A unified framework for modeling hysteresis in ferroic materials

This paper addresses the development of a unified framework for quantifying hysteresis and constitutive nonlinearities inherent to ferroelectric, ferromagnetic and ferroelastic materials. Because the mechanisms which produce hysteresis vary substantially at the microscopic level, it is more natural to initiate model development at the mesoscopic, or lattice, level where the materials share common energy properties along with analogous domain structures. In the first step of the model development, Helmholtz and Gibbs energy relations are combined with Boltzmann theory to construct mesoscopic models which quantify the local average polarization, magnetization and strains in ferroelectric, ferromagnetic and ferroelastic materials. In the second step of the development, stochastic homogenization techniques are invoked to construct unified macroscopic models for nonhomogeneous, polycrystalline compounds exhibiting nonuniform effective fields. The combination of energy analysis and homogenization techniques produces low-order models in which a number of parameters can be correlated with physical attributes of measured data. Furthermore, the development of a unified modeling framework applicable to a broad range of ferroic compounds facilitates material characterization, transducer development, and model-based control design. Attributes of the models are illustrated through comparison with piezoceramic, magnetostrictive and shape memory alloy data and prediction of material behavior.     

An elasto-plastic model for granular materials with microstructural consideration

In this paper, we have extended the granular mechanics approach to derive an elasto-plastic stress–strain relationship. The deformation of a representative volume of the material is generated by mobilizing particle contacts in all orientations. Thus, the stress–strain relationship can be derived as an average of the mobilization behavior of these local contact planes. The local behavior is assumed to follow a Hertz–Mindlin’s elastic law and a Mohr–Coulomb’s plastic law. Essential features such as continuous displacement field, inter-particle stiffness, and fabric tensor are discussed. The predictions of the derived stress–strain model are compared to experimental results for sand under both drained and undrained triaxial loading conditions. The comparisons demonstrate the ability of this model to reproduce accurately the overall mechanical behavior of granular media and to account for the influence of key parameters such as void ratio and mean stress. A part of this paper is devoted to the study of anisotropic specimens loaded in different directions, which shows the model capability of considering the influence of inherent anisotropy on the stress–strain response under a drained triaxial loading condition.     

An approximate explicit constitutive relation for non-flowing granular materials

An approximate stress-strain relation is derived for a granular material composed of spherical elastic granules in contact. The material is assumed to be statistically homogeneous so that the effective stress tensor can be obtained by a volume average. The resulting stress-strain relation is markedly non-linear and begins with the term $\text{∥ε∥}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, where ε is the classical infinitesimal strain tensor. Some simple deformation fields are worked out.     

A unified framework for compressible plasticity

A framework for compressible elastoplasticity is cultivated to proficiently characterize various loading states of an isotropic solid undergoing a full range of deformation, including both hardening and softening behavior. By employing Legendre's dual transformation, it is shown that Drucker's path independence in the small and the linearity of rate change between stress and strain are characteristics of the additive-decomposition of strain rates. The normality property of the plastic strain rate to a loading function is established on the basis of path independence in the small of the potential functions used in the dual transformation. Work-hardening is a sufficient but not a necessary condition for ensuring stability in the small. Within this framework, a compressible elastoplasticity theory is generalized to cover the behavior of softening and elastoplastic coupling. Analysis results indicate that plastic instability for a void-contained aggregate is independent of the hydrostatic component of a stress tensor.     

Bifurcation conditions for ideal fibre-reinforced materials

A bifurcation of an equilibrium state for ideal fibre-reinforced material is discussed. It is assumed that the material is elastic, locally transversely isotropic, incompressible and inextensible in the direction of fibres. On a finite state of strain an arbitrary field of small displacements is superposed and a set of governing equations for the perturbed state is derived.As an example a stability problem of a rectangular block. Objected to a finite, homogeneous deformation is considered. A discussion of the results is focused on the influence on the stability of the pressure applied in the direction of fibres.Due to the assumption of inextensibility this pressure has no influence on the state of strain, but it is shown that it may cause a loss of stability.     

An elastoplastic model for granular materials taking into account grain breakage

Granular materials are constituted of an assembly of particles. In spite of the simplicity of this assembly, its mechanical behaviour is complex. In the first stage we propose a framework to establish correlations between parameters of the supposedly continuous medium and grain properties which are assumed to be constant. However, this hypothesis is no longer valid in the case where physical (shape, size…) or mechanical properties (Young modulus E_g, Poisson's ratio ν_g…) of grains evolve during loading, causing the behaviour of the assembly to modify. We study the influence of the physical and mechanical parameters on grain breakage. We subsequently propose a way to model the influence of the grain breakage on granular materials and we introduce this influence in an elastoplastic constitutive model. Validations are made on two kinds of sands under isotropic and triaxial loading. Since the results of numerical computations corresponded well with the experimental data, we believe that the new model is capable of accurately simulating the behaviour of granular materials under a wide range of stresses and of taking into account, through new parameters, the individual strength of grains.     

