Thermodynamic Admissibility Research Articles

Hysteretic models with damage and flexibility increase

The need to accurately describe nonlinear degrading phenomena in both modern and classical complex materials that exhibit hysteretic behavior is a relevant task for the analysis of their response. In the present study an enrichment to an existing formulation to describe strength and stiffness degradation is proposed. The description of damage, that reduces the hysteretic force, is then accompanied by the introduction of flexibility increase, which increments the elastic displacement experienced by the system. Both effects influence the energy dissipated by the system. The condition of thermodynamic admissibility in presence of damage and flexibility increase is derived, entailing constraints on system parameters. Additional constraints arise from consistency conditions on the transformation of the pure hysteretic system into the system with damage and flexibility increase. A numerical solution involving the forward Euler method is presented to evaluate the response quantities. The formulation is eventually applied to the elastic–plastic and Bouc Wen hysteresis models, and the influence of the degrading parameters is presented. The presented Bouc-Wen model with damage and flexibility increase is applied to study the cyclic response of a masonry panel experimentally tested to validate the numerical model and prove its capability of describing the structural response.

Multiscale Thermodynamics-Informed Neural Networks (MuTINN) towards fast and frugal inelastic computation of woven composite structures

The complex behavior of inelastic woven composites stems primarily from their inherent heterogeneity. Achieving accurate predictions of their linear and nonlinear responses, while considering their microstructures, appears feasible through the application of multi-scale modeling approaches. However, effectively incorporating these methodologies into real-scale applications, particularly within FE2 analyses, remains challenging due to the significant computational requirements they entail. To overcome this issue, while considering the scale effects, this study introduces an alternative approach based on Artificial Neural Networks (ANNs) to perform a macroscopic surrogate model of composites. This model, referred to as Multiscale Thermodynamics Informed Neural Networks (MuTINN), is founded on thermodynamic principles and introduces specific quantities of interest that serve as internal state variables at the macroscopic level. This captures efficiently the state and evolution laws governing the history-dependent behavior of these composites while retaining the thermodynamic admissibility and the physical interpretability of their overall responses. Moreover, to facilitate its numerical implementation within a FE code, a Meta-UMat has been developed, streamlining the application of multiscale FE×MuTINN approach for composite structure computations. The prediction capabilities of the proposed approach is demonstrated across the material scales, exemplified through diverse instances of woven composite structures. These applications account for anisotropic yarn damage and an elastoplastic polymer matrix behavior. The numerical results and the related comparison with experimental findings and FE computations demonstrate remarkable consistency across a wide range of non-proportional loading paths. This promises a potential solution to alleviate the computational challenges associated with multiscale simulations of large-scale composite structures.

Modeling the rheological behavior of silica filled rubber compounds

The rheological behavior of styrene–butadiene rubber (SBR) compounds filled with silica is investigated as a function of silica volume fraction. To predict the mechanical response, a continuum model for entangled polymer melts filled with nanoparticles is herein introduced. This model is capable of describing the rheological response in both the linear and nonlinear viscoelastic regimes in the context of non-equilibrium thermodynamics to guarantee its thermodynamic admissibility. The constitutive model describes the polymer nanocomposite melts at a mesoscopic level of description by considering the conformation tensor between successive entanglement points, and the orientation tensor for the, in general, spheroidal nanoparticles that describes their average orientation. Evolution equations are developed for nanoparticles with an arbitrary shape but are eventually specified to the case of spherical ones. The multimode version of the new constitutive model provides a very accurate prediction of the rheological behavior of the processability range of SBR/silica nanocomposites. Thus, the new model is a tool able to provide answers to the several difficulties that rubber-producing manufacturers face when processing rubber compounds.

The choice of a thermodynamic formulation dramatically affects modelled chemical zoning in minerals

Quantifying natural processes that shape our planet is a key to understanding the geological observations. Many phenomena in the Earth are not in thermodynamic equilibrium. Cooling of the Earth, mantle convection, mountain building are examples of dynamic processes that evolve in time and space and are driven by gradients. During those irreversible processes, entropy is produced. In petrology, several thermodynamic approaches have been suggested to quantify systems under chemical and mechanical gradients. Yet, their thermodynamic admissibility has not been investigated in detail. Here, we focus on a fundamental, though not yet unequivocally answered, question: which thermodynamic formulation for petrological systems under gradients is appropriate—mass or molar? We provide a comparison of both thermodynamic formulations for chemical diffusion flux, applying the positive entropy production principle as a necessary admissibility condition. Furthermore, we show that the inappropriate solution has dramatic consequences for understanding the key processes in petrology, such as chemical diffusion in the presence of pressure gradients.

