Equations Of Continuum Mechanics Research Articles

Multiscale homogenized constrained mixture model of the bio-chemo-mechanics of soft tissue growth and remodeling.

Constrained mixture models have successfully simulated many cases of growth and remodeling in soft biological tissues. So far, extensions of these models have been proposed to include either intracellular signaling or chemo-mechanical coupling on the organ-scale. However, no version of constrained mixture models currently exists that includes both aspects. Here, we propose such a version that resolves cellular signal processing by a set of logic-gated ordinary differential equations and captures chemo-mechanical interactions between cells by coupling a reaction-diffusion equation with the equations of nonlinear continuum mechanics. To demonstrate the potential of the model, we present 2 case studies within vascular solid mechanics: (i) the influence of angiotensin II on aortic growth and remodeling and (ii) the effect of communication between endothelial and intramural arterial cells via nitric oxide and endothelin-1.

Investigation the plastic flows in the metal stamping-drawing process at the die corner

The present investigation covers the study of the influence of metal-plastic flow on the die corner for the deformed state of the blank and energy-power parameters during stamping-drawing. The influence of workpiece size and material on stress intensity and maximum deformation force has also been investigated both experimentally and theoretically. In addition, simulation of the stamping and drawing process was also presented. One type of die, with variations in diameter are considered different in the drawn part size and for each standard size four types of workpiece materials have been considered. The stamping process continued until the part failed structurally. The predicted test results show a dependence of the maximum load and stress intensity on both the diameter of the blank and the yield strength of the alloys and these data are in agreement with the actual measured values. Then, the method for calculating the processes of plastic deformation of metals based on a closed set of equations of continuum mechanics is proposed for the theoretical study of energy-power parameters of the technological processes. Comparison of the results of a full-scale experiment, simulation of metal flow at the die corner and theoretical calculations shows that the proposed theoretical model gives satisfactory results and can be effectively used for engineering calculations.

On Λ-Fractional Wave Propagation in Solids

Wave propagation in solids is discussed, based upon inherently non-local Λ-fractional analysis. Following the governing equations of Λ-fractional continuum mechanics, the Λ-fractional wave equations are derived. Since the variational procedures are only global, in the present Λ-fractional analysis, various jumpings, either in the strain or the stress, may be shown. The proposed theory is applied to impact-induced transitions in two-phase elastic materials and viscoelastic materials.

An arbitrary-order immersed interface method for the two-dimensional propagation of acoustic and elastic waves

This paper proposes an arbitrary-order immersed interface method for simulating the two-dimensional propagation of acoustic and elastic waves through fluid/solid interfaces. The present technique involves two main ingredients: (1) the linearized equations of continuum mechanics are simulated through an ADER (Arbitrary high-order schemes using DERivatives) scheme of arbitrary-order in both space and time [Schwartzkopff et al., J. Comput. Phys. 197(2), 532–539 (2004)]; (2) the jump conditions along the material interfaces are taken into account through the “explicit simplified interface method” (ESIM) derived by Lombard and Piraux [J. Comput. Phys. 195(1), 90–116, 2004]. To implement the ESIM, arbitrary-order spatial derivatives of the interface conditions must be calculated. To this end, an algorithm not requiring their explicit analytical expressions is developed for their numerical computation. Two numerical experiments involving flat and curved interfaces are finally discussed. When increasing the order of both the ADER scheme and of the interface treatment, the improvement of the convergence and of the accuracy of the numerical method is more specifically demonstrated by comparing the numerical results with analytical solutions.

Time-dependent modelling of thin poroelastic films drying on deformable plates

AbstractUnderstanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. One way to study these stresses is through the cantilever experiment, whereby a thin film is deposited onto the surface of a thin plate that is clamped at one end to a wall. The stresses that are generated in the film during drying are transmitted to the plate and drive bending. Mathematical modelling enables the film stress to be inferred from measurements of the plate deflection. The aim of this paper is to present simplified models of the cantilever experiment that have been derived from the time-dependent equations of continuum mechanics using asymptotic methods. The film is described using nonlinear poroelasticity and the plate using nonlinear elasticity. In contrast to Stoney-like formulae, the simplified models account for films with non-uniform thickness and stress. The film model reduces to a single differential equation that can be solved independently of the plate equations. The plate model reduces to an extended form of the Föppl-von Kármán (FvK) equations that accounts for gradients in the longitudinal traction acting on the plate surface. Consistent boundary conditions for the FvK equations are derived by resolving the Saint-Venant boundary layers at the free edges of the plate. The asymptotically reduced models are in excellent agreement with finite element solutions of the full governing equations. As the Péclet number increases, the time evolution of the plate deflection changes from $t$ to $t^{1/2}$ , in agreement with experiments.