Computational multiscale methods for granular materials

The fine-scale heterogeneity of granular material is characterized by its polydisperse microstructure with randomness and no periodicity. To predict the mechanical response of the material as the microstructure evolves, it is demonstrated to develop computational multiscale methods using discrete particle assembly-Cosserat continuum modeling in micro- and macro- scales, respectively. The computational homogenization method and the bridge scale method along the concurrent scale linking approach are briefly introduced. Based on the weak form of the Hu-Washizu variational principle, the mixed finite element procedure of gradient Cosserat continuum in the frame of the second-order homogenization scheme is developed. The meso-mechanically informed anisotropic damage of effective Cosserat continuum is characterized and identified and the microscopic mechanisms of macroscopic damage phenomenon are revealed.     

Multi-mechanism anisotropic model for granular materials

By representing the assembly by a simplified column model, a constitutive theory, referred to as sliding–rolling theory, was recently developed for a two-dimensional assembly of rods subjected to biaxial loading, and then extended to a three-dimensional assembly of spheres subjected to triaxial (equibiaxial) loading. The sliding–rolling theory provides a framework for developing a phenomenological constitutive law for granular materials, which is the objective of the present work. The sliding–rolling theory provides information concerning yield and flow directions during radial and non-radial loading. In addition, the theory provides information on the role of fabric anisotropy on the stress–strain behavior and critical state shear strength. In the present paper, a multi-axial phenomenological model is developed within the sliding–rolling framework by utilizing the concepts of critical state, classical elasto-plasticity and bounding surface. The resulting theory involves two yield surfaces and falls within the definition of the multi-mechanism models. Computational issues concerning the solution uniqueness for stress states at the corner of yield surfaces are addressed. The effect of initial and induced fabric anisotropy on the constitutive behavior is incorporated. It is shown that the model is capable of simulating the effect of anisotropy, and the behavior of loose and dense sands under drained and undrained loading.     

Volume-weighted mixture theory for granular materials

In the present work we treat granular materials as mixtures composed of a solid and a surrounding void continuum, proposing then a continuum thermodynamic theory for it. In contrast to the common mass-weighted balance equations of mass, momentum, energy and entropy for mixtures, the volume-weighted balance equations and the associated jump conditions of the corresponding physical quantities are derived in terms of volume-weighted field quantities here. The evolution equations of volume fractions, volume-weighted velocity, energy, and entropy are presented and explained in detail. By virtue of the second law of thermodynamics, three dissipative mechanisms are considered which are specialized for a simple set of linear constitutive equations. The derived theory is applied to the analysis of reversible and irreversible compaction of cohesionless granular particles when a vertical oscillation is exerted on the system. In this analysis, a hypothesis for the existence of a characteristic depth within the granular material in its closely compacted state is proposed to model the reversible compaction.     

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part II – the formulation of elastoplastic coupling at large strain

The two key phenomena occurring in the process of ceramic powder compaction are the progressive gain in cohesion and the increase of elastic stiffness, both related to the development of plastic deformation. The latter effect is an example of ‘elastoplastic coupling’, in which the plastic flow affects the elastic properties of the material, and has been so far considered only within the framework of small strain assumption (mainly to describe elastic degradation in rock-like materials), so that it remains completely unexplored for large strain. Therefore, a new finite strain generalization of elastoplastic coupling theory is given to describe the mechanical behaviour of materials evolving from a granular to a dense state.The correct account of elastoplastic coupling and of the specific characteristics of materials evolving from a loose to a dense state (for instance, nonlinear – or linear – dependence of the elastic part of the deformation on the forming pressure in the granular – or dense – state) makes the use of existing large strain formulations awkward, if even possible. Therefore, first, we have resorted to a very general setting allowing general transformations between work-conjugate stress and strain measures; second, we have introduced the multiplicative decomposition of the deformation gradient and, third, employing isotropy and hyperelasticity of elastic response, we have obtained a relation between the Biot stress and its ‘total’ and ‘plastic’ work-conjugate strain measure. This is a key result, since it allows an immediate achievement of the rate elastoplastic constitutive equations. Knowing the general form of these equations, all the specific laws governing the behaviour of ceramic powders are finally introduced as generalizations of the small strain counterparts given in Part I of this paper.     

