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We present a study of smart growth in layered cylindrical structures. We start from the characterization of a compatible growth field in an anisotropic growing tube with the aim to show a small perturbation in the compatible growth field that may produce a controlled deprivation of compatibility and localization of elastic energy storage in a composite structure made up of anisotropic growing tubes. [ABSTRACT FROM AUTHOR]

Our goal is to reproduce the key features of carangiform swimming by modeling muscle functioning using the notion of active distortions, thus emphasizing the kinematical role of muscle, the generation of movement, rather than the dynamical one, the production of force. This approach, already proposed to model the action of muscles in different contexts, is here tested again for the problem of developing an effective and reliable framework to model and simulate swimming. A proper undulatory movement of a fish-like body is reproduced by defining a pattern of distortions, tuned in both space and time, meant to model the muscles activation which produce the flexural motion of body fish; eventually, interactions with the surrounding water yields the desired thrust. Carangiform swimmers have a relatively inflexible anterior body section and a generally flat, flexible posterior section. Because of this configuration, undulations sent rearward along the body attain a significant amplitude only in the posterior section. We compare the performances of different swimming gaits, and we are able to find some important relations between key parameters such as frequencies, wavelength, tail amplitude, and the achieved swim velocity, or the generated thrust, which summarize the swimming performance. In particular, an interesting relation is found between the Strouhal number and the wavelength of muscles activation. We highlight the muscle function during fish locomotion describing the activation of muscles and the relation between the force production and the shortening–lengthening cycle of muscle. We found a great accordance between results and empirical relations, giving an implicit validation of our models. [ABSTRACT FROM AUTHOR]

We model cardiac muscle contractions in the framework of finite elasticity with large distortions and couple a mechanical model with reaction–diffusion equations representing electrophysiological activity. Both models are implemented using anisotropic constitutive relations: we use stress–strain relations for fiber-reinforced materials, and anisotropic diffusion tensors for both the membrane potential and calcium ions. The effects of these choices on the electromechanical behavior are presented and discussed. [ABSTRACT FROM PUBLISHER]

Aim We tested the hypothesis that the evolutionary fates of two sister groups (Alligatoroidea and Crocodyloidea) are differently constrained by phylogenetic and ecological (functional) factors in the face of climatic change. Location Global. Methods We quantified disparity in skull rostrum shape by means of geometric morphometrics. Mechanical performance of the rostrum was analyzed by applying beam theory calculations to morphological data and experimentally measured bite force. The phylogeny was expressed in the form of principal coordinates, the first ones of which were used as a set of explanatory variables. Extents of species occurrence were computed using species distribution maps. Finally, species maximum skull size were measured and considered as a proxy of maximum body size. We performed variation partitioning analyses in order to compare differential contributions of phylogenetic and ecological factors in Alligatoroidea and Crocodyloidea. Results Alligatoroidea show higher ‘pure’ historical components than Crocodyloidea in explaining both rostrum shape and extent of occurrence (after controlling for body size). On the contrary, geometric variation of skull rostra of Crocodyloidea unequivocally shows a higher ‘pure’ functional component (linked to performance on prey capture) and a higher phylogenetically structured environmental variation than those found in Alligatoroidea. Results obtained for body size variation are consistent with these patterns. In Alligatoroidea, body size variation contains a higher phylogenetic signal than in Crocodyloidea. Main Conclusions Our results suggest that Crocodyloidea and Alligatoroidea may react differently when faced with significant environmental changes. We predict that global climatic changes will have a more important effect on Crocodyloidea than in Alligatoroidea by (1) promoting trait shift, adaptation to the new diet and speciation and (2) modifying the geographical range distribution of species (which may track favourable ecological conditions). [ABSTRACT FROM AUTHOR]

In order to account for the in vivo conditions of blood vessels, we investigate the mechanical behavior of a stressed tube-like membrane when small deformations are superimposed on large deformations: the latter simulate the stretches present in the in vivo arteries while the superimposed deformations account for the small — but essential for the blood propagation — deformations due to the pulsatile nature of the blood flow. Our aim is to discuss how a stress state influence the response of the vessel-blood system. [ABSTRACT FROM AUTHOR]

