Terms Of Constitutive Equations Research Articles

Modeling of rubber-cord layers under quasi-static loading

Modeling a pneumatic tire with a strong change in shape causes the problem of choosing an adequate model for rubber-cord plies. Generally, the classical method of asymptotic homogenization is not suitable due to physical and geometrical nonlinearity for the strain up to 15 %. The known models used for simulation the entire tire as well as rubber-cord plies are analyzed. A choice is made in favor of modeling the plies using the stress anisotropic potential, which is an anisotropic function of the strain tensor invariants. The relationship of such a constitutive law with a quasi-linear constitutive equation in terms of stress and strain differentials is indicated. A convenience of these two types of constitutive equations in terms of numerical implementation is also given. A modification of the effective properties definition for an inhomogeneous layer is explained. The difference from the standard effective moduli definition is clarified. The arrangement of the cords is supposed to be approximately periodic and all cords to be equivalent to the effective fiber. Two models of a rubber-cord plies are described under such assumption. These are a model of an orthotropic material and a model of a transversely isotropic material. Computational experiments, which make it possible to determine the material parameters of anisotropic potentials, are pointed out. Real tests with a sample of rubber ply under slow quasi-static loading were conducted. Significant hysteresis was detected. It is shown that an additive model combining a hyper elastic material with Maxwell viscoelastic model provides good accuracy in stress dependence on the strain rate. The numerical procedure developed to calculate solution to the quasi-static problem of tire deformation is described. It is implemented in home-made computer code. A numerical example on the tire simulation is given. That is so-called breaking test, in which strong deformation is achieved and the developed model is applied to.

The Oldroyd-B fluid in elastic instabilities, turbulence and particle suspensions

The Oldroyd-B fluid has become the starting point for almost all complex flow calculations and analysis involving the behavior of dilute polymer solutions. The reasons for this are clear: based on the kinetic theory of high molecular weight “phantom” chains near equilibrium, the resulting model, in its simplest form, produces a single mode description of the solution constitutive equation in terms of a single symmetric dyad order parameter. The physical connection to the individual chains is therefore unambiguous through the end-to-end configuration tensor. The flow of the fluid resulting from the mathematical solution of the constitutive equation coupled to Cauchy’s equation of motion and continuity has been examined for decades and its strengths and weaknesses in predicting the features of non-linear viscometric and non-viscometric polymer solution flows are well known. Moreover, “simple” extensions (e.g. the FENE-P model) can ameliorate the most serious of these shortcomings. For example, the multi-mode Oldroyd-B model can provide quantitative predictions of the linear viscoelasticity of a wide range of polymer solutions. Remarkably, even with all its shortcomings, the Oldroyd-B fluid has been invaluable in predicting at least qualitatively, features as complex as purely elastic instabilities, turbulent drag modification, and even the effective rheology of particles suspended in polymer solutions. While remaining mindful of the model’s shortcomings, any student of these diverse fields may find the Oldroyd-B fluid an important starting point.

A pseudo-strain energy density function for mechanical behavior modeling of visco-hyperelastic materials

In this paper, a constitutive phenomenological framework is proposed for the mechanical behavior of the visco-hyperelastic materials. This phenomenological framework is based on the concept of developing the schemas of the classical linear theory to the nonlinear regime. For this purpose, at first, the structure of the constitutive law of a viscoelastic material is constructed in the linear behavior case. In this construction, it can be observed that the stress can be decomposed into the sum of the elastic part and viscous part with a time function. In generalization of the rebuilt framework of the linear theory to a nonlinear regime to model the mechanical behavior of visco-hyperelastic materials, a pseudo-strain energy density function including elastic and viscous parts is assumed. The hyperelastic behaviors of the developed nonlinear model is organized via a Hookean-type constitutive equation in terms of the power law and exponential strain measures which have a high flexibility for capturing the mechanical behaviors with different trends from the S-shaped to J-shaped forms. The time part of the developed framework is proposed such that it can model the long-term memory fading principle. For this purpose, the new weight functions are used such that they can capture the rate behavior and recognize the relaxation curves with different starting points from stretch and rate of stretch. Finally, in order to do the comprehensive evaluation of the performance of the proposed model, the high strain rate tensile and compression tests, separately and simultaneously; and also, the relaxation tests at different levels from strain and strain rate beside the moderate rate tensile tests are examined to calibrate the model for different mechanical behaviors. It is observed that there is a good accordance between the test data and predictions of the model for the mechanical behavior of visco-hyperelastic materials.

