An empirical constitutive law for concentrated colloidal suspensions in the approach of the glass transition

Concentrated, non-crystallizing colloidal suspensions in their approach of the glass state exhibit distinct dynamics patterns. These patterns suggest a powerlaw rheological constitutive model for near-glass viscoelasticity, as presented here. The rheological parameters used for this model originate in the mode-coupling theory. The proposed constitutive model provides explicit expressions for the steady shear viscosity, the steady normal stress coefficient, the modulus-compliance relation, and the α peak of G″. The relaxation pattern distinctly differs from gelation.     

The site model theory and the standard linear solid

Summary The site model theory (SMT) is shown to lead to the same deformation behaviour as that displayed by the standard linear solid (SLS), group I, for all loading conditions. If a second deformation mechanism (inter-molecular slip) is introduced the result is the same as that obtained with the standard linear solid, group II, and models the behaviour of a polymer melt near to the solidification temperature.

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Zusammenfassung Es wird gezeigt, daß ein einfaches Platzwechsel-Modell (site model theory) bei allen Belastungsbedingungen das gleiche Deformationsverhalten voraussagt wie der lineare Drei-Parameter-Festkörper (standard linear solid, group I). Wenn ein weiterer Deformationsmechanismus (zwischenmolekulare Gleitung) eingeführt wird, entspricht das Verhalten dagegen demjenigen einer linearen Drei-Parameter-Flüssigkeit (standard linear solid, group II), welche das Verhalten einer Polymerschmelze in der Nähe der Schmelztemperatur beschreibt.

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a = ₁₂ ⁰ + ₂₁ ⁰ , see eq. [1] - b =N ₁ ⁰ ₁₂ ⁰ (V ₁₂ +V ₂₁), see eq. [1] - c = 2N _s ⁰ V _s see eq. [6] - k Boltzmann constant - t time - E,E ₁,E ₂ spring constants, see figures 1 and 3 - E _u unrelaxed modulus - N ₁ ⁰ site 1 equilibrium population in the unstressed state - N _s number of units available for slip - N(t) decrease in site 1 population - N _s (t) net number of slip jumps in the stressaided direction - T temperature (K) - V _i,j activation volume for jumps in directioni j - V _s activation volume for the slip process - strain - strain rate - incremental change in strain per unit change in site population - µ,µ ₁,µ ₂ dashpot constants, see figures 1 and 3 - applied stress - ₀ initial applied stress, (stress relaxation) =(t) (creep) - incremental change in stress per unit change in site population - ⁰ jump rate for slip in the unstressed state - _i,j ⁰ jump rate in the directioni j in the unstressed state With 3 figures and 3 tables     

Applicability of the free volume concept on relaxation phenomena in the glass transition range

Conclusion Except the discrepancies mentioned with respect to pressure investigations, which need clarification in the future, we can conclude in a general way, as follows.As far as only average parameters of macroscopic samples are considered (complex moduli, or dielectric constants, volume or heat content etc....), the free volume concept can relate the variations of molecular mobility to the changes of an average free volume in a semiquantitative way. This average free volume can no longer fully caracterize the wide variety of molecular motions involved in the kinetics of redistribution of holes in the liquid during the recovery experiments. These kinetic processes involve a wide distribution of retardation times, which may be associated with the local distribution of holes, or with that of cooperating groups of molecules, or molecular segments.On the other hand, free volume is not necessarily the fundamental molecular parameter which controls the rate of configurational changes, characterized by the variation of entropy of the liquid (14). Even if this is the case, most of the above discussion may be applied to any other average excess parameter, as far as theDoolittle equation is formally adopted, in which (f/b) is expressed in terms of the new parameter, rather than that of free volume. However, since the relaxational free volume, as determined from the W.L.F. equation, and the independently measured volume changes are often in close agreement, this means that the variations of the excess entropy (12), or those of the configurational free energy (13), and the changes in volume are closely related. Therefore, the free volume concept remains still a valuable tool for unifying different kinds of rate processes from both a theoretical and an experimental point of view, especially in the glass transition range.     