A direct theory of affine rods is developed from first principles. To concentrate on the central aspects of the model, we use an axiomatic format and tools from Lie group theory. To facilitate comparisons with other theories, we propose an identification procedure to derive the constitutive relations of the affine rod from those of a rod modeled as a three-dimensional body. [Copyright &y& Elsevier]

• We investigated the stress response of hydrogels to multiple and consecutive planar extensions. • We showed explicitly how the stress relaxation at a given extension depends on the number of incremental steps that lead to such extension. • We showed explicitly how the stress relaxation depends on the magnitude of incremental steps that lead to such extension. We investigated the stress response of hydrogels to multiple and consecutive planar extensions using the standard Flory–Rehner thermodynamics. We show explicitly how the stress relaxation at a given extension depends on both the number and magnitude of incremental steps that lead to such extension. These results have the potential to impact the design of hydrogel-based actuators where a prescribed extension can be achieved through several consecutive extensions. [ABSTRACT FROM AUTHOR]

Highlights • The dichotomy motor-brake is used to explain the diastolic functioning of the heart in a concise and comprehensible way. • The notion of active stretch is used to describe muscle contraction and to argue that a muscle can both produce (motor) and hamper (brake) motion. • The contractions in the left ventricle and atrium have distinguishing features in each phase of the cardiac cycle. • The time course of the contraction brings to light the active role of the ventricle during the filling phase. • Despite its simplicity, our model clearly prompts at the existence of an active role of the ventricles during the filling phase. Abstract The controversy between passive and active ventricular filling has been debated for decades and the question about the existence of an active diastole remains open. In this work, we advocate the model of active diastole by considering the heart as a suction pump and we add some more clues to support this point of view by the analysis of the pressure-volume (PV) loops of the left heart, comprising of the left ventricle (LV) and atrium (LA). Our working hypothesis is based on the dichotomy motor-brake: the cardiac muscle can act as a motor, when shortening against a load, or as a brake, when lengthening to a load. We discuss our hypothesis by means of a lumped model of the left heart, where both chambers are considered as hollow spherical shells. The notion of active stretch, introduced to describe the contractile behavior of the muscle fibers, plays a major role in our model. Then, the contraction of the muscle is related to the pressure and volume of the chamber through a nonlinear hyperelastic energy density function. Despite its simplicity, the model enlightens some important features of the LV-LA coupling and of the pumping function of the heart. Based on experimental PV data of the left heart of a normal human subject, it is shown that the contraction patterns of the LV and LA are synchronized with each other and have distinguishing features in each phase of the cardiac cycle. These results highlight the interplay between the two chambers and support the idea that the heart may act as a suction pump functioning in turn as a motor or a brake in order to meet specific demands in each phase of the cardiac cycle. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The piston problem for a hyperelastic hyperbolic conservative model where the stored energy is given in separable form is studied. The eigenfields corresponding to the hyperbolic system are of three types: linearly degenerate fields (corresponding to the contact characteristics), the fields which are genuinely nonlinear in the sense of Lax (corresponding to longitudinal waves), and, finally, nonlinear fields which are not genuinely nonlinear (corresponding to transverse waves). Taking the initial state free of stresses, we presented possible auto-similar solutions to the piston problem. In particular, we have shown that the equations admit transverse shock waves having a remarkable property: the solid density is decreasing through such a shock, it is thus a 'rarefaction' shock. [ABSTRACT FROM AUTHOR]