Fluctuation dynamics in a relativistic fluid with a critical point

To describe dynamics of bulk and fluctuations near the QCD critical point we develop general relativistic fluctuation formalism for a fluid carrying baryon charge. Feedback of fluctuations modifies hydrodynamic coefficients including bulk viscosity and conductivity and introduces nonlocal and noninstantaneous terms in constitutive equations. We perform necessary ultraviolet (short-distance) renormalization to obtain cutoff-independent deterministic equations suitable for numerical implementation. We use the equations to calculate the universal nonanalytic small-frequency dependence of transport coefficients due to fluctuations (long-time tails). Focusing on the critical mode we show how this general formalism matches existing $\text{Hydro+}$ description of fluctuations near the QCD critical point and nontrivially extends it inside and outside of the critical region.

Fractional Derivatives and Dynamical Systems in Material Instability

Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a non-local strain appears. In such a way, various versions of non-localities are obtained by using various types of fractional derivatives. The study aims for constitutive modeling via instability phenomena, that is, by observing the way of loss of stability of material, we can be informed about the proper form of its mathematical model.

Oscillatory shear flow between two parallel plates for viscoelastic constitutive model of distributed-order derivative

PurposeThe purpose of this study is to investigate viscoelastic properties for the constitutive equation in terms of distributed-order derivatives.Design/methodology/approachThe authors considered the steady oscillatory shear flow between two parallel plates (one is fixed and another oscillates in its own plane), and then examined the effects of different forms of the order-weight functions.FindingsThe constitutive equation in terms of distributed-order derivatives can characterize viscoelastic properties. The order-weight functions can effectively describe viscoelasticity.Originality/valueModel the viscoelastic constitutive equation in terms of distributed-order derivatives, where order-weight functions can select to respond viscoelastic properties.

Heat and peristaltic propagation of water based nanoparticles with variable fluid features

In present article a peristaltic analysis has been carried out to examine the nanofluid flow through the symmetric vertical channel. Three types of nanoparticles, copper oxide (), silver (Ag) and silicon dioxide (SiO2), are incorporated with base fluid as pure water. For this persistence the aspects of mixed convection and heat generation/absorption are accounted to observe the modifications in flow field as well as in fluid temperature. Viscosity is also considered which is determined by temperature. The large wavelength approximation with small Reynolds number is used to simplify the involved nonlinear terms in constitutive equations. The arising expressions for axial velocity, temperature profile, pumping characteristics and phenomenon of trapping are evaluated analytically under established physical conditions at boundaries. The solutions thus acquire is utilized to describe the impacts of various emerging parameters on important phenomena related to peristaltic motion. Results found that axial velocity depicts opposite variations in central and around the walls. Moreover, impact of heat generation parameter has shown increasing trend. It is also noticed that in pumping region, the pressure rise per wavelength increases (decreases) by increasing Grashof number, viscosity parameter and nanoparticles volume fraction. A comparative study is also explored for CuO, SiO2 and Ag nanofluids and pure water.

Linear viscoelastic responses and constitutive equations in terms of fractional operators with non-singular kernels

The article investigates implementation of fractional operators with non-singular memories (Caputo-Fabrizio) and Atangana-Baleanu (ABC) derivatives in response to functions and constitutive equations of linear viscoelastic models. The analysis focuses on the adequate selection of fractional operators with the strong requirement that stress relaxation function (approximating experimental data) and memory function of the operator coincide. As a basic step in the implementation of the non-singular fractional operators in linear viscoelastic models, both the Prony decomposition of stress relaxation curves and the approximation by the Mittag-Leffler function of one parameter are discussed. It is demonstrated that the Prony decomposition naturally leads to application of Caputo-Fabrizio operators in the constitutive equations of hereditary type. The resulting equations are two-fractional order models with constructions similar to the Bagley-Torvik equation. In addition, application of the Caputo-Fabrizio and ABC derivatives to ordinary Maxwell, Kelvin-Voigt and Zener models, by formalistic fractionalization and determination of the stress relaxation moduli and creep compliances applying formal convolution interconversion, are discussed.

Linear Viscoelastic Responses: The Prony Decomposition Naturally Leads Into the Caputo-Fabrizio Fractional Operator

The study refers the physical background and modelling of linear viscoelastic response functions and their reasonable relationships to the Caputo-Fabrizio fractional operator via the Prony (Dirichlet series) series decomposition. The problem of interconversion with power-law and exponential (single and multi-term functions) has been discussed. Special attentions have been paid on the Prony series decomposition approach, the related interconversion problems and the expression of the viscoelastic constitutive equations in terms of Caputo-Fabrizio fractional operator.

