Continuum Mechanics Model Research Articles

Damping of vibrations of smart-systems incorporating piezoelectric elements and shunt circuits with parameters derived from the models of continuum and discrete mechanics

The paper considers tasks related to the search for the parameters of smart-systems with piezoelectric elements, to the electroded surfaces to which elements of electrical circuits are connected. The required parameters should provide the maximum damping properties of the simulated systems at free and steady-state forced vibrations. The paper presents the results of calculations of smart-system parameters providing maximum damping properties at free and forced vibrations, which were performed in the framework of continuum and discrete mechanics models. The results demonstrate variants for which the parameters found based on continuum mechanics models provide higher vibration damping rates.

Mechanics of cell-cell junctions

Tissue cells in epithelial or endothelial monolayers are connected through cell-cell junctions, which are stabilized by transmembrane E-cadherin bonds and intracellular actin filaments. These bonds and junctions play a crucial role in maintaining the barrier function of epithelia and endothelia and are believed to transmit forces between cells. Additionally, E-cadherin bonds can impact the shape of cell-cell junctions. In this study, we develop a continuum mechanical model of the cell-cell junction by explicitly incorporating the cell membrane, distributions of E-cadherin bonds, cytoplasmic fluid pressure, and F-actin dynamics. The static force-balanced version of the model is able to analyze the influences of cell cortical tension, actin dynamics, and cytoplasmic pressure on the junction shape and E-cadherin bonds. Furthermore, an extended model that incorporates fluid flow, across the cell boundary as well as around the cell, is also examined. This model can couple cell-shape changes with cell cortical tension and fluid flow, and predicts the additional effect of fluid motion on cell-cell junction mechanics. Taken together, our models serve as an intermediate link between molecular-scale models of cell-junction molecules and cell-scale models of tissue and epithelia.

Stiffness sensing by smooth muscle cells: Continuum mechanics modeling of the acto-myosin role

Aortic smooth muscle cells (SMCs) play a vital role in maintaining homeostasis in the aorta by sensing and responding to mechanical stimuli. However, the mechanisms that underlie the ability of SMCs to sense and respond to stiffness change in their environment are still partially unclear. In this study, we focus on the role of acto-myosin contractility in stiffness sensing and introduce a novel continuum mechanics approach based on the principles of thermal strains. Each stress fiber satisfies a universal stress-strain relationship driven by a Young's modulus, a contraction coefficient scaling the fictitious thermal strain, a maximum contraction stress and a softening parameter describing the sliding effects between actin and myosin filaments. To account for the inherent variability of cellular responses, large populations of SMCs are modeled with the finite-element method, each cell having a random number and a random arrangement of stress fibers. Moreover, the level of myosin activation in each stress fiber satisfies a Weibull probability density function. Model predictions are compared to traction force measurements on different SMC lineages. It is demonstrated that the model not only predicts well the effects of substrate stiffness on cellular traction, but it can also successfully approximate the statistical variations of cellular tractions induced by intercellular variability. Finally, stresses in the nuclear envelope and in the nucleus are computed with the model, showing that the variations of cytoskeletal forces induced by substrate stiffness directly induce deformations of the nucleus which can potentially alter gene expression. The predictability of the model combined to its relative simplicity are promising assets for further investigation of stiffness sensing in 3D environments. Eventually, this could contribute to decipher the effects of mechanosensitivity impairment, which are known to be at the root of aortic aneurysms.

Objective molecular dynamics investigation of dissociation and recombination kinetics in high-temperature nitrogen

In this study, we propose the use of the novel approach of objective molecular dynamics (OMD) simulating far-from-equilibrium gas dynamics problems with chemical reactions. The OMD method has an exact relation to models in continuum mechanics and can be used to improve those models. We provide a detailed molecular dynamics investigation of chemically reacting nitrogen gas in a space-homogeneous adiabatic reactor. The analysis is based on a first-principles derived reactive ReaxFF potential energy surface, which captures the relevant processes of rovibrational relaxation, dissociation, and exchange as well as recombination in a gas evolving under non-equilibrium conditions. We examine the evolution of the internal mode population distribution of all the molecules as well as the rovibrational probability distribution of the pre-collision dissociating and post-collision recombined N2 molecules to investigate the microscopic selectivity of various reactive processes. Subsequently, we make comparisons with results obtained by means of an alternative modeling approach called direct molecular simulation. The current work illustrates the application of the method of OMD to study the compression and expansion kinetics of dissociation-recombination nitrogen mixture relevant to normal shock wave and nozzle expansion.

