through the elastic energy, between the target metric and the actual one. Morphing through distortions might have a paramount feature: the resulting configurations might be stress-free; if this is the case, the distortions field is called compatible. We maintain that the morphing through compatible distortions is a key strategy exploited by many soft biological materials, which can exhibit very large shape-change in response to distortions controlled by stimuli such as chemicals or temperature changes, while keeping their stress state almost null. Thus, the study of compatible distortions, and of the related shape-changes, is quite important. Here, we show a blueprint for stress-free morphing based on the notions of metric tensor and of Riemann curvature which can be used to design large morphing of three-dimensional objects.

Studying the changes of shape is a common concern in many scientific fields. We address here two problems: (1) quantifying the deformation between two given shapes and (2) transporting this deformation to morph a third shape. These operations can be done with or without point correspondence, depending on the availability of a surface matching algorithm, and on the type of mathematical procedure adopted. In computer vision, the re-targeting of emotions mapped on faces is a common application. We contrast here four different methods used for transporting the deformation toward a target once it was estimated upon the matching of two shapes. These methods come from very different fields such as computational anatomy, computer vision and biology. We used the large diffeomorphic deformation metric mapping and thin plate spline, in order to estimate deformations in a deformational trajectory of a human face experiencing different emotions. Then we use naive transport (NT), linear shift (LS), direct transport (DT) and fanning scheme (FS) to transport the estimated deformations toward four alien faces constituted by 240 homologous points and identifying a triangulation structure of 416 triangles. We used both local and global criteria for evaluating the performance of the 4 methods, e.g., the maintenance of the original deformation. We found DT, LS and FS very effective in recovering the original deformation while NT fails under several aspects in transporting the shape change. As the best method may differ depending on the application, we recommend carefully testing different methods in order to choose the best one for any specific application.

In shape analysis, the interpolation of shapes’ trajectories is often performed by means of geodesics in an appropriate Riemannian Shape Space. Over the past several decades, different metrics and shape spaces have been proposed, including Kendall shape space, LDDMM based approaches, and elastic contour, among others. Once a Riemannian space is chosen, geodesics and parallel transports can be used to build splines or piecewise geodesics paths. In a recent paper, we introduced a new Riemannian shape space named TPS Space based on the Thin Plate Spline interpolant and characterized by an appropriate metric and parallel transport rule. In the present paper, we further explore the geometry of the TPS Space by characterizing the properties of its geodesics. Several applications show the capability of the proposed formulation to conserve important physical properties of deformation, such as local strains and global elastic energy.

In the problem of the synthesis of metamaterials, the pantographic architecture revealed remarkable potentialities. Indeed it allowed for the synthesis of second gradient 2D (nonlinear) continua: i.e. 2D shells whose deformation energy depends also on the second derivatives of displacements in the tangent directions to the reference configuration. Moreover, pantographic architecture seems to be able to produce metamaterials whose macroscopic elongations are large, albeit remaining in the elastic regime. The theoretically shown potentialities have started to become of ≫practical≪ interest thanks to a series of experiments, which were made possible by the recent 3D additive manufacturing. The actual construction of pantographic architecture has been based on the design of two arrays of beams interconnected by small cylinders, whose behavior can be modeled in different ways: if they are very short they can be regarded as clamps, while if they are short enough as elastic (or inelastic for large rotations) cylindrical hinges connecting the beams of different arrays. Otherwise, they must be modeled as elastic (or inelastic) elements allowing for relative rotations and displacements. In this paper, we focus on this particular case and we introduce, after a homogenization based on heuristic arguments, a 2D generalized continuum model whose kinematics is characterized by two placement and rotation fields (one for each array of beams) and whose deformation energy depends on relative displacements and rotations. The offset between the two beams arrays is proven to be an essential tool for defining effective invariant kinematical deformation measures. In facts, one wants to postulate a deformation energy for the introduced 2D generalized continuum which gives predictions in agreement with those given by the more refined 3D model where the pantographic architecture is described with its maximum geometric complexity and where the constituting material is assumed to be modelable as a standard 3D first gradient continuum. In the present paper, in order to arrive at the correct conjecture for the postulated energy, we consider the concept of averages of rotations in SO(3) Lie group. The used enriched kinematics is seen to be a possible alternative to the adoption of second gradient 2D models. Some rather surprising deformation processes are studied, where interesting non-symmetric post-buckling phenomena are observed in both the models used. Mentioned post-buckling has been observed experimentally.