Implementation of isotension ensemble in molecular dynamics

The implementation of the new type of ensemble, with constant tension force in the molecular dynamic simulations, is considered. In contrast to the isostress NtT ensemble, in which the stress components act on the surface and the volume elements, the new NfT ensemble is determined by the tension forces acting on each particle in the extended space. Correspondingly, instead of the stress–strain relation used in conventional theory, a new type of constitutive equation in terms of stretch and tension becomes an objective of the molecular dynamics simulations. Comparison with the Parrinello–Rahman method is made for the general external load and the temperature applied. Results for the aluminum lattice show that the suggested approach gives an unbiased sampling of the isostress–isothermal ensemble in the LAMMPS environment. Additionally, the problem of atomistic boundary condition was renovated, due to almost identical tension and traction forces. It was shown on several examples of strains that the tension forces properly represent the type and magnitude of the atomistic external load applied to the surface, despite the absence of any quantities related to the area in the constitutive equations. An analog of Cauchy’s stress theorem in the new setting is formulated and verified by numerical simulations.

Stokes’ second problem of viscoelastic fluids with constitutive equation of distributed-order derivative

The steady-state periodic flow of Stokes’ second problem for viscoelastic fluids with constitutive equation in terms of the distributed-order derivative was considered. The distributed-order derivative involves an integration with respect to the order of fractional derivative, and the order is associated with a weight function p(α). With a general weight function p(α), the flow velocity was obtained. The amplitude, the penetration depth and the wavelength were given analytically. Results of Newtonian fluid and single fractional constitutive equation were derived as special cases of weight function p(α). Also we considered other three cases of weight function p(α): linear combination of Dirac delta functions, constant and parameterized exponential function.

Some continuous dependence results about high-order time differential thermoelastic models

ABSTRACTThis article aims to contribute to the investigation of the well-posedness question for three different heat conduction thermoelastic models, obtained starting from the dual-phase-lag (DPL) constitutive equation in its most general time differential formulation and considering Taylor expansion orders higher than those most commonly studied in literature so far. It is necessary to emphasize right from now that the investigation of such thermomechanical models, although they originate in terms of constitutive equations from suitable Taylor series expansions, has to be properly interpreted not as an attempt to emulate the delayed behavior characteristic of the original (i.e. not expanded) DPL model, but rather with the awareness of deepening the well-posedness question for three different stand-alone and self-consistent models. In particular, three estimates are obtained (one for each of the considered models) able to show the continuous dependence of the solutions of the related initial boundary value problems with respect to the supply terms and to the initial given data. All the continuous dependence results are obtained without the need to impose particular restrictions involving the delay times, except for the requirement that they are strictly positive.

A Viscoplastic Constitutive Equation from Kolsky Bar Data for Two Steels

Kolsky bar data for viscoplastic materials are usually presented as true stress versus true strain curves at nearly constant values of engineering strain rates. However, constitutive models for numerical simulations require the true axial rate of deformation which is true strain rate. We present a procedure that uses existing data with constant engineering strain rate to obtain a constitutive equation in terms of true strain rate for 304 stainless steel and 4340 Rc 45 steel.

Mathematical modelling of fractional order circuit elements and bioimpedance applications

In this work a classical derivation of fractional order circuits models is presented. Generalised constitutive equations in terms of fractional Riemann–Liouville derivatives are introduced in the Maxwell’s equations for each circuit element. Next the Kirchhoff voltage law is applied in a RCL circuit configuration. It is shown that from basic properties of Fractional Calculus, a fractional differential equation model with Caputo derivatives is obtained. Thus standard initial conditions apply. Finally, models for bioimpedance are revisited.

Fast Plastic Integration Algorithm for Damage Prediction in Forming Process Simulations

The iterative Return Mapping Algorithm (RMA) is widely used for the plastic integration owing to its accuracy and efficiency, but it is CPU time consuming and may cause divergence problems in case of large strain increments. This paper presents a fast plastic integration method called Direct Scalar Algorithm (DSA) for the damage prediction in forming process simulations. A simplified three-dimensional (3D) strain-based damage model is coupled with the plasticity and implemented into the DSA which does not need iterative solution to make the plastic integration very fast and robust even for very large strain increments. The basic idea is to transform the constitutive equations in terms of the unknown stress vectors into a scalar equation in terms of the equivalent stresses which can be determined by using the experimental tensile curve; thus, the plastic multiplier ∆λ can be directly calculated. The DSA is as accurate as RMA but much faster for the plastic integration.

Elastic strain energy density decomposition in failure of ductile materials under combined torsion-tension

The constitutive behavior and failure of ductile materials are described in the present work for a general case of loading in terms of the secant moduli, which depend on the first (dilatational) and second (deviatoric) strain invariants. This approach exposes the distinct behavior of materials to the equivalent normal and shear stresses. The secant moduli enable the establishment of two (instead of one) constitutive equations necessary for the complete description of these materials. Emphasis is given in the accuracy of the resulting constitutive equations in terms of their predictions relative to actual experimental data for two materials systems. Failure predictions, according to T-criterion, are derived for two materials under combined torsion and tension, which are in good agreement with experimental data. Finally, the associated failure surfaces in a stress space are presented as well.