Balanced-Viscosity Solutions to Infinite-Dimensional Multi-Rate Systems

We consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable u having a viscous damping with relaxation time varepsilon ^alpha and an internal variable z with relaxation time varepsilon we obtain different limits for the three cases alpha in (0,1), alpha =1 and alpha >1. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics.

Stabilized non-ordinary state-based peridynamics with irregular nodal distribution

Peridynamics is a nonlocal continuum mechanics model that is employed in fracture simulation, and its numerical applicability can be extended using irregular arrangements in the nodal distribution. This study investigates the effect of irregular grids on the numerical accuracy of stabilized non-ordinary state-based peridynamics to establish an appropriate range of peridynamic geometric parameters for irregular grids with geometrically nonlinear formulations. To this end, the peridynamic geometric parameters of irregular grids are estimated by interpolating a piecewise graph of the parameter versus the number of neighbor nodes in the regular grid. In addition, a direct tensile test is performed on a two-dimensional square plate, and the displacement results are compared with the analytical solution to determine the appropriate range of the geometric parameters. Using the parameter range, a large deflection of a beam is simulated in the explicit dynamic analysis. Overall, the results of the horizontal and vertical displacements are consistent with those obtained using the finite element method. The numerical models are verified to be generalizable for large displacements and rotations.

A continuum mechanical porous media model for vertebroplasty: Numerical simulations and experimental validation.

The outcome of vertebroplasty is hard to predict due to its dependence on complex factors like bone cement and marrow rheologies. Cement leakage could occur if the procedure is done incorrectly, potentially causing adverse complications. A reliable simulation could predict the patient-specific outcome preoperatively and avoid the risk of cement leakage. Therefore, the aim of this work was to introduce a computationally feasible and experimentally validated model for simulating vertebroplasty. The developed model is a multiphase continuum-mechanical macro-scale model based on the Theory of Porous Media. The related governing equations were discretized using a combined finite element-finite volume approach by the so-called Box discretization. Three different rheological upscaling methods were used to compare and determine the most suitable approach for this application. For validation, a benchmark experiment was set up and simulated using the model. The influence of bone marrow and parameters like permeability, porosity, etc., was investigated to study the effect of varying conditions on vertebroplasty. The presented model could realistically simulate the injection of bone cement in porous materials when used with the correct rheological upscaling models, of which the semi-analytical averaging of the viscosity gave the best results. The marrow viscosity is identified as the crucial reference to categorize bone cements as 'high- 'or 'low-' viscosity in the context of vertebroplasty. It is confirmed that a cement with higher viscosity than the marrow ensures stable development of the injection and a proper cement interdigitation inside the vertebra.

Continuum Mechanical Modeling of the Inelastic Response of Woven Textile Membranes

AbstractThis work discusses the derivation and numerical implementation of an elastoplastic model of textile membranes at finite strains. The mechanical model accounts for the anisotropic hyperelastic response [1], reached in the saturation region. The plastic anisotropy is modeled by implementing the yield surface and the flow rule as a function of the structural tensor [2]. Furthermore, the capability of the model to simulate the behavior of textile membranes under non‐proportional cyclic loading is investigated. For this purpose, a fitting procedure to identify the elastic and plastic parameters is presented.

Hyperelastic Modeling and Validation of Hybrid-Actuated Soft Robot with Pressure-Stiffening.

