Methods Of Continuum Mechanics Research Articles

Relation between mechanical properties and microstructure of human fetal membranes: An attempt towards a quantitative analysis

Objective We sought to measure the mechanical baseline behavior of fetal membranes in order to determine constitutive mechanical model parameters for fetal membranes, and to examine their relation to molecular correlates for mechanical function, i.e. collagen and elastin. Study design The uniaxial stress–strain response of nine human term fetal membranes was measured. Methods of nonlinear continuum mechanics were applied for the analysis of the stress–strain curves. Thickness of amnion and chorion were determined from histologic sections for each fetal membrane sample. Complementary biochemical analysis was performed to quantify the soluble collagen and soluble elastin components for each sample. Results We report a straightforward histologic modality for measurements of amnion and chorion thickness. Average thickness of the amnion and chorion layers were 111 ± 78 μm, and 431 ± 113 μm, respectively, which are about twice larger than previously reported. The average content of acid-soluble elastin was 2.1% of wet weight and the one of pepsin/acetic acid-soluble collagen was 10.5% of dry weight. Our data show an inverse proportionality between soluble elastin and soluble collagen content. The low strain elastic modulus ranged between 10 and 25 kPa. Correlations were found between biochemical data and mechanical parameters: there is clearly a direct proportionality between small strain elastic modulus and elastin content. Further, a (less pronounced) direct correlation was observed also between soluble collagen content and the parameter governing the increase in stiffness at larger strains in the nonlinear mechanical model. The mechanical tests revealed a relatively low variability for samples from the same membrane but a large variation between donors. The proposed nonlinear model provides a good fit of the experimental data, with a coefficient of determination, R 2, typically in the range of 0.94. Membranes failure originated at the clamping points thus impairing the quantification of ultimate stress and strain. Thus, no correlation was found between maximum stress and collagen or elastin content. Conclusions This study provides a starting point for comprehensive quantitative analysis of the relationship between fetal membranes microstructure and their nonlinear deformation behavior. These insights could become useful in identifying potential medical interventions to prevent membranes rupture.

Simulation of fracture propagation subjected to compressive and shear stress field using virtual multidimensional internal bonds

The virtual multidimensional internal bonds (VMIB) model and its predecessor, the virtual internal bond (VIB) model, belong to a special class of continuum mechanics methods that are aimed to simulate fractures with no additional fracture criterion apart from the constitutive laws of the material. However, in the simulation of the fracture subjected to compressive and shear stress field, the contact and friction effect of fracture faces is significant. To resolve this problem, the present paper, taking advantage of the geometrical characteristics of the three-node triangular element (TTE), develops a three-node contact element (TCE) based on the finite element method. A TCE consists of two contact pairs, through which the stiffness matrix of TCE is derived. The pre-existing fracture is represented by transforming the TTE intersected by fracture into the TCE, which allows the meshing procedure to proceed regardless of the pre-existing fractures. During the simulation process, the failure of an intact TTE yields a ‘micro’ fracture at the element level. These ‘micro’ fractures align to form a ‘macro’ fracture. To represent the newly generated fracture, the TTEs identified to fail are automatically transformed into the TCEs. As the TCE shares the common nodes with the intact TTE, the transformation from TTE to TCE is accomplished without any modification on the original mesh format, which makes the fracture simulation highly efficient and simple. The simulation examples indicate that the present method can efficiently simulate the fracture propagation subjected to shear and compressive stress.

Modelling of the Nanosystems Formation by the Molecular Dynamics, Mesodynamics and Continuum Mechanics Methods

The problems of simulation of the formation and development of nanosystems with time were considered at different structural levels. Main theoretical concepts were systematized, which allowed successive modelling of a nanosystem with the use of the molecular dynamics, mesodynamics and continuum mechanics methods. Practical problems such as the calculation of dependence of the elasticity modulus of a nanoparticle on its size, the investigation of the formation and movement of nanoparticles in gas phase were discussed.