Fast Plastic Integration Algorithms for Forming Process Simulations

This paper presents a fast plastic integration algorithm and compares it with other algorithms for forming process simulations. The iterative Return Mapping Algorithm (RMA) is widely used owing to its accuracy and efficiency, but it is still time consuming and may cause divergence problems. Another algorithm based on the Incremental Deformation Theory (IDT) was proposed, using the deformation theory of plasticity by piecewise; it is very fast but could not well consider the loading history, leading to notable errors. The new Direct Scalar Algorithm (DSA) based on the flow theory of plasticity is proposed in this paper. The basic idea is to transform the constitutive equations in terms of the unknown stress vectors into a scalar equation in terms of the equivalent stresses which can be determined by using the experimental tensile curve; thus, the plastic multiplier λ can be directly calculated without iterative solution. The DSA is a fast and robust plastic integration algorithm. The comparison of the results obtained by using the three algorithms shows the accuracy and efficiency of the DSA.

Constitutive equations optimized for determining strengths of metallic alloys

We investigate compatibilities of three constitutive equations, the Hollomon, the Swift, and the Voce equations for determination of yield and ultimate tensile strengths based on tensile true stress–strain curves of 27 metal alloys including those with power-law type and linear-type strain-hardening. We analyze each constitutive equation in terms of yield strength determined by the intercept of the linear elastic loading curve and plastic flow curve and ultimate tensile strength evaluated by the concept of instability in tension. We found that the describing plastic flow is very sensitive in determination of the yield strength and tensile strength from parameters of constitutive equation. Voce equation gives estimate yield strength and tensile strength better than Hollomon and Swift equations.

Quantitative phase-field modeling for boiling phenomena

A phase-field model is developed for quantitative simulation of bubble growth in the diffusion-controlled regime. The model accounts for phase change and surface tension effects at the liquid-vapor interface of pure substances with large property contrast. The derivation of the model follows a two-fluid approach, where the diffuse interface is assumed to have an internal microstructure, defined by a sharp interface. Despite the fact that phases within the diffuse interface are considered to have their own velocities and pressures, an averaging procedure at the atomic scale, allows for expressing all the constitutive equations in terms of mixture quantities. From the averaging procedure and asymptotic analysis of the model, nonconventional terms appear in the energy and phase-field equations to compensate for the variation of the properties across the diffuse interface. Without these new terms, no convergence towards the sharp-interface model can be attained. The asymptotic analysis also revealed a very small thermal capillary length for real fluids, such as water, that makes impossible for conventional phase-field models to capture bubble growth in the millimeter range size. For instance, important phenomena such as bubble growth and detachment from a hot surface could not be simulated due to the large number of grids points required to resolve all the scales. Since the shape of the liquid-vapor interface is primarily controlled by the effects of an isotropic surface energy (surface tension), a solution involving the elimination of the curvature from the phase-field equation is devised. The elimination of the curvature from the phase-field equation changes the length scale dominating the phase change from the thermal capillary length to the thickness of the thermal boundary layer, which is several orders of magnitude larger. A detailed analysis of the phase-field equation revealed that a split of this equation into two independent parts is possible for system sizes ranging from micrometers and above. The split of the phase-field equation leads to two equations controlling independently the phase change and the shape of the phase-field profile across the diffuse interface. A new algorithm is presented to solve a fully coupled model, including advection, phase change, and surface tension effects. Good agreement is found among numerical simulations, analytical solutions, and experimental results.

In-plane shear behaviour of thin low reinforced concrete panels for earthquake re-construction

Low reinforced thin concrete panels have been used for the re-construction of living buildings in the devastated zone of L’Aquila. A structural characterization of these types of panels is presented in this paper, paying particular attention to the fact that these panels are subjected mainly to shear forces. Refined compression-field theory (RCFT) has recently been proposed in order to better predict the behaviour of reinforced concrete members subjected to in-plane shear and axial stresses. This theory is based on continuum mechanics, i.e. satisfying compatibility, equilibrium and formulating the constitutive equations in terms of average (i.e. “smeared”) stresses and strains. The improvement of RCFT in comparison with the two most famous theories for reinforced concrete member subjected to shear [i.e. the modified compression-field theory (MCFT), and the rotating-angle softened-truss model (RA-STM)], deals with an embedded bar model based on the tension stiffening model in concrete. After an ad-hoc calibration procedure, the RCFT is employed in order to reproduce the envelope of the experimental load-deflection response of three full-scale thin low reinforced concrete panels subjected to cyclic loading. The predictions provided by RCFT are compared with the experimental data as well as with those provided by MCFT and RA-STM. This paper presents the necessary parameters for the design of thin low reinforced concrete panels using the RCFT. The preliminary numerical validations seem very promising. However, additional experimental data are required for calibrating and validating the parameters of the proposed RCFT theory.