Nonlinear Constitutive Equations Research Articles

Nuclear Fuel Assembly Deformation, Reduced Mechanical Model Dedicated to FSI Simulation

In the pressure vessel of a PWR the core is constituted of a number of fuel assemblies. Under operation, they all suffer from various phenomena, among them, initial trapped efforts, thermal expansion, irradiation growth, creeping, tightening relaxation and rod slipping. Each fuel assembly may develop contacts with its neighbors and strongly interacts with the fluid. It then undergoes vertical as well as lateral fluid forces, which are partly coupled with its instantaneous shape. As a consequence the fuel assembly tends to get distorted during and still after operation. The first concern is economic: at the end of a cycle, when changing the assembly position in the core, the contact interactions with its neighbors due to its distorted shape sometimes delay the handling. The second concern is that the shutdown rods have to slide into the deformed shape of the guide tube and might be slowed due to excessive friction forces. In order to precisely assess these situations it is necessary to simulate numerically a complete core, taking into account all of the previously mentioned phenomena. For that purpose, we propose a light but relevant reduced mechanical model of a fuel assembly, partially based on a POD analysis performed on a detailed fuel assembly model. We then show that a rather simple model, still based on classical finite elements can be used and simplifies the integration of the local nonlinear constitutive equations. We then compare both of the detailed and reduced model on typical situations and show excellent agreement. The CPU time ratio between them, for a full nonlinear simulation of a single fuel assembly under operation, is over 1000, which is competitive against separation of variables based reduced order methods such as POD, PGD and even APHR. So, yet still with a simplified FSI approach, the drastic memory and CPU reductions obtained now enable us to complete a 3D core simulation representing at least 4 years under operation (4 cycles), with as much as 241 individual fuel assemblies, in a reasonable time of a few hours.

Long‐term strength determination and nonlinear creep damage constitutive model of salt rock based on multistage creep test: Implications for underground natural gas storage in salt cavern

AbstractLong‐term mechanical property is critical for underground energy storage in a salt cavern. Multistage loading has been widely used in creep tests to save test time and cost. If the number of loading stages is too small or the loading time of each stage is short, the long‐term strength of salt rock is difficult to determine. For this reason, this paper carried out a ten‐stage loading creep test on the salt rock. The loading time of each stage is about 14 days, and the total creep time is more than 5 months. The stress‐strain curve and the stress‐steady‐state creep rate curve obtained from the test results can be well described by an exponential function. Based on the two‐stage power function model, a new method for determining the long‐term strength is proposed. The method is very similar to the results determined by the isochronous stress‐strain curve method. Moreover, considering that the rock is damaged during the accelerated creep phase, the stress will no longer be a constant value during the accelerated creep phase, but a slow loading process. In this paper, the accelerated creep phase is a coupled process of loading and creep. The total strain in the accelerated creep phase is equal to the sum of the loaded strain and the creep strain, and a new nonlinear creep damage constitutive equation is derived. The model was verified by the creep test results. The results show that the model can well describe the whole process of salt rock creep. It can theoretically explain the reason why the accelerated creep phase of salt rock exhibits nonlinear acceleration characteristics.

A Weak Form Nonlinear Model for Thermal Sensitivity of Love Wave Mode on Layered Structures.

A precise theoretical model for the thermal sensitivity of Love wave mode is significant in the structure design, temperature compensation, and the prediction of thermal behavior. This article proposes a weak form nonlinear model to calculate the thermal sensitivity of Love waves on arbitrary layered structures. The third-order material constants, as well as the thermal stress and strain tensors between the substrate, electrodes, and wave-guiding layer, are considered in the model. The 9 ×9 effective elastic and the 3 ×9 effective piezoelectric matrixes are imported into the nonlinear constitutive equations and boundary conditions using weak form expressions. A temperature-compensated Love wave mode resonator on a layered ZnO/interdigital transducer (IDT)/quartz structure is obtained. The theoretical model is verified through the comparison of experimental and analytical results. The model is beneficial for the design of Love wave devices and sensors.