Elucidating the rheological implications of adding particles in blood.

In the past few decades, nanotechnology has been employed to provide breakthroughs in the diagnosis and treatment of several diseases using drug-carrying particles (DCPs). In such an endeavor, the optimal design of DCPs is paramount, which necessitates the use of an accurate and trustworthy constitutive model in computational fluid dynamics (CFD) simulators. We herein introduce a continuum model for elaborating on the rheological implications of adding particles in blood. The model is developed using non-equilibrium thermodynamics to guarantee thermodynamic admissibility. Red blood cells are modeled as deformed droplets with a constant volume that are able to aggregate, whereas particles are considered rigid spheroids. The model predictions are compared favorably against rheological data for both spherical and non-spherical particles immersed in non-aggregating blood. It is expected that the use of this model will allow for the testing of DCPs in virtual patients and for their tailor-design in treating various diseases.Graphical abstract Supplementary InformationThe online version contains supplementary material available at 10.1007/s00397-021-01289-x.

Simple, Accurate and User-Friendly Differential Constitutive Model for the Rheology of Entangled Polymer Melts and Solutions from Nonequilibrium Thermodynamics.

In a recent reformulation of the Marrucci-Ianniruberto constitutive equation for the rheology of entangled polymer melts in the context of nonequilibrium thermodynamics, rather large values of the convective constraint release parameter βccr had to be used in order for the model not to violate the second law of thermodynamics. In this work, we present an appropriate modification of the model, which avoids the splitting of the evolution equation for the conformation tensor into an orientation and a stretching part. Then, thermodynamic admissibility simply dictates that βccr ≥ 0, thus allowing for more realistic values of βccr to be chosen. Moreover, and in view of recent experimental evidence for a transient stress undershoot (following the overshoot) at high shear rates, whose origin may be traced back to molecular tumbling, we have incorporated additional terms into the model accounting, at least in an approximate way, for non-affine deformation through a slip parameter ξ. Use of the new model to describe available experimental data for the transient and steady-state shear and elongational rheology of entangled polystyrene melts and concentrated solutions shows close agreement. Overall, the modified model proposed here combines simplicity with accuracy, which renders it an excellent choice for managing complex viscoelastic fluid flows in large-scale numerical calculations.

An enriched Bouc-Wen model with damage

This paper presents an enriched hysteresis model with damage relying on the Bouc-Wen formulation, in which a single scalar variable is introduced to reproduce effects of strength and stiffness degradation emerging for damaging materials. First, some acknowledged limits of the original Bouc-Wen model are discussed, focusing on its thermodynamic admissibility and compatibility with Drucker's plasticity postulate, and highlighting the effects of the parameters β and γ. Then, a reformulated expression of the dissipated energy is presented, which is consistent from the thermodynamic point of view. This formulation is extended to the case of degrading Bouc-Wen models. Uniaxial validation analyses are proposed, together with some considerations on the physical and thermodynamic admissibility of the model.The effect of pinching is also introduced by means of the arrangement in parallel of a non-dissipative Bouc-Wen and a degrading (damaging) device, resulting in a varied set of hysteresis shapes.Finally, the proposed model is adopted to analyse the cyclic response of masonry structural elements under in-plane horizontal loads. Within the framework of equivalent frame formulation, masonry panels are modelled by an elastic beam arranged in series with nonlinear flexural hinges lumped at the ends and a shear link, whose constitutive relationships are described by means of the enriched hysteretic model with damage.

Stability analysis for a thermodynamically consistent model of relativistic fluid dynamics

In relativistic fluid mechanics, positive entropy production is known to be insufficient for guaranteeing stability. Much stronger criteria for thermodynamic admissibility have become available in nonequilibrium thermodynamics. We here perform a linear stability analysis for a model of relativistic hydrodynamics that is based on the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework of nonequilibrium thermodynamics. Assuming a quadratic entropy function near equilibrium, we find stability for the entire range of physically meaningful model parameters for relativistic fluid dynamics based on GENERIC. The search for thermodynamic admissibility moreover reveals a fundamental difference between liquids and gases in relativistic fluid dynamics.

