Continuum Mechanics Model Research Articles

A unified HTC multiphase model of continuum mechanics

In this paper, we present a unified nonequilibrium model of continuum mechanics for compressible multiphase flows. The model, which is formulated within the framework of Hyperbolic Thermodynamically Compatible (HTC) equations, can describe an arbitrary number of phases that can be heat conducting inviscid and viscous fluids, as well as elastoplastic solids. The phases are allowed to have different velocities, pressures, temperatures, and shear stresses, while the material interfaces are treated as diffuse interfaces with the volume fraction playing the role of the interface field. To relate our model to other multiphase approaches, we reformulate the novel HTC governing equations in terms of the phase state parameters and put them in the form of Baer-Nunziato-type models. It is the Baer-Nunziato form of the HTC equations which is then solved numerically using a robust second-order path-conservative MUSCL-Hancock finite volume method on Cartesian meshes. Due to the fact that the obtained governing equations are very challenging we restrict our numerical examples to a simplified version of the model for three-phase mixtures. To address the stiffness properties of the relaxation source terms present in the model, the implemented scheme incorporates a semi-analytical time integration method specifically designed for the non-linear stiff source terms governing the strain relaxation. The validation process involves a wide range of benchmarks and several applications to compressible multiphase problems. Notably, results are presented for multiphase flows in several relaxation limit cases of the model, including inviscid and viscous Newtonian fluids, as well as non-linear hyperelastic and elastoplastic solids. In all cases, the numerical results demonstrate good agreement with established models, including the Euler or Navier-Stokes equations for fluids and the classical hypo-elastic model with plasticity for solids. Importantly, however, this approach achieves these results within a unified multiphase framework of continuum mechanics.

Evolution of local relaxed states and the modeling of viscoelastic fluids

We introduce a class of continuum mechanical models aimed at describing the behavior of viscoelastic fluids by incorporating concepts originated in the theory of solid plasticity. Within this class, even a simple model with constant material parameters is able to qualitatively reproduce a number of experimental observations in both simple shear and extensional flows, including linear viscoelastic properties, the rate dependence of steady-state material functions, the stress overshoot in incipient shear flows, and the difference in shear and extensional rheological curves. Furthermore, by allowing the relaxation time of the model to depend on the total strain, we can reproduce some experimental observations of the non-attainability of steady flows in uniaxial extension and link this to a concept of polymeric jamming or effective solidification. Remarkably, this modeling framework helps in understanding the interplay between different mechanisms that may compete in determining the rheology of non-Newtonian materials.

Mechanical modeling of the petiole-lamina transition zone of peltate leaves

Plant leaves have to deal with various environmental influences. While the mechanical properties of petiole and lamina are generally well studied, only few studies focused on the properties of the transition zone joining petiole and lamina. Especially in peltate leaves, characterised by the attachment of the petiole to the abaxial side of the lamina, the 3D leaf architecture imposes specific mechanical stresses on the petiole and petiole-lamina transition zone. Several principles of internal anatomical organisation have been identified. Since the mechanical characterisation of the transition zone by direct measurements is difficult, we explored the mechanical properties and load-bearing mechanisms by finite-element simulations. We simulate the petiole-lamina transition zone with five different fibre models that were abstracted from CT data. For comparison, three different load cases were defined and tested in the simulation. In the proposed model, the fibres are represented in a smeared sense, where we considered transverse isotropic behavior in elements containing fibres. In a pre-processing step, we determined the fibre content, direction, and dispersion and fed them into our model. The simulations show that initially, matrix and fibres carry the load together. After relaxation of the stresses in the matrix, the fibres carry most of the load. Load dissipation and stiffness differ according to fibre arrangement and depend, among other things, on orientation and cross-linking of the fibres and fibre amount. Even though the presented method is a simplified approach, it is able to show the different load-bearing capacities of the presented fibre arrangements. Statement of significanceIn plant leaves, the petiole-lamina transition zone is an important structural element facilitating water and nutrient transport, as well as load dissipation from the lamina into the petiole. Especially in peltate leaves, the 3D leaf architecture imposes specific mechanical stresses on the petiole-lamina transition zone. This study aims at investigating its mechanical behavior using finite-element simulations. The proposed continuum mechanical anisotropic viscoelastic material model is able to simulate the transition zone under different loads while also considering different fibre arrangements. The simulations highlight the load-bearing mechanisms of different fibre organisations, show the mechanical significance of the petiole-lamina transition zone and can be used in the design of a future biomimetic junction in construction.

