Instantaneous Relaxation Research Articles

Уравнения вязкоупругости не полностью отвержденного эпоксидного связующего при малых деформациях

In this paper, the mechanical state of an epoxy binder during incomplete curing is studied. The degree of curing is described by a three-parameter kinetic equation of conversion with the material parameters and their temperature dependence determined using mathematical optimization methods based on isothermal conversion data. Mechanical properties of the incompletely cured polymer are obtained with the use of experimental data on uniaxial loading of reference samples according to a program assuming stretching to a specified strain at a given rate and holding at a fixed strain during a specified period of time. The physical equations for the polymer under study are assumed to be linear viscoelastic Volterra equations under the condition that the volumetric deformation is elastic. A method for determining the material parameters of the equations, i.e., instantaneous elastic constants and relaxation kernels, is proposed. Curing of the epoxy binder in vacuum is accompanied by foaming due to the presence of air bubbles in the binder. This phenomenon is shown to be prevented by pre-curing up to the "barrier" level in atmospheric conditions. The research results can be used when calculating the technological processes of manufacturing structures made of composites.

On the computation of compressible multiphase flows with heat and mass transfer in elastic pipelines

The present work is motivated by the engineering need to simulate the multiphase flows in the course of the start-up of a long-distance crude oil pipeline. We propose a model for compressible multiphase flows with heat and mass transfer and diffusion processes in elastic pipelines. The model is derived with two approaches, i.e., (a) the spatial averaging procedure of a single phase model and (b) the Arbitrary Lagrangian Eulerian (ALE) formulation of the Baer-Nunziato model with quasi-1D approximation. The diffusion processes include the viscous dissipation, wall friction, heat conduction, and heat exchange with external environment. In particular, the wall friction consists of steady friction term calculated by Darcy-Weisbach formula, and the unsteady friction determined by the instantaneous-acceleration-based (IAB) model. The unsteady friction modifies the characteristic structure of the hyperbolic part of the model. For the solution of the hyperbolic part, we propose a three-wave approximate Riemann solver incorporating the unsteady friction term. Mass and heat transfer are realized via instantaneous relaxations of the chemical potential and temperature, respectively. Efficient iterative relaxation procedures for N-phase flows have been proposed. We have validated the proposed model and numerical methods against some benchmark multiphase problems and applied the model to calculate the start-up of a realistic liquid pipeline with intermediate pump station boundary condition.

A nonlinear creep model of rocks based on memory-dependent derivative

The creep behavior of soil and rock exhibits significant time-dependence and non-linearity, which are difficult to capture using traditional composite element models. In the framework of memory-dependent derivative theory, a memory-dependent visco-pot element with non-linear creep behavior, no instantaneous deformation, elastic after-effect, and stress relaxation is proposed. By connecting a memory-dependent visco-pot element with strain triggering to the Nishihara model, a memory-dependent derivative creep model is developed and its constitutive equation is derived. The creep test data pertaining to coal, salt, sandstone, and olivine rock are used to validate the model, which can describe the initial creep, steady-state creep, and non-linear accelerated creep deformation characteristics of various soil and rock materials. Comparison with fractional-order models, improved Nishihara models, and Hehai models shows that our model has better fitting results and is more advantageous in characterizing the non-linear features of soil and rock during the accelerated creep stage.

Waiting time dependence of aging

Aging phenomena have been observed in many non-equilibrium systems such as polymers and glasses, where physical properties depend on the waiting time between the starting time of observation and the time when the temperature is changed. The aging is classified into two types on the basis of the waiting time dependence of an instantaneous relaxation time: When the relaxation time is always an increasing function of the waiting time, the aging is called Type I and when it depends on the protocol of the temperature change, the aging is called Type II. Aging of a random walk in three dimensions is investigated when the free energy landscape controlling the jump rate responds to temperature change with a delay. It is shown that the intermediate scattering function of the random walk model exhibits Type II aging. It is also shown that the relaxation time of the free energy landscape can be deduced from the waiting time dependence of the instantaneous relaxation time.