High-frequency robust position control of a nonlinear piezoelectric bending actuator

Piezoelectric bending actuators have been widely used in a variety of micro- and nano-applications, including atomic force microscopy, micro assembly, cell manipulation, and in general, micro electromechanical systems. However, their control algorithms at low frequencies suffer from nonlinearities such as hysteresis in high voltages and creep in long-time static applications. Also, in high-frequency applications, especially near the actuator natural frequencies, the actuator dynamic is greatly affected by the material nonlinearity. Therefore, the control approaches based on the linear dynamic modeling cannot be effective at high frequencies. Thus, the position control of the foregoing actuators become challenging, and it has been of researchers’ interests in the last decade. In this article, the robust position control of a bimorph piezoelectric bending actuators is investigated. In this regard, based on the nonlinear constitutive equations and the Euler–Bernoulli beam theory, a nonlinear dynamic model is presented. Then, to track a desired motion trajectory, an observer-based robust position control algorithm is proposed. The proposed control methodology is able to accommodate parametric uncertainties and other un-modeled dynamics. Also, it ensures the elimination of the position tracking error in the presence of the estimated states. Finally, the tracking ability of the controller is demonstrated in an experimental study. The experimental results show that the identification of the system is properly conducted with the average error of 5.5%. Also, the efficiency of the robust controller is proved with the error of 3.7% and 4.9% in the position tracking of the actuator inside and outside of the identified region, respectively.

Creep Mechanics of the High‐Stress Soft Rock under Grade Unloading

In order to study the creep behavior of deep soft rock, gritstone was chosen as the research subject, and a rock triaxial rheometer (Rock 600‐50) and acoustic emission (AE) system (SH‐II) were used to carry out the grade unloading confining pressure creep test under a high‐stress level. The test results showed that the lateral creep behavior of the gritstone was more prominent than the axial creep under the initial high confining pressure. Under the same confining pressure, the creep strain rate (the direction the same as strain) of the gritstone decreases with the increase in axial pressure. As shown by the AE count, AE signals were generated throughout the entire test process, indicating that the creep was a “microdynamic” process. The creep behavior was characterized by a significant confining pressure effect. As the confining pressure was decreased, the degree of creep increases significantly. During the test, the AE energy increased on the whole but decreases during the creep phase. During the entire test process, the overall energy in the constant deviatoric stress grade unloading of the confining pressure was 45% higher than that in the constant axial pressure grade unloading. The degree of failure of the rock was different in these two unloading creep tests, and the constant axial pressure grade unloading of the confining pressure entails greater damage than the constant deviatoric stress grade unloading of the confining pressure. The main reason was that the former had a lower confining pressure level and longer creep process than the latter, and the sample was mainly characterized by creep damage and large cumulative damage, while the latter features mainly unloading damage. Through the inversion of the Burgers constitutive model and nonlinear damage constitutive model for the creep test curve, the nonlinear constitutive equation can better fit the accelerated creep stage, which suggested that this model can describe the accelerated creep characteristics of the high‐stress soft rock.

On Lagrangian and non-Lagrangian conformal-invariant nonlinear electrodynamics

A general approach is presented to describe nonlinear classical Maxwell electrodynamics with conformal symmetry in flat spacetime. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functions of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.

Peridynamic model for nonlinear viscoelastic creep and creep rupture of Polypropylene

This paper presents the peridynamic numerical method for nonlinear viscoelastic creep behaviour which consists of primary, secondary, tertiary creep stages and creep rupture. A nonlinear viscoelastic creep constitutive equation based on internal state variable (ISV) theory which covers four creep stages is examined. The viscoelastic equation is substituted into material parameter in the peridynamic equation to derive a new peridynamic method with two time parameters i.e. numerical time and real time. The parameters of the viscoelastic equation is analyzed and evaluated. In validating this peridynamic method, a comparison is made between numerical and experimental data. The peridynamic method for nonlinear viscoelastic creep behaviour (VE-PD) is approved by the good similarity between numerical and experimental creep strain curves with overall difference of 10.67%. The nonlinearity of experimental and numerical data is adequately similar as the error between experimental and numerical curves of secondary stage strain rate against load is 8.022%. The shapes of fractured numerical specimen show good resemblance with the experimental result as well.

Dual ideal shear strengths for chiral single-walled carbon nanotubes

By considering a single-walled carbon nanotube (SWCNT) as a two-dimensional elastica obtained from the roll-up of a graphene sheet into a circular tube, the present paper develops a precise well-posed continuum theory for describing the entire torsional behavior of SWCNTs from an initial unloaded state through their ultimate levels of loading. In addition, the proposed approach can capture the dual ideal shear strengths as well as the asymmetrical behavior of chiral tubes with respect to the direction of the applied torsional loading. The theory incorporates a highly nonlinear constitutive equation which provides information about the nanoscopic morphological parameters of the tubes. As it will be shown, the longitudinal and twist deformations are coupled. Accordingly, a parameter referred to as the torsion-induced extension will be defined.

