Conventional Continuum Approaches Research Articles

Capturing sclera anisotropy using direct collagen fiber models. Linking microstructure to macroscopic mechanical properties.

Because of the crucial role of collagen fibers on soft tissue mechanics, there is great interest in techniques to incorporate them in computational models. Recently we introduced a direct fiber modeling approach for sclera based on representing the long-interwoven fibers. Our method differs from the conventional continuum approach to modeling sclera that homogenizes the fibers and describes them as statistical distributions for each element. At large scale our method captured gross collagen fiber bundle architecture from histology and experimental intraocular pressure-induced deformations. At small scale, a direct fiber model of a sclera sample reproduced equi-biaxial experimental behavior from the literature. In this study our goal was a much more challenging task for the direct fiber modeling: to capture specimen-specific 3D fiber architecture and anisotropic mechanics of four sclera samples tested under equibiaxial and four non-equibiaxial loadings. Samples of sclera from three eyes were isolated and tested in five biaxial loadings following an approach previously reported. Using microstructural architecture from polarized light microscopy we then created specimen-specific direct fiber models. Model fiber orientations agreed well with the histological information (adjusted R2's>0.89). Through an inverse-fitting process we determined model characteristics, including specimen-specific fiber mechanical properties to match equibiaxial loading. Interestingly, the equibiaxial properties also reproduced all the non-equibiaxial behaviors. These results indicate that the direct fiber modeling method naturally accounted for tissue anisotropy within its fiber structure. Direct fiber modeling is therefore a promising approach to understand how macroscopic behavior arises from microstructure.

A Review of Nano and Microscale Heat Transfer: An Experimental and Molecular Dynamics Perspective

Significant progress in the development of micro and nanoscale devices has been observed for the past three decades. The thermal transportation in these small-length scales varies significantly, and it is difficult to explain the underlying physics using the pre-existing theoretical formulations. When the bulk dimension of a system is comparable to or smaller than the mean free path (MFP) of the thermal carriers, classical theories, such as Fourier’s Law of heat conduction, are unable to accurately explain the system energy dynamics. The phenomena of energy transit and conversion at the micro to nanoscale is an interesting topic of research due to the substantial changes in behavior that are documented when compared to those at the macro size. This review article is broadly divided into two parts. Initially, the recent development in the field of molecular dynamic (MD) simulations is emphasized. Classical MD simulation is such a powerful tool that provides insight into the length scales where the conventional continuum approaches cease to be valid. Several examples of recent developments in the applicability of MD simulations for micro and nanoscale thermal transportation are reviewed. However, there are certain limitations of the MD simulations where the results deviate from experimental validation due to the lack of knowledge of the appropriate force fields. Hence the experimental development of micro and nanoscale thermal transportation processes is briefly reviewed and discussed in the other section of this review article.

A direct fiber approach to model sclera collagen architecture and biomechanics

Sclera collagen fiber microstructure and mechanical behavior are central to eye physiology and pathology. They are also complex, and are therefore often studied using modeling. Most models of sclera, however, have been built within a conventional continuum framework. In this framework, collagen fibers are incorporated as statistical distributions of fiber characteristics such as the orientation of a family of fibers. The conventional continuum approach, while proven successful for describing the macroscale behavior of the sclera, does not account for the sclera fibers are long, interwoven and interact with one another. Hence, by not considering these potentially crucial characteristics, the conventional approach has only a limited ability to capture and describe sclera structure and mechanics at smaller, fiber-level, scales. Recent advances in the tools for characterizing sclera microarchitecture and mechanics bring to the forefront the need to develop more advanced modeling techniques that can incorporate and take advantage of the newly available highly detailed information. Our goal was to create a new computational modeling approach that can represent the sclera fibrous microstructure more accurately than with the conventional continuum approach, while still capturing its macroscale behavior. In this manuscript we introduce the new modeling approach, that we call direct fiber modeling, in which the collagen architecture is built explicitly by long, continuous, interwoven fibers. The fibers are embedded in a continuum matrix representing the non-fibrous tissue components. We demonstrate the approach by doing direct fiber modeling of a rectangular patch of posterior sclera. The model integrated fiber orientations obtained by polarized light microscopy from coronal and sagittal cryosections of pig and sheep. The fibers were modeled using a Mooney-Rivlin model, and the matrix using a Neo-Hookean model. The fiber parameters were determined by inversely matching experimental equi-biaxial tensile data from the literature. After reconstruction, the direct fiber model orientations agreed well with the microscopy data both in the coronal plane (adjusted R2 = 0.8234) and in the sagittal plane (adjusted R2 = 0.8495) of the sclera. With the estimated fiber properties (C10 = 5746.9 MPa; C01 = −5002.6 MPa, matrix shear modulus 200 kPa), the model's stress-strain curves simultaneously fit the experimental data in radial and circumferential directions (adjusted R2's 0.9971 and 0.9508, respectively). The estimated fiber elastic modulus at 2.16% strain was 5.45 GPa, in reasonable agreement with the literature. During stretch, the model exhibited stresses and strains at sub-fiber level, with interactions among individual fibers which are not accounted for by the conventional continuum methods. Our results demonstrate that direct fiber models can simultaneously describe the macroscale mechanics and microarchitecture of the sclera, and therefore that the approach can provide unique insight into tissue behavior questions inaccessible with continuum approaches.