Plane and linear blow-up and source

Nonrelativistic equations of continuum mechanics are invariant with respect to space translations and Galilean translations. These symmetries form the 6-parameter group. The 3-dimensional subgroup of the general type containing the Galilean subgroup produces invariant solutions with the blow-up and the instantaneous source. The subgroup generating the solutions with the plane and linear blow-up and source or with both singularities together is considered. Gas particles move along a straight line for the invariant solutions and along plane curvilinear trajectories for the partially invariant solutions with special state equations. The classification of the state equations for these solutions with constant entropy is obtained. Simple solutions for the polytropic gas and the specific gas are found.

Data-driven discovery of the governing equationsfor transport in heterogeneous media by symbolic regression and stochastic optimization.

With advances in instrumentation and the tremendous increase in computational power, vast amounts of data are becoming available for many complex phenomena in macroscopically heterogeneous media, particularly those that involve flow and transport processes, which are problems of fundamental interest that occur in a wide variety of physical systems. The absence of a length scale beyond which such systems can be considered as homogeneous implies that the traditional volume or ensemble averaging of the equationsof continuum mechanics over the heterogeneity is no longer valid and, therefore, the issue of discovering the governing equationsfor flow and transport processes is an open question. We propose a data-driven approach that uses stochastic optimization and symbolic regression to discover the governing equationsfor flow and transport processes in heterogeneous media. The data could be experimental or obtained by microscopic simulation. As an example, we discover the governing equationfor anomalous diffusion on the critical percolation cluster at the percolation threshold, which is in the form of a fractional partial differential equation, and agrees with what has been proposed previously.

Formulations of the elastodynamic equations in anisotropic and multiphasic porous media from the principle of energy conservation

Abstract Elastodynamic equations have been formulated with Newton's second law of motion, Lagrange's equation, or Hamilton's principle for over 150 years. In this work, contrary to classical continuum mechanics, a novel strategic methodology is proposed for formulating general mechanical equations using the principle of energy conservation. First, based on Hamilton's principle, Hamilton's equations, Lagrange's equation, and the elastodynamic equation of motion are derived in arbitrarily anisotropic and multiphasic porous elastic media, for the first time. Secondly, these equations are all formulated using the principle of energy conservation for the related media. Both formulation results using the two kinds of principles are compared and validated by each other. The advantages of our methodology lie in that the elastodynamic equation of motion, Lagrange's equation, and Hamilton's equations in continuum mechanics are directly formulated using a simple constraint of energy conservation without introducing variational concepts. It is easy to understand and has clear physical meanings. Our methodology unlocks the physics essences of Hamilton's principle in continuum mechanics, which is a consequence of the principle of energy conservation. Although the linear stress–strain constitutive relation is considered, our methodology can still be used in a nonlinear dynamical system. The methodology also paves an alternative way of treating other complex continuous dynamical systems in a broad sense. In addition, as an application, the continuity conditions at various medium interfaces are also revisited and extended using our proposed approach, which explains the law of reflections and refractions.

The scaled boundary finite element method for dispersive wave propagation in higher‐order continua

AbstractThe classical theory of elasticity relies on the perfect structure of matter. No matter how small a medium is, it always stays homogeneous—but the reality is different. Even though the classical continuum mechanics suffices well for describing phenomena that evolve at super‐microscopic scales, it ceases to reproduce reasonable responses when the size effects are pronounced. It is due to the local constitutive equations of classical continuum mechanics, which are devoid of any length scales associated with the underlying microstructure. The gradient‐dependent theory of elasticity redresses this shortcoming by incorporating the kinematic quantities of the corresponding representative volume element by enriching the differential equations with higher‐order spatial and temporal derivatives. In this study, the scaled boundary finite element method is formulated for the equations of motion with higher‐order inertia terms. To this end, the semi‐discretized scaled boundary finite element equations of the medium are derived by introducing the scaled boundary transformation of geometry to the gradient‐enriched equations of motion and applying the Galerkin method of weighted residuals. It is shown that the well‐established available solution methods are incapable of handling the frequency‐domain representation of the derived formulation. Accordingly, a numerical solution method based on the shooting technique is proposed. The solution procedure is formulated for general numerical integration schemes via the infinite Taylor series. The evolution of the impedance‐diffusion matrix and contribution of inter‐subdomain forces and tractions are extracted. Four different numerical integration methods are employed to describe the solution procedure. In addition, a comparison regarding their computational efficiency is presented. For the sake of verification, the scaled boundary finite element solutions of three numerical examples are compared with reference solutions that are obtained using finite element models with extremely fine meshes. The convergence trends are also presented for both ‐ and ‐refinement methods. The results demonstrate the capability of the proposed formulation in reproducing accurate responses.