Housekeeping and excess entropy production for general nonlinear dynamics

We propose a housekeeping/excess decomposition of entropy production for general nonlinear dynamics in a discrete space, including chemical reaction networks and discrete stochastic systems. We exploit the geometric structure of thermodynamic forces to define the decomposition; this does not rely on the notion of a steady state, and it even applies to systems that exhibit multistability, limit cycles, and chaos. In the decomposition, distinct aspects of the dynamics contribute separately to entropy production: the housekeeping part stems from a cyclic mode that arises from external driving, generalizing Schnakenberg's cyclic decomposition to nonsteady states, while the excess part stems from an instantaneous relaxation mode that arises from conservative forces. Our decomposition refines previously known thermodynamic uncertainty relations and speed limits. In particular, it not only improves an optimal-transport-theoretic speed limit, but it also extends the optimal transport theory of discrete systems to nonlinear and nonconservative settings.

Cell mechanical responses to subcellular perturbations generated by ultrasound and targeted microbubbles

The inherently dynamic and anisotropic microenvironment of cells imposes not only global and slow physical stimulations on cells but also acute and local perturbations. However, cell mechanical responses to transient subcellular physical signals remain unclear. In this study, acoustically activated targeted microbubbles were used to exert mechanical perturbations to single cells. The cellular contractile force was sensed by elastic micropillar arrays, while the pillar deformations were imaged using brightfield high-speed video microscopy at a frame rate of 1k frames per second for the first 10s and then confocal fluorescence microscopy. Cell mechanical responses are accompanied by cell membrane integrity changes. Both processes are determined by the perturbation strength generated by microbubble volumetric oscillations. The instantaneous cellular traction force relaxation exhibits two distinct patterns, correlated with two cell fates (survival or permanent damage). The mathematical modeling unveils that force-induced actomyosin disassembly leads to gradual traction force relaxation in the first few seconds. The perturbation may also influence the far end subcellular regions from the microbubbles and may propagate into connected cells with attenuations and delays. This study carefully characterizes the cell mechanical responses to local perturbations induced by ultrasound and microbubbles, advancing our understanding of the fundamentals of cell mechano-sensing, -responsiveness, and -transduction. Statement of significanceSubcellular physical perturbations commonly exist but haven't been fully explored yet. The subcellular perturbation generated by ultrasound and targeted microbubbles covers a wide range of strength, from mild, intermediate to intense, providing a broad biomedical relevance. With µm2 spatial sensing ability and up to 1ms temporal resolution, we present spatiotemporal details of the instantaneous cellular contractile force changes followed by attenuated and delayed global responses. The correlation between the cell mechanical responses and cell fates highlights the important role of the instantaneous mechanical responses in the entire cellular reactive processes. Supported by mathematical modeling, our work provides new insights into the dynamics and mechanisms of cell mechanics.

Exact and Numerical Solutions of the Riemann Problem for a Conservative Model of Compressible Two-Phase Flows

In this work we study the solution of the Riemann problem for the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) two-phase flow model introduced in Romenski et al. (J Sci Comput 42(1):68, 2009, Quart Appl Math 65(2):259–279, 2007). All characteristic fields are carefully studied and explicit expressions are derived for the Riemann invariants and the Rankine–Hugoniot conditions. Due to the presence of multiple characteristics in the system under consideration, non-standard wave phenomena can occur. Therefore we briefly review admissibility conditions for discontinuities and then discuss possible wave interactions. In particular we will show that overlapping rarefaction waves are possible and moreover we may have shocks that lie inside a rarefaction wave. In contrast to nonconservative two phase flow models, such as the Baer–Nunziato system, we can use the advantage of the conservative form of the model under consideration. Furthermore, we show the relation between the considered conservative SHTC system and the corresponding barotropic version of the nonconservative Baer–Nunziato model. Additionally, we derive the reduced four equation Kapila system for the case of instantaneous relaxation, which is the common limit system of both, the conservative SHTC model and the non-conservative Baer–Nunziato model. Finally, we compare exact solutions of the Riemann problem with numerical results obtained for the conservative two-phase flow model under consideration, for the non-conservative Baer–Nunziato system and for the Kapila limit. The examples underline the previous analysis of the different wave phenomena, as well as differences and similarities of the three systems.

