Absence Of External Loads Research Articles

Reconstruction of the local contractility of the cardiac muscle from deficient apparent kinematics

Active solids are a large class of materials, including both living soft tissues and artificial matter, that share the ability to undergo strain even in absence of external loads. While in engineered materials the actuation is typically designed a priori, in natural materials it is an unknown of the problem. In such a framework, the identification of inactive regions in active materials is of particular interest. An example of paramount relevance is cardiac mechanics and the assessment of regions of the cardiac muscle with impaired contractility. The impossibility to measure the local active forces directly suggests us to develop a novel methodology exploiting kinematic data from clinical images by a variational approach to reconstruct the local contractility of the cardiac muscle. By finding the stationary points of a suitable cost functional we recover the contractility map of the muscle. Numerical experiments, including severe conditions with added noise to model uncertainties, and data knowledge limited to the boundary, demonstrate the effectiveness of our approach. Unlike other methods, we provide a spatially continuous recovery of the contractility map without compromising the computational efficiency.

Reconstructing relaxed configurations in elastic bodies: Mathematical formulations and numerical methods for cardiac modeling

Modeling the behavior of biological tissues and organs often necessitates the knowledge of their shape in the absence of external loads. However, when their geometry is acquired in-vivo through imaging techniques, bodies are typically subject to mechanical deformation due to the presence of external forces, and the load-free configuration needs to be reconstructed. This paper addresses this crucial and frequently overlooked topic, known as the inverse elasticity problem (IEP), by delving into both theoretical and numerical aspects, with a particular focus on cardiac mechanics. In this work, we extend Shield’s seminal work to determine the structure of the IEP with arbitrary material inhomogeneities and in the presence of both body and active forces. These aspects are fundamental in computational cardiology, and we show that they may break the variational structure of the inverse problem. In addition, we show that the inverse problem might have no solution even in the presence of constant Neumann boundary conditions and a polyconvex strain energy functional. We then present the results of extensive numerical tests to validate our theoretical framework, and to characterize the computational challenges associated with a direct numerical approximation of the IEP. Specifically, we show that this framework outperforms existing approaches both in terms of robustness and optimality, such as Sellier’s iterative procedure, even when the latter is improved with acceleration techniques. A notable discovery is that multigrid preconditioners are, in contrast to standard elasticity, not efficient, where a one-level additive Schwarz and generalized Dryja–Smith–Widlund provide a much more reliable alternative. Finally, we successfully address the IEP for a full-heart geometry, demonstrating that the IEP formulation can compute the stress-free configuration in real-life scenarios where Sellier’s algorithm proves inadequate.

Mixed Virtual Element approximation of linear acoustic wave equation

Abstract We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In the absence of external loads, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical $\theta $-method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing an optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering applications of the method.

Thermo-mechanical analysis of evacuated honeycomb structures. Application to ultra-high temperature thermal insulation and structural elements

Thermal energy storage is one of the key vectors to achieve a full decarbonisation of the energy sector in order to mitigate the effects of climate change. Specifically, ultra-high temperature (>900 K) thermal storage is of particular importance as it unlocks greater energy densities and round-trip efficiencies, among other benefits. However, increased thermal losses and reduced structural strength of insulation materials constitute significant challenges in that temperature range, and insulating structures are required which retain thermal efficiency and mechanical robustness at elevated temperatures. In this paper, a thermo-mechanical analysis of evacuated honeycomb structures at ultra-high temperature is proposed, and the impact of mechanical stresses on their structural integrity and thermal efficiency is investigated. In particular, plastic yield, brittle fracture, buckling, sagging and creep are considered for mechanical failure modes. The structures are then optimised for thermal insulation efficiency whilst avoiding mechanical failure.It is found that mechanical stresses must be considered even in the absence of external loads as such structures may collapse under their own weight over time. However it is also shown that, in the absence of external loads, structurally-sound honeycomb structures at elevated temperatures (>1200 K) remain 3-14 times more thermally efficient than other non load-bearing and structurally-ideal market-available technologies, while the presence of an external load inevitably reduces their thermal efficiency. But wall thicknesses as low as 1μm may be required, which is below today’s manufacturing capacities and highlights the importance of further research into manufacturing processes beyond what is achievable today.