Forced convective heat transfer of nonlinear viscoelastic flows over a circular cylinder at low Reynolds inertia regime

In this paper, the forced convective heat transfer of viscoelastic fluid flow around a circular cylinder at a low Reynolds number is studied numerically. The effect of viscous dissipation on the problem is modeld which is the main innovative aspect of present study. The Phan–Thien–Tanner (PTT) model is used as the nonlinear constitutive equation. To avoid divergence and stabilize the numerical process in high elastic cases, the log-conformation approach is used. The results indicated that the Nusselt number (Nu) monotonically increases with increasing elasticity number (El), retardation ratio (β), Prandtl number (Pr), and Brinkman number (Br) in a wide range. Both drag reduction (for El < 0.025) and drag enhancement (for El > 0.025) were seen in the numerical results. In high elastic flow regime (El > 10), due to the increased storing nature of viscoelastic fluid compared to the dissipative nature, drag coefficient (CD) and Nu become constant, which causes the fluid to behave like a Newtonian fluid. The temperature-dependent properties were also investigated for all of fluid property, and the large effects of viscosity, and relaxation time were seen in hydrodynamic and thermal parameters. The contribution of normal stresses in viscous dissipation (VD) is much lower than shear stress, due to the lower values of corresponding components in deformation tensor. The elasticity influences the heat transfer by two mchanisms of convection empowering as an indirect way and viscous dissipation as a direct way. Increasing El, β, and ξ increases Nu via the convection mechanism (even at Br = 0), while Nu is decreased with ε. However, Nu increases with the variations of all the parameters, in presence of viscous dissipation. It was also seen that the heat transfer direction changes for Br > 90. Finally, the effects of El, β, ε, ξ, Pr, and Br on the distribution of stress components, velocity, temperature, Nu, and CD were investigated. Two accurate correlations were proposed for the variations of CD and Nu against El, which were validated for a wide range of El.

Simulation-Based Approach for Probing Rheology-Processing-Structure Relationships in Foam Blow Molding

AbstractThe broad objective of this work was to demonstrate a modelling and simulation framework for foam blow molding using commercially available simulation software. The simulation framework would have to account for the initial morphology of the foam, the relationship between the morphology and the rheological and deformation characteristics of the foam at high temperatures and high strains that are typically encountered during blow molding, and correlate the strains developed during blow molding to the morphological aspects in the resulting blow molded part. These aspects are addressed in this paper using simulations of uniaxial tensile deformation of a virtual representative volume element of a foam microstructure (rendered in DIGIMAT-FE) to derive the nonlinear tensile response of the foam at high temperatures (using ABAQUS). The resulting simulated stress-strain curve is employed to parameterize a nonlinear rheological constitutive equation. These parameters are then employed for the homogenized representation of the foam in the blow molding simulation carried out in B-SIM, a commercially available simulation software for blow molding. The regions where the simulated parison has undergone primarily uniaxial elongation are then mapped back to the expected local foam morphology using the transfer functions derived from the RVE simulations. These steps result in a preliminary and simple demonstration of the simulation framework, and offer a template that can be detailed further with experimental rheological information on actual foamed parisons, and more detailed post-processing algorithms to correlate multiaxial elongations with microstructure.

Dimensional changes during shear without normal tractions (the Poynting effect) in nonlinear viscoelastic fiber-reinforced solids

When a rectangular block of a nonlinear material is subjected to a simple shearing deformation, specific normal tractions are required to ensure that the distances between the faces of the block, i.e. its dimensions, do not change. This work investigates the time-dependent dimensional changes during shear in the absence of these normal tractions (the Poynting effect) that occur in a block composed of an incompressible nonlinearly viscoelastic fiber-reinforced solid. The material is modeled using the Pipkin–Rogers nonlinear single integral constitutive equation for viscoelasticity. This constitutive equation is used because (1) it exhibits the essential features of nonlinear viscoelasticity; (2) it is straightforward to include the material symmetry restrictions due to the reinforcing fibers. A system of nonlinear Volterra integral equations is formulated for the dimensional changes in the block. Numerical solutions are presented for the case when the standard reinforcing model for nonlinearly elastic fiber-reinforced materials is incorporated in the Pipkin–Rogers constitutive framework. The results illustrate how the time-dependent dimensional changes depend on the fiber orientation and the viscoelastic properties of the fibers relative to those of the matrix.