Soft robots have gained popularity, especially in intraluminal applications, because their soft bodies make them safer for surgical interventions than flexures with rigid backbones. This study investigates a pressure-regulating stiffness tendon-driven soft robot and provides a continuum mechanics model for it towards using that in adaptive stiffness applications. To this end, first, a central single-chamber pneumatic and tri-tendon-driven soft robot was designed and fabricated. Afterward, the classic Cosserat's rod model was adopted and augmented with the hyperelastic material model. The model was then formulated as a boundary-value problem and was solved using the shooting method. To identify the pressure-stiffening effect, a parameter-identification problem was formulated to identify the relationship between the flexural rigidity of the soft robot and internal pressure. The flexural rigidity of the robot at various pressures was optimized to match theoretical deformation and experiments. The theoretical findings of arbitrary pressures were then compared with the experiment for validation. The internal chamber pressure was in the range of 0 to 40 kPa and the tendon tensions were in the range of 0 to 3 N. The theoretical and experimental findings were in fair agreement for tip displacement with a maximum error of 6.40% of the flexure's length.

Mode specific dynamics for the acoustic vibrations of a gold nanoplate

The vibrational modes of semiconductor and metal nanostructures occur in the MHz to GHz frequency range, depending on dimensions. These modes are at the heart of nano-optomechanical devices, and understanding how they dissipate energy is important for applications of the devices. In this paper ultrafast transient absorption microscopy has been used to examine the breathing modes of a single gold nanoplate, where up to four overtones were observed. Analysis of the frequencies and amplitudes of the modes using a simple continuum mechanics model shows that the system behaves as a free plate, even though it is deposited onto a surface with no special preparation. The overtones decay faster than the fundamental mode, which is not predicted by continuum mechanics calculations of mode damping due to radiation of sound waves. Possible reasons for this effect include frequency dependent thermoelastic effects in the nanoplate, and/or flow of acoustic energy out of the excitation region.

Trends in the 40 + -year development of advanced structural acoustic and vibration computational modeling

Over the course of the past 40-45 years of computational modeling in the fields of structures, structural acoustics, and vibrations, there have been extreme advancements made in numerous areas that have greatly increased and improved the size, accuracy, efficiency, and complexity of numerical solutions possible. Most of this progress has been made through a combination of improvements in the distinct areas of computer processing approaches/speed and available memory, primarily evolutionary enhancements in detailed numerical material and continuum mechanics models, development of entirely new physics-based numerical vibroacoustic modeling approaches, etc. This paper presents a historical review of the detailed progress made in computational structural acoustics and vibration from the “early days” in the 1980s through current state-of-the-art advanced numerical approaches, much of which would not have been dreamed possible back four decades ago! While most of the examples utilized in this talk are pulled from the speaker’s specific technical area of undersea vehicle/ system structural acoustics, the overall trends presented are certainly represented across the broad spectrum of vibro-acoustics modeling applications.

Solutions of Some Wave Mechanics Models

We consider one-dimensional second order partial differential equations describing waves in inhomogeneous and nonlinear media. Contact transformations and Euler differential substitution are used to construct general solutions. General and partial solutions of some nonstationary continuum mechanics models are found.

Incorporation of thermal effects into the energy‐momentum consistent elastic model for fiber‐bending stiffness in composites

AbstractOwing to the wide acceptance of fiber reinforced composites in the lightweight structure industry, the need to study its thermo‐elastic behavior in dynamical multi‐body systems still remains an open question. Standard continuum models, by virtue of first‐order gradient of deformation, cannot capture curvature effects and hence appropriate extensions of the standard modeling techniques are indispensable to model the accurate physical stiffness. The fiber‐bending curvature are captured by introducing a higher‐order gradient of the deformation as an independent field in the standard continuum. This work explores in detail, a further extension of this continuum to understand the thermal effects on the bending stiffness of fiber bundle in fiber‐reinforced composites. Our proposed model incorporates the temperature of the composite and its bidirectional dependency with the higher‐order gradients of the deformation into the extended mechanical continuum model. The implementation of the model is pursued in the context of multi‐field mixed finite element method and dynamic variational principle. The principle of virtual power enables our extended non‐isothermal continuum to derive an energy‐momentum scheme of higher order. The resulting time‐integrator exhibits physically consistent locking‐ free numerical behavior in dynamical simulations. Our novel approach is illustrated by simple transient dynamic simulations with a hyperelastic, transversely isotropic, polyconvex material behavior. We investigate the conservation of total energy and total momentum in the framework of thermo‐elasticity and study the spatial and temporal convergence for long‐term dynamic simulations with coarser grids and increased accurate numerical solutions.