Mathematical modeling of transport phenomena in compressible multicomponent flows

The present article proposes a diffuse interface model for compressible multicomponent flows with transport phenomena of mass, momentum and energy (i.e., mass diffusion, viscous dissipation and heat conduction). The model is reduced from the seven-equation Baer-Nuziato type model with asymptotic analysis in the limit of instantaneous mechanical relaxations. The main difference between the present model and the Kapila's five-equation model consists in that different time scales for pressure and velocity relaxations are assumed, the former being much smaller than the latter. Thanks to this assumption, the velocity disequilibrium is retained to model the mass diffusion process. Aided by the diffusion laws, the final model still formally consists of five equations. The proposed model satisfy two desirable properties: (1) it respects the laws of thermodynamics, (2) it is free of the spurious oscillation problem in the vicinity of the diffused interface zone. For solution of the model governing equations, we implement the fractional step method to split the model into five physical steps: the hydrodynamic step, the viscous step, the heat transfer step, the heat conduction step and the mass diffusion step. The split governing equations for the hydrodynamic step formally coincide with the Kapila's five-equation model and are solved with the Godunov finite volume method. The mass diffusion, viscous dissipation and heat conduction processes contribute parabolic partial differential equations that are solved with the Chebyshev method of local iterations. Numerical results show that the proposed model maintains pressure, velocity and temperature equilibrium near the diffused interface. Convergence tests demonstrate that the numerical methods achieve second order in space and time. The proposed model and numerical methods are applied to simulate the hydrodynamic instability problems in the inertial confinement fusion. Good agreement with experimental results is observed.

Diffuse interface relaxation model for two-phase compressible flows with diffusion processes

The paper addresses a two-temperature model for simulating compressible two-phase flow taking into account diffusion processes related to the heat conduction and viscosity of the phases. This model is reduced from the two-phase Baer-Nunziato model in the limit of complete velocity relaxation and consists of the phase mass and energy balance equations, the mixture momentum equation, and a transport equation for the volume fraction. Terms describing effects of mechanical relaxation, temperature relaxation, and thermal conduction on volume fraction evolution are derived and demonstrated to be significant for heat conduction problems. The thermal conduction leads to instantaneous thermal relaxation so that the temperature equilibrium is always maintained in the interface region with meeting the entropy relations. A numerical method is developed to solve the model governing equations that ensures the pressure-velocity-temperature (PVT) equilibrium condition in its high-order extension. We solve the hyperbolic part of the governing equations with the Godunov method with the HLLC approximate Riemann solver. The non-linear parabolic part is solved with an efficient Chebyshev explicit iterative method without dealing with large sparse matrices. To verify the model and numerical methods proposed, we demonstrate numerical results of several numerical tests such as the multiphase shock tube problem, the multiphase impact problem, and the planar ablative Rayleigh–Taylor instability problem.

Experimental investigation and micromechanics-based damage modeling of the stress relaxation mechanical properties in gray sandstone

The geomechanical behaviors of the gray sandstones under dry and fully saturated conditions are investigated by both of experimental and numerical studies in this paper. Conventional triaxial compression tests and single-stage stress relaxation tests are performed on cylindrical specimens under these two different states. The experimental results show that both peak strength and elastic modulus are reduced due to the presentence of the pore water which is contributed the weakening. While the typical incomplete attenuation characteristics of the two states are only having slight difference, the final stress relaxation extent under the statured condition is larger than the dry condition when the strain ratio is the same. Based on the pioneering work, a new micromechanical damage–friction coupling constitutive model with containing fewer parameters and each being physically meaningful for the stress relaxation is proposed here. Its chief novelty lies in considering both the instantaneous plastic internal variables and relaxation load to build the relationship between model’s parameter and experimental data. The single-stage relaxation tests of the sandstones are well-benchmarked with this model, which indicates that the proposed new model can capture the incomplete attenuation features of stress relaxation behavior of gray sandstones.