Emergence of lobed wakes during the sedimentation of spheres in viscoelastic fluids

The motion of rigid particles in complex fluids is ubiquitous in natural and industrial processes. The most popular toy model for understanding the physics of such systems is the settling of a solid sphere in a viscoelastic fluid. There is general agreement that an elastic wake develops downstream of the sphere, causing the breakage of fore-and-aft symmetry, while the flow remains axisymmetric, independent of fluid viscoelasticity and flow conditions. Using a continuum mechanics model, we reveal that axisymmetry holds only for weak viscoelastic flows. Beyond a critical value of the settling velocity, steady, non-axisymmetric disturbances develop peripherally of the rear pole of the sphere, giving rise to a four-lobed fingering instability. The transition from axisymmetric to non-axisymmetric flow fields is characterized by a regular bifurcation and depends solely on the interplay between shear and extensional properties of the viscoelastic fluid under different flow regimes. At higher settling velocities, each lobe tip is split into two new lobes, resembling fractal fingering in interfacial flows. For the first time, we capture an elastic fingering instability under steady-state conditions, and provide the missing information for understanding and predicting such instabilities in the response of viscoelastic fluids and soft media.

Modeling and Estimation of Continuous Flexible Structure Using Theory of Functional Connections

This paper presents a novel method for modeling and estimating the dynamics of a continuous structure based on a limited number of noisy measurements. The goal is reached using a Kalman filter in synergy with the recently developed mathematical framework known as the Theory of Functional Connections (TFC). The TFC allows deriving a functional expression capable of representing the entire space of the functions that satisfy a given set of linear and, in some cases, nonlinear constraints. The proposed approach exploits the possibilities offered by the TFC to derive an approximated dynamical model for the flexible system using the Lagrangian mechanics. The result is a representation of the structural dynamics using a finite number of states, in contrast to the infinite-dimensional model that would be obtained by application of the traditional continuum mechanics models that are based on sets of partial differential equations. The limited number of states enables the application of the well-known Kalman filter framework to improve the estimation of the displacements and displacement velocities. In addition, the continuous displacement field of the structure can be reconstructed with high fidelity. The theoretical development of the method is presented in relation to the case of an Euler–Bernoulli beam. Finally, the obtained model is used to carry out a simulation campaign aimed at assessing the accuracy, efficiency, and robustness of the proposed method.

The peeling behavior of compliant nano-films in adhesive contact with a planar rigid substrate: Insights from molecular dynamics and continuum mechanics

Peeling compliant nano-films from supporting substrates is crucial in the mechanical exfoliation and transfer processes. However, the peeling behavior, especially concerning the peeling stiffness and peak peeling force, exhibits intricate interplay with the geometric and material properties of nano-films, as well as interfacial interactions, which have yet to be fully elucidated. In this work, both classical molecular dynamics (MD) simulations and continuum analysis are adopted to investigate the entire peeling process of compliant nano-films from a planar rigid substrate. Considering the atomic structure and van der Waals (vdW) interactions at the interface, we establish a continuum mechanics model to describe the entire peeling process, encompassing the initial, transitional, steady-state, and unstable peel-off stages. The theoretical predictions are reasonably consistent with the results obtained by MD simulations. The effects of film length and interface toughness on the peeling process, the peeling stiffness and peak peeling force, are thoroughly investigated, and a phase diagram for the peeling deformation modes is quantitatively constructed. Finally, dimensional analysis yields scaling relations for the peak peeling force in terms of the length and bending stiffness of compliant nano-films, as well as the governing parameters for interfacial vdW interactions. These results contribute to a better understanding of the peeling mechanics of various two-dimensional nano-films (e.g., graphene, hexagonal boron nitride, and molybdenum disulfide) adhered to substrates.

Computationally Efficient Emulation of Spheroidal Elastic Deformation Sources Using Machine Learning Models: A Gaussian‐Process‐Based Approach

AbstractElastic continuum mechanical models are widely used to compute deformations due to pressure changes in buried cavities, such as magma reservoirs. In general, analytical models are fast but can be inaccurate as they do not correctly satisfy boundary conditions for many geometries, while numerical models are slow and may require specialized expertise and software. To overcome these limitations, we trained supervised machine learning emulators (model surrogates) based on parallel partial Gaussian processes which predict the output of a finite element numerical model with high fidelity but >1,000× greater computational efficiency. The emulators are based on generalized nondimensional forms of governing equations for finite non‐dipping spheroidal cavities in elastic halfspaces. Either cavity volume change or uniform pressure change boundary conditions can be specified, and the models predict both surface displacements and cavity (pore) compressibility. Because of their computational efficiency, using the emulators as numerical model surrogates can greatly accelerate data inversion algorithms such as those employing Bayesian Markov chain Monte Carlo sampling. The emulators also permit a comprehensive evaluation of how displacements and cavity compressibility vary with geometry and material properties, revealing the limitations of analytical models. Our open‐source emulator code can be utilized without finite element software, is suitable for a wide range of cavity geometries and depths, includes an estimate of uncertainties associated with emulation, and can be used to train new emulators for different source geometries.