New look at an old principle: an alternative formulation of the theorem of minimum entropy production

We formulate a direct generalization of the Prigogine’s principle of minimum entropy production, according to a new isoperimetric variation principle by classical non-equilibrium thermodynamics. We focus our attention on the possible mathematical forms of constitutive equations. Our results show that the Onsager’s reciprocity relations are consequences of the suggested variation principle. Furthermore, we show by the example of the thermo-diffusion such reciprocity relations for diffusion tensor, which are missing in Onsager’s theory. Our theorem applied to the non-linear constitutive equations indicates the existence of dissipation potential. We study the forms of general reciprocity with the dissipation potential. This consideration results in a weaker condition than Li-Gyarmati-Rysselberhe reciprocity has. Furthermore, in the case of electric conductivity in the magnetic field, our theorem shows the correct dependence of the Onsager’s kinetic coefficient by the axial vector of magnetic induction. We show in general that the evolution criterion of the global entropy production is a Lyapunov-function, and so the final stationer state is independent of the initial, time-independent boundary conditions.

Fitting thermal conductivity and optimizing thermoelectric efficiency in [formula omitted] nanowires

We consider a thermoelectric energy generator constituted by a Si∕Ge nanowire of length L. The dependence on composition and temperature of its thermal conductivity is analyzed in view of three series of experimental data obtained at the constant temperatures T=300K, T=400K, and T=500K. The best-fit curve is determined by a nonlinear regression method (NLRM). Then, under the hypothesis of nonlinear constitutive equation for the heat flux, we investigate the thermoelectric efficiency of the system as function of the composition of the nanowire and of the difference of temperature applied to its ends. For each temperature we calculate the value of the composition which realizes the optimal efficiency of the thermoelectric energy conversion. The corresponding value of the thermal conductivity is determined as well.

Possibility to Describe the Alternating and Nonmonotonic Time Dependence of Poisson’s Ratio during Creep Using a Nonlinear Maxwell-Type Viscoelastoplasticity Model

Abstract—A physically nonlinear constitutive equation with four arbitrary material functions for isotropic rheonomic materials is studied to determine the rheological effects simulated by it and the spheres of influence of the material functions. The properties of the volumetric, axial, and lateral creep curves generated by this equation and the dependence of the lateral strain coefficient (Poisson’s ratio) on the material functions, time, and stress during creep are analytically investigated. A general exact estimate of the upper and lower bounds of the Poisson’s ratio and criteria of its negativeness, increase, decrease, and constancy are obtained. The Poisson’s ratio of the model is proved not to be a nonmonotonic function of time.

A nonintrusive distributed reduced‐order modeling framework for nonlinear structural mechanics—Application to elastoviscoplastic computations

SummaryIn this work, we propose a framework that constructs reduced‐order models for nonlinear structural mechanics in a nonintrusive fashion and can handle large‐scale simulations. Three steps are carried out: (i) the production of high‐fidelity solutions by commercial software, (ii) the offline stage of the model reduction, and (iii) the online stage where the reduced‐order model is exploited. The nonintrusivity assumes that only the displacement field solution is known, and the proposed framework carries out operations on these simulation data during the offline phase. The compatibility with a new commercial code only needs the implementation of a routine converting the discretized solution into our in‐house data format. The nonintrusive capabilities of the framework are demonstrated on numerical experiments using commercial versions of Z‐set and Ansys Mechanical. The nonlinear constitutive equations are evaluated by using an external plugin. The large‐scale simulations are handled using domain decomposition and parallel computing with distributed memory. The features and performances of the framework are evaluated on two numerical applications involving elastoviscoplastic materials: the second one involves a model of high‐pressure blade, where the framework is used to extrapolate cyclic loadings in 6.5 hours, whereas the reference high‐fidelity computation would take 9.5 days.

Forced Convection Heat Transfer of a Giesekus Fluid in Circular Micro-Channels Subjected to a Constant Wall Temperature

Abstract In this paper, an exact analytical solution for forced convective heat transfer of nonlinear viscoelastic fluid in isothermal circular micro-channel is presented. The nonlinear Giesekus constitutive equation is used to model the Giesekus fluid heat transfer in micro-channel with constant wall temperature, which is the main innovative aspect of the current study. This constitutive equation is a powerful tool and able to model the fractional viscometric functions, extensional viscosity, and elastic property. The solution of temperature profile and Nusselt number is obtained based on the Frobenius method. The effects of Weissenberg number, mobility factor, slip coefficient, and Navier index on temperature distribution, velocity profile, and Nusselt number are investigated in detail. The results show that the increases in both slip coefficient and Navier index cause the increases in slip velocity and maximum dimensionless temperature at the wall and the micro-channel center, respectively. Moreover, the Nusselt number has an upward trend with increases in slip coefficient and Navier index parameters. The results are indicated that the flow and temperature fields have a complex relation with mobility factor which controls the level of the nonlinearity of the Giesekus model. Additionally, three correlations for Nusselt number of Giesekus flow in micro-channel are presented.

Method for Calibration of Left Ventricle Material Properties Using Three-Dimensional Echocardiography Endocardial Strains.