Mesomechanics of multiwall carbon nanotubes and nanowhiskers

The paper considers the initial (from nanometers) scales of multiscale solid structures using well-known carbon nanotubes (hollow nanorods) and nanowhiskers (solid nanorods) as an example. It is shown that structural anisotropy contributes significantly to the mechanical properties of nanotubes and nanowhiskers. It is demonstrated that the rod systems, beginning with those of nanometer cross-sections, can be described by conventional methods of continuum mechanics. Analysis is performed to elucidate the initial stages of mechanical instability and structural rearrangement of nanotubes. The paper also discusses the general peculiarities of multiscale rearrangement of solids at the mesolevel in terms of the development of instabilities and the formation of a mesoscale structural cascade. Consideration is given to plausible ways in which stochastic and regular mechanisms of the formation of structural hierarchy of solids show up under deformation and fracture. Stochastically common regularities and possible differences of the developed structural cascade and strains in solids and of the developed turbulence in fluid flows are noted. The deterministic mechanisms are discussed in terms of the formation of ordered discontinuity systems (fracture structures) responsible for possible scale-to-scale fracture transition in which both scale enlargement and refinement of the fracture structures take place.

A multilevel computer simulation of friction and wear by numerical methods of discrete mechanics and a phenomenological theory

A procedure for a multilevel computer simulation of friction and wear is proposed where numerical calculations at local contact scales by the method of movable cellular automata and a macroscopic phenomenological friction theory are chained together within a unified concept. For a mesoscopic simulation, use is made of the method of movable cellular automata in combination with numerical methods of continuum mechanics. Results obtained from a simulation of local contacts in contacting pairs of heterogeneous friction materials are reported. The effect of the initial surface microroughness on a quasi-fluid layer in the contact zone of plastic materials is examined. The macroscopic phenomenological model for friction describes spalling, reintegration of wear debris into the wear surfaces and mass transfer from one contacting body to the other, using stochastic differential equations. The parameters of the phenomenological equations are found solely by means of microscopic models for processes involved in a tribological contact. It is suggested that a simulation by the method of movable cellular automata be used to this end. The procedure developed enables the effective diffusion coefficient of the friction layer to be estimated. Identification of the parameters of the macroscopic model is made possible by simulating the processes operative at the nanoscale level thus bridging nano- and macroscale levels.

Deformation field of the soft substrate induced by capillary force

Prediction on the deformation of a soft substrate induced by capillary force has been widely paid attention in the broad range of applications, such as metallurgy, material science, astronavigation, micro/nano-technology, etc., which is also a supplementary result to the classical Young's equation. We quantitatively analyzed the deformation of an elastic substrate under capillary force by means of the energy principle and the continuum mechanics method. The actual drop's morphology was investigated and was compared with that calculated based on the classical spherical shape assumption of the droplet. The displacement field of the substrate was obtained, especially, its singularity at the droplet edge was also discussed. The results are beneficial to engineering application and micro/nano-measurement.

Fatigue failure investigation on anti-vibration springs

Fatigue failure investigation on anti-vibration springs, involving both metal and rubber materials, is presented. Rubber-to-metal bonded springs are widely used in industry as anti-vibration components giving many years of service. Recently a need to improve time and cost efficiencies has caused an unexpected early fatigue failure of the component with no immediate explanation. The required total fatigue life was 1.25 million cycles but only 0.7 million cycles achieved. There was an urgent need to investigate the causes of the fatigue failure and to modify the component design accordingly to meet the customer requirement and the supply schedule. The investigation, based on the actual fatigue loads, is carried out on these failed and modified products using a method of continuum mechanics. To simplify the simulation, a non-linear quasi-static analysis is carried out and then the residual stresses are superimposed to obtain the effective stress range to predict the metal crack initiation. For the rubber parts a three-dimensional effective stress criterion is employed to predict the fatigue crack initiation. The fatigue failure is taken as visual crack observation (normally 1–2 mm). The fatigue crack initiation for the metal parts of the failed component is predicted at 225 K cycles under specified fatigue load against total metal broken at 700 K cycles from the test. For the modified part the minimum total fatigue life for the metal parts of the component, estimated conservatively, is 2.1 million cycles against 1.75 million cycles from the test without any crack observed. The rubber fatigue crack initiation is predicted at 90 K cycles against crack onset around 79 K cycles and crack length 40 mm at 145 K cycles from the test. From design point of view it is important to optimize the rubber profile under this very tight allowable space to provide the maximum support of the metal interleaves and at the same time to meet the minimum requirements of the manufacture process. It is shown that this approach can be employed at a design stage for both metal and rubber fatigue evaluations on anti-vibration springs.