Damage identification in scale sensitive beam-like nano-joints in the framework of space-fractional Euler–Bernoulli theory

In the paper, a fundamental question about damage identification in scale-sensitive bodies is posed. For this purpose, nano-scale beam-like joints are analyzed using dedicated computational schemes. The theoretical description is led in the framework of non-local space-fractional Euler–Bernoulli (s-FEBB) theory — a special case of space-Fractional Continuum Mechanical model generalizing classical continuum mechanics utilizing fractional calculus. The problem of damage identification is stated as an optimization task where the original numerical strategy is implemented. What is crucial, the s-FEBB model assumes constitutive parameters identified and validated for the silver nanobeam in previous studies. Based on tens of thousands of analyses, we arrive at the general conclusion that localization of fracture/damage/deterioration in material bodies whose dimensions are of the order of intrinsic microstructure is possible but requires advanced modeling — otherwise, damage can be identified in a wrong place or more places, or local damage can become distributed one and/or with wrong intensity.

Multi-time-step coupling of peridynamics and classical continuum mechanics for dynamic brittle fracture

Peridynamics (PD), as a nonlocal theory, is well-suited for solving problems with discontinuities, such as cracks. However, the nonlocal effect of peridynamics makes it computationally expensive for dynamic fracture problems in large-scale engineering applications. As an alternative, this study proposes a multi-time-step (MTS) coupling model of PD and classical continuum mechanics (CCM) based on the Arlequin framework. Peridynamics is applied to the fracture domain of the structure, while continuum mechanics is applied to the rest of the structure. The MTS method enables the peridynamic model to be solved at a small time step and the continuum mechanical model is solved at a larger time step. Consequently, higher computational efficiency is achieved for the fracture domain of the structure while ensuring computational accuracy, and this coupling method can be easily applied to large-scale engineering fracture problems.

Strength-induced Peridynamic model for the dynamic failure of porous materials

Predicting the dynamic failure process of porous materials is a challenging task due to their complex structure. To minimize the use of time-consuming peridynamic (PD) models and to avoid surface effect issues in the dynamic failure of complex porous materials, this paper proposes a strength-induced PD model. The paper first presents the dynamic formula and relevant finite element discrete equation of the coupled PD and classical continuum mechanics (PD-CCM) model based on the Morphing method. The Morphing function is implemented to control the material parameters and enable the free transformation of PD and CCM models. Based on the coupled PD-CCM model, the strength-induced PD model is established to adaptively expand the PD subdomain in porous materials by controlling the Morphing function value through the strength state of the structure. This model enables the PD subdomain to appear automatically when the porous materials reach the critical stress state. The proposed model accurately predicts the location of crack initiation and path while minimizing computational costs and improving efficiency. Three two-dimensional numerical examples are used to verify the effectiveness, efficiency, and accuracy of the model. The results of the simulation suggest that the location where the crack initiates in the porous materials is strongly influenced by the amplitude of the dynamic load. Cracking is dependent on the pores and typically occurs through them.

First-principles study on thermal expansion of W-Re sigma and chi phases

We investigate how the Re content affects the coefficient of thermal expansion (CTE) of the non-stoichiometric W-based σ and χ phases, forming upon neutron irradiation of W, to explore and quantify its mismatch between precipitates (W-Re) and matrix (W). To this end, we have conducted first-principles calculations using two approaches: the Debye-Grüneisen (DG) model and the quasi-harmonic approximation (QHA). The two approaches yield different results: the QHA, which is deemed to be the most accurate of the two, predicts substantial changes with Re content, while the acoustic-modes based DG model does not. The CTE of the σ and χ at stable Re contents is compared to experimental values for bcc-W and bcc-W-Re containing 25 at.% Re. Taking bcc-W as a reference, we find a significant mismatch in CTE of up to 37% and 62% for σ and χ, respectively, which may contribute to thermal stress buildup in the material at elevated temperatures. The mismatch is shown to increase with the temperature and Re content for both phases. The produced data are used to fit a temperature and Re concentration-dependent analytical function of the CTE for both phases, which can be employed as input for continuum mechanical modeling.

