Elastic Solid Model Research Articles

On prescribing the number of singular points in a Cosserat-elastic solid

Abstract In a geometrically non-linear Cosserat model for micro-polar elastic solids, we prove that critical points of the Cosserat energy functional with an arbitrary large (finite) number of singularities do exist, whereas Cosserat energy minimizers are known to be locally Hölder continuous. To reach that goal, we first develop a technique to insert dipole pairs of singularities into smooth maps while controlling the amount of Cosserat energy needed to do so. We then use this method to force an arbitrary number of singular points into (weak) Cosserat-elastic solids by prescribing smooth boundary data. The boundary data themselves are given in such a way, that they contain no topological obstruction to regularity. Throughout this paper, we often exploit connections between harmonic maps and Cosserat-elastic solids, so that we are able to adapt and incorporate ideas of R. Hardt and F.-H. Lin for harmonic maps with singularities, as well as of F. Béthuel for dipole pairs of singularities.

Synthetic, self-oscillating vocal fold models for voice production researcha).

Sound for the human voice is produced by vocal fold flow-induced vibration and involves a complex coupling between flow dynamics, tissue motion, and acoustics. Over the past three decades, synthetic, self-oscillating vocal fold models have played an increasingly important role in the study of these complex physical interactions. In particular, two types of models have been established: "membranous" vocal fold models, such as a water-filled latex tube, and "elastic solid" models, such as ultrasoft silicone formed into a vocal fold-like shape and in some cases with multiple layers of differing stiffness to mimic the human vocal fold tissue structure. In this review, the designs, capabilities, and limitations of these two types of models are presented. Considerations unique to the implementation of elastic solid models, including fabrication processes and materials, are discussed. Applications in which these models have been used to study the underlying mechanical principles that govern phonation are surveyed, and experimental techniques and configurations are reviewed. Finally, recommendations for continued development of these models for even more lifelike response and clinical relevance are summarized.

A novel oblique impact model for elastic solids

Hertzian contact theory is a key step in modeling low-velocity impact on structures. This study presents an oblique contact model for simulating the oblique impact contact between two elastic solids. The presented model can be used for analyzing contact responses between rigid and non-rigid bodies. The contact force and indentation predicted by the presented model were compared to available oblique impact experimental results in the literature. The results demonstrated high accuracy in predicting the impact response of a plane subjected to an impactor with different obliquely velocities. The numerical results show that with increase in the incident angle, the contact force and local indentation decrease significantly, while the contact time increases. The elastic modulus ratio between the indenter and the plane significantly affects the prediction of the contact force and indentation. Hence, it is essential to choose an impact contact model that will facilitate accurate evaluation of its impact response. The present results represent a useful method to assess the safety of different materials. In addition, the oblique impact model can also provide a theoretical framework for evaluating the obliquely low-velocity impacting the performance of beam, plate, and shell structural elements.

A collection of wet beam models for wave–ice interaction

Abstract. Theoretical models for the prediction of decay rate and dispersion process of gravity waves traveling into an integrated ice cover expanded over a long way are introduced. The term “wet beam” is chosen to refer to these models as they are developed by incorporating water-based damping and added mass forces. Presented wet beam models differ from each other according to the rheological behavior considered for the ice cover. Two-parameter viscoelastic solid models accommodating Kelvin–Voigt (KV) and Maxwell mechanisms along with a one-parameter elastic solid model are used to describe the rheological behavior of the ice layer. Quantitative comparison between the landfast ice field data and model predictions suggests that wet beam models, adopted with both KV and Maxwell mechanisms, predict the decay rate more accurately compared to a dry beam model. Furthermore, the wet beam models, adopted with both KV and Maxwell mechanisms, are found to construct decay rates of disintegrated ice fields, though they are built for a continuous ice field. Finally, it is found that wet beam models can accurately construct decay rate curves of freshwater ice, though they cannot predict the dispersion process of waves accurately. To overcome this limitation, three-parameter solid models, termed standard linear solid (SLS) mechanisms, are suggested to be used to re-formulate the dispersion relationship of wet beam models, which were seen to construct decay rates and dispersion curves of freshwater ice with an acceptable level of accuracy. Overall, the two-parameter wet beam dispersion relationships presented in this research are observed to predict decay rates and dispersion process of waves traveling into actual ice covers, though three-parameter wet beam models were seen to reconstruct the those of freshwater ice formed in a wave flume. The wet beam models presented in this research can be implemented in spectral models on a large geophysical scale.