Anisotropic continuum damage constitutive model to describe the cyclic response of quasi-brittle materials: The regularized unilateral effect

Within the framework of damage mechanic, numerous anisotropic damage models have been proposed in the literature with the aim to represent the anisotropic degradation of quasi-brittle materials. The benefits from such models arise from the fact they are consistent with the principles of the continuum mechanics enabling easy numerical implementation in the majority of finite element codes. Despite the wealth of anisotropic models in the literature, further developments are needed to simulate correctly the responses involving phenomena related to crack closure. The present paper proposes a new class of anisotropic damage models characterized by its capabilities to describe non linear progressive stiffness recovery with the possibility to introduce permanent strains. The theoretical framework takes benefits from some results of the operator function theory. Further mathematical features are established for sets of functions, which are termed opening (closure) cracking functions. These features are useful to control the material behavior when the tensile or the compressive strains are activated (deactivated) with more or less smoothness. The thermodynamical admissibility condition is fulfilled, as long as the damage variable and the cracking functions satisfy further conditions. The robustness associated with the time integration of the proposed class of models is illustrated by a structural case study.

Beyond equilibrium thermodynamics in the low temperature plasma processor

Low temperature plasmas are open driven thermodynamic systems capable of increasing the free energy of the mass that flows through them. An interesting thing about low temperature plasmas is that different species have different temperatures at the same location in space. Since thermal equilibrium cannot be assumed, many of the familiar results of equilibrium thermodynamics cannot be applied in their familiar form to predict, e.g., the direction of a chemical reaction. From the perspective of classical processing governed by thermal equilibrium, examples of highly unexpected gas-phase chemical reactions (CO2 dissociation, NO, N2H4, O3 synthesis) and solid material transformations (surface activation, size-focusing, and hyperdoping) promoted by low temperature plasmas are presented. The lack of a known chemical reaction equilibrium criterion prevents assessment of predictive kinetics models of low temperature plasmas, to ensure that they comply with the laws of thermodynamics. There is a need for a general method to predict chemical reaction equilibrium in low temperature plasmas or an alternative method to establish the thermodynamic admissibility of a proposed kinetics model. Toward those ends, two ideas are explored in this work. The first idea is that chemical reactions in low temperature plasmas proceed toward a thermal equilibrium state at an effective temperature intermediate between the neutral gas temperature and the electron temperature. The effective temperature hypothesis is simple, and surprisingly is adequate for elucidation in some systems, but it lacks generality. The general equation for nonequilibrium reversible–irreversible coupling (GENERIC) is a general beyond equilibrium thermodynamics framework that can be used to rigorously establish the thermodynamic admissibility of a set of dynamic modeling equations, such as a kinetic model, without knowledge of the final state that the system is tending toward. The use of GENERIC is described by way of example using a two-temperature hydrodynamic model from the literature. The conclusion of the GENERIC analysis presented in this work is that the concept of superlocal equilibrium is thermodynamically admissible and may be applied to describe low temperature plasmas, provided that appropriate terms are included for exchange of internal energy and momentum between different species that may have different temperatures and bulk velocities at the same location in space. The concept of superlocal equilibrium is expected to be of utility in future work focused on deriving equilibrium criteria for low temperature plasmas.

On intrinsic time measure concept with application to the modeling of smart materials

AbstractIn this paper, the cyclic behavior of a Nitinol cubic block is described by using the Bouc‐Wen model [2] coupled to the intrinsic time measure, other than the clock time which governs the behavior of the materials [4]. The modified Bouc‐Wen model consists from the classical Bouc‐Wen model coupled to the intrinsic time measure, other than the clock time which governs the behaviour of the materials. As a consequence, the thermodynamic admissibility of the Bouc–Wen model is provided by the endochronic theory of plasticity. The role of the intrinsic time measure is described by capturing the stiffness and strength degradation and the opposite phenomena. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Passivity of Hysteretic Systems with Double Hysteretic Loops.

The Bouc–Wen hysteresis model is widely employed to mathematically represent the dynamical behavior of several physical devices, materials and systems such as magnetorheological dampers, lanthanide zirconium or aluminum oxides, mechanical structures or biomedical systems. However, these mathematical models must account for different properties such as the bounded-input bounded-output stability, asymptotic motion, thermodynamic admissibility or passivity in order to be physically consistent with the systems they represent. The passivity of a system is related to energy dissipation. More precisely, a system is passive if it does not generate energy but only dissipates it. The objective of this paper is to prove that two different double-loop Bouc–Wen models are passive under a particular set of model parameters.