Theory of 4D-media with stationary dislocations

In earlier studies, the authors showed that an application of classical methods of mechanics of deformable media to the study of properties of 4D-space-time continuum permit stating consistent models of nonholonomic media mechanics consistent with the first and second laws of thermodynamics. In the present paper, we show that the classical methods of continuum mechanics are also promising when modeling physical processes. It is shown that, just as in the three-dimensional theory of stationary dislocations, there exist dislocations of three types for a generalized 4D-medium. They correspond to the decomposition of the free distortion tensor into a spherical tensor, a deviator tensor, and a pseudotensor of rotations. We interpret several particular models, thus showing that the proposed model describes the spectrum of known physical interactions: electromagnetic, strong, weak, and gravitational. We show that the resolving equations include the Maxwell equations of electrodynamics and the Yukawa equations for strong interactions as subsystems.

An efficient approach to converting three-dimensional image data into highly accurate computational models

Image-based meshing is opening up exciting new possibilities for the application of computational continuum mechanics methods (finite-element and computational fluid dynamics) to a wide range of biomechanical and biomedical problems that were previously intractable owing to the difficulty in obtaining suitably realistic models. Innovative surface and volume mesh generation techniques have recently been developed, which convert three-dimensional imaging data, as obtained from magnetic resonance imaging, computed tomography, micro-CT and ultrasound, for example, directly into meshes suitable for use in physics-based simulations. These techniques have several key advantages, including the ability to robustly generate meshes for topologies of arbitrary complexity (such as bioscaffolds or composite micro-architectures) and with any number of constituent materials (multi-part modelling), providing meshes in which the geometric accuracy of mesh domains is only dependent on the image accuracy (image-based accuracy) and the ability for certain problems to model material inhomogeneity by assigning the properties based on image signal strength. Commonly used mesh generation techniques will be compared with the proposed enhanced volumetric marching cubes (EVoMaCs) approach and some issues specific to simulations based on three-dimensional image data will be discussed. A number of case studies will be presented to illustrate how these techniques can be used effectively across a wide range of problems from characterization of micro-scaffolds through to head impact modelling.

Evolution of Microstructure and Texture during Hot Torsion of Ferritic Stainless Steels Stabilized by Nb and Ti

The study was carried out to understand the mechanisms occurring during dynamic recrystallization of hot deformed 11% chromium stabilized ferritic stainless steels and to compare the behaviour induced by various types of stabilization. The experimental temperatures ranged from 800 to 1150°C and strain rates from 10−2 to 15 s−1. The development of the textures and microstructures was analysed using EBSD maps. It was observed that continuous dynamic recrystallization occurs in all materials starting at the onset of straining. Niobium has a more pronounced influence on hardening than titanium during hot deformation. The D2 component was found as the major texture component at the steady state for the torsion tests conducted along the negative shear direction. It was likely to be formed by the combination of straining and growth of the grains exhibiting both low stored energy and low rotation rate of the crystallographic axes. The texture evolution was the main reason for the flow stress behavior during the hot torsion tests.A complementary study was carried out to understand the texture formation occurring during the direct (negative shear direction) and reversed (positive shear direction) hot torsion. After reversion of the shear direction, i.e. during positive shear, the above major texture component is gradually changed into the D1 component. Using the method of Continuum Mechanics of Textured Polycrystals, the stress evolution is explained by the volume fraction changes of each component at various strains, associated with their respective Taylor factors. This simplified approach leads to a good agreement with experimental results.