Closed-form solution of Euler–Bernoulli frames in the frequency domain

This paper presents the frequency domain formulation of the Green’s Functions Stiffness Method (GFSM) for plane Euler–Bernoulli frames subjected to arbitrary external loads (forces and bending moments). The GFSM is a generalization of the Stiffness Method (SM) or the Finite Element Method (FEM) that computes the closed-form analytical structural response by decomposing it into a homogeneous and a fixed or particular responses. The homogeneous response is obtained from the displacements at the element ends in the absence of external loads, while the fixed response is calculated using the Green’s functions of fixed elements through a superposition integral that includes the external loads. The GFSM can be categorized as a mesh reduction method that generalizes the Spectral Element Method (SEM), in which the structural response is first obtained in the frequency domain and then converted to the time domain using the Fast Fourier Transform algorithm. Compared to the SEM, the significant advantage of the GFSM is that the frequency domain structural response is exact and avoids the need for dense meshes when external loads have complex patterns. Three examples are presented to demonstrate the effectiveness of the GFSM, each of which shows excellent agreement when compared to the FEM solution even using far fewer elements.

How does prestrain in the tympanic membrane affect middle-ear function? A finite-element model study in rabbit

Experiments have shown that prestrain exists in the rabbit tympanic membrane (TM), also in the absence of external loads. To date, it is unclear how prestrain influences the vibration response of the middle ear (ME). In this study, a detailed 3D finite-element model of the rabbit ME was constructed based on experimentally validated material properties. The model incorporates different degrees of prestrain in the TM and simulates the ME vibration response to sound as a linear harmonic perturbation around the prestressed reference state. To account for finite deformations associated with large prestrains, a framework was developed that iteratively updates the initial unstrained geometry until the prestrained geometry is in agreement with the given reference geometry. After validating the model using quasi-static and acoustic measurement data, it was shown that small levels of prestrain already have a substantial impact on the normal umbo and footplate response due to a phenomenon known as prestress stiffening. Although the approach is not preferable, it was possible to replicate the effect of prestrain in the normal ME by appropriately scaling the elastic moduli and damping factors in the base model. To evaluate the effect of possible changes in TM prestrain when the normal state of the ear is altered due to pathological modifications in the ME structure, we created a model with a perforation in the TM. It was shown that the change in vibration response after perforation is affected at low frequencies by a release of TM prestrain. In future studies, it may be necessary to incorporate prestrain in ME models to better understand the function of the diseased or reconstructed ME, which may be relevant for the development of reconstructive tissue grafts in the middle ear.

Prestrain in the rabbit eardrum measured by digital image correlation and micro-incisions

Prestrain in the absence of external loads can have an important effect on the vibrational behavior of mechanical systems such as the middle ear. Studies that measure tympanic membrane (TM) prestrain are scarce, however, and provide no conclusive answer on the existence and nature of the prestrain. In this study, prestrain is measured in the TM of cadaveric rabbit ears by stereo digital image correlation. To release the prestrain, straight incisions of 0.33 mm are made on different locations in the TM with a direction parallel to either the radial or circular fibers in the membrane. The effect of sample dehydration during different stages in the experimental procedure is assessed and eliminated by rehydrating the samples directly before each measurement. The measurements demonstrate average prestrain values around the incisions between 3.52±2.34% and 13.62±7.92% for the different locations, with a noise floor of 0.07%. No clear differences were found between the prestrain values obtained for radial and circular incisions. Observed local variations in TM prestrain could not be clearly related to specific locations on the TM. The results suggest that TM prestrain may need to be considered in future studies of middle-ear function if the findings can be confirmed in human ears.

A novel method for determining the feasible integral self-stress states for tensegrity structures

Abstract The form-finding analysis is a crucial step for determining the stable self-equilibrated states for tensegrity structures, in the absence of external loads. This form-finding problem leads to the evaluation of both the self-stress in the elements and the shape of the tensegrity structure. This paper presents a novel method for determining feasible integral self-stress states for tensegrity structures, that is self-equilibrated states consistent with the unilateral behaviour of the elements, struts in compression and cables in tension, and with the symmetry properties of the structure. In particular, once defined the connectivity between the elements and the nodal coordinates, the feasible self-stress states are determined by suitably investigating the Distributed Static Indeterminacy (DSI). The proposed method allows for obtaining feasible integral self-stress solutions by a unique Singular Value Decomposition (SVD) of the equilibrium matrix, whereas other approaches in the literature require two SVD. Moreover, the proposed approach allows for effectively determining the Force Denstiy matrix, whose properties are strictly related to the super-stability of the tensegrity structures. Three tensegrity structures were studied in order to assess and discuss the efficiency and accuracy of the proposed innovative method.