Numerical modelling of the quasi-brittle behaviour of refractory ceramics by considering microcracks effect

In the steelmaking industry, the inner lining of ladles is made of refractory ceramics, which are constantly subjected to thermal shocks during their service. Experimentally, it is observed that pre-existing microcracks could significantly increase the thermal shock resistance of these ceramics. The presence of such microcracks network within the refractory microstructure could lead to a non-linear quasi-brittle mechanical behaviour.To model this quasi-brittle behaviour, a suitable numerical approach is the Discrete Element Method (DEM), which can circumvent the limitations of more conventional continuum approaches in capturing microstructural effects required to simulate multi-fracture propagation.Here, it is aimed to simulate such quasi-brittle behaviour by initial well-distributed damages, with a strength dispersion following a Weibull distribution. In this way, the microcracks effect on the quasi-brittle behaviour of a numerical sample under uniaxial and cyclic tensile tests is investigated. Ultimately, a quantitative DEM model to simulate such a complex behaviour is proposed.

Molecular level simulations of combustion processes using the DSMC method

A technique called the Direct Simulation Monte Carlo (DSMC) method is used for simulating laminar one-dimensional hydrogen-air flames with an aim to establish the feasibility of molecular-level simulations for combustion using the Quantum-Kinetic (QK) reaction model. In DSMC, simulation particles are employed which represent many molecules with similar properties. The DSMC method effectively solves the Boltzmann equation by decoupling the motion of molecules into two phases: a Move phase and a Collision phase. Chemistry is treated using the Quantum-Kinetic model in which the total energy exchange resulting from molecular collision determines the outcome of the chemical reactions. A major advantage of DSMC is that it avoids using continuum Arrhenius reaction rates or simplified gradient laws for the diffusivities of mass, momentum and heat. Instead, any such laws and their parameters emerge naturally from the DSMC results. Simulations of one-dimensional hydrogen-air flames using the DSMC method with a detailed chemical scheme show promising results for molecular-level simulations of combustion. An adjustment to the activation energy of a key reaction is made in order to achieve the correct flame speed. The species and temperature profiles from DSMC show reasonable agreement with those obtained from Direct Numerical Simulation (DNS) using a conventional continuum approach. Likewise, the molecular diffusivity of hydrogen and oxygen also show reasonable agreement with the DNS results, although differences in the results still exist, especially for oxygen diffusivity. These are expected to improve with further development in the DSMC method for combustion.

Viscous heating and temperature profiles of liquid water flows in copper nanochannel

Understanding nanoscale fluidic transport becomes increasingly important due to the rapid development of nanotechnology and nanofabrication. By using molecular dynamics (MD) simulations, we investigated the viscous heating of water flows in copper nanochannels. The two scenarios that were studied are Couette flows and Poiseuille flows. We observed the scale effects on the distribution of fluid density, streaming velocity, fluid viscosity, and temperature across the channel. The results revealed the significant effects of surface forces on causing a large deviation between simulation results and classical hypothesis. We found that the energy equation coupled with the thermal-slip boundary conditions still fails to predict the temperature distributions. Hereby, further scale effects are taken into account, which leads to better predictions. The model that we developed in this study shows the relative deviation to the simulation data within 5 %, which is small compared to the conventional continuum approach (i.e., up to 51 %).

Slip avalanche in nanoscratching of metallic glasses

Slip avalanches, similar to discrete earthquake events, of Zr-, Co-, and Ce-based metallic glasses during nanoscratching were investigated. Differing from the conventional continuum approach, mean-field theory, which is an inherently-discrete model, was applied to analytically compute intermittent slip avalanches. Mean-field theory was first connected with the potential energy barrier and concentration of free volume in order to study the stick-slip behavior. The results suggest that the motion behavior of free volume affects the critical slip avalanche size.

Multiresolution modeling of ductile reinforced brittle composites

Tungsten carbide–cobalt (WC–Co) is an important ductile reinforced brittle composite used in a range of important applications. The relationship between microstructure and mechanical properties of WC–Co is truly multiscale; micromechanical processes interact at different scales, resulting in permanent plastic deformation, damage accumulation and final failure of the composite. The goal of the current paper is to develop a continuum-based model, which captures the progressively finer scales of strain localization observed in WC–Co composites during plastic deformation and failure. This is achieved via a set of multiresolution governing equations; a microstress is introduced at each scale of strain localization, which represents the resistance to inhomogeneous strain localization at that scale. The extra constitutive models associated with these microstresses can be elucidated from the average response of separate computational cell models of a representative microstructure. The final multiresolution continuum model is capable of capturing the important length scales of deformation during the plastic stage of deformation without resorting to modeling microstructural scale features directly. The result is a more realistic continuum model; in particular the fracture toughness prediction is more physical when these length scales are incorporated compared to a conventional continuum approach.