On the volumetric part of strain-energy functions used in the constitutive modeling of slightly compressible solid rubbers

A popular model for the finite element simulation of slightly compressible solid rubber-like materials assumes that the strain-energy function can be additively decomposed into a volumetric part and a deviatoric part. Based on mathematical convenience, the volumetric part is usually assumed to be a finite polynomial in the volume change. Experimental evidence suggests that for solid rubbers in compression, this polynomial can be taken to be a simple quadratic for moderate deformations and that this function also adequately models the volume change and the stress/stretch relation for materials in simple tension, up to stretches of order 100%. For larger tensile deformations, however, experimental data suggest that the Cauchy stress-volume change relation has an increasingly large slope and therefore a truncated Taylor series expansion is not the most appropriate. A rational function approach is proposed here as an alternative.     

Mapping the stiffness-temperature-moisture relationship of solid biomaterials at and around their glass transition

Mechanical changes in biomaterials at and around their glass transition are key factors in their functionality and/or stability. They are described in terms of a relationship betwen a relative stiffness R(T,M) defined as the ratio between a modulus or storage modulus at a temperature T, and moisture M, and its magnitude in the glassy state. The relationship, in turn, is expressed by the model R(T,M) = 1/[1+exp [{T-T _c,(M))/a(M)]} where T _c(M) is a critical temperature identifying the transition temperature range and a(M) a constant representing the relationship's slope. The proposed model correctly accounts for the downward concavity of the stiffness vs temperature relationship at the transition onset. Published data on biosolids indicate that T _c(M) can be described by a single exponential decay term, and so most probably also a(M). Incorporation of these terms into the model enables the creation of realistic three-dimensional maps of the relative stiffness-temperature-moisture relationship at and in the neighborhood of the glass transition region. In principle, the same method can also be used to describe the effect of plasticizers other than water if their influence on T _cand the steepness parameter can be formulated as an algebraic expression.     

A thermo-mechanically-coupled large-deformation theory for amorphous polymers in a temperature range which spans their glass transition

Amorphous thermoplastic polymers are important engineering materials; however, their non-linear, strongly temperature- and rate-dependent elastic-viscoplastic behavior is still not very well understood, and is modeled by existing constitutive theories with varying degrees of success. There is no generally agreed upon theory to model the large-deformation, thermo-mechanically-coupled, elastic-viscoplastic response of these materials in a temperature range which spans their glass transition temperature. Such a theory is crucial for the development of a numerical capability for the simulation and design of important polymer processing operations, and also for predicting the relationship between processing methods and the subsequent mechanical properties of polymeric products. In this paper we extend our recently published theory [Anand, L., Ames, N. M., Srivastava, V., Chester, S. A., 2009. A thermo-mechanically-coupled theory for large deformations of amorphous polymers. Part I: formulation. International Journal Plasticity 25, 1474–1494; Ames, N. M., Srivastava, V., Chester, S. A., Anand, L., 2009. A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: applications. International Journal of Plasticity 25, 1495–1539] to fill this need.     

A constitutive model in linear thermoviscoelasticity of polymers based on the concept of cooperative relaxation

New constitutive relations are derived for amorphous glassy polymers based on the concept of cooperative relaxation. A polymer is treated as a system of rearranging regions (flow units) embedded into a homogeneous elastic matrix. The viscoelastic (time-dependent) response of a medium is explained by rearrangements of segments of long chains in relaxing regions which occur at random instants. The kinetics of rearrangement is described in the framework of the Eyring concept of thermally activated processes, whereas the energy of any flow unit is assumed to randomly change at the instant of its reformation. Based on experimental data, phenomenological formulas are proposed for material functions. Adjustable parameters are found by fitting observations for mixtures of nylon with lithium halides in isothermal tensile relaxation tests. The thermoviscoelastic response in other tests is studied numerically. It is demonstrated that the material behavior predicted by the constitutive model in non-isothermal tests substantially differs from that predicted by conventional models whose adjustable parameters are determined by using the same experimental data. Received September 30, 1998     

Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws

A physically sound three-dimensional anisotropic formulation of the standard linear viscoelastic solid with integer or fractional order rate laws for a finite set of the pertinent internal variables is presented. It is shown that the internal variables can be expressed in terms of the strain as convolution integrals with kernels of Mittag–Leffler function type. A time integration scheme, based on the Generalized Midpoint rule together with the Grünwald algorithm for numerical fractional differentiation, for integration of the constitutive response is developed. The predictive capability of the viscoelastic model for describing creep, relaxation and damped dynamic responses is investigated both analytically and numerically. The algorithm and the present general linear viscoelastic model are implemented into the general purpose finite element code Abaqus. The algorithm is then used together with an explicit difference scheme for integration of structural responses. In numerical examples, the quasi-static and damped responses of a viscoelastic ballast material that is subjected to loads simulating the overrolling of a train are investigated.     