A Structure-Preserving Finite Volume Scheme for a Hyperbolic Reformulation of the Navier–Stokes–Korteweg Equations

In this paper, we present a new explicit second-order accurate structure-preserving finite volume scheme for the first-order hyperbolic reformulation of the Navier–Stokes–Korteweg equations. The model combines the unified Godunov-Peshkov-Romenski model of continuum mechanics with a recently proposed hyperbolic reformulation of the Euler–Korteweg system. The considered PDE system includes an evolution equation for a gradient field that is by construction endowed with a curl-free constraint. The new numerical scheme presented here relies on the use of vertex-based staggered grids and is proven to preserve the curl constraint exactly at the discrete level, up to machine precision. Besides a theoretical proof, we also show evidence of this property via a set of numerical tests, including a stationary droplet, non-condensing bubbles as well as non-stationary Ostwald ripening test cases with several bubbles. We present quantitative and qualitative comparisons of the numerical solution, both, when the new structure-preserving discretization is applied and when it is not. In particular for under-resolved simulations on coarse grids we show that some numerical solutions tend to blow up when the curl-free constraint is not respected.

Formation of lamellar domains in liquid crystal elastomers under compression

Liquid crystal elastomers (LCEs) are lightly cross-linked polymer materials containing liquid crystal (LC) molecules with unique strain-induced director reorientation properties. Compressive studies on LCEs have been very little given the difficulty synthesizing sufficiently large monodomain block samples. Based on the continuum mechanics model of LCEs, the compressive behaviors of monodomain LCE blocks in plane stain with different initial directors between two rigid plates are systematically studied in this paper. The rotation of the director, especially the formation of lamellar domains, significantly changes the characteristics of the compressive stress-strain curves. Compression along the initial director will induce the formation of lamellar domains orthogonal to the loading axis due to the alternating rotation of the director after loading to a critical strain. The rotation of the director also leads to the stress plateau on the stress-strain curves. When the initial director is nearly parallel to the compression axis, the lamellar domains can still be observed in the central region of the sample bonded to the rigid plates, which is also related to the deformed morphology of the stress-free boundaries. If the initial director is slightly deviated from the loading axis, almost uniform rotation occurs in the central region of the sample, while the stress increases monotonically with strain hardening. The formation of lamellar domains is not only related to the angles between the initial director and the compressive loading axis, but also strongly depends on the geometrical and material parameters of the samples. For large aspect ratios or small anisotropy constants, the formation of lamellar domains may require compression sufficiently parallel to the initial director. Our work provides a deeper understanding of the behavior of LCEs under compression.

Implementing multiphysics models in FEniCS: Viscoelastic flows, poroelasticity, and tumor growth

The open-source finite element code FEniCS is considered as an alternative to commercial finite element codes for evaluating complex constitutive models of multiphysics phenomena. FEniCS deserves this consideration because it is well-suited for encoding weak forms corresponding to partial differential equations arising from the fundamental balance laws and constitutive equations. It is shown how FEniCS can be adopted for solving boundary-value problems describing viscoelastic flows, poroelasticity, and tumor growth. Those problems span a wide range of models of continuum mechanics, and involve Eulerian, Lagrangian, and combined Eulerian-Lagrangian descriptions. Thus it is demonstrated that FEniCS is a viable computational tool capable of transcending traditional barriers between computational fluid and solid mechanics. Furthermore, it is shown that FEniCS implementations are straightforward, and do not require advanced knowledge of finite element methods and/or coding skills.

Mechanical investigations of the peltate leaf of Stephania japonica (Menispermaceae): Experiments and a continuum mechanical material model.