Two-Phase Flow Modeling for Bed Erosion by a Plane Jet Impingement

This paper presents experimental and numerical studies on the erosion of a horizontal granular bed by a two-dimensional plane vertical impinging jet to predict the eroded craters’ size scaling (depth and width). The simulations help understand the microscopic processes that govern erosion in this complex flow. A modified jet-bed distance, accounting for the plane jet virtual origin, is successfully used to obtain a unique relationship between the crater size and a local Shields parameter. This work develops a two-phase flow numerical model to reproduce the experimental results. The numerical techniques are based on a finite volume formulation to approximate spatial derivatives, a projection technique to calculate the pressure and velocity for each phase, and a staggered grid to avoid spurious oscillations. Different options for the sediment’s solid-to-liquid transition during erosion are proposed, tested, and discussed. One model is based on unified equations of continuum mechanics, others on modified closure equations for viscosity or momentum transfer. A good agreement between the numerical solutions and the experimental measurements is obtained.

TECHNOLOGICAL EQUIPMENT FOR GEARWORKING OF HARDENED WHEELS

The relationship between the theoretical dependence of the cutting speed on various technological factors, the physical and mechanical properties of the tool materials and the hardened gear allows the choice of rational cutting conditions in each specific production case, taking into account the required indicators of tool life, processing productivity, accuracy and quality of the surface layer. Cutting schemes and technological equipment were developed for intensive and high-quality gear cutting using the method of numerical simulation of the surface layer shaping process, which made it possible to obtain the stress-strain state of the surface layer of the gear being machined, tool and chips, and recommendations were developed for choosing the technological processing schedule. The numerical simulation method is based on solving a system of continuum mechanics equations (equations of motion, continuity, energy). The constitutive relations of the theory of elasticity, plasticity, and fracture are used as closing equations for the system.

Numerical modeling of low-frequency scattering from targets buried under rough seafloors

The acoustic backscattering of elastic bodies lying across the seafloor is a problem of great interest in many underwater applications, such as detecting and classifying buried objects. In this work, the backscattered wavefield of meter-sized targets is investigated numerically in the 1–50 kHz frequency band. The simulator solves the three-dimensional linearized equations of continuum mechanics. The computations are performed through a multi-GPU solver based on a multi-grid staggered finite-difference time-domain method (cf. Sabatini et al., 181st ASA Meeting, Seattle, Washington, 2021). The effects of the seafloor roughness and the target burial depth on the spectrum of the scattered field are more specifically analyzed. This study demonstrates the usefulness of direct numerical simulations of acoustic scattering from realistic elastic objects in complex underwater environments to improve the capabilities of sonar systems in detecting and identifying buried objects.

Simultaneously measurement of strain field and Poisson’s ratio by using an off-axis phase-sensitive optical coherence elastography

Phase-sensitive optical coherence elastography (PhS-OCE) is a novel functional imaging modality capable of mapping strain fields inside semi-transparent materials. In this work, an off-axis PhS-OCE was further developed to measure strain field and Poisson’s ratio simultaneously. Based on the intrinsic equations of continuum mechanics, the relationship between the elastic parameters of the measured material and the physical quantity (i.e. optical path difference) of PhS-OCE was first established. For validation, the depth-resolved strain field and Poisson’s ratio of a silicone rubber film were quantitatively measured during tensile tests. The experimental results, such as the estimates for Poisson’s ratio, agreed with the reference values. Moreover, phase difference maps of bilayer composites were discussed, indicating the effectiveness and potential of the proposed off-axis measurement method.

Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem

Physics-informed neural networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs) and are in principle capable of modeling a large variety of differential equations. PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that develop highly localized nonlinear shock waves. Learning solutions of PDEs with dominant hyperbolic character is a challenge for current PINN approaches, which rely, like most grid-based numerical schemes, on adding artificial dissipation. Here, we address the fundamental question of which network architectures are best suited to learn the complex behavior of non-linear PDEs. We focus on network architecture rather than on residual regularization. Our new methodology, called physics-informed attention-based neural networks (PIANNs), is a combination of recurrent neural networks and attention mechanisms. The attention mechanism adapts the behavior of the deep neural network to the non-linear features of the solution, and break the current limitations of PINNs. We find that PIANNs effectively capture the shock front in a hyperbolic model problem, and are capable of providing high-quality solutions inside the convex hull of the training set.