Arbitrary-rate relaxation techniques for the numerical modeling of compressible two-phase flows with heat and mass transfer

We describe compressible two-phase flows by a single-velocity six-equation flow model, which is composed of the phasic mass and total energy equations, one volume fraction equation, and the mixture momentum equation. The model contains relaxation source terms accounting for volume, heat and mass transfer. The equations are numerically solved via a fractional step algorithm, where we alternate between the solution of the homogeneous hyperbolic portion of the system via a HLLC-type wave propagation scheme, and the solution of a sequence of three systems of ordinary differential equations for the relaxation source terms driving the flow toward mechanical, thermal and chemical equilibrium. In the literature often numerical relaxation procedures are based on simplifying assumptions, namely simple equations of state, such as the stiffened gas one, and instantaneous relaxation processes. These simplifications of the flow physics might be inadequate for the description of the thermodynamical processes involved in various flow problems. In the present work we introduce new numerical relaxation techniques with two significant properties: the capability to describe heat and mass transfer processes of arbitrary relaxation time, and the applicability to a general equation of state. We show the effectiveness of the proposed methods by presenting several numerical experiments.

Energy relaxation approximation for compressible multicomponent flows in thermal nonequilibrium

This work concerns the numerical approximation with an explicit first-order finite volume method of inviscid, nonequilibrium, high-temperature flows in multiple space dimensions. It is devoted to the analysis of the numerical scheme for the approximation of the hyperbolic system in homogeneous form. We derive a general framework for the design of numerical schemes for this model from numerical schemes for the monocomponent compressible Euler equations for a polytropic gas. Under a very simple condition on the adiabatic exponent of the polytropic gas, the scheme for the multicomponent system enjoys the same properties as the one for the monocomponent system: discrete entropy inequality, positivity of the partial densities and internal energies, discrete maximum principle on the mass fractions, and discrete minimum principle on the entropy. Our approach extends the relaxation of energy (Coquel and Perthame in SIAM J. Numer. Anal. 35:2223–2249, 1998) to the multicomponent Euler system. In the limit of instantaneous relaxation we show that the solution formally converges to a unique and stable equilibrium solution to the multicomponent Euler equations. We then use this framework to design numerical schemes from three schemes for the polytropic Euler system: the Godunov exact Riemann solver, and the HLL and pressure relaxation based approximate Riemann solvers. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and convergence of the schemes.

Effects of interactions, structure formation, and polydispersity on the dynamic magnetic susceptibility and magnetic relaxation of ferrofluids

Linear response theory relates the decay of equilibrium magnetisation fluctuations in a ferrofluid to the frequency-dependent response of the magnetisation to a weak ac external magnetic field. The characteristic relaxation times are strongly affected by interactions between the constituent particles. Similarly, the relaxation of an initially magnetised system towards equilibrium in zero field occurs on a range of timescales depending on the structure of the initial state, and the interactions between the particles. In this work, ferrofluids are modelled as colloidal suspensions of spherical particles carrying point dipole moments, and undergoing Brownian motion. Recent theoretical and simulation work on the relaxation and linear response of these model ferrofluids is reviewed, and the effects of interactions, structure formation, and polydispersity on the characteristic time scales are outlined. It is shown that: (i) in monodisperse ferrofluids, the timescale characterising the collective response to weak fields increases with increasing interaction strength and/or concentration; (ii) in monodisperse ferrofluids, the initial, short-time decay is independent of interaction strength, but the asymptotic relaxation time is the same as that characterising the collective response to weak fields; (iii) in the strong-interaction regime, the formation of self-assembled chains and rings introduces additional timescales that vary by orders of magnitude; and (iv) in polydisperse ferrofluids, the instantaneous magnetic relaxation time of each fraction varies in a complex way due to the role of interactions.

