Rheological Schemes Research Articles

A comparison of finite strain viscoelastic models based on the multiplicative decomposition

The constitutive equations of several finite strain viscoelastic models, based on the multiplicative decomposition of the deformation gradient tensor and formulated in a thermodynamically consistent framework, are reviewed to demonstrate their similarities and differences. The proposed analysis shows that dissipation formulations, which may appear different, are similar when expressed in the same configuration, enabling the definition of a unified general model. The ability of this general model to reproduce the main features of the behavior of rubbers is then explored. First, its responses are compared to those of finite linear viscoelastic models commonly implemented in commercial finite element codes. Cases of monotonic uniaxial tension and simple shear, relaxation, and sinusoidal simple shear are considered. Second, a comparison is made between a classic generalized Maxwell rheological scheme and a Zener one with a non-constant viscosity, exploring the relevance of both options within the general model’s constitutive equations.

Finite elastic-viscoplastic deformation behaviour of PEO-based solid polymer electrolyte: Experimental investigation and constitutive modelling

Polyethene oxide (PEO) has emerged as a preferred candidate for solid electrolytes due to its excellent solubility towards lithium salts, stable chemical properties, high flexibility, and low cost. However, the poor knowledge of the mechanical behaviours of PEO electrolytes leads to the road facing vast challenges in developing all-solid-state lithium batteries. In this study, we conducted the uniaxial tensile tests on PEO-LiTFSI films at 20–40 °C, and the strain rate ranges from 0.001 s−1 to 0.01 s−1 while investigating the potential effect mechanisms. Based on a rheological scheme of semi-crystalline polymers, we developed a new constitutive model for simulating the stress-strain curves of PEO-based electrolytes. The results indicate that PEO-LiTFSI exhibits the finite viscoelastic-plastic deformation characteristic with a minimal elastic strain range, low elastic modulus and yield strength. Increasing LiTFSI content and temperature can reduce the crystallinity of the PEO, resulting in significant softening of the polymer electrolyte; the opposite is true with raising the strain rate. Notably, the present model can capture the mechanical response of PEO electrolytes across a range of temperatures and strain rates, and the predicted tensile curves align closely with the experimental results. In addition, the simulation reveals that subpar mechanical performance would occur for PEO-LiTFSI under the recommended electrochemical operation conditions (0.05–5 C and 50 °C).

Dynamic magnetic birefringence in a viscoelastic ferrocolloid.

A model is developed to describe the oscillations of optical anisotropy induced in a viscoelastic ferrocolloid (nanodispersion of magnetic particles) by an AC magnetic field. The viscoelasticity of the matrix (carrier medium) is assumed to obey the Jeffreys rheological scheme, whose advantage is that with the aid of just two viscous parameters and a single one for elasticity it enables one to vary the retarded mechanical response of the carrier from a weakly Maxwellian fluid to a medium with the rheology of a Kelvin gel. As the orientational motion of the particles driven by the AC field is always strongly affected by thermal motion, the occurring process is described with the aid of a kinetic (Fokker-Planck type) equation that combines diffusional and drift terms. On this basis, an exact evolution equation for the macroscopic optical anisotropy of a ferrocolloid is derived that is, however, just one link in an infinite chain of equations for statistical moments. The solution is obtained by applying effective field approximation: reducing the number of moment equations to their minimum and closing the chosen set. This solution is substituted to the scheme of a standard polarimetric set-up, and it is demonstrated how the peculiarities imparted by viscoelasticity should manifest themselves on the intensity of the light transmitted through the set up containing a ferrocolloid sample. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

A Novel Approach to the Analysis of the Soil Consolidation Problem by Using Non-Classical Rheological Schemes

The paper presents classical and non-classical rheological schemes used to formulate constitutive models of the one-dimensional consolidation problem. The authors paid special attention to the secondary consolidation effects in organic soils as well as the soil over-consolidation phenomenon. The systems of partial differential equations were formulated for every model and solved numerically to obtain settlement curves. Selected numerical results were compared with standard oedometer laboratory test data carried out by the authors on organic soil samples. Additionally, plasticity phenomenon and non-classical rheological elements were included in order to take into account soil over-consolidation behaviour in the one-dimensional settlement model. A new way of formulating constitutive equations for the soil skeleton and predicting the relationship between the effective stress and strain or void ratio was presented. Rheological structures provide a flexible tool for creating complex constitutive relationships of soil.