In modern shape analysis, deformation is quantified in different ways depending on the algorithms used and on the scale at which it is evaluated. While global affine and non-affine deformation components can be decoupled and computed using a variety of methods, the very local deformation can be considered, infinitesimally, as an affine deformation. The deformation gradient tensor F can be computed locally using a direct calculation by exploiting triangulation or tetrahedralization structures or by locally evaluating the first derivative of an appropriate interpolation function mapping the global deformation from the undeformed to the deformed state. A suitable function is represented by the thin plate spline (TPS) that separates affine from non-affine deformation components. F, also known as Jacobian matrix, encodes both the locally affine deformation and local rotation. This implies that it should be used for visualizing primary strain directions (PSDs) and deformation ellipses and ellipsoids on the target configuration. Using C = FTF allows, instead, one to compute PSD and to visualize them on the source configuration. Moreover, C allows the computation of the strain energy that can be evaluated and mapped locally at any point of a body using an interpolation function. In addition, it is possible, by exploiting the second-order Jacobian, to calculate the amount of the non-affine deformation in the neighborhood of the evaluation point by computing the body bending energy density encoded in the deformation. In this contribution, we present (i) the main computational methods for evaluating local deformation metrics, (ii) a number of different strategies to visualize them on both undeformed and deformed configurations, and (iii) the potential pitfalls in ignoring the actual three-dimensional nature of F when it is evaluated along a surface identified by a triangulation in three dimensions. © Copyright © 2020 Piras, Profico, Pandolfi, Raia, Di Vincenzo, Mondanaro, Castiglione and Varano.

Transporting deformations from a template to a different one is a typical task of the shape analysis. In particular, it is necessary to perform such a kind of transport when performing group-wise statistical analyses in Shape or Size and Shape Spaces. A typical example is when one is interested in separating the difference in function from the difference in shape. The key point is: given two different templates (Formula presented.) and (Formula presented.) both undergoing their own deformation, and describing these two deformations with the diffeomorphisms (Formula presented.) and (Formula presented.), then when is it possible to say that they are experiencing the same deformation? Given a correspondence between the points of (Formula presented.) and (Formula presented.) (i.e. a bijective map), then a naïve possible answer could be that the displacement vector (Formula presented.), associated to each corresponding point couple, is the same. In this manuscript, we assume a different viewpoint: two templates undergo the same deformation if for each corresponding point couple of the two templates the condition (Formula presented.) holds or, in other words, the local metric (non linear strain) induced by the two diffeomorphisms is the same for all the corresponding points.

A nonlinear staged construction analysis is adopted here to simulate the forming process of timber gridshell. In this respect, particular attention is paid to the modelling of the cylindrical joints connecting the orthogonal laths. Cylindrical joints enable the initially flat lattice to be formed into a doubly curved gridshell shape; therefore, accurate modelling of the joints’ mechanical behaviour is a crucial aspect when simulating the forming process of timber gridshells. Another important parameter influencing the gridshell shape is the orientation of laths making up the initially flat lattice. Using an equivalent continuum definition, the final geometry of the gridshells, obtained from different laths orientations, is analysed in here to assess how laths orientation affects the structural shape in terms of funicularity. The Relaxed Funicularity Ellipse method presented in Gabriele et al. (2018) is used for such purpose.

We exploit the possibility of deforming a solid volume, or a surface as well, by assigning a target metric. As well known, an elastic solid may change its shape under two different kinds of actions: one are the loadings, the other one are the distortions, aka, the pre-strains. Actually, a distortion prescribes the target metric sought by the solid, and the metric effectively realized is the one that minimize the distance, measured through an elastic energy, between the target metric and the actual one. The challenge of our work is to use the metric information taken from a pair of configurations X ¯ , X of a given body B x , to deform a different body B y , so that its deformation be as similar as possible to the one suffered by B x . In particular, we assess the metric change between the two configurations X ¯ , X of the template B x ; then, we use this information to build a target metric for B y , exploiting the notion of distortions: given a source configuration Y ¯ , we find a target configuration Y by minimizing an appropriate elastic energy. The proposed method is very effective in deforming solids as well as surfaces, and in reproducing similar deformations even when applied to different bodies.

Between the ’50s and the ’70s of the last century a class of analytical forms, the so called parabolic velaroid surfaces, were widely used to design thin reinforced concrete shells due to their ability to equi-librate design loads by pure membrane actions. These forms are obtained as an approximate analytical solution to the differential equation corresponding to a compressed membrane subjected to a uniformly distributed vertical load. The same equation furnishes an analytical solution in the form of a series. The membrane equation is also solved numerically by an iterative implementation of the finite difference method. These numerical solutions are compared in terms of funicular efficiency by evaluating the generalized eccentricity and estimating their R-funicularity.