Residual stress relaxation in the carburized case of austenitic stainless steel under alternating loading

Low-temperature gaseous carburization is an effective thermochemical surface modification process for austenitic stainless steels. The treatment results in the development of a carbon-enriched case with high hardness, improved localized corrosion performance and high compressive residual stress, that in principle improves the fatigue performance. Residual stress relaxation occurs during cyclic loading and the residual stress relaxation in carburized case on 316L austenitic stainless steel under cyclic loading was investigated with rotating bending fatigue testing. Two stages of residual stress relaxation in the surface region were identified: an instantaneous and large relaxation in the first cycle followed by gradual relaxation with the number of fatigue cycles. Depth-resolved analysis of residual stress showed that relaxation mainly occurs in the surface-near region of the carburized case. The rate of stress relaxations depends on the amplitude of the alternating load. The occurrence of plastic accommodation provoked by lattice expansion and the additional plastic deformation imposed during loading is decisive for stress relaxation. A model of surface residual stress evolution for carburized case under cyclic loading was established, which could accurately describe the relaxation.

Two-phase hyperbolic model for porous media saturated with a viscous fluid and its application to wavefields simulation

A new hyperbolic two-phase model of a porous deformable medium saturated with a viscous fluid is presented and some of its features or performances are discussed. The governing equations are derived in the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems. The model accounts for such dissipative mechanisms as interfacial friction and viscous dissipation of the saturating fluid. Both dissipative mechanisms are modeled with hyperbolic equations with relaxation type source terms. The linearized version of the presented model is derived under the assumption of instantaneous relaxation of the phase pressures and its application to modeling propagation of small-amplitude waves in a porous medium saturated with a viscous fluid is discussed. We observe that accounting of the fluid viscosity may significantly affect the dispersion of fast and slow compression waves. Additionally, the shear wave attenuates rapidly due to the viscosity of the saturating fluid which makes it difficult to observe this wave in most of the test cases presented. However, some test cases are presented in which shear waves can be observed in the vicinity of interfaces between regions with different porosity.

Instantaneous wheat dough relaxation by alternating current electric fields

Wheat dough exhibits poor processibility immediately after a mechanical strain, such as mixing or laminating. For proper dough processing, the relaxation of gluten network must occur during a process interrupting resting time of usually 10–25 min. The ability of electric fields to induce modifications in biological structures within seconds, makes an application of this technique reasonable for accelerating the structural relaxation during dough resting. Applying alternating current electric fields (AC-fields) to wheat dough, causes viscoelastic properties corresponding to those of rested dough: AC-fields (110–260 V, 1–5 s) increased extensibility and softness of dough corresponding to resting times up to 25 or rather 50 min. Based on this significant acceleration of structural relaxation, an almost continuously dough processing after a mechanical stress by a simple and short post-processing step without changing dough's gas release (ΔVtotal ≤ 3.0%) as well as holding capacity (Δgas retention ≤ 1.47%) and the textural properties of the product appears feasible.

A reduced model for compressible viscous heat-conducting multicomponent flows

In the present paper we propose a reduced temperature non-equilibrium model for simulating multicomponent flows with inter-phase heat transfer, diffusion processes (including the viscosity and the heat conduction) and external energy sources. We derive three equivalent formulations for the proposed model. The first formulation consists of balance equations for partial densities, the mixture momentum, the mixture total energy, and phase volume fractions. The second formulation is symmetric and obtained by replacing the equations for the mixture total energy and volume fractions in the first formulation with balance equations for the phase total energy. Replacing one of the phase total energy equation of the second formulation with the mixture total energy equation gives the third formulation. All the three formulations assume velocity and pressure equilibrium across the material interface. These equivalent forms provide different physical perspectives and numerical conveniences. Temperature equilibration and continuity across the material interfaces are achieved with the instantaneous thermal relaxation. Temperature equilibrium is maintained during the heat conduction process. The proposed models are proved to respect the thermodynamical laws. For numerical solution, the model is split into a hyperbolic partial differential equation (PDE) system and parabolic PDE systems. The former is solved with the high-order Godunov finite volume method that ensures the pressure–velocity–temperature (PVT) equilibrium conduction. The parabolic PDEs are solved with both the implicit and the explicit locally iterative method (LIM) based on Chebyshev parameters. Numerical results are presented for several multicomponent flow problems with diffusion processes. Furthermore, we apply the proposed model to simulate the target ablation problem that is of significance to inertial confinement fusion. Comparisons with one-temperature models in literature demonstrate the ability to maintain the PVT property and superior convergence performance of the proposed model in solving multicomponent problems with diffusions.