Vibrational modes of free nanoparticles: From atomic to continuum scales

Vibration analysis of free standing silicon nanoparticles, with sizes ranging from 0.732 to 4.223 nm, are calculated using two different methods: molecular mechanics and classical continuum elasticity. Three different geometries are studied: cubes, spheres, and tetrahedrons. Continuum mechanics methods provide good estimates of the lowest natural frequency of particles having at least 836 (R>1.5 nm) and 800 (R>1.28 nm) atoms for cube- and tetrahedron-shaped nanostructures, respectively. Equations for vibrational frequencies of smaller particles as a function of size are proposed. The vibrational modes obtained from both methods were practically the same for the sphere- and tetrahedron-shaped particles with a large number of atoms. However, for the cube geometry only the shape of the modes corresponding to the lowest couple of frequencies occur in the same order. In general, vibrational modes shapes obtained using both methods are the same although the order in which they appear may be shifted.

Multiscale coupling schemes spanning the quantum mechanical, atomistic forcefield, and continuum regimes

Strategies for coupling quantum mechanical (QM), molecular mechanical (MM), and continuum mechanical (CM) methods are described. For QM/MM coupling, we consider two overlapping domain schemes: the widely-used ONIOM method that involves full overlap, and a new minimal-overlap scheme denoted as the “quantum to molecular mechanical overlapping domain” (QtMMOD) method. In the QtMMOD method, the part of the region that is treated by quantum mechanics does not require MM calculations, which makes the method suitable for systems containing localized regions that cannot be modeled by available empirical potentials. We describe how the QtMMOD method can in turn be coupled to a finite element model by the bridging domain method, another overlapping domain method, yielding a QM/MM/CM model. We also show how the QtMMOD scheme can be used to couple a finite element model directly with a QM model. These coupling methods were used to calculate the fracture properties of graphene sheets containing defects. We obtained good agreement between stress–strain curves and fracture strengths calculated by the three methods and benchmark results of a strictly QM calculation. An extension of an atomistic strain calculation method based on a moving least squares scheme, which is applicable to the treatment of material interfaces, is given.

Coupled Quantum–Atomistic and Quantum–Continuum Mechanics Methods in Materials Research

AbstractThe interface of quantum mechanics methods with classical atomistic simulation techniques, such as molecular dynamics and Monte Carlo, continues to be an area of considerable promise and interest. Such coupled quantum–atomistic approaches have been developed and employed, for example, to gain a comprehensive understanding of the energetics, kinetics, and dynamics of chemical processes involving surfaces and interfaces of hard materials. More recently, it has become possible to directly couple first-principles electronic structure techniques to continuum solid mechanics, either on the fly with feedback between length scales or by information passing between length scales. We discuss, with tutorial examples, the merging of quantum mechanics with molecular dynamics and Monte Carlo simulations, as well as quantum–continuum coupled techniques. We illustrate the opportunities offered by incorporation of information from quantum mechanics (reducing assumptions in higher length-scale models) and outline the challenges associated with achieving full predictive capability for the behavior of materials.

New Collective Mode in the Fractional Quantum Hall Liquid

We apply the methods of continuum mechanics to the study of the collective modes of the fractional quantum Hall liquid. Our main result is that at long-wavelength, there are two distinct modes of oscillations, while previous theories predicted only one. The two modes are shown to arise from the internal dynamics of shear stresses created by the Coulomb interaction in the liquid. Our prediction is supported by recent light scattering experiments, which report the observation of two long-wavelength modes in a quantum Hall liquid.

Classification of the models of materials in continuum mechanics

The classification of the models of materials in continuum mechanics proposed by the author on the basis of the general theory of Noll constitutive relations is developed by using the methods of rational continuum mechanics.