Entropy productions in granular materials

Granular materials display more abundant dissipation phenomena than ordinary materials. In this paper, a brief energy flow path with irreversible processes is illustrated, where the concept of granular temperature T_g, initially proposed for dilute systems, is extended to dense systems in order to quantify disordered force chain configurations. Additionally, we develop the concept of conjugate granular entropy s_g and its production equation. Our analyses find out that the granular entropy significantly undermined the elastic contact between particles, seriously affecting the transport coefficients in granular materials and creating new transport processes.     

A unified energy-based control framework for tethered spacecraft deployment

Nonlinear Dynamics - This paper proposed a unified energy-based control framework for fast, stable, and precision deployment of underactuated TSS. The tension controller with partial state feedback...     

A generalized continuum theory for granular materials

A generalized continuum theory for granular media is formulated by allowing for the possibility of rotation of granules. The basic balance laws are presented and based on thermodynamical consideration a set of constitutive equations are derived. The theory naturally gives rise to the generation of antisymmetric stress tensor and existence of couple stresses. The basic equations of motion are derived and it is shown that the theory contains Mohr-Coulomb criterion of limiting equilibrium as a special case. The problem of coupled porosity and microrotational wave propagation is investigated and the rectilinear shear flow of granular materials is discussed.     

Analysis and calibration of a three-invariant plasticity model for granular materials

Summary In this paper we briefly review issues related to the characterization of properties of granular materials subjected to micro-gravity and one-gravity conditions at very low effective stress levels. We describe the development of a three-invariant plasticity model that resembles the model devised by Lade. An inverseidentification scheme where the analysis tools are used to extract constitutive model parameters from experiments is also discussed.

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Untersuchung und Kalibrierung eines mit drei Spannungsinvarianten formulierten Plastizitätsmodells für granulare Stoffe

Zusammenfassung Übersicht: Über Resultate, die mit der Charakterisierung der Eigenschaften von granularen Stoffen in schwachen Schwerefeldern bzw. dem der Erde bei sehr kleinen Spannungen zusammenhängen, wird ein kurzer Überblick gegeben. Beschrieben wird ein mit drei Spannungsinvarianten formuliertes plastisches Stoffgesetz, welches dem von Lade entwickelten Modell ähnlich ist. Weiterhin wird ein inverses Identifikationsschema diskutiert, bei dem mit analytische Methoden benutzt werden, um die Parameter des Stoffmodells zu gewinnen.

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Presented at the workshop on Limit Analysis and Bifurcation Theory, held at the University of Karlsruhe (FRG), February 22–25, 1988     

Change of scale in granular materials

This paper analyzes relationships allowing global variables to be deduced from local ones in granular materials. This change of scale is now clear for static variables but rather confusing for kinematic variables. This paper compares the various established formulations for kinematic variables and also proposes original formulations. By running numerical simulations all the formulations in question are compared and the basic premises to define the change of scale for kinematic variables in granular media put forward.     

A microstructural elastoplastic model for unsaturated granular materials

The homogenization technique is used to obtain an elastoplastic stress–strain relationship for dry, saturated and unsaturated granular materials. Deformation of a representative volume of material is generated by mobilizing particle contacts in all orientations. In this way, the stress–strain relationship can be derived as an average of the mobilization behavior of these local contact planes. The local behavior is assumed to follow a Hertz–Mindlin’s elastic law and a Mohr–Coulomb’s plastic law. For the non-saturated state, capillary forces at the grain contacts are added to the contact forces created by an external load. They are calculated as a function of the degree of saturation, depending on the grain size distribution and on the void ratio of the granular assembly. Numerical simulations show that the model is capable of reproducing the major trends of a partially saturated granular assembly under various stress and water content conditions. The model predictions are compared to experimental results on saturated and unsaturated samples of silty sands under undrained triaxial loading condition. This comparison shows that the model is able to account for the influence of capillary forces on the stress–strain response of the granular materials and therefore, to reproduce the overall mechanical behavior of unsaturated granular materials.     

A unified energy approach to a class of micromechanics models for composite materials

Several micromechanics models for the determination of composite moduli are investigated in this paper, including the dilute solution, self-consistent method, generalized self-consistent method, and Mori-Tanaka's method. These micromechanical models have been developed by following quite different approaches and physical interpretations. It is shown that all the micromechanics models share a common ground, the generalized Budiansky's energy-equivalence framework. The difference among the various models is shown to be the way in which the average strain of the inclusion phase is evaluated. As a bonus of this theoretical development, the asymmetry suffered in Mori-Tanaka's method can be circumvented and the applicability of the generalized self-consistent method can be extended to materials containing microcracks, multiphase inclusions, non-spherical inclusions, or non-cylindrical inclusions. The relevance to the differential method, double-inclusion model, and Hashin-Shtrikman bounds is also discussed. The application of these micromechanics models to particulate-reinforced composites and microcracked solids is reviewed and some new results are presented.