Mechanochemical properties of human myosin-1C are modulated by isoform-specific differences in the N-terminal extension

Myosin-1C is a single-headed, short-tailed member of the myosin class I subfamily that supports a variety of actin-based functions in the cytosol and nucleus. In vertebrates, alternative splicing of the MYO1C gene leads to the production of three isoforms, myosin-1C0, myosin-1C16, and myosin-1C35, that carry N-terminal extensions of different lengths. However, it is not clear how these extensions affect the chemomechanical coupling of human myosin-1C isoforms. Here, we report on the motor activity of the different myosin-1C isoforms measuring the unloaded velocities of constructs lacking the C-terminal lipid-binding domain on nitrocellulose-coated glass surfaces and full-length constructs on reconstituted, supported lipid bilayers. The higher yields of purified proteins obtained with constructs lacking the lipid-binding domain allowed a detailed characterization of the individual kinetic steps of human myosin-1C isoforms in their productive interaction with nucleotides and filamentous actin. Isoform-specific differences include 18-fold changes in the maximum power output per myosin-1C motor and 4-fold changes in the velocity and the resistive force at which maximum power output occurs. Our results support a model in which the isoform-specific N-terminal extensions affect chemomechanical coupling by combined steric and allosteric effects, thereby reducing both the length of the working stroke and the rate of ADP release in the absence of external loads by a factor of 2 for myosin-1C35. As the large change in maximum power output shows, the functional differences between the isoforms are further amplified by the presence of external loads.

The Geometrical Arrangement of Knee Constraints That Makes Natural Motion Possible: Theoretical and Experimental Analysis.

The study of the knee natural motion, namely the unresisted motion that the knee exhibits in the absence of external loads, provides insights into the physiology of this articulation. The natural motion represents the baseline condition upon which deformations of its passive structures (i.e., ligaments and cartilage) take place when loads are applied. Moreover, during natural motion, the strain energy density stored within ligaments and cartilage is minimized. This reduces the chance of microdamage occurrences and the corresponding metabolic cost for tissue repairing. The study of the knee natural motion is thus fundamental in understanding the joint physiology. This paper shows that the line of action of resultant forces of all the knee constraints provided by the passive structures must intersect the instantaneous helical axis (IHA) to make the knee natural motion possible. In other words, the lines of action of all these constraints must cross the same line at each flexion angle to guarantee the natural motion of the joint. This geometrical property is first proven theoretically and then verified in four in vitro and one in vivo experiments. The geometrical characterization of the knee natural motion presented in this study provides a fundamental property that must be satisfied to allow the correct joint mobility. The knowledge of this property may thus allow the definition of better models, treatments, and devices.

Modeling Restrained Shrinkage Induced Cracking in Concrete Rings Using the Thick Level Set Approach

Modeling restrained shrinkage-induced damage and cracking in concrete is addressed herein. The novel Thick Level Set (TLS) damage growth and crack propagation model is used and adapted by introducing shrinkage contribution into the formulation. The TLS capacity to predict damage evolution, crack initiation and growth triggered by restrained shrinkage in absence of external loads is evaluated. A study dealing with shrinkage-induced cracking in elliptical concrete rings is presented herein. Key results such as the effect of rings oblateness on stress distribution and critical shrinkage strain needed to initiate damage are highlighted. In addition, crack positions are compared to those observed in experiments and are found satisfactory.

Spontaneous morphing of equibiaxially pre-stretched elastic bilayers: The role of sample geometry

An elastic bilayer, consisting of an equibiaxially pre-stretched sheet bonded to a stress-free one, spontaneously morphs into curved shapes in the absence of external loads or constraints. Using experiments and numerical simulations, we explore the role of geometry for square and rectangular samples in determining the equilibrium shape of the system, for a fixed pre-stretch. We classify the observed shapes over a wide range of aspect ratios according to their curvatures and compare measured and computed values, which show good agreement. In particular, as the bilayer becomes thinner, a bifurcation of the principal curvatures occurs, which separates two scaling regimes for the energy of the system. We characterize the transition between these two regimes and show the peculiar features that distinguish square from rectangular samples. The results for our model bilayer system may help explaining morphing in more complex systems made of active materials.