An energy-equivalent model on studying the mechanical properties of single-walled carbon nanotubes

In this paper, an energy-equivalent model is put forward for studying the mechanical properties of single-walled carbon nanotubes (SWCNTs). The equivalent Young's modulus and shear modulus for both armchair and zigzag SWCNTs are derived by combining the methods of molecular mechanics and continuum mechanics. On the one hand, based on the principle of molecular mechanics, the total system potential energy associated with both stretching and angular variations is obtained. On the other hand, considering SWCNT as a thin cylinder subjected to an axial or torsion loading, the strain energy can be obtained based on continuum mechanics. Equating the total system potential energy to the strain energy, one derives the equivalent Young's modulus, shear modulus and Poisson's ratio. Finally, computations of the mechanical properties reveal that the elastic constants exhibit a strong dependence on the diameter of nanotubes. Young's modulus and shear modulus for both armchair and zigzag nanotubes increase with increasing tube diameter, but the variation trend of Poisson's ratios is reverse. The present results agree well with existing results and approach to those of carbon graphite when the diameter is large. Therefore, the method presented here is valid for both carbon nano-tubes and carbon graphite.

Simulation of adatom clustering on a stepped surface

Study of the interaction and evolution behaviour of adatoms is fundamental to many surface phenomena. In the present paper, a continuum mechanics method, based on the force dipole concept, is presented to study the evolution of adatoms adsorbed on a smooth or stepped surface. The effects of elastic and van der Waals interactions are examined. It is found that the elastic interaction among adatoms usually serves as the main driving force for the clustering behaviour of adatoms and that the presence of surface defects, such as steps, may significantly influence the evolutionary arrangement of interacting adatoms.

An atomistic simulation method combining molecular dynamics with finite element technique

Abstract A numerical method that combines molecular dynamics simulation and finite element analysis to simulate the mechanical behaviors of materials and structures at nano-scale is proposed. In this combined method, the initial atomistic model is transformed to continuum model, and an approximate solution is first obtained with the finite element method for the system under the specified boundary conditions and external loadings. Then the deformed continuum model is transformed back to form a new atomistic model, and molecular dynamics simulation is performed to quickly reach the final stable equilibrium state. An example is presented to demonstrate that the combination procedure is valid and efficient. This method can take advantages of both the efficiency of continuum mechanics method and the accuracy of atomistic simulation method.

Bending and shear moduli of single-walled carbon nanotubes

Elastic properties of single-walled carbon nanotubes (SWCNT) obtained experimentally and computationally are reviewed. Attention is paid particularly on the evaluation of Young's and shear moduli of SWCNT. A finite element method (FEM) previously presented by Li and Chou is modified with the Poisson's ratio effect included in the determination of Young's modulus and shear modulus of SWCNT. It is pointed out that within the linear regime, the Young's and shear moduli of SWCNT have been estimated to be constant. This is in contrast to those appeared in the literature in which a wide range of Young's moduli of SWCNT has been reported. The important implications are that the classical theory of elasticity can be employed in the determination of Young's and shear moduli of elasticity of SWCNT, and that nanosensors, nanodevices, and advanced materials embedded with SWCNT can be designed without resort to adopting the uncertain Young's and shear moduli of SWCNT. The modified FEM of treating the SWCNT as a spatial cylindrical frame structure is very simple to use, very economical, and efficient computationally compared with those based on particle or molecular dynamics (MD) methods, simulation and continuum mechanics (CM) methods where huge computational resources are necessary.

Continuum Mechanics Approach and Computational Modelling of Submicrocrystalline and Nanoscale Materials

The material volumes containing internal submicrocrystalline and nano structures are investigated by numerical methods of continuum and quantum mechanics and some computational tools have been developed. Purpose of the research is numerical multiscale modeling and computer simulation of nanoparticle‐reinforced materials and carbon‐like nanostructures.