A continuum viscoelastic model of Timoshenko NSGT nanobeams

Developed and solved in this article, for the first time, is a coupled continuum viscoelastic model for Timoshenko nonlocal strain gradient theory (NSGT)-based nanobeams; this is performed using a finite difference analysis (FDA). The viscosity of infinitesimal elements of the Timoshenko NSGT nanobeams is incorporated via the Kelvin–Voigt scheme (as a two-parameter rheological scheme) for both the transverse/longitudinal/rotational motions. The NSGT for the normal/shear stress fields is used for ultrasmall size influences on the continuum model. Rotary inertia is automatically present due to the Timoshenko-type rotation. The viscosity in the Timoshenko NSGT nanobeam is responsible for energy dissipation formulated via negative work. Hamilton’s balance scheme is utilised and resulted in the coupled continuum viscoelastic dynamic model for Timoshenko NSGT nanobeams, which is solved via a FDA for nonlinear mechanics. A nonlinear bending test is conducted via development of a finite element method model in the absence of the NSGT.

Parametric Space for Elastic Waves in Porous Media with Complex Rheology

We evaluate the influence of mechanical parameters on the decay rate of elastic waves in fluid-saturated media with bubbles. The parameters describe the complex rheological scheme adopted as a model of the grainy porous medium. Areas of validity of the model in the space of mechanical parameters are determined.

The role of rheology in modelling elastic waves with gas bubbles in granular fluid-saturated media

Elastic waves in fluid-saturated granular media depend on the rheology which includes elements representing the fluid and, if necessary, gas bubbles. We investigated the effect of different rheological schemes, including and excluding the bubbles, on the linear Frenkel–Biot waves of P1 type. For the wave with the bubbles the scheme consists of three segments representing the solid continuum, fluid continuum, and a bubble surrounded by the fluid. We derived the Nikolaevskiy-type equations describing the velocity of the solid matrix in the moving reference system. The equations are linearized to yield the decay rate lambda as a function of the wave number k. We compared the lambda(k)-dependence for the cases with and without the bubbles, using typical values of the input mechanical parameters. For the both cases, the lambda(k)-curve lies entirely below zero, which is in line with the notion of the elastic wave being an essentially passive system. We found that the increase of the radius of the bubbles leads to faster decay, while the increase in the number of the bubbles leads to slower decay of the elastic wave.

Modelling of single degree of freedom SMA oscillators by using rheological schemes

Introduction. The interest to the SMA elements in civil engineering is constantly increasing. Regarding the field of civil engineering, the devices incorporating SMA elements mostly find applications in mitigation of natural disaster hazards, such as earthquakes. The promising results of applications are possible due to unique properties of SMA, such as shape memory effect (recovering of relatively high strains while material is heated) and superelasticity (recovering of strains upon load removal). The most common approach to the formulation of SMAs constitutive relations is a thermomechanical modelling, in which constitutive equations are dependent on internal state variables. Materials and methods. The article describes the phenomenological approach to modelling of single degree of freedom SMA oscillators by using rheological schemes. Results. Certain sets of rheological components are presented and their influence on the oscillator response is examined. As an example of numerical application of presented approach, the simple one-dimensional oscillator is used in order to solve the case of forced vibrations of a cantilever with embedded SMA reinforcement. Conclusions. One of the advantages of the phenomenological modelling approach presented in the article is a possibility of formulation of constitutive relationships as a set of explicit differential equations. Such system of equations can be easily implemented in mathematical software or in the commercial FEM codes as a user’s subroutines.

The Effect of Rheology with Gas Bubbles on Linear Elastic Waves in Fluid-Saturated Granular Media

Elastic waves in fluid-saturated granular media depend on the grain rheology, which can be complicated by the presence of gas bubbles. We investigated the effect of the bubble dynamics and their role in rheological scheme, on the linear Frenkel-Biot waves of P1 type. For the wave with the bubbles the scheme consists of three segments representing the solid continuum, fluid continuum and bubbles surrounded by the fluid. We derived the Nikolaevskiy-type equation describing the velocity of the solid matrix in the moving reference system. The equation is linearized to yield the decay rate λ as a function of the wave number <i>k</i>. We compared the λ (<i>k</i>) -dependence for the cases with and without the bubbles, using typical values of the input mechanical parameters. For both the cases, the λ(<i>k</i>) curve lies entirely below zero, which implies a global decay of the wave. We found that the increase of the radius of the bubbles leads to a faster decay, while the increase in the number of the bubbles leads to slower decay of the wave.

Time\u2013temperature superposition for viscoelastic materials with application to asphalt\u2013aggregate mixes

This paper presents a new approach to characterisation of asphalt mixtures constitutive models, with application to pavement structures. It is well-known fact that temperature, frequency and time of loading have a great influence on the mechanical properties of bituminous mixtures. For this reason, the properties are usually presented in the frequency domain as complex numbers having real and imaginary parts. This convenient representation in terms of complex modulus and phase angle leads to the dynamic modulus master curve. The effective use of the theory of linear viscoelasticity to characterise constitutive model of asphaltic material is shown in this paper. Viscoelastic constitutive model is represented by a combination of rheological schemes, and its identification is based on both laboratory tests results and mixture composition. The temperature–frequency or temperature–time superposition principle being applied in order to produce master curves of mechanical properties is illustrated with real experimental data. Further, the process of identification of Huet–Sayegh parameters is carried out using best-fitting methods implemented in MATLAB. Fractional rheological model is used as it needs only a small number of elements to fully characterise the response of asphalt materials.