Convexity and Quasiconvexity in a Cosserat Model for Fiber-Reinforced Elastic Solids

Convexity and Quasiconvexity in a Cosserat Model for Fiber-Reinforced Elastic Solids

Biomechanical characterization and modeling of human mesenchymal stem cells under compression

The application of microelectromechanical systems (MEMS) in biomedical devices has expanded vastly over the last few decades, with MEMS devices being developed to measure different characteristics of cells. The study of cell mechanics offers valuable understanding of cell viability and functionality. Cell biomechanics approaches also facilitate the characterization of important cell and tissue behaviors. In particular, understanding of the biological response of cells to their biomechanical environment would enhance the knowledge of how cellular responses correlate to tissue level characteristics and how some diseases, such as cancer, grow in the body. This study focuses on viscoelastic modeling of the behavior of a single suspended human mesenchymal stem cell (hMSC). Mechanical properties of hMSC cells are particularly important in tissue engineering and research for the treatment of cardiovascular diseases. We evaluated the elastic and viscoelastic properties of hMSC cells using a miniaturized custom-made BioMEMS device. Our results were compared to the elastic and viscoelastic properties measured by other methods such as atomic force microscopy (AFM) and micropipette aspiration. Different approaches were applied to model the experimentally obtained force data, including elastic and Standard Linear Solid (SLS) constitutive models, and the corresponding constants were derived. These values were compared to the ones in literature that were based on micropipette aspiration and AFM methods. We then utilized a tensegrity approach to model major parts of the internal structure of the cell and treat the cell as a network of viscoelastic microtubules and microfilaments, as opposed to a simple spherical blob. The results predicted from the tensegrity model were similar to the recorded experimental data.

Development of the scaled boundary finite element method for crack propagation modeling of elastic solids subjected to coupled thermo-mechanical loads

This study presents the development of the scaled boundary finite element method to model discrete crack propagation induced by thermal loads. The SBFEM excels in modeling stress singularities at sharp crack tips with high accuracy. Polygon meshes are used so that a robust local re-meshing algorithm can be utilized to propagate the crack. The scaled boundary finite element formulation for steady-state thermal stress analysis is presented. Following a scaled boundary finite element analysis of a given thermal problem, the effect of initial strains due to temperature is taken into account semi-analytically in a subsequent stress analysis. Several numerical examples are presented to validate the technique and illustrate its salient features.

A Short Review on Numerical Modelling Approaches for Seismic Evaluation Performance of Nuclear Power Plant Structures

Nuclear power plant (NPP) structures play a crucial role in protecting the safety of the whole plant. Overall, NPP structures have complex shapes and large dimensions. Therefore, the decision of an appropriate finite element model for seismic response analysis is important. This study presents a brief review of various numerical modelling approaches for seismic evaluation performance of NPP structures. Different conventional models, i.e. lumped-mass stick model (LMSM), full three-dimensional finite element model (3D FEM), elastic solid element model (ESM), and multi-layer shell model (MLSM), which have been applied in modelling nuclear structures, are introduced. Also, the advantages and drawbacks of those models are analysed. Furthermore, a new model namely, beam-truss model (BTM), which is recently proposed, is highlighted. It reveals that LMSM is the most simplified approach for structural modelling of NPP structures. However, it is normally used for linear analyses and not able to simulate the local behaviours and vertical responses of the complex NPP structures. Even though 3D FEM is the most sufficient method for nonlinear seismic response analyses, this approach is very time-consuming and costly computation. MLSM and BTM are recommended as practical and efficient models for nonlinear analyses of NPP structures.

Периодические контактные и смешанные задачи теории упругости (обзор)