Microscopic density-functional approach to nonlinear elasticity theory

Starting from a general classical model of many interacting particles we present a well defined step by step procedure to derive the continuum-mechanics equations of nonlinear elasticity theory with fluctuations which describe the macroscopic phenomena of a solid crystal. As the relevant variables we specify the coarse-grained densities of the conserved quantities and a properly defined displacement field which describes the local translations, rotations, and deformations. In order to stay within the framework of the conventional density-functional theory we first and mainly consider the isothermal case and omit the effects of heat transport and warming by friction where later we extend our theory to the general case and include these effects. We proceed in two steps. First, we apply the concept of local thermodynamic equilibrium and minimize the free energy functional under the constraints that the macroscopic relevant variables are fixed. As results we obtain the local free energy density and we derive explicit formulas for the elastic constants which are exact within the framework of density-functional theory. Second, we apply the methods of nonequilibrium statistical mechanics with projection-operator techniques. We extend the projection operators in order to include the effects of coarse-graining and the displacement field. As a result we obtain the time-evolution equations for the relevant variables with three kinds of terms on the right-hand sides: reversible, dissipative, and fluctuating terms. We find explicit formulas for the transport coefficients which are exact in the limit of continuum mechanics if the projection operators are properly defined. By construction the theory allows the diffusion of particles in terms of point defects where, however, in a normal crystal this diffusion is suppressed.

Enhancing efficiency and scope of first-principles quasiharmonic approximation methods through the calculation of third-order elastic constants

A novel computational strategy is presented to calculate from first principles the coefficient of thermal expansion and the elastic constants of a material over meaningful intervals of temperature and pressure. This strategy combines a novel implementation of the quasiharmonic approximation to calculate the isothermal-isochoric linear and nonlinear elastic constants of a material, with elementary equations of nonlinear continuum mechanics. Our implementation of the quasiharmonic approximation relies on finite deformations, the use of nonprimitive supercells to describe a material, a recently proposed technique to calculate generalized mode Gr\"uneisen parameters, and the numerical differentiation of the stress tensor to calculate both second- and third-order elastic constants. The combination of this method with nonlinear continuum mechanics is shown to yield accurate predictions of lattice parameters and linear elastic constants of a material over finite intervals of temperature and pressure, at the cost of calculating isothermal second- and third-order elastic constants for a single reference state. Here, the validity and limits of our novel methods are assessed by carrying out calculations of MgO based on classical interatomic potentials. To demonstrate potential, our methods are then used in conjunction with a density functional theory approach to calculate thermal expansion and elastic properties of silicon, lithium hydrate, and graphite.

Variational principles for two kinds of non-linear geophysical KdV equation with fractal derivatives

It is an important and difficult inverse problem to construct variational principles from complex models directly, because their variational formulations are theoretical bases for many methods to solve or analyze the non-linear problems. At first, this paper extends two kinds of non-linear geophysical KdV equations in continuum mechanics to their fractional partners in fractal porous media or with irregular boundaries. Then, by designing skillfully, the trial-Lagrange functional, variational principles are successfully established for the non-linear geophysical KdV equation with Coriolis term, and the high-order extended KdV equation with fractal derivatives, respectively. Furthermore, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations.

A three-dimensional multi-grid staggered finite-difference time-domain method for underwater target acoustic scattering

Solving the linearized equations of continuum mechanics to model the acoustic scattering of targets close to the seafloor is a challenging task. Among the traditional algorithms, staggered second-order finite-difference time-domain (FDTD) schemes combine simplicity and ease of implementation and parallelization, especially on modern GPUs. They have been successfully applied in two-dimensional configurations. Their major drawback is their inherent low resolution, due to their formal low order and the stair-step representation of the interfaces between media (e.g., target/water). To avoid excessive computational costs in three-dimensional configurations, while maintaining high numerical accuracy, we propose a FDTD multi-grid method. The mesh is refined locally near the interfaces, while a coarse grid is employed in homogeneous regions. In the fine grids, the aforesaid FDTD is employed, whereas, in the coarse grid, higher-order staggered schemes are used to compensate for the decrease in resolution due to the larger spacing. High-order interpolation allows matching the solutions at the grid interfaces. In this presentation, the performances of the new method on classical benchmark tests are analyzed and discussed. The numerical and computational capabilities are then illustrated by applying the proposed technique to the scattering of buried objects.

On the Correspondence between the Variational Principles in the Eulerian and Lagrangian Descriptions

A relation between variational principles for equations of continuum mechanics in Eulerian and Lagrangian descriptions is considered. It is shown that for a system of differential equations in Eulerian variables corresponding Lagrangian description is related to introducing nonlocal variables. The connection between these descriptions is obtained in terms of differential coverings. The relation between variational principles of a system of equations and its symplectic structures is discussed. It is shown that if a system of equations in Lagrangian variables can be derived from a variational principle then there is no corresponding variational principle in Eulerian variables.

On variational principles in Euler’s and Lagrange’s descriptions

The correspondence between the variational principles of the equations of continuum mechanics in Euler’s and Lagrange’s descriptions is considered. It is shown that the introduction of Lagrange’s variables can be considered as the construction of a differential covering by introducing the potential into the mass conservation law. A correspondence between the variational principles of the l-normal equations of continuum mechanics in the Euler’s and Lagrange’s descriptions in terms of symplectic structures is proposed.