Modelling of single degree of freedom SMA oscillators by using rheological schemes

The article describes the approach to modelling of single degree of freedom SMA oscillators by using rheological schemes. Certain sets of rheological components are presented and their influence on the oscillator response is examined. Regarding the field of civil engineering, the devices incorporating SMA elements mostly find applications in mitigation of natural disaster hazards, such as earthquakes. The promising results of applications are possible due to unique properties of SMA, such as shape memory effect (recovering of relatively high strains while material is heated) and superelasticity (recovering of strains upon load removal). The most common approach to the formulation of SMAs constitutive relations is a thermomechanical modelling, in which constitutive equations are dependent on internal state variables. One of the advantages of the phenomenological modelling approach presented in the article is a possibility of formulation of constitutive relationships as a set of explicit differential equations. Such system of equations can be easily implemented in mathematical software or in the commercial FEM codes as a user's subroutines. As an example of numerical application of presented approach, the simple one-dimensional oscillator is used in order to solve the case of forced vibrations of a cantilever with embedded SMA reinforcement.

Mathematical Formulation of Soft‐Contact Problems for Various Rheological Models of Damper

The paper deals with analysis of selected soft‐contact problems in discrete mechanical systems. Elastic‐dissipative rheological schemes representing dampers as well as the notion of unilateral constraints were used in order to model interaction between colliding bodies. The mathematical descriptions of soft‐contact problems involving variational inequalities are presented. The main finding of the paper is a method of description of soft‐contact phenomenon between rigid object and deformable rheological structure by the system of explicit nonlinear differential‐algebraic equations easy for numerical implementation. The results of simulations, that is, time histories of displacements and contact forces as well as hysteretic loops, are presented.

Modeling of wave processes in blocky media with porous and fluid-saturated interlayers

The wave processes in blocky media are analyzed by applying different mathematical models, wherein the elastic blocks interact with each other via pliant interlayers with the complex mechanical properties. Four versions of constitutive equations are considered. In the first version, an elastic interaction between the blocks is simulated within the framework of linear elasticity theory, and the model of elastic–plastic interlayers is constructed to take into account the appearance of irreversible deformation of interlayers at short time intervals. In the second one, the effects of viscoelastic shear in the interblock interlayers are taken into the consideration using the Poynting–Thomson rheological scheme. In the third option, the model of an elastic porous material is used in the interlayers, where the pores collapse if an abrupt compressive stress is applied. In the fourth case, the model of a fluid-saturated material with open pores is examined based on Biot's equations. The collapse of pores is modeled by the generalized rheological approach, wherein the mechanical properties of a material are simulated using four rheological elements. Three of them are the traditional elastic, viscous and plastic elements, the fourth element is the so-called rigid contact, which is used to describe the behavior of materials with the different resistance to tension and compression. It was shown that the thermodynamically consistent model is provided, which means that the energy balance equation is fulfilled for an entire blocky structure, where the kinetic and potential energy of the system is the sum of the kinetic and potential energies of the blocks and interlayers. Under numerical implementation of the interlayers models, the dissipationless finite difference Ivanov's method was used. The splitting method by spatial variables in the combination with the Godunov gap decay scheme was applied in the blocks. As a result, robust and stable computational algorithms are built and tested. Using MPI technology, the parallel software was designed for the modeling of wave processes in 2D setting. The numerical results are presented, discussed and future studies are outlined.

Secondary consolidation modelling by using rheological schemes

The paper presents the possibility of the secondary consolidation modelling in soil mechanics by using rheological schemes. Non-elastic rheological models represent time-dependent behaviour of the soil skeleton and should be expressed by appropriate constitutive relationships in the form of ordinary differential equations. On the other hand, the primary consolidation phenomenon caused by dissipation of the excess pore pressure is modelled by applying the well-known formulation proposed by Terzaghi and expressed by the partial differential equation. The resulting initial boundary value problem was solved numerically for three selected rheological schemes. Algorithms implemented in Mathematica were used for this purpose.

Characterization of the Complex Moduli for Asphalt-aggregate Mixtures at Various Temperatures

Constitutive properties of asphalt-aggregate mixtures are modelled very often applying viscoelastic models represented by rheological schemes. The parameters of these schemes are identified using cyclic test at various temperatures. The procedure leads to the dynamic modulus master-curve. The sigmoidal model is applied in order to represent the master-curve usually, but such a model does not have real physical meaning. The objective of the paper is to demonstrate the method of characterization of the complex moduli for bituminous mixtures using classical and fractional rheological models. Fractional models give the possibility to characterize the response of asphalt materials with a small number of elements. In order to find the parameters of rheological schemes, the optimization problem is formulated. The solution is obtained using least-squares methods implemented in MATLAB software.