The Resolution of d'Alembert's Paradox and Thermodynamic Admissibility

The d'Alembert paradox, annunciated in 1752, was established after it was shown that the result of a net zero drag, obtained by applying potential theory to the incompressible flow past a sphere, was in contradiction with experimental results which showed a positive drag. Interpreting the result as a flaw in the theory, resulted in the declaration of the paradox. Following d'Alembert, we assume a potential motion, and proceed to analyze the consequences of this assumption using the global principles of continuum mechanics. We show that if the fluid is inviscid, the potential motion is thermodynamically admissible, the drag is zero, and the motion can persist indefinitely. Although no conventional fluid is available to either falsify (or validate), this result experimentally, in principle, the theory could be tested by using a superfluid, such as liquid Helium. If the fluid is viscous, we show that the potential irrotational motion is thermodynamically inadmissible, it is in violation of the second law of thermodynamics, and hence such a motion cannot persist. Indeed, observations show that when a rigid body is impulsively set into motion, an irrotational motion may exist initially but does not persist. Any flow property which is derived from a thermodynamically inadmissible motion cannot be expected to be in agreement with experimental data. As an illustration we show that the drag, calculated from the viscous potential solution of the flow past a sphere, is zero. We submit that since the theory of continuum mechanics predicts that no agreement between results obtained from viscous potential theory and experimental data can be expected, there is no room for a paradox once a contradiction is in fact observed. In nature, or under experimental conditions, the nonpersistence of the thermodynamically inadmissible motion proceeds in a breakup of the irrotational motion which transforms into a rotational and obviously admissible motion. We show that by selecting boundary conditions, required in the solution of the differential equations of motion, such that they satisfy the Clausius–Duhem jump conditions inequality, the thermodynamic admissibility of the solution is a priori assured. We also show the vorticity distribution at the wall associated with the particular choice of boundary condition.

On intrinsic time measure in the modeling of cyclic behavior of a Nitinol cubic block

In this paper, the cyclic behavior of a superelastic-plastic nitinol cubic block is described by using the Bouc–Wen model coupled to an intrinsic time measure other than clock time, which governs the behavior of the materials. As a consequence, the thermodynamic admissibility of the Bouc–Wen model is provided by the endochronic theory of plasticity. The role of the intrinsic time measure is described by capturing the stiffness and strength degradation and the opposite phenomena. Such behavior is due to the permanent-strain addition of residual martensite and alterations in the properties of the texture during phase transformation.

Berthierine formation in reservoir rocks from the Siri oilfield (Danish North Sea) as result of fluid–rock interactions: Part III. Determining mineral stability and CO2-sequestering capacity of glauconitic sandstones

Berthierine was proven as one of the most important products of glauconite alteration in the Siri oilfield (Danish North Sea). However, there is an ongoing debate regarding the main product of glauconite dissolution: siderite, berthierine, or berthierine as a precursor of siderite and/or magnesium-bearing carbonate. In order to investigate the consequences of glauconite dissolution in view of thermodynamic admissibility and the resulting CO2-sequestering capacity, a hydrogeochemical model, which is based on thermodynamics of chemical equilibrium, was developed. Calculating various modeling scenarios helps to conclude on the pH-EH conditions of glauconite dissolution as well as of berthierine formation and dissolution in generic, aqueous systems under elevated temperature-pressure conditions.Our modeling results highlight that carbonate formation cannot be triggered exclusively by CO2 addition into glauconitic sandstones. The injection of pure CO2 into glauconitic sandstones leads to acidic and anoxic oxidizing conditions under which glauconite remains stable. To intensify glauconite alteration by CO2 injection, glauconitic sandstones have to be in contact with degradable organic matter, or, alternatively, reducing agents have to be co-injected with CO2. Sufficient electron transfer to ferric iron bound in glauconite is the ultimate control for intense glauconite alteration and for subsequent berthierine precipitation.Once formed, berthierine remains stable over a broad pH range and is not transformed to any carbonate under reducing conditions. Thus, CO2 injection into glauconitic sandstones under reducing conditions mainly leads to formation of berthierine instead of iron- and magnesium-carbonates. However, hydrogeochemical conditions in the subsurface can affect CO2 sequestration via glauconite dissolution and the resulting carbonate formation, including the pH-EH conditions, the chemical composition of glauconite, and the overall mineralogical composition of glauconitic sandstones.