We explored the morphological organization of the skull within Crocodylidae, analyzing functional and phylogenetic interactions between its 2 constituent functional modules: the rostrum and the postrostrum. We used geometric morphometrics to identify localized shape changes, focusing on the differences between the major clades of the crown-group Crocodylia: Alligatoridae and Crocodylidae. We used published bite performance data to correlate rostral function with postrostral morphology. The skull modules appear more integrated within Alligatoridae than within Crocodyliade. Phylogenetic effects on shape variation are more evident in Alligatoridae than in Crocodylidae, where functional parameters concerning the rostral morphology are proportionally more important than phylogeny. Long-snouted species are characterized by low structural performance, which is significantly associated with a reduction of the pterygoid-quadrate cranial nipper, suggesting that the nipper is important for the ingestion of large food items in generalist species. This functional association is coupled with a significant evolutionary allometry at the clade level, while Alligatoridae and Crocodylidae show different degrees of evolutionary allometry for their entire shape and rostrum. The postrostrum is more conservative than the rostrum in terms of morphospace occupation, evolutionary allometry and phylogenetic signal. [ABSTRACT FROM AUTHOR]

Cat-like carnivorous mammals represent a relatively homogeneous group of species whose morphology appears constrained by exclusive adaptations for meat eating. We present the most comprehensive data set of extant and extinct cat-like species to test for evolutionary transformations in size, shape and mechanical performance, that is, von Mises stress and surface traction, of the mandible. Size and shape were both quantified by means of geometric morphometrics, whereas mechanical performance was assessed applying finite element models to 2D geometry of the mandible. Additionally, we present the first almost complete composite phylogeny of cat-like carnivorans for which well-preserved mandibles are known, including representatives of 35 extant and 59 extinct species of Felidae, Nimravidae, and Barbourofelidae. This phylogeny was used to test morphological differentiation, allometry, and covariation of mandible parts within and among clades. After taking phylogeny into account, we found that both allometry and mechanical variables exhibit a significant impact on mandible shape. We also tested whether mechanical performance was linked to morphological integration. Mechanical stress at the coronoid process is higher in sabertoothed cats than in any other clade. This is strongly related to the high degree of covariation within modules of sabertooths mandibles. We found significant correlation between integration at the clade level and per-clade averaged stress values, on both original data and by partialling out interclade allometry from shapes when calculating integration. This suggests a strong interaction between natural selection and the evolution of developmental and functional modules at the clade level. [Comparative methods; felidae mandible; finite element analysis; geometric morphometrics; morphological integration; sabertooth; structural performance] [ABSTRACT FROM PUBLISHER]

We investigate the swelling dynamics driven by solvent absorption in a hydrogel sphere immersed in a solvent bath, through a computational model and a numerical study. We extensively describe the transient process from dry to wet and discuss the onset of surface instabilities through a measure of the lack of smoothness of the outer surface and a morphological pattern of that surface with respect to the two material paremeters driving the swelling dynamics. [ABSTRACT FROM AUTHOR]

Abstract: We are interested in developing a simple model to investigate blood–vessel interactions in a finite arterial segment of the cardiovascular tree. For this purpose, we developed a continuum model for a vascular segment, and we coupled it with a discrete model for the remaining systemic circulation. In working out the modeling, we addressed some main issues, such as the nonlinearity of blood flow, the compliance of the vessel and the prestress state of the artery walls, that is always present aside from the filling of blood. Moreover, we set a discrete model capable of providing appropriate boundary conditions to the continuum model, by reproducing the proper waveforms entering the vessel and avoiding spurious reflections. [Copyright &y& Elsevier]

• Bulk contraction of active gels is modeled in the perspective of continuum mechanics. • Species diffusion, polymer deformation and material remodeling are strongly coupled. • A geometrical visualization of steady stress-free states in a state space is presented. • Not uniform bulk contraction generates local stress/strain actuators. In active gels, liquid redistribution, network deformation and material remodeling due to bulk activation, mimicking the presence of molecular motors, are strongly coupled. We present a consistent mathematical model capable to gain a deep understanding of the phenomenon in both steady and transient conditions. With explicit reference to active gel spheres, we evidence the role that not uniform bulk activation may have in generating local stress or strain actuators based on liquid redistribution. [ABSTRACT FROM AUTHOR]

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. 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. [ABSTRACT FROM AUTHOR]

The article discusses study of muscle contraction and pumping action in the left heart through pressure-volume (PV) relationships for the both left ventricle (LV) and the left atrium. Topics discussed mathematical models representing heart function, quantitative understanding of ventricular-atrial coupling, and closure of the mitral valve.