An energy-momentum co-rotational formulation for nonlinear dynamics of planar beams

This article presents an energy-momentum integration scheme for the nonlinear dynamic analysis of planar Euler-Bernoulli beams. The co-rotational approach is adopted to describe the kinematics of the beam and Hermitian functions are used to interpolate the local transverse displacements. In this paper, the same kinematic description is used to derive both the elastic and the inertia terms. The classical midpoint rule is used to integrate the dynamic equations. The central idea, to ensure energy and momenta conservation, is to apply the classical midpoint rule to both the kinematic and the strain quantities. This idea, developed by one of the authors in previous work, is applied here in the context of the co-rotational formulation to the first time. By doing so, we circumvent the nonlinear geometric equations relating the displacement to the strain which is the origin of many numerical difficulties. It is rigorously shown that the proposed method conserves the total energy of the system and, in absence of external loads, the linear and angular momenta remain constant. The accuracy and stability of the proposed algorithm, especially in long term dynamics with a very large number of time steps, is assessed through four numerical examples.

Highly loaded behavior of kinesins increases the robustness of transport under high resisting loads.

Kinesins are nano-sized biological motors which walk by repeating a mechanochemical cycle. A single kinesin molecule is able to transport its cargo about 1 μm in the absence of external loads. However, kinesins perform much longer range transport in cells by working collectively. This long range of transport by a team of kinesins is surprising because the motion of the cargo in cells can be hindered by other particles. To reveal how the kinesins are able to accomplish their tasks of transport in harsh intracellular circumstances, stochastic studies on the kinesin motion are performed by considering the binding and unbinding of kinesins to microtubules and their dependence on the force acting on kinesin molecules. The unbinding probabilities corresponding to each mechanochemical state of kinesin are modeled. The statistical characterization of the instants and locations of binding are captured by computing the probability of unbound kinesin being at given locations. It is predicted that a group of kinesins has a more efficient transport than a single kinesin from the perspective of velocity and run length. Particularly, when large loads are applied, the leading kinesin remains bound to the microtubule for long time which increases the chances of the other kinesins to bind to the microtubule. To predict effects of this behavior of the leading kinesin under large loads on the collective transport, the motion of the cargo is studied when the cargo confronts obstacles. The result suggests that the behavior of kinesins under large loads prevents the early termination of the transport which can be caused by the interference with the static or moving obstacles.

The Stress and Strain States of the Posterior Annulus Under Flexion

Experimental measurement of bovine caudal annulus fibrosus (AF) biaxial stress-strain states for unloaded motion segments and those loaded in flexion combined with axial compression. To measure AF biaxial stress-strain states for motion segments free of external loads and those loaded in flexion combined with axial compression. In vitro, cyclically loaded flexion is associated with the complete rupture of the AF in acute cases. However, currently little is known about the tissue-level stress and strain states in this damaging load configuration. Surface strains of the posterior annulus were measured, during loading of intact bovine caudal motion segments in flexion combined with axial compression, using the unloaded motion segment as a reference. Planar annulus specimens were excised from the motion segments, and the measured strains from the unloaded and flexed intact discs were applied to the excised specimens in a planar biaxial stress-strain device to determine the stress state in each condition. The annulus strain state of the flexed disc was input into a previously developed nonlinear strain energy function with additive terms that directly represent the structural features of the AF. The AF posterior surface strain for motion segments in flexion combined with compression was -0.1087 (circumferential) and 0.1051 (axial) on average. This strain state corresponded to average biaxial stresses in the annulus of 0.046 MPa (circumferential) and 0.224 MPa (axial). The planar AF specimens contracted after sectioning, resulting in small negative strains. In the absence of external loads on a motion segment, the annulus was approximately in a state of equibiaxial tensile stress. The theoretical analysis indicated that collagen crosslinks store a greater portion of the strain energy than the matrix or collagen fibers. In flexion combined with compression, the posterior AF stress state is biaxial and is much larger in the axial direction in bovine caudal discs.