Results are reviewed collected in the investigations of periodic contact and mixed problems of the plane, axially symmetric and spatial elasticity theory. Among mixed problems, cut (crack) problems are focused integral equations of which are connected with those for contact problems. The periodic contact problems stimulate research of the discrete contact of rough (wavy) surfaces. Together with classical elastic domains (half-plane, half-space, plane and full space), we consider periodic problems for cylinder, layer, cone and spatial wedge. Most publications including fun-damental ones by Westergaard and Shtaerman deals with plane periodic problems of the elasticity theory. Here, one can mention approaches based on complex variable functions, Fourier series, Green’s functions and potential func-tions. A fracture mechanics approach to the plane periodic contact problem was developed. Methods and approaches are considered which allow us to take friction forces, adhesion and wear into account in the periodic contact. For spatial periodic and doubly periodic contact and properly mixed problems, we describe such methods as the localiza-tion method, the asymptotic methods, the method of nonlinear boundary integral equations, the fast Fourier trans-form. The half-space is the simplest model for elastic solids. But for the simplest straight-line periodic punch system, some three-dimensional contact problems (normal contact or tangential contact for shifted cohesive coatings) turn out to be incorrect because their integral equations contain divergent series. Considering three-dimensional periodic problems, I.G. Goryacheva disposes circular punches in special way (circular orbits, polar coordinated are used for centers of the punches), in this case one can prove convergence of the series in the integral equation (it is important that the punches are circular). For the periodic problems for an elastic layer, V.M. Aleksandrov has shown that the series in integral equations converge but the kernels become more complicated. In the present paper, we demonstrate that for the straight-line periodic punch system of arbitrary form the contact problem for a half-space turns out to be correct in case of more complicated boundary conditions. Namely, it can be sliding support or rigid fixation of a half-plane on the half-space boundary, the half-plane boundary should be parallel to the straight-line (the punch system axis) for arbitrary finite distance between the parallel lines. On this way, for sliding support, the kernel of the period-ic problem integral equation kernel is free of integrals, it consists of single convergent series (normal contact, the kernel is given in two equivalent forms). We consider classical percolation (how neighboring contact domains pene-trate one to another, investigated by K.L. Johnson, V.A. Yastrebov with co-authors) for the three-dimensional periodic contact amplification as well as percolation for the straight-line punch system. A similar approach is suggested for the case of periodic tangential contact (coatings system cohesive with a half-space boundary shifted along its axis or perpendicular to it). Here, one can separate out unique solutions of auxiliary problems because the line of changing boundary conditions on the half-space boundary can provoke non-uniqueness. The method proposed opens possibility to consider more complicated three-dimensional periodic contact problems for straight-line punch systems with changing boundary conditions inside the period.

Bridging between material properties of proteins and the underlying molecular interactions.

In this work, we develop a novel method that bridges between material properties of proteins, particularly the modulus of elasticity, and the underlying molecular interactions. To this end, we employ both an all-atom normal mode analysis (NMA) model with the CHARMM force field and an elastic solid model for proteins and protein interfaces. And the "bridge" between the two models is a common physical property predictable by both models: the magnitude of thermal vibrations. This connection allows one to calibrate the Young's moduli of proteins and protein interface regions. We find that the Young's moduli of proteins are in the range of a few Gpa to 10 Gpa, while the Young's moduli of the interface regions are several times smaller. The work is significant as it represents the first attempt to systematically compute the elastic moduli of proteins from molecular interactions.

A study on the Gurtin–Murdoch model for spherical solids with surface tension

An investigation is made on the Gurtin–Murdoch (GM) model for soft spherical elastic solids with surface tension-induced bulk residual stress. Unlike the original GM model which treats the surface tension as a finite value but the surface tension-induced bulk residual stress as infinitesimal, the present model treats both consistently as finite values. Detailed comparisons between the present model and the GM model show that for a solid sphere (for which the surface tension-induced bulk residual stress remains uniform in the entire body), the GM model deviates considerably from the present model. Conversely, for a small cavity in an infinite body (for which the surface tension-induced bulk residual stress is significant only nearby the cavity’s surface but quickly diminishes with radial distance), the deviation between the present model and the original GM model is negligible. It is concluded that the surface tension-induced bulk residual stress, which is addressed in the present paper but ignored in the GM model, can have a significant effect on deformation and vibration of soft elastic solids when its magnitude does not vanishingly diminish within the body.

Integral representation for energies in linear elasticity with surface discontinuities

Abstract In this paper we prove an integral representation formula for a general class of energies defined on the space of generalized special functions of bounded deformation ( GSBD p {\mathrm{GSBD}^{p}} ) in arbitrary space dimensions. Functionals of this type naturally arise in the modeling of linear elastic solids with surface discontinuities including phenomena as fracture, damage, surface tension between different elastic phases, or material voids. Our approach is based on the global method for relaxation devised in [G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation, Arch. Ration. Mech. Anal. 145 1998, 1, 51–98] and a recent Korn-type inequality in GSBD p {\mathrm{GSBD}^{p}} , cf. [F. Cagnetti, A. Chambolle and L. Scardia, Korn and Poincaré–Korn inequalities for functions with a small jump set, preprint 2020]. Our general strategy also allows to generalize integral representation results in SBD p {\mathrm{SBD}^{p}} , obtained in dimension two [S. Conti, M. Focardi and F. Iurlano, Integral representation for functionals defined on SBD p \mathrm{SBD}^{p} in dimension two, Arch. Ration. Mech. Anal. 223 2017, 3, 1337–1374], to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation ( GSBV p {\mathrm{GSBV}^{p}} ).