New findings What is the central question of this study? Can impaired deformational indicators for genotype positive for hypertrophic cardiomyopathy in subjects that do not exhibit a left-ventricular wall hypertrophy condition (G+LVH-) be determined using non-invasive 3D echocardiography? What is the main finding and its importance? Using 3D-STE and modern shape analysis, peculiar deformational impairments can be detected in G+LVH- subjects that can be classified with good accuracy. Moreover, the patterns of impairment are located mainly on the apical region in agreement with other evidence coming from previous biomechanical investigations. Abstract We propose a non-invasive procedure for predicting genotype positive for hypertrophic cardiomyopathy (HCM) in subjects that do not exhibit a left-ventricular wall hypertrophy condition (G+LVH-); the procedure is based on the enhanced analysis of medical imaging from 3D speckle tracking echocardiography (3D-STE). 3D-STE, due to its low quality images, has not been used so far to detect effectively the G+LVH- condition. Here, we post-processed echocardiographic images exploiting the tools of modern shape analysis, and we studied the motion of the left ventricle (LV) during an entire cycle. We enrolled 82 controls, 21 HCM patients and 11 G+LVH- subjects. We followed two steps: (i) we selected the most impaired regions of the LV by analysing its strains; and (ii) we used shape analysis on these regions to classify the subjects. The G+LVH- subjects showed different trajectories and deformational attributes. We found high classification performance in terms of area under the receiver operating characteristic curve (∼90), sensitivity (∼78) and specificity (∼79). Our results showed that (i) G+LVH- subjects present important deformational impairments relative to healthy controls and (ii) modern shape analysis can efficiently predict genotype by means of a non-invasive and inexpensive technique such as 3D-STE.

The geometric morphometric (GM) analysis of complex anatomical structures is an ever more powerful tool to study biological variability, adaptation and evolution. Here, we propose a new method (combinland), developed in R, meant to combine the morphological information contained in different landmark coordinate sets into a single dataset, under a GM context. combinland builds a common ordination space taking into account the entire shape information encoded in the starting configurations. We applied combinland to a Primate case study including 133 skulls belonging to 14 species. On each specimen, we simulated photo acquisitions converting the 3D landmark sets into six 2D configurations along standard anatomical views. The application of combinland shows statistically negligible differences in the ordination space compared to that of the original 3D objects, in contrast to a previous method meant to address the same issue. Hence, we argue combinland allows to correctly retrieve 3D-quality statistical information from 2D landmark configurations. This makes combinland a viable alternative when the extraction of 3D models is not possible, recommended, or too expensive, and to make full use of disparate sources (and views) of morphological information regarding the same specimens. The code and examples for the application of combinland are available in the Arothron R package. © 2018 Associazione Teriologica Italiana.

In a previous contribution, a new Riemannian shape space, named TPS space, was introduced to perform statistics on shape data. This space was endowed with a Riemannian metric and a flat connection, with torsion, compatible with the given metric. This connection allows the definition of a Parallel Transport of the deformation compatible with the three-fold decomposition in spherical, deviatoric, and non-affine components. Such a parallel transport also conserves the Γ-energy, strictly related to the total elastic strain energy stored by the body in the original deformation. A new approach is here presented in order to calculate the bending energy on the body alone (body bending energy) and to restrict it exclusively within physical boundaries of objects involved in the deformation analysis. The novelty of this new procedure resides in the fact that we propose a new metric to be preserved during the TPS direct transport. This allows transporting the shape change more coherently with the mechanical meaning of the deformation. The geometry of the TPS space is then further discussed in order to better represent the relationship between the Γ-energy, the strain energy, and the so-called bending energy densities.

A mechanics–based analysis of data from three–dimensional speckle tracking echocardiography is proposed, aimed at investigating deformations in myocardium and at assessing shape and function of distinct strain lines corresponding to the principal strain lines of the cardiac tissue. The analysis is based on the application of a protocol of measurement of the endocardial and epicardial principal strain lines, which was already tested on simulated left ventricles. In contrast with similar studies, it is established that endocardial principal strain lines cannot be identified with any structural fibers, not even along the systolic phase and is suggested that it is due to the capacity of the endocardial surface to contrast the dilation of the left ventricle. A mechanics-based analysis of data from three-dimensional speckle tracking echocardiography is proposed, aimed at investigating deformations in myocardium and at assessing shape and function of distinct strain lines corresponding to the principal strain lines of the cardiac tissue. The analysis is based on the application of a protocol of measurement of the endocardial and epicardial principal strain lines, which was already tested on simulated left ventricles. In contrast with similar studies, it is established that endocardial principal strain lines cannot be identified with any structural fibers, not even along the systolic phase and is suggested that it is due to the capacity of the endocardial surface to contrast the dilation of the left ventricle.

We present and discuss a method to infer noninvasively information on the fibre architecture in real human left ventricle walls. The method post-processes the echocardiographic data acquired by three-dimensional speckle tracking echocardiography through a MatLab-based protocol, already presented, discussed and validated. Our results indicate the difference between the role of endocardial and epicardial principal strain lines, at the systolic peak, and set the bases for possible future investigations aimed to analyse the onset of specific cardiac diseases through noninvasive analysis of LV fibre architecture. Moreover, we set a rational statistical analysis to investigate the reliability of the proposed method.