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In this work, an equivalent continuum of a barrel gridshell is introduced. Constitutive identification procedures based on periodic homogenization are provided in the literature for this purpose, based on a flat Representative Element Volume (REV), notwithstanding that the geometry of the structures concerned is curved. Therefore, the novelty of the present study is the selection of a curved REV to obtain the equivalent elastic constants. The numerical validation of the identification procedure is made comparing gridshell response to that of the equivalent shell under homogeneous load conditions. Finally, in order to highlight the effect of the curved geometry on the constitutive law of the continuum, the response of the proposed model is also compared to that of a continuum obtained from a flat REV.

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 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.

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.

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.

The study of new design methods targeted to minimize the use of materials is a theme of great relevance nowadays; structural designers pursue structural solutions characterized by efficiency, sustainability and optimization. Funicular systems adopt the “right” shape in accordance with the applied load and are ideally able to act without introducing bending. In this work an effective and easy-to-read method to study and quantify the funicularity is presented and applied to structural shells obtained using form finding, and analyzed under different static loads. In order to formulate the new method, the classical funicularity concept has been extended and the definition of Relaxed Funicularity (R-Funicularity) introduced. The parameter used to define the funicularity is the eccentricity and a structural shell is called R-Funicular when the eccentricity is included into an admissibility interval.

Left ventricle and left atrium are and have been practically always analyzed separately in common clinically and non-clinically oriented cardiovascular investigations. Both classic and speckle tracking echocardiographic data contributed to the knowledge about deformational impairments occurring in systo-diastolic differences. Recently new trajectory based approaches allowed a greater awareness about the entire left ventricle or left atrium revolution and on their deficiencies that take place in presence of hypertrophic cardiomyopathy. However, surprisingly, the concomitant function of the two left heart chambers has not been analyzed for their geometrical/mechanical relationship. For the first time we study here, by acquiring left ventricle and left atrial geometries on the same heartbeat, the trajectory attributes of the entire left heart treated as a whole shape and the shape covariation of its two subunits. We contrasted healthy subjects with patients affected by hypertrophic cardiomyopathy. We found impaired left heart trajectory mainly in terms of orientation and size. More importantly, we found profound differences in the direction of morphological covariation of left ventricle and left atrium. These findings open to new perspectives in pathophysiological evaluation of different diseases by allowing the appreciation of concomitant functioning of both left heart whole geometry and of its two chambers.

Modern shape analysis allows the fine comparison of shape changes occurring between different objects. Very often the classic machineries of generalized Procrustes analysis and principal component analysis are used in order to contrast the shape change occurring among configurations represented by homologous landmarks. However, if size and shape data are structured in different groups thus constituting different morphological trajectories, a data centering is needed if one wants to compare solely the deformation representing the trajectories. To do that, inter-individual variation must be filtered out. This maneuver is rarely applied in studies using simulated or real data. A geometrical procedure named parallel transport, that can be based on various connection types, is necessary to perform such kind of data centering. Usually, the Levi Civita connection is used for interpolation of curves in a Riemannian space. It can also be used to transport a deformation. We demonstrate that this procedure does not preserve some important characters of the deformation, even in the affine case. We propose a novel procedure called ‘TPS Direct Transport’ which is able to perfectly transport deformation in the affine case and to better approximate non affine deformation in comparison to existing tools. We recommend to center shape data using the methods described here when the differences in deformation rather than in shape are under study.

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.

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.

In this paper a procedure for constructing a 1D continuum equivalent to a Koiter plate, is presented. The result is used to build-up a coarse model of a Thin Walled Beam. The model results to be a nonlinear hyperelastic 1D second gradient continuum, endowed with an internal microstructure described by seven scalar kinematic parameters. The kinematic parameters are able to describe, although in a rough way, the deformation of the TWB in the plane of its cross-section. The nonlinear behaviour of a simply supported strut has been analyzed and the results compared with those given by the commercial finite element code ABAQUS.

SUMMARY Ontogeny is usually studied by analyzing a deformation series spanning over juvenile to adult shapes. In geometric morphometrics, this approach implies applying generalized Procrustes analysis coupled with principal component analysis on multiple individuals or multiple species datasets. The trouble with such a procedure is that it mixes intra- and inter-group variation. While MANCOVA models are relevant statistical/mathematical tools to draw inferences about the similarities of trajectories, if one wants to observe and interpret the morphological deformation alone by filtering inter-group variability, a particular tool, namely parallel transport, is necessary. In the context of ontogenetic trajectories, one should firstly perform separate multivariate regressions between shape and size, using regression predictions to estimate within-group deformations relative to the smallest individuals. These deformations are then applied to a common reference (the mean of per-group smallest individuals). The estimation of deformations can be performed on the Riemannian manifold by using sophisticated connection metrics. Nevertheless, parallel transport can be effectively achieved by estimating deformations in the Euclidean space via ordinary Procrustes analysis. This approach proved very useful in comparing ontogenetic trajectories of species presenting large morphological differences at early developmental stages.