Shape-preserving elastic solid models of macromolecules.

Mass-spring models have been a standard approach in molecular modeling for the last few decades, such as elastic network models (ENMs) that are widely used for normal mode analysis. In this work, we present a vastly different elastic solid model (ESM) of macromolecules that shares the same simplicity and efficiency as ENMs in producing the equilibrium dynamics and moreover, offers some significant new features that may greatly benefit the research community. ESM is different from ENM in that it treats macromolecules as elastic solids. Our particular version of ESM presented in this work, named αESM, captures the shape of a given biomolecule most economically using alpha shape, a well-established technique from the computational geometry community. Consequently, it can produce most economical coarse-grained models while faithfully preserving the shape and thus makes normal mode computations and visualization of extremely large complexes more manageable. Secondly, as a solid model, ESM’s close link to finite element analysis renders it ideally suited for studying mechanical responses of macromolecules under external force. Lastly, we show that ESM can be applied also to structures without atomic coordinates such as those from cryo-electron microscopy. The complete MATLAB code of αESM is provided.

Physics-constrained local convexity data-driven modeling of anisotropic nonlinear elastic solids

Abstract As characterization and modeling of complex materials by phenomenological models remains challenging, data-driven computing that performs physical simulations directly from material data has attracted considerable attention. Data-driven computing is a general computational mechanics framework that consists of a physical solver and a material solver, based on which data-driven solutions are obtained through minimization procedures. This work develops a new material solver built upon the local convexity-preserving reconstruction scheme by He and Chen (2020) A physics-constrained data-driven approach based on locally convex reconstruction for noisy database. Computer Methods in Applied Mechanics and Engineering 363, 112791 to model anisotropic nonlinear elastic solids. In this approach, a two-level local data search algorithm for material anisotropy is introduced into the material solver in online data-driven computing. A material anisotropic state characterizing the underlying material orientation is used for the manifold learning projection in the material solver. The performance of the proposed data-driven framework with noiseless and noisy material data is validated by solving two benchmark problems with synthetic material data. The data-driven solutions are compared with the constitutive model-based reference solutions to demonstrate the effectiveness of the proposed methods.

How vision governs the collective behaviour of dense cycling pelotons.

In densely packed groups demonstrating collective behaviour, such as bird flocks, fish schools or packs of bicycle racers (cycling pelotons), information propagates over a network, with individuals sensing and reacting to stimuli over relatively short space and time scales. What remains elusive is a robust, mechanistic understanding of how sensory system properties affect interactions, information propagation and emergent behaviour. Here, we show through direct observation how the spatio-temporal limits of the human visual sensory system govern local interactions and set the network structure in large, dense collections of cyclists. We found that cyclists align in patterns within a ± 30° arc corresponding to the human near-peripheral visual field, in order to safely accommodate motion perturbations. Furthermore, the group structure changes near the end of the race, suggesting a narrowing of the used field of vision. This change is consistent with established theory in psychology linking increased physical exertion to the decreased field of perception. Our results show how vision, modulated by arousal-dependent neurological effects, sets the local arrangement of cyclists, the mechanisms of interaction and the implicit communication across the group. We furthermore describe information propagation phenomena with an analogous elastic solid mechanics model. We anticipate our mechanistic description will enable a more detailed understanding of the interaction principles for collective behaviour in a variety of animals.

The stress-strain relationship of liquid marbles under compression

Liquid marbles can be characterized using elastic solid models consisting of a liquid surrounded by a soft solid membrane. The elastic properties of liquid marbles determine the amount of compression under a given external force. This is an important property as the elasticity of liquid marbles determines their morphology under a given stress. We show that the stress-strain relationship of liquid marbles can be described by σ*Bo=0.6[1/(1−εhro)2−1], where Bo is the Bond number, σ* is the normalised stress, and εhr0 is the strain measured with respect to the equivalent radius of the liquid marble. This stress-strain relationship could pave the way for the development of microfluidic devices with robust liquid marbles.