The article discusses muscle contraction and pressure-volume loops in the left-heart. It discusses establishment of pressure-volume analysis as a robust way to characterize the cardiac function. It also discusses quantitative understanding of ventricular-atrial coupling and its response to complex load alterations.

Abstract Wood is a natural engineering material that has traditionally been exploited in design for a wide variety of applications. The recent demand for sustainable material and construction processes in the construction industry has triggered a renewed interest and research in the inherent properties of wood and their derived applications, and specifically for developing low-tech architectural adaptive systems. This paper focuses on the physical and computational modeling of the morphing behavior of wood through hygroscopic expansion or contraction to a high degree of precision. The amount of stress related to the hygroscopic shrinking or swelling ranges from almost zero to high values, and its prediction is fundamental to alleviate any fatigue challenges. The capability of designing wood composite whose stress state remains limited under changes of the environmental humidity is beneficial for any engineering application subjected to a repeated reversal of loading such as adaptive systems. In this paper, a mechanical model, together with its numerical implementation, is presented; the model is benchmarked against some prototypical experiments, performed by using real material parameters. The control parameter in the model is the relative moisture change in wood that determines the orthotropic swelling/de-swelling phenomenon, and is coupled with the elastic behavior of wood. This model is integrated into a programmable matter design approach that combines physical and computational exploration. The approach is illustrated for a hygro-morphic building façade panel. The approaches and algorithms presented in this paper have further applications for computer-aided design of smart materials and systems with interchanging functionalities. Highlights • Physical exploration of the effect of wood characteristics on its morphing behavior. • Generalized orthotropic model for both elastic and hygroscopic behavior of wood. • Physical and numerical programmable matter approach for hygroscopic adaptive systems. [ABSTRACT FROM AUTHOR]

In landmarks-based Shape Analysis size is measured, in most cases, with Centroid Size. Changes in shape are decomposed in affine and non affine components. Furthermore the non affine component can be in turn decomposed in a series of local deformations (partial warps). If the extent of deformation between two shapes is small, the difference between Centroid Size and m-Volume increment is barely appreciable. In medical imaging applied to soft tissues bodies can undergo very large deformations, involving large changes in size. The cardiac example, analyzed in the present paper, shows changes in m-Volume that can reach the 60%. We show here that standard Geometric Morphometrics 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 space tangent to the shape space in affine and non affine components is enriched to include also the change in size, in order 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 m-Volume rather than change in Centroid Size. This leads to a redefinition of some aspects of the Kendall’s size-and-shape space without losing Kendall’s original formulation. This new formulation is discussed by means of simulated examples using 2D and 3D platonic shapes as well as a real example from clinical 3D echocardiographic data. We demonstrate that our decomposition based approaches discriminate very effectively healthy subjects from patients affected by Hypertrophic Cardiomyopathy. [ABSTRACT FROM AUTHOR]

Ceratopsidae represents a group of quadrupedal herbivorous dinosaurs that inhabited western North America and eastern Asia during the Late Cretaceous. Although horns and frills of the cranium are highly variable across species, the lower jaw historically has been considered to be relatively conservative in morphology. Here, the lower jaws from 58 specimens representing 21 ceratopsoid taxa were sampled, using geometric morphometrics and 2D finite element analysis (FEA) to explore differences in morphology and mechanical performance across Ceratopsoidea (the clade including Ceratopsidae, Turanoceratops and Zuniceratops). Principal component analyses and non-parametric permuted manovas highlight Triceratopsini as a morphologically distinct clade within the sample. A relatively robust and elongate dentary, a larger and more elongated coronoid process, and a small and dorso-ventrally compressed angular characterize this clade, as well as the absolutely larger size. By contrast, non-triceratopsin chasmosaurines, Centrosaurini and Pachyrhinosaurini have similar morphologies to each other. Zuniceratops and Avaceratops are distinct from other taxa. No differences in size between Pachyrhinosaurini and Centrosaurini are recovered using non-parametric permuted anovas. Structural performance, as evaluated using a 2D FEA, is similar across all groups as measured by overall stress, with the exception of Triceratopsini. Shape, size and stress are phylogenetically constrained. A longer dentary as well as a long coronoid process result in a lower jaw that is reconstructed as relatively much more stressed in triceratopsins. [ABSTRACT FROM AUTHOR]