Analyze the temperature-dependent elastic properties of single-walled boron nitride nanotubes by a modified energy method

The elastic properties of boron nitride nanotubes (BNNTs) were investigated utilizing an enhanced energy method. By considering small deformations and applying the principle of minimum potential energy, the variations in atomic bonds and bond angles within the nanotube structure were determined. The modified model incorporated the contribution of inversion energy to the overall potential energy of the system, leading to the derivation of analytical expressions for the Young's modulus, shear modulus, and strain energy of both armchair and zigzag BNNTs under varying temperatures. The results indicate that compared to zigzag BNNTs, the impact of inversion energy on the elastic constants of armchair BNNTs is more significant, especially at small diameters (<1 nm). In thermal environment, this study demonstrates that the change in Young's modulus of BNNTs is lower than that of carbon nanotubes (CNTs), confirming the superior thermal stability of BNNTs over CNTs. Furthermore, molecular structure mechanics (MSM) and continuum mechanics models were employed to analyze the strain energy of BNNTs. The effects of different bonds, bond angles, and inversion angles on strain energy were analyzed in a thermal environment, revealing distinct differences between the two types of BNNTs. These findings provide more accurate theoretical guidance for thermal applications based on the stretching of BNNTs.

Mechanical deformation in lithium-ion battery electrodes: modelling and experiment

Abstract Models that can accurately describe deformation and stress in lithium-ion batteries are required to inform new device designs that can better withstand mechanical fatigue. Developing such models is particularly challenging because (i) there is a need to capture several different materials including, active materials, binders, current collectors and separators, and (ii) the length scales of interest are highly disparate (ranging from a few microns, relevant to active material particles, up to centimeters, relevant to whole devices). In this study we present a continuum mechanical model that resolves individual active material particles of a nickel-manganese-cobalt-oxide cathode, and predicts the mechanical response of the cathode coating as a whole. The model is validated by comparison with experimental tests which mimic industrial-scale electrode calendaring, and then a parametric study is conducted to provide insight into the roles of the material and geometric properties of the electrode's constituents on the cathode's overall behavior.

Coupled dislocations and fracture dynamics at finite deformation: model derivation, and physical questions

A continuum mechanical model of coupled dislocation based plasticity and fracture at finite deformation is proposed. Motivating questions and target applications of the model are sketched.

Vibrational analysis of graphyne-based nanoplates using a hybrid nonlocal strain gradient-atomistic simulation model

By employing a hybrid atomistic-continuum model, we can combine the advantages of atomic and continuum models. In this approach, resonance frequencies are matched to those obtained from the continuum model using the size effect and Young’s modulus as fit parameters. Our proposed method examines how radius affects the elastic modulus, size effect parameters, and resonance frequency values of the graphyne family, taking into account both macro and nano-scale models. The non-local strain gradient model has been calibrated to achieve optimal error. In contrast, the error is highest when the non-local strain gradient effect is not considered in the continuum mechanics model.

Towards improved online dissolution evaluation of Pt-alloy PEMFC electrocatalysts via electrochemical flow cell - ICP-MS setup upgrades

Electrochemical flow cell coupled with an inductively coupled plasma mass spectrometer (EFC-ICP-MS) is a powerful electroanalytical technique to monitor in-situ dissolution of metallic electrocatalysts and to understand mechanism of degradation under operating conditions. Its utilisation has witnessed a notable increase in the electrocatalyst field in the last decade where it has been extensively used to study the stability of platinum group metals (PGMs) under oxygen reduction and oxygen evolution reaction conditions. Online ICP-MS has allowed the scientific and industrial community to optimise the activity and stability of PGMs thanks to a better understanding of the complex metal corrosion processes. Among the different setups, the electrochemical flow cell design is the most common as it is based on a commercially available design. Nonetheless, besides different materials and different electrochemical protocols, the impact of the geometry and various parameters of the setup on the recorded dissolution signal has not been studied until now. Such parameters can influence the results obtained with an EFC-ICP-MS and thus the interpretation of the dissolution mechanism and/or stability assessment. Hereby, we demonstrate that the length of the tubing between the outlet of the cell and the inlet of the ICP-MS impacts the resolution of the PtCo catalyst dissolution peaks. This, in turn, facilitates studies where the detection of extremely low concentrations is necessary, such as under a very narrow potential window. Similarly, a reduced internal volume of the cell restricts Pt redeposition, contributing to a more precise evaluation of stability. These claims were supported by dynamic continuum mechanics modelling of the ion concentration in a model EFC. Finally, we provide guidelines and advice to properly measure dissolution with an electrochemical cell coupled with ICP-MS.