Peridynamics via finite element analysis

Peridynamics is a recently developed theory of solid mechanics that replaces the partial differential equations of the classical continuum theory with integral equations. Since the integral equations remain valid in the presence of discontinuities such as cracks, the method has the potential to model fracture and damage with great generality and without the complications of mathematical singularities that plague conventional continuum approaches. Although a discretized form of the peridynamic integral equations has been implemented in a meshless code called EMU, the objective of the present paper is to describe how the peridynamic model can also be implemented in a conventional finite element analysis (FEA) code using truss elements. Since FEA is arguably the most widely used tool for structural analysis, this implementation may hasten the verification of peridynamics and significantly broaden the range of problems that the practicing analyst might attempt. Also, the present work demonstrates that different subregions of a model can be solved with either the classical partial differential equations or the peridynamic equations in the same calculation thus combining the efficiency of FEA with the generality of peridynamics. Several example problems show the equivalency of the FEA and the meshless peridynamic approach as well as demonstrate the utility and robustness of the method for problems involving fracture, damage and penetration.

Predicting the Hall–Petch effect in fcc metals using non-local crystal plasticity

Many conventional continuum approaches to solid mechanics do not address the size sensitivity of deformation to microstructural features like grain boundaries, and are therefore unable to capture much of the experimentally observed behavior of polycrystal deformation. We propose a non-local crystal plasticity model, in which the geometrically necessary dislocation (GND) density is calculated using a non-local integral approach. The model is based on augmented F e F p kinematics, which account for the initial microstructure (primarily grain boundaries) present in the polycrystal. With the augmented kinematics, the initial GND and the evolving GND state are determined in a consistent manner. The expanded kinematics and the non-local crystal plasticity model are used to simulate the tensile behavior in copper polycrystals with different grain sizes ranging from 14 μm to 244 μm. The simulation results show a grain size dependence on the polycrystal’s yield strength, which are in good agreement with the experimental data.

310 格子ボルツマン法によるマイクロ流れの数値シミュレーション

Application of Lattice Boltzmann Method (LBM) to micro-CFD is discussed. Incompressible 2-D and 3-D micro-chanenel flows are solved by LBM. The results are compared with those of the conventional Navier-Stokes equations. It is found that comparable results are obtained with less computer time in LBM. LBM is found to be an alternative tool to conventional continuum approach in CFD, especially for micro flow problems.

APPLICATION OF THE DISCRETE ELEMENT MODELLING IN AIR DRYING OF PARTICULATE SOLIDS

ABSTRACT The Discrete Element Method (DEM) has been widely used as a mathematical tool for the study of flow characteristics involving particulate solids. One distinct advantage of this fast developing technique is the ability to compute trajectories of discrete particles. This provides the opportunity to evaluate the interactions between particle, fluid and boundary at the microscopic level using local gas parameters and properties, which is difficult to achieve using a continuum model. To date, most of these applications focus on the flow behaviour. This paper provides an overview of the application of DEM in gas–solids flow systems and discusses further development of this technique in the application of drying particulate solids. A number of sub-models, including momentum, energy and mass transfer, have been evaluated to describe the various transport phenomena. A numerical model has been developed to calculate the heat transfer in a gas–solids pneumatic transport line. This implementation has shown advantages of this method over conventional continuum approaches. Future application of this technique in drying technology is possible but experimental validation is crucial.

CHIRAL FERMIONS ON THE LATTICE AND INDEX RELATIONS

Comparing recent lattice results on chiral fermions and old continuum results for the index puzzling questions arise. To clarify this issue we start with a critical reconsideration of the results on finite lattices. We then work out various aspects of the continuum limit. After determining bounds and norm convergences we obtain the limit of the anomaly term. Collecting our results the index relation of the quantized theory gets established. We then compare in detail with the Atiyah–Singer theorem. Finally we analyze conventional continuum approaches.

Stochastic aggregation model (SAM) for DNAPL–water displacement in porous media

Source zone characterization of DNAPL in the subsurface is a necessary part of managing environmental risks. Modeling DNAPL–water displacement processes at the pore-scale can lead to greater insight and understanding of lateral spreading, the extent of vertical migration, and the final fluid distributions of dense non-aqueous phase liquids. A two-dimensional stochastic aggregation model is developed to study DNAPL–water displacement in porous media. The model is a modified diffusion limited aggregation model, and uses essential properties governing front stability as model input: DNAPL–water viscosity differences, DNAPL–water density differences, intrinsic permeability of the porous media, flow rate, and the inclination angle of the porous media from the horizontal. Due to the simplicity of the algorithm, this modeling technique can produce realistic DNAPL–water configurations much faster than conventional continuum approaches and allows for a Monte Carlo approach to be utilized. The model is validated through comparison to laboratory experiments involving 1,2-Dichloroethane, carbon tetrachloride, tetrachloroethylene, and mobile pyrogard 53. These experiments were performed at varying flow rates and angles of inclination from the horizontal.