Luxuriant, bushy antlers, bizarre crests, and huge, twisting horns and tusks are conventionally understood as products of sexual selection. This view stems from both direct observation and from the empirical finding that the size of these structures grows faster than body size (i.e., ornament size shows positive allometry).We contend that the familiar evolutionary increase in the complexity of ornaments over time in many animal clades is decoupled from ornament size evolution. Increased body size comes with extended growth. Since growth scales to the quarter power of body size, we predicted that ornament complexity should scale according to the quarter power law as well, irrespective of the role of sexual selection in the evolution and function of the ornament. To test this hypothesis, we selected three clades (ammonites, deer, and ceratopsian dinosaurs) whose species bore ornaments that differ in terms of the importance of sexual selection to their evolution.We found that the exponent of the regression of ornament complexity to body size is the same for the three groups and is statistically indistinguishable from0.25.We suggest that the evolution of ornament complexity is a by-product of Cope's rule. We argue that although sexual selection may control size in most ornaments, it does not influence their shape. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we are presenting a method for achieving strain analysis of cardiac tissues from 3D ultrasound images. First of all, the left ventricle was segmented by an iterative snake algorithm which exploits the gradient vector flow field estimated from the 3D images. Once the ventricle was segmented, a set of points located on the external surface of the ventricle was selected. The movement of all these points along the entire cardiac cycle was calculated by using speckle tracking techniques. The strain of all the segments connecting two of the considered points was computed, together with the rotation of the points along the long axis of the ventricle. The method was tested on images acquired from 10 subjects: three with heart pathologies, five healthy subjects, and two patients with mild hypertrophy. Preliminary results showed the feasibility of characterizing healthy subjects and patients with well-defined heart pathologies by using the outcomes achieved with the strain analysis. [ABSTRACT FROM AUTHOR]

The study aims to highlight the differences between growth-by-diffusion, based on the elastic stretch of the polymeric network, and growth-by-activity, based on the remodeling of the network, in polymer gels. The study is based on a multiphysics continuum model, which allows to describe the interactions between mechanics, diffusion and growth through a system of nonlinear partial differential equations. We discuss the differences between growth-by-diffusion and growth-by-activity and evidence the role of the corresponding characteristic times in the case of homogeneous activation source. We also show as a huge variety of three-dimensional shapes can be generated by controlling the distribution of the activation sources. [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate human left ventricular heart morphological changes in time among 17 healthy subjects. Preliminarily, 2 patients with volumetric overload due to aortic insufficiency were added to our analyses. We propose a special strategy to compare the shape, orientation and size of cardiac cycle’s morphological trajectories in time. We used 3D data obtained by Speckle Tracking Echocardiography in order to detect semi-automated and homologous landmarks clouds as proxies of left ventricular heart morphology. An extended Geometric Morphometrics toolkit in order to distinguish between intra- and inter-individual shape variations was used. Shape of trajectories with inter-individual variation were compared under the assumption that trajectories attributes, estimated at electrophysiologically homologous times are expressions of left ventricular heart function. We found that shape analysis as commonly applied in Geometric Morphometrics studies fails in identifying a proper morpho-space to compare the shape of morphological trajectories in time. To overcome this problem, we performed a special type of Riemannian Parallel Transport, called “linear shift”. Whereas the two patients with aortic insufficiency were not differentiated in the static shape analysis from the healthy subjects, they set apart significantly in the analyses of motion trajectory’s shape and orientation. We found that in healthy subjects, the variations due to inter-individual morphological differences were not related to shape and orientation of morphological trajectories. Principal Component Analysis showed that volumetric contraction, torsion and twist are differently distributed on different axes. Moreover, global shape change appeared to be more correlated with endocardial shape change than with the epicardial one. Finally, the total shape variation occurring among different subjects was significantly larger than that observable across properly defined morphological trajectories. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]