Thermodynamic admissibility of continuum damage representations

Representations of continuum damage mechanics introduce internal damage parameters into the constitutive stress–strain relations to account for the growth of microscopic flaws and the loss of material strength. Nominally elastic materials are characterized by their elasticity parameters, such as Young’s modulus and Poisson’s ratio, which become dependent on the internal damage parameters. An increase of internal damage involves an energy release similar to the well-known energy release rate in fracture mechanics. This energy release must satisfy the second law of thermodynamics. The implications for the representations of the elasticity parameters in terms of the internal damage parameters are investigated. Necessary conditions for the thermodynamic admissibility of the continuum damage representations are derived based on theoretical models. Several possibilities are given for materials with isotropic or with orthotropic elastic behaviour in the damaged state.

Continuum Model for the Phase Behavior, Microstructure, and Rheology of Unentangled Polymer Nanocomposite Melts

We introduce a continuum model for polymer melts filled with nanoparticles capable of describing in a unified and self-consistent way their microstructure, phase behavior, and rheology in both the linear and nonlinear regimes. It is based on the Hamiltonian formulation of transport phenomena for fluids with a complex microstructure with the final dynamic equations derived by means of a generalized (Poisson plus dissipative) bracket. The model describes the polymer nanocomposite melt at a mesoscopic level by using three fields (state variables): a vectorial (the momentum density) and two tensorial ones (the conformation tensor for polymer chains and the orientation tensor for nanoparticles). The dynamic equations are developed for nanoparticles with an arbitrary shape but then they are specified to the case of spherical ones. Restrictions on the parameters of the model are provided by analyzing its thermodynamic admissibility. A key ingredient of the model is the expression for the Helmholtz free energy A ...

Thermo-Solutal Modelling of Microstructure Formation during Multiphase Alloy Solidification - a New Approach

This paper shows how to move from a specification of free energy for the solidification of a binary alloy to the dynamical equations using the elegance of a dissipative bracket analogous to the Poisson bracket of Hamiltonian mechanics. A key new result is the derivation of the temperature equation for single-phase thermal-solutal models, which contains generalisations and extra terms which challenge standard models. We also present, for the first time, the temperature equation for thermal multi-phase field models. There are two main ingredients: one, the specification of the free energy in terms of the time and space dependent field variables: $n$-phases $\phi_i$, a concentration variable $c$, and temperature $T$; two, the specification of the dissipative bracket in terms of these variables, their gradients and a set of diffusion parameters, which may themselves depend on the field variables. The paper explains the method within this context and demonstrates its thermodynamic admissibility.

On the thermodynamic requirement of elastic stiffness anisotropy in isotropic materials

In general, thermodynamic admissibility requires isotropic materials develop reversible deformation induced anisotropy (RDIA) in their elastic stiffnesses. Taking the elastic potential for an isotropic material to be a function of the strain invariants, isotropy of the elastic stiffness is possible under distortional loading if and only if the bulk modulus is independent of the strain deviator and the shear modulus is constant. Previous investigations of RDIA have been limited to applications in geomechanics where material non-linearity and large deformations are commonly observed. In the current paper, the degree of RDIA in other materials is investigated. It is found that the resultant anisotropy in materials whose strength does not vary appreciably with pressure, such as metals, is negligible, but in materials whose strength does vary with pressure, the degree of RDIA can be significant. Algorithms for incorporating RDIA in a classical elastic–plastic model are provided.

A Ductile Damage Nonlocal Model of Integral-type at Finite Strains: Formulation and Numerical Issues

This contribution is devoted to the formulation and numerical implementation of a ductile damage constitutive model enriched with a thermodynamically consistent nonlocal theory of integral type. In order to describe ductile deformation, the model takes finite strains into account. To model elasticity, a Hencky-like hyperelastic free energy potential coupled with nonlocal damage is adopted. The thermodynamic consistency of the model is ensured by applying the first and second thermodynamical principles in the global form and the dissipation inequality can be re-written in a local form by incorporating a nonlocal residual that accounts for energy exchanges between material points of the nonlocal medium. The thermodynamically consistent nonlocal model is compared with its associated classical formulation (in which nonlocality is merely incorporated by averaging the damage variable without resorting to thermodynamic potentials) where the thermodynamical admissibility of the classical formulation is demonstrated. Within the computational scheme, the nonlocal constitutive initial boundary value problem is discretized over pseudo-time where it is shown that well established numerical integration strategies can be straightforwardly extended to the nonlocal integral formulation. A modified Newton-Raphson solution strategy is adopted to solve the nonlinear complementarity problem and its numerical implementation, regarding the proposed nonlocal constitutive model, is presented in detail. The results of two-dimensional finite element analyses show that the model is able to eliminate the pathological mesh dependence inherently present under the softening regime if the local theory is considered.