An axisymmetric ordinary state-based peridynamic model for linear elastic solids

In this study, a new axisymmetric ordinary state-based peridynamic (PD) model for axisymmetric problems of linear elastic solids is presented. A fracture criterion based on the PD bond energy density is proposed. Adaptive dynamic relaxation (ADR) method is adopted to obtain equilibrium solutions, and a viable fictitious density of the model is derived and proven to be valid for implementation in the ADR method. Performance and validity of the proposed axisymmetric PD model are demonstrated by three kinds of numerical problems, i.e., compression tests, pull-out deformation, and indentation fracture. In the compression tests with focus on constant-strain deformations, the displacements predicted by the present model are compared with classical analytical solutions, and the results show good agreements. Both m- convergence and δ-convergence behaviors under four influence functions are investigated and discussed based on a thorough error analysis in different compression cases. The recovery of the Poisson’s ratios in the model is tested in detail as well. The capability of capturing general non-uniform axisymmetric deformation by the proposed model is verified in the pull-out analysis. The equilibrium displacement fields predicted by the present model agree very well with those by the finite element method. The peridynamic evaluation of strains and stresses also show good match with the finite element ones. The proposed fracture criterion is validated by the third example of indentation cracking which is compared with the available experimental data. The model developed can effectively be used to analyze axisymmetric problems of linear elastic solids in the framework of the ordinary state-based peridynamics.

The elastic solid solution model for minerals at high pressures and temperatures

Non-ideality in mineral solid solutions affects their elastic and thermodynamic properties, their thermobaric stability, and the equilibrium phase relations in multiphase assemblages. At a given composition and state of order, non-ideality in minerals is typically modelled via excesses in Gibbs free energy which are either constant or linear with respect to pressure and temperature. This approach has been extremely successful when modelling near-ideal solutions. However, when the lattice parameters of the solution endmembers differ significantly, extrapolations of thermodynamic properties to high pressures using these models may result in significant errors. In this paper, I investigate the effect of parameterising solution models in terms of the Helmholtz free energy, treating volume (or lattice parameters) rather than pressure as an independent variable. This approach has been previously applied to models of order–disorder, but the implications for the thermodynamics and elasticity of solid solutions have not been fully explored. Solid solution models based on the Helmholtz free energy are intuitive at a microscopic level, as they automatically include the energetic contribution from elastic deformation of the endmember lattices. A chemical contribution must also be included in such models, which arises from atomic exchange within the solution. Derivations are provided for the thermodynamic properties of n-endmember solutions. Examples of the use of the elastic model are presented for the alkali halides, pyroxene, garnet, and bridgmanite solid solutions. Elastic theory provides insights into the microscopic origins of non-ideality in a range of solutions, and can make accurate predictions of excess enthalpies, entropies, and volumes as a function of volume and temperature. In solutions where experimental data are sparse or contradictory, the Helmholtz free energy approach can be used to assess the magnitude of excess properties and their variation as a function of pressure and temperature. The formulation is expected to be useful for geochemical and geophysical studies of the Earth and other planetary bodies.

A discrete element model for the investigation of the geometrically nonlinear behaviour of solids

A three-dimensional discrete element model for elastic solids with large deformations is presented. Therefore, an discontinuum approach is made for solids. The properties of elastic material are transferred analytically into the parameters of a discrete element model. A new and improved octahedron gap-filled face-centred cubic close packing of spheres is split into unit cells, to determine the parameters of the discrete element model. The symmetrical unit cells allow a model with equal shear components in each contact plane and fully isotropic behaviour for Poisson’s ratio above 0. To validate and show the broad field of applications of the new model, the pin–pin Euler elastica is presented and investigated. The thin and sensitive structure tends to undergo large deformations and rotations with a highly geometrically nonlinear behaviour. This behaviour of the elastica can be modelled and is compared to reference solutions. Afterwards, an improved more realistic simulation of the elastica is presented which softens secondary buckling phenomena. The model is capable of simulating solids with small strains but large deformations and a strongly geometrically nonlinear behaviour, taking the shear stiffness of the material into account correctly.

Novel formulations of microscopic boundary-value problems in continuous multiscale finite element methods

This article explores the use of a homogenization method inspired by the classical Irving–Kirkwood procedure in the continuous multiscale modeling of elastic solids within the context of the finite element method. This homogenization method gives rise to a broad range of allowable boundary conditions for the RVE, which, in turn, yield a rich spectrum of estimates of the macroscopic stress response for two representative problems involving heterogeneous elastic bodies.