Martensitic-like microstructures across the isostructural phase transitions in Cerium

The isostructural gamma–alpha phase transition in elemental cerium is an electronic transition caused by a delocalization of the 4f electrons. This affects the bonding properties of Ce atoms and leads to a large volume collapse reaching ∼17% in the low pressure regime (¡ 2 GPa). While great attention has been drawn on the electronic description of this transition, attempts to understand the mesoscale mechanisms of this structural transition and their consequences in terms of microstructure remain scarce. We have investigated this transition by means of combined X-ray Computed Tomography and Energy Dispersive X-ray Diffraction on polycrystalline samples. Our experimental observations reveal a platelet-shape microstructure across the transition and up to the critical point, which have been associated to displacive mechanisms. Based on continuum mechanics modeling and ab initio calculations, we propose that here, this microstructure is initiated by elastic instability and shear anisotropy in the ¡100¿ directions.

Nonlinear bending of a soft slab subjected to vertical compression: A continuum mechanics model

The small bending behavior of linear elastic materials has been widespread in engineering in the history of mechanics. With the occurrence of soft materials, finite deformation bending is of great interest due to the high nonlinearity in large bending deformation. Previous theories primarily focus on the pure bending assumption, which is hard to achieve in practical bending experiments of soft materials because compressive forces cannot be neglected. In this paper, we propose a continuum mechanics-based nonlinear bending theory that incorporates an initial compression process to align with real-world experiments. We compare experimental data and simulation to our theoretical prediction to verify the accuracy of our theoretical model. The results demonstrated good agreement between our theory, experimental data, and finite element simulation. Furthermore, we compare our nonlinear bending theory with the classical Euler–Bernoulli theory to understand their respective behaviors better. This paper advances our understanding of bending phenomena by considering the effects of initial compression and incorporating a nonlinear approach. It provides valuable insights for practical applications of soft materials subjected to large bending deformation.

Role of foliation occurrence and stress orientation on asymmetric failure in foliated rock tunnel: insight from numerical simulation

The failure problem of deep-buried foliated rock tunnel is very prominent, which often presents asymmetry. Moreover, this failure is closely related to foliation occurrence and in-situ stress, and its mechanism is still unclear at present. In this study, a deep buried foliated rock tunnel in a hydropower station was taken as a case, and its failure mechanism was studied by using CASRock software embedded with three-dimensional equivalent continuous mechanical model of foliated rock. On this basis, the relationship and mechanism between the failure of deep-buried foliated rock tunnel and foliation occurrence, in-situ stress and tunnel axis were systematically studied. The results indicate that when the angle between the foliation strike and the tunnel axis (γ) is small, the surrounding rock will experience a larger range of failure. As the γ increases, the depth and degree of failure will significantly decrease. The dip angle of foliation (θ) and the orientation of maximum principal stress determine the failure location of tunnel section. When designing foliated rock tunnels, the angle between the foliation strike and the tunnel axis (γ) should be minimized as much as possible. When the γ is small, more attention should be paid to the orientation of the stress concentration area and its angle with the foliation on the tunnel section after excavation, and support should be strengthened for the small angle parts. This study has important guiding significance for the design, understanding of failure mechanisms, and disaster prevention and control of foliated rock tunnels.

Modeling of shock wave propagation in porous magnesium based on artificial neural network

Data transfer from the level of atomistic simulations to the level of continuum modeling is one of the urgent problems in the mechanics of materials. In this work, we perform multiple molecular dynamics (MD) simulations of the uniaxial (along different crystallographic directions) and volumetric (isotropic) compression of porous HCP magnesium single crystals in order to study the atomistic processes of the plastic deformation of anisotropic porous material and to create a reference dataset (strain dependencies of stresses, porosity and dislocation density at different temperatures and initial porosities). The MD data are used to train two artificial neural networks (ANNs): the first ANN is a surrogate of the constitutive equation of porous magnesium, while the second ANN describes the dislocation nucleation/emission as the onset of plastic flow. In contrast to the previously considered case of porous FCC aluminum, the present approach takes into account the anisotropy of the matrix material. The ANN-based constitutive equation is successfully used within the continuum mechanics model for numerical study on the shock wave structure in porous magnesium in comparison with direct MD simulation of the shock wave problem.