Statistical Shape Analysis, that includes classic Geometric Morphometrics (GM), is often based on landmarks and frequently used to describe shape and shapes changes in biological studies. In this context changes in shape are typically analyzed separately from changes in size measured, in most cases, with centroid size (CS) [1]. Changes in shape are projected in a common linear space: the tangent space to the consensus and decomposed in affine and non affine components. The non affine component can be in turn de-composed in a series of local deformations (partial warps). This approach relies on the assumption that shapes are limitedly scattered in the shape space. In these conditions the difference between centroid size and m-volume is barely appreciable. In medical image, and in general in soft tissues, bodies can undergo very large deformations, involving also large changes in size strictly coupled with the change in shape from mechanical point of view. The cardiac example, analyzed in the present paper, shows changes in volume that can reach the 60% when comparing systole and diastole, coupled with severe longitudinal, radial and torsional strains. Because ventricle’s volume, together with its pressure, is one of the most important descriptors of the pumping heart function, it is natural considering the size change as volumes change. In fact, when dealing with such large size differences, CS and volume behave differently. In the last years the emerging disciplines of Diffeomorphometry and the related Functional Anatomy apply to replace classic GM through the use of diffeomorphisms, more suitable to describe soft tissues changes [2, 3]. On the other hand, these descriptions are very sophisticated but ignore some synthetic properties of the decomposition of deformation in some significant aspects. The goal of the present work is to show that standard GM tools (landmarks, Thin Plate Spline, and related decomposition of the deformation) can be generalized to better describe the very large deformations of biological tissues, without losing a synthetic description. In particular, the classical decomposition of the tangent space to the shape space in affine and non affine components [4] is enriched to include also the change in size, in such a way to give a complete description of the tangent space to the size-and-shape space. The proposed generalization is formulated by means of a new Riemannian metric describing the change in size as change in volume rather than change in CS. This lead to a redefinition of some aspect of the Kendalls size-and-shape space without abandoning Kendalls original formulation. This new formulation is discussed by analyzing 3D heart ventricular shapes coming from 3D Speckle Tracking Echocardiography. We evaluate the performances of different methods in recognizing Control (healthy) subjects from patients affected by Hypertrophic Cardiomyopathy.

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 investigate the relationships between pressure, volume and contraction for both the left ventricle and the left atrium, considered as a coupled system, by using a lumped model to describe the geometry of the two chambers and the functioning of muscles. Our long-term goal is to upscale the mechanical model here proposed to a full, detailed 3D model capable of describing the heart function.

The heart is a specialized muscle acting as the body’s circulatory pump. The effectiveness of the pumping action is commonly assessed through different measurements; among the most important are th...

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.

"An L frame made up by beam and column having channel cross sections, has been analyzed in a previous. work by two of the authors [14]. Depending on the aspect ratio and the joint configuration, it has been. proved that the structure can exhibit two simultaneous buckling modes. Here using the asymptotic. theory of elastic bifurcation that takes into account mode interaction, the initial slope of the bifurcated. paths has been determined. Three cases of joint configurations, which are the more common used in. welded connections, have been considered. For each case three admissible bifurcated paths have been. found. Two of them show a slope having the same order of magnitude of the ones found in the absence of. mode interaction while the remaining exhibits a slope largely steepest. Selecting, for each joint case, the. bifurcated path with the higher slope and between them the smallest one, it is found that it is associated. to the path which bifurcates at the higher critical load. This means that the stiffer structure is also the less. imperfection sensitive. Finally for each one of the cases studied, the effect of initial imperfection has been. considered and the real load carrying capacity of the frames has been determined. Finally some results. have been compared with those obtained using the FE code ABAQUS."

The analysis of full Left Atrium (LA) deformation and whole LA deformational trajectory in time has been poorly investigated and, to the best of our knowledge, seldom discussed in patients with Hypertrophic Cardiomyopathy. Therefore, we considered 22 patients with Hypertrophic Cardiomyopathy (HCM) and 46 healthy subjects, investigated them by three–dimensional Speckle Tracking Echocardiography, and studied the derived landmark clouds via Geometric Morphometrics with Parallel Transport. Trajectory shape and trajectory size were different in Controls versus HCM and their classification powers had high AUC (Area Under the Receiving Operator Characteristic Curve) and accuracy. The two trajectories were much different at the transition between LA conduit and booster pump functions. Full shape and deformation analyses with trajectory analysis enabled a straightforward perception of pathophysiological consequences of HCM condition on LA functioning. It might be worthwhile to apply these techniques to look for novel pathophysiological approaches that may better define atrio–ventricular interaction.