We sought to calibrate mechanical properties of left ventricle (LV) based on three-dimensional (3D) speckle tracking echocardiographic imaging data recorded from 16 segments defined by American Heart Association (AHA). The in vivo data were used to create finite element (FE) LV and biventricular (BV) models. The orientation of the fibers in the LV model was rule based, but diffusion tensor magnetic resonance imaging (MRI) data were used for the fiber directions in the BV model. A nonlinear fiber-reinforced constitutive equation was used to describe the passive behavior of the myocardium, whereas the active tension was described by a model based on tissue contraction (Tmax). isight was used for optimization, which used abaqus as the forward solver (Simulia, Providence, RI). The calibration of passive properties based on the end diastolic pressure volume relation (EDPVR) curve resulted in relatively good agreement (mean error = -0.04 ml). The difference between the experimental and computational strains decreased after segmental strain metrics, rather than global metrics, were used for calibration: for the LV model, the mean difference reduced from 0.129 to 0.046 (circumferential) and from 0.076 to 0.059 (longitudinal); for the BV model, the mean difference nearly did not change in the circumferential direction (0.061) but reduced in the longitudinal direction from 0.076 to 0.055. The calibration of mechanical properties for myocardium can be improved using segmental strain metrics. The importance of realistic fiber orientation and geometry for modeling of the LV was shown.

Geometrically Nonlinear Constitutive Equations of the Plastic Flow Theory in Terms of Asymmetric Stress and Strain Measures

A formulation of the geometrically nonlinear plastic flow theory (PFT) based on asymmetric measures of stress and strain states is proposed. A main emphasis is placed on the physically reasonable decomposition of the deformation gradient into three components: elastic distortions, which determine stresses, an orthogonal tensor characterizing the quasi-rigid motion of a material and the plastic strain gradient. The quasi-rigid motion of the material is defined by introducing for a representative volume element a generalized lattice, which represents its symmetry elements. The hypoelastic anisotropic law is introduced in terms of the movable coordinate system associated with the material. The rate of plastic deformations is determined by the associated law of plastic flow. As a result, the closed system of constitutive equations of the geometrically nonlinear PFT of is obtained.

An Integrated Approach for Friction and Wear Simulation of Tire Tread Rubber. Part I: Friction Test, Characterization, and Modeling

ABSTRACT:To simulate tire tread friction and wear, a variety of factors have to be taken into account. Among them, the local nonlinear constitutive equation for friction and wear in the contact interface and the related solving strategies, including the wear increment optimization and worn mesh update, are very critical to the predicting methodology. This two-part contribution addresses an integrated approach for friction and wear simulation of tire tread rubber. In Part I, a modified friction test scheme of rubber wheels with the Laboratory Abrasion and Skid Tester (LAT 100) is proposed, along with numerical verification, which greatly improved the distribution uniformities of the contact pressure and sliding velocity. In order to investigate the friction characteristics of tire tread rubber, various contact conditions were conducted, and then a unified friction model was put forward to describe the nonlinear relationship of rubber friction with contact pressure and sliding velocity. Based on the established frictional contact model, the locked traction and cornering rolling were simulated, and the calculated friction forces and lateral forces agree with the experimental results on the whole.

Effect of cross-diffusion on the stability of a triple-diffusive Oldroyd-B fluid layer

The onset and stability of a triple cross-diffusive viscoelastic fluid layer is investigated. The rheology of viscoelastic fluid is approximated by the nonlinear Oldroyd-B constitutive equation which encompasses Maxwell and Newtonian fluid models as special cases. By performing the linear instability analysis, analytical expression for the occurrence of stationary and oscillatory convection is obtained. The numerical results show that the elasticity and cross-diffusion effects reinforce together in displaying complex dynamical behavior on the system. The presence of cross-diffusion is found to either stabilize or destabilize the system depending on the strength of species concentration as well as elasticity of the fluid and also alters the nature of convective instability. The disconnected closed oscillatory neutral curve lying well below the stationary neutral curve is observed to be convex in its shape in contrast to quasiperiodic bifurcation from the quiescent basic state noted in the case of Newtonian fluids. This striking feature is attributed to the viscoelasticity of the fluid. By performing a weakly nonlinear stability analysis, the stability of bifurcating solution is discussed. It is worth reporting that the viscoelastic parameters significantly influence the stability of stationary bifurcation though the stationary onset is unaffected by viscoelasticity. Besides, subcritical instability is occurs and the critical Rayleigh number at which such an instability is possible decreases in the presence of cross-diffusion terms. The results of Maxwell and Newtonian fluids are delineated as particular cases from the present study.