Mathematical description of rheological properties of asphalt-aggregate mixes

Abstract The procedure of the formulation of constitutive equations for asphalt-aggregate mixes is based very often on rheological schemes composed of classical elastic, plastic and viscous elements. The parameters of these schemes can be obtained based on laboratory experiments. In order to obtain better curve fitting results one can use non-classical viscoelastic elements described by fractional derivatives. In this paper we present the characteristics of the fractional viscoelastic Huet-Sayegh model as well as the characteristics of an original simplified fractional model. The results have been obtained using algorithms of numerical calculation of inverse Laplace transforms. Then the proposal of an original rheological model including plasticity has been given. The non-linear differential constitutive relationships of such a model are presented in the paper. The results of computer simulations are also visualized. Finally, 3D viscoelasticplastic models of asphalt aggregatemixes are proposed. The models are based on a generalized macroscopic theory taking into account the effect of pressure-dependency on yielding

Numerical modelling of catastrophic events produced by mud or debris flows

Mud and debris fl ows are natural phenomena representing serious hazard for population and structures in mountain zones, because of their rapid occurrence and the diffi culty in forecasting the phenomena initiation. Numerical models can however be useful in predicting the peak discharge and the strength of fl owing mass, helping administrations in preparing risk mitigation measures. In this work, a numerical model for hyperconcentrated fl ows is presented. It is based on shallow water equations, with a particular source terms treatment which translates into an increased numerical stability and makes the model highly versatile. The test case applications focus on some fundamental characteristics necessary for debris- and mud-fl ow representation. In particular, classic dam-break problems have been used to test wave celerity and wet-dry fronts propagation, while a mud-fl ow dam-break problem has been chosen to investigate model sensibility to different rheological schemes. Then, the model has been applied to two real events that occurred in Northern Italy. The fi rst one is a debris fl ow which took place at Acquabona, near Cortina d’Ampezzo. This event is extensively documented, since it has been observed by a monitoring station prepared by the University of Padua. The second one is a tragic event, during which the little town of Stava has been stricken by a destructive mud fl ow caused by the collapse of two earth dams.

Elastoplastic model for steel buckling-restrained braces

Buckling-restrained braces (BRBs) permit exploiting the hysteretic properties of their core material both in tension and compression. However, the global behaviour of the BRB does not fully replicate the local behaviour of the core material and important differences have been observed in experimental tests, such as a higher bearing capacity in compression and a significant role of isotropic hardening. These observations have thus promoted the use of specific constitutive models for BRBs. Besides a proper description of behavioural aspects, a BRB model should include some highly desired requirements: explicit computation of the plastic component of the deformation as required in BRB capacity models, smoothness of the elastic–plastic transition to improve convergence rate, simple implementation and limited number of input data parameters to facilitate its implementation and use. In order to address the above behavioural and modelling issues, an elastoplastic model for steel BRBs is presented in this paper following a consistent approach based on a rheological scheme that leads to a quite simple constitutive law involving only one internal variable. The adopted formulation allows a straightforward physical interpretation and identification of the constitutive parameters of the model as well as explicit computation of response quantities related to failure and dissipated energy. Response results obtained using the proposed model are compared to the outcomes of experimental tests available in the literature and involving different BRBs under various strain histories, showing good predictions of the experimental behaviour.

Mathematical modelling of rate-independent pseudoelastic SMA material

The paper presents a procedure for the formulation of constitutive equations for rate-independent pseudoelastic SMA material models. The procedure applies a rheological scheme representing mechanical properties of the material. An additive decomposition of strains into two parts is proposed. The first part describes strains of a perfectly elastic body while the second part may be represented by a combination of a rigid perfectly elastic body and a rigid perfectly plastic body. It is demonstrated that the key problem of formulation of constitutive relationships is to derive the 1st order differential equation with respect to the tensor describing the second part of the strain field. This equation may be obtained in explicit form starting from the variational inequalities defining non-elastic parts of rheological model. The uniqueness of the obtained differential equation has been proved. A numerical implementation of the constitutive relationships of SMA material was done through the user subroutine module VUMAT within the FE commercial code ABAQUS/Explicit. As an example we analyzed the problem of vibration of a simple 3D structure made of SMA.

Rheological analysis of geomechanical processes

The modified variable module method, based on the a priori information about the intensifying daylight surface subsidence in time, is developed. The practical application of the proposed rheological analysis scheme is considered for the chamber mining of potash series.