We present a procedure that detects myocardial infarction by analyzing left ventricular shapes recorded at end-diastole and end-systole, involving both shape and statistical analyses. In the framework of Geometric Morphometrics, we use Generalized Procrustes Analysis, and optionally an Euclidean Parallel Transport, followed by Principal Components Analysis to analyze the shapes. We then test the performances of different classification methods on the dataset. Among the different datasets and classification methods used, we found that the best classification performance is given by the following workflow: full shape epicardium+endocardium analyzed in the Shape Space i.e. by scaling shapes at unit size; successive Parallel Transport centered toward the Grand Mean, in order to detect pure deformations; final statistical analysis via Support Vector Machine with radial basis Gaussian function. Healthy individuals show both a stronger contraction and a shape difference in systole with respect to pathological subjects. Moreover, endocardium clearly presents a larger deformation when contrasted with epicardium. Eventually, the solution for the blind test dataset is given. When using Support Vector Machine for learning from the whole training dataset and for successively classifying the 200 blind test dataset, we obtained 96 subjects classified as normal and 104 classified as pathological. After the disclosure of the blind dataset this resulted in 95i¾?% of total accurracy with sensitivity at 97i¾?% and specificity at 93i¾?%.

A thin rectangular plate is modelled as an (initially flat) shell. Following Koiter, the two fundamental forms of the deformed middle surface are then used to define the strain measures of the body. On the middle surface of the 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. It is shown that by modelling each side of a thin walled beam as a 1D continuum, the entire assembly can be reduced to a 1D model as well. This procedure gives rise to an hyperelastic 1D beam model in which at least the warping effect is taken into account. The main features of the model are shown by means of some simple applications.

We present a novel system for the markerless shape analysis of bicipital contraction. The study of the soft tissue deformation due to the biceps muscular activity is achieved by analysing video sequences obtained from a RGB-D sensor and applying geometric morphometrics and parallel transport algorithms. In particular, we tested for differences between genders in soft tissue deformation during isometric contraction. Tests on 20 subjects (10 males, 10 females) in biceps brachii isometric contraction have been performed. The obtained results, in terms of size and shape deformations, are in accordance with biomechanical and physiological studies. The performance of the proposed method is particularly encouraging for its application to elderlies, in the bio-medical investigations as well as in rehabilitation of the upper limb.

We provide a quantitative description of the helicoid--to--spiral transition in thin ribbons of nematic elastomers using an elementary calculation based on a Koiter-type plate with incompatible reference configuration. Our calculation confirms that such transition is ruled by the competition between stretching energy and bending energy.

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 corresponding 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.

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.

The aim of this paper is to emphasise the role of the primary strain-line patterns in a human left ventricle (LV) within the complex system that is the heart. In particular, a protocol is proposed for the measurement of the principal strain lines (PSL) in the walls of the LV; this protocol is tested by means of a computational model which resembles a human LV. When the analysis is focused on the epicardial surface, PSL can be used to derive information on the directions of muscle fibres during the entire cardiac cycle, not only the systolic phase.

We studied the time evolution problem driven by growth for a non-Euclidean ball which at its initial state is equipped with a non-compatible distortion field. The problem is set within the framework of non-linear elasticity with large growing distortions. No bulk accretive forces are considered, and growth is only driven by the stress state. We showed that, when stress-driven growth is considered, distortions can evolve along different trajectories which share a common attracting manifold; moreover, they eventually attain a steady and compatible form, to which there corresponds a stress-free state of the ball.

We present recent investigations on the state of strain in human left ventricle based on the synergy between continuum mechanics and echocardiographic imaging. When data from three-dimensional Speckle Tracking Echocardiography are available, special strain directions can be detected on the epicardial and endocardial surfaces, which are well-known in continuum mechanics as principal strain lines (PSLs), further classified into primary and secondary strain lines. An appropriate investigation on PSLs can help to identify lines where strains are largest as primary and smallest as secondary. As PSLs change when cardiac diseases appear, the challenge is that the analysis may allow for the identification of new indicators of cardiac function.