Active Stress vs. Active Strain in Mechanobiology: Constitutive Issues

Many biological tissues exhibit a non-standard continuum mechanics behavior: they are able to modify their placement in absence of external loads. The activity of the muscles is usually represented in solid mechanics in terms of an active stress, to be added to the standard one. A less popular approach is to introduce a multiplicative decomposition of the tensor gradient of deformation in two factors: the passive and the active one. Both approaches should satisfy due mathematical properties, namely frame indifference and ellipticity of the total stress. At the same time, the constitutive laws should reproduce the observed physiological behavior of the specific living tissue. In this paper we focus on cardiac contractility. We review some constitutive examples of active stress and active strain taken from the literature and we discuss them in terms of precise mathematical and physiological properties. These arguments naturally suggest new possible models.

Spontaneous Cavitation in Growing Elastic Membranes

One of the possible effects of volumetric growth in elastic materials is the creation of residual stresses. These stresses are known to change many of the classical properties of the material and have been studied extensively in the context of volumetric growth in biomechanics. Here we consider the problem of elastic cavitation in a growing compressible elastic membrane. Growth is taken to be homogeneous but anisotropic, and the membrane is assumed to remain axisymmetric during growth and deformation. We prove that neo-Hookean membranes cannot cavitate, but for Varga elastic materials we find conditions under which the material exhibits spontaneous cavitation in the absence of external loads, in marked distinction from the cavitation problem without growth.

Characterisation of residual stresses in a single fibre composite with FBG sensor

Residual stresses are self-equilibrated stresses supported by a body in the absence of external loads. They play a significant role in the dimensional stability and mechanical performance of a composite structure. Optical fibre Bragg grating (FBG) sensors are excellent non-destructive tools for internal strain measurements in composite materials especially when are used simultaneously as reinforcement and sensor. In this study, long-gauge FBGs are introduced in single fibre cylindrical specimens of epoxy. The strain response of the fibre due to shrinkage of the epoxy during polymerisation is investigated using optical low-coherence reflectometry which provides a direct reconstruction of the axial strain field along the reinforcing optical fibre. In addition, Bragg wavelength distributions have been determined as functions of the depth of circumferential cracks in the radial direction of the specimen and normal to the fibre. These results are combined with a modified crack compliance method to obtain the radial dependence of the residual strains in the matrix. The experimental data are coupled with numerical simulations of an equivalent thermo-elastic matrix material to obtain the entire residual stress field in the epoxy. The disturbance of the residual stress field due to the presence of the fibre extends up to 4 fibre radii. The results at the center of the specimen, where end effects are negligible, match very well with a theoretical model of a fibre in an infinite matrix. The shear stress evolution at the entry point of the fibre is very well described by using a sufficiently refined radial mesh. The redistribution of the residual stresses due to the introduction of a longitudinal crack has also been investigated. The resulting axial stress relaxation is well below the one obtained with the transversal circular crack and no birefringence in the fibre is detected.

Internal Stresses in a Solid with Dislocations

Formulas for calculating internal stresses in a material, generated by continuously distributed dislocations, are found on the basis of the gauge theory of defects. It is shown that internal stresses are self‐balanced and satisfy the equilibrium equations and boundary conditions in the absence of external loads.

Modeling failure of metallic glasses due to hydrogen embrittlement in the absence of external loads

A model is developed to describe the expansion of high-pressure hydrogen bubbles and propagation of cracks between them in the absence of external loads. The focus is on cracks that form during electrochemical hydrogen charging of amorphous Fe 80B 11Si 9 ribbons. A coupled diffusion/fracture mechanics approach is developed, allowing determination of the time to failure. Finite element analyses are carried out to determine the values of the stress intensity factor for cracks of different lengths, assuming linear elasticity. In addition, the volume of a bubble with edge cracks is related to the internal pressure. The relation between critical pressure and crack length is obtained from an appropriate value of the fracture toughness, K Ic. A criterion is proposed to obtain the pressure and volume as a function of the number of hydrogen moles within the bubble with edge cracks. Finite element analyses are also used to calculate the hydrogen concentration and hydrogen diffusion flux as a function of crack length and time. The time to failure as predicted from this model is of the same order of magnitude as that observed experimentally.