Stephania japonica is a slender climbing plant with peltate, triangular-ovate leaves. Not many research efforts have been devoted to investigate the anatomy and the mechanical properties of this type of leaf shape. In this study, displacement driven tensile tests with three cycles on different displacement levels are performed on petioles, venation and intercostal areas of the Stephania japonica leaves. Furthermore, compression tests in longitudinal direction are performed on petioles. The mechanical experiments are combined with light microscopy and X-ray tomography. The experiments show, that these plant organs and tissues behave in the finite strain range in a viscoelastic manner. Based on the results of the light microscopy and X-ray tomography, the plant tissue can be considered as a matrix material reinforced by fibers. Therefore, a continuum mechanical anisotropic viscoelastic material model at finite deformations is proposed to model such behavior. The anisotropy is specified as the so-called transverse isotropy, where the behavior in the plane perpendicular to the fibers is assumed to be isotropic. The model is obtained by postulating a Helmholtz free energy, which is split additively into an elastic and an inelastic part. Both parts of the energy depend on structural tensors to account for the transversely isotropic material behavior. The evolution equations for the internal variables, e.g. inelastic deformations, are chosen in a physically meaningful way that always fulfills the second law of thermodynamics. The proposed model is calibrated against experimental data, and the material parameters are identified. The model can be used for finite element simulations of this type of leaf shape, which is left open for the future work.

Three-dimensional representative volumetric elements for the construction of composite materials with improved mechanical properties

A methodological and software tool was developed for the effective assessment of the properties of three-dimensional composites of the matrix type and periodically structured composite materials using modern methods of boundary and finite elements. Continuum mechanics models have been modified to adequately describe the behavior of composites, numerical simulation models and software have been adapted to these models for solving static problems and problems in the frequency domain. The novelty, advantages and efficiency of the developed computer technology are provided by theoretically grounded models for the analysis of composites of three-dimensional configuration with non-canonical particle shapes, improvement of modern boundary and finite element methods and coupled algorithmic schemes to achieve high accuracy of calculation analysis; suitability for the analysis of both static and dynamic elastic fields in the considered structures. Calculation models and software tools have been developed that allow to take into account special forms of individual and interacting inhomogeneities in a three-dimensional matrix, in particular, with an emphasis on short fibers and tubes. These models and methods are applied to the study of static and dynamic characteristics of launch vehicle tanks made of various composite materials. For the studying free and forced oscillations of shell structures with compartments containing liquid, the method of natural modes has been developed. Composites with an aluminum matrix with steel spherical inclusions and with steel and carbon fiber inclusions were considered as tank materials. The proposed models and computer simulation algorithms allow to carry out the numerical analysis of three-dimensional static problems under classical conditions of contact with the matrix; three-dimensional problems regarding the static interaction of a finite number of inclusions in the matrix; research of ordered and stochastically distributed inclusions; study of inclusions in the form of spheres, cylinders, carbon tubes and inclusions in the form of hollow spheres and to determine the effective elastic parameters of three-dimensional matrix composites with spherical inclusions, fibers and tubes. The results of the calculations showed strengthening of the obtained composite materials with a simultaneous decrease in density

A comparative study of analytical and numerical models for the elastic behavior of composites reinforced by coated unidirectional fibers

A number of continuum mechanical models for the macroscopic and microscopic linear elastic behavior of composites reinforced by continuous, unidirectional, coaxially coated fibers are critically compared. For the study, the interphase is assumed to be of finite thickness and uniform. Selected analytical mean-field methods and Finite-Element-based periodic homogenization models are applied to obtaining predictions for the effective elastic tensors and moduli of generic composites of this type. In addition, the phase-averaged microscopic stresses are evaluated for some macroscopic load cases. By comparing these results modeling schemes capable of providing high-quality predictions are identified.

Molecular dynamics study on friction of the iron-aluminum alloy

Iron-aluminum alloy has been widely used in aerospace, nuclear power, electronics and other fields. Taking into account the limitations of the continuum mechanics model at the microscopic scale, it is critical to use nanoscale simulation methods like molecular dynamics (MD) to have a deeper understanding of the friction and wear behavior. Through MD simulation, this thesis mainly studied the friction and wear process of iron aluminum alloy under different conditions. The main conclusions are as follows: (1) The frictional force first experiences a period of growth, and then stabilizes with some fluctuations due to the release of stress-energy caused by deformation. (2) Temperature and friction speed has no significant effect on friction force. (3) The higher the friction speed is, the more wear debris it produces, and the more severely the material is damaged. (4) The force and coefficient of friction will increase with the rise of aluminum content, and the plastic deformation of the alloy is strengthened, while the elastic deformation decreases. (5) The frictional force increases with the wear depth since the accumulation phenomenon of atoms becomes more obvious.