The assessment of left ventricular shape changes during cardiac revolution may be a new step in clinical cardiology to ease early diagnosis and treatment. To quantify these changes, only point registration was adopted and neither Generalized Procrustes Analysis nor Principal Component Analysis were applied as we did previously to study a group of healthy subjects. Here, we extend to patients affected by hypertrophic cardiomyopathy the original approach and preliminarily include genotype positive/phenotype negative individuals to explore the potential that incumbent pathology might also be detected. Using 3D Speckle Tracking Echocardiography, we recorded left ventricular shape of 48 healthy subjects, 24 patients affected by hypertrophic cardiomyopathy and 3 genotype positive/phenotype negative individuals. We then applied Generalized Procrustes Analysis and Principal Component Analysis and inter-individual differences were cleaned by Parallel Transport performed on the tangent space, along the horizontal geodesic, between the per-subject consensuses and the grand mean. Endocardial and epicardial layers were evaluated separately, different from many ecocardiographic applications. Under a common Principal Component Analysis, we then evaluated left ventricle morphological changes (at both layers) explained by first Principal Component scores. Trajectories’ shape and orientation were investigated and contrasted. Logistic regression and Receiver Operating Characteristic curves were used to compare these morphometric indicators with traditional 3D Speckle Tracking Echocardiography global parameters. Geometric morphometrics indicators performed better than 3D Speckle Tracking Echocardiography global parameters in recognizing pathology both in systole and diastole. Genotype positive/phenotype negative individuals clustered with patients affected by hypertrophic cardiomyopathy during diastole, suggesting that incumbent pathology may indeed be foreseen by these methods. Left ventricle deformation in patients affected by hypertrophic cardiomyopathy compared to healthy subjects may be assessed by modern shape analysis better than by traditional 3D Speckle Tracking Echocardiography global parameters. Hypertrophic cardiomyopathy pathophysiology was unveiled in a new manner whereby also diastolic phase abnormalities are evident which is more difficult to investigate by traditional ecocardiographic techniques.

We are going to analyze the interval solution of an elastic beam under uncertain boundary conditions. Boundary conditions are defined as rotational springs presenting interval stiffness. Developments occur according to the interval analysis theory, which is affected, at the same time, by the overestimation of interval limits (also known as overbounding, because of the propagation of the uncertainty in the model). We suggest a method which aims to reduce such an overestimation in the uncertain solution. This method consists in a reparameterization of the closed form Euler-Bernoulli solution and set intersection.

"In this paper a procedure for obtaining a beam model from a thin plate theory, is introduced. The starting point is the theory proposed by Koiter which describes the deformation of the plate in a kinematically exact way. By assuming some fair constraints on the plate motion, its deformation measures can be described by means of four scalar parameters that are a measure of the elongation, two bending terms and the torsion of a 1D continuum. Given that the plate is thin and made up of a material that is isotropic elastic and linear in the 2D finite strains, its strain energy is written in terms of the 1D strains. The resulting function is then given a power expansion. Taking the derivative of this expression with respect to the 1D strain measures, the constitutive functions of the corresponding stress measures are obtained. Some samples show how the procedure can be used to study the mechanical behaviour of TWBs with open cross sections."

The mechanical behavior of soft matter is characterized by large shape changes, often accompanied by small changes of elastic energy; non-linear elasticity, with large, inhomogeneous, and anisotropic distortions, that may evolve in time, proved to be an effective modeling tool for many of such soft materials. Here, we deal with the definition of an appropriate strain measure, called the elastic metric, upon which the elastic energy density can be defined. Moreover, we discuss two key issues about distortions: one deals with the notion of compatible distortions, that is, distortion fields yielding a global configuration without any change of the elastic energy; the other concerns the symmetries of the material responses. We also present few selected examples of non-linear, anisotropic, elastic response where distortions play a key role.

"The analysis of two L-frames made up by channel sections subjected to various constraint. conditions has been performed in order to determine their buckling and postbuckling. behaviour. The analysis has been carried out in the framework of the asymptotic. bifurcation theory, using a one-dimensional nonlinear elastic beam model able. to account for warping. For each case considered, the first two buckling loads have. been determined, together with the associated eigenmodes. For the frames showing. an asymmetric postbuckling behaviour only the initial slope of the bifurcated path has. been determined. In case of symmetric behaviour, also the secondary modes and the. initial curvatures of the bifurcated path have been evaluated. The effects of small initial. imperfections have also been studied in order to estimate the real load carrying. capacity of the frames for all the cases examined"

A question which has not been addressed so far in the analysis of the twisting motion of the heart, relates to the existence of any advantages in energetic expenditure due to differential contractile properties across the wall of the ventricles. The question is addressed in this paper by a highly simplified analytical model of the ventricular cavity, based on a cylindrical geometry and set in the context of the linear theory of elasticity; however, the anisotropy of contraction is also taken into account. It is concluded, that when oppositely directed spirals of fibres in the internal and external layers of the cylinder keep the torsion within suitable limits, i.e. mimicking the biological context, a smaller energetic expenditure is related to a transmural pattern of contraction which is not uniform, and presents a larger epicardial contraction.

We present a simplified heart model with the aim of introducing a mechanical point of view in the interpretation of the pressure–volume loops. In particular, we propose a mechanical description of muscle contraction, and discuss its implications with reference to a specific pressure–volume loop representative of a normal human patient.