EXPERIMENTAL RESEARCH ON VISCOELASTICITY PROPERTY OF DIFFERENT LAYERS PERIODONTAL LIGAMENT UNDER COMPRESSION

The periodontal ligament (PDL) exhibits different material mechanical properties along the long axis of the teeth. To explore the creep and the relaxation effects of dissimilar layers of PDL, this paper took the central incisors of porcine mandibular as experimental subjects and divided them perpendicular to the teeth axis into five layers. Creep experiments and relaxation experiments on five layers were conducted to obtain the creep compliance and relaxation modulus at different layers. Linear elastic model, generalized Kelvin model, and generalized Maxwell model were used to describe the major characteristics of the PDL: Instantaneous elasticity, creep and relaxation. Fitting accuracy of three-parameter, five-parameter, and seven-parameter of the model was compared, and the constitutive equations of different layers were established by the least square method. The results presented that the creep strain and the relaxation stress of PDL were exponentially correlated with time under different loading conditions. Different layers showed a significant effect on the creep strain and relaxation stress of PDL. Along the long axis of the teeth, the changing rule of the creep compliance and relaxation modulus of each layer showed quite the contrary, and the instantaneous elastic modulus first decreased to the minimum, then increased to the maximum. Higher instantaneous elastic modulus led to lower creep compliance and higher relaxation modulus. The generalized Kelvin model and the generalized Maxwell model well characterized the creep and relaxation properties of PDL. Fitting accuracy increased with the number of model parameters. The relaxation time of PDL was about one order of magnitude shorter than the creep retardation time, which indicated that the relaxation effect lasted shorter than the creep effect.

A homogeneous relaxation low mach number model

In the present paper, we investigate a new homogeneous relaxation model describing the behaviour of a two-phase fluid flow in a low Mach number regime, which can be obtained as a low Mach number approximation of the well-known HRM. For this specific model, we derive an equation of state to describe the thermodynamics of the two-phase fluid. We prove some theoretical properties satisfied by the solutions of the model, and provide a well-balanced scheme. To go further, we investigate the instantaneous relaxation regime, and prove the formal convergence of this model towards the low Mach number approximation of the well-known HEM. An asymptotic-preserving scheme is introduced to allow numerical simulations of the coupling between spatial regions with different relaxation characteristic times.

Osteoarthritis-Related Degeneration Alters the Biomechanical Properties of Human Menisci Before the Articular Cartilage.

An exact understanding of the interplay between the articulating tissues of the knee joint in relation to the osteoarthritis (OA)-related degeneration process is of considerable interest. Therefore, the aim of the present study was to characterize the biomechanical properties of mildly and severely degenerated human knee joints, including their menisci and tibial and femoral articular cartilage (AC) surfaces. A spatial biomechanical mapping of the articulating knee joint surfaces of 12 mildly and 12 severely degenerated human cadaveric knee joints was assessed using a multiaxial mechanical testing machine. To do so, indentation stress relaxation tests were combined with thickness and water content measurements at the lateral and medial menisci and the AC of the tibial plateau and femoral condyles to calculate the instantaneous modulus (IM), relaxation modulus, relaxation percentage, maximum applied force during the indentation, and the water content. With progressing joint degeneration, we found an increase in the lateral and the medial meniscal instantaneous moduli (p < 0.02), relaxation moduli (p < 0.01), and maximum applied forces (p < 0.01), while for the underlying tibial AC, the IM (p = 0.01) and maximum applied force (p < 0.01) decreased only at the medial compartment. Degeneration had no influence on the relaxation percentage of the soft tissues. While the water content of the menisci did not change with progressing degeneration, the severely degenerated tibial AC contained more water (p < 0.04) compared to the mildly degenerated tibial cartilage. The results of this study indicate that degeneration-related (bio-)mechanical changes seem likely to be first detectable in the menisci before the articular knee joint cartilage is affected. Should these findings be further reinforced by structural and imaging analyses, the treatment and diagnostic paradigms of OA might be modified, focusing on the early detection of meniscal degeneration and its respective treatment, with the final aim to delay osteoarthritis onset.