The SPH approach to the process of container filling based on non-linear constitutive models

In this work,the transient free surface of container filling with non-linear constitutive equation’s fluids is numerically investigated by the smoothed particle hydrodynamics(SPH) method.Specifically,the filling process of a square container is considered for non-linear polymer fluids based on the Cross model.The validity of the presented SPH is first verified by solving the Newtonian fluid and OldroydB fluid jet.Various phenomena in the filling process are shown,including the jet buckling,jet thinning,splashing or spluttering,steady filling.Moreover,a new phenomenon of vortex whirling is more evidently observed for the Cross model fluid compared with the Newtonian fluid case.     

Remarks on ellipticity for the generalized Blatz-Ko constitutive model for a compressible nonlinearly elastic solid

One of the most widely used constitutive models for compressible isotropic nonlinearly elastic solids is the generalized Blatz-Ko material for foam-rubber and its various specializations. For this model, a unified derivation of necessary and sufficient conditions for ellipticity of the governing three-dimensional displacement equations of equilibrium is provided. When the parameterf occurring in the generalized Blatz-Ko model is in the range 0f<1, it is shown that ellipticity is always lost at sufficiently large stretches, while forf=1, the equilibrium equations are globally elliptic. The implications of these results for a variety of physical problems are discussed.     

Mechanically alloyed nanocrystalline iron and copper mixture: behavior and constitutive modeling over a wide range of strain rates

A comprehensive study on the response of a nanocrystalline iron and copper mixture (80% Fe and 20% Cu) to quasi-static and dynamic loading is performed. The constitutive model developed earlier by Khan, Huang & Liang (KHL) is extended to include the responses of nanocrystalline metallic materials. The strain rate and grain size dependent behaviors of porous nanocrystalline iron-copper mixture were determined experimentally for both static and dynamic loading. A viscoplastic model is formulated by associating the modified KHL model (representing the fully dense matrix behavior), and Gurson's plastic potential which provides the yield criteria for porous material. Simulations of uniaxial compressive deformations of iron-copper mixture with different initial porosity, grain size and at a wide range of strain rate (10⁻⁴ to 10³ s⁻¹) are made. The numerical results correlate well with the experimental observations.     

On a thermodynamical consistent constitutive law based on the concept of internal variables

A multiaxial constitutive law describing the behaviour of polycrystalline ice in a phenomenological way will be introduced. Internal variables represent the hardening and softening of the material and also a modified inelastic work. The equations are implemented in a thermodynamical frame.To derive the unknown functions and parameters a strategy is developed, facilitating the fitting in a simple and fast manner.Finally, the comparison between computations and experiments shows a good agreement.     

The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach

After recalling the constitutive equations of finite strain poroelasticity formulated at the macroscopic level, we adopt a microscopic point of view which consists of describing the fluid-saturated porous medium at a space scale on which the fluid and solid phases are geometrically distinct. The constitutive equations of poroelasticity are recovered from the analysis conducted on a representative elementary volume of porous material open to fluid mass exchange. The procedure relies upon the solution of a boundary value problem defined on the solid domain of the representative volume undergoing large elastic strains. The macroscopic potential, computed as the integral of the free energy density over the solid domain, is shown to depend on the macroscopic deformation gradient and the porous space volume as relevant variables. The corresponding stress-type variables obtained through the differentiation of this potential turn out to be the macroscopic Boussinesq stress tensor and the pore pressure. Furthermore, such a procedure makes it possible to establish the necessary and sufficient conditions to ensure the validity of an ‘effective stress’ formulation of the constitutive equations of finite strain poroelasticity. Such conditions are notably satisfied in the important case of an incompressible solid matrix.     

On the impossibility of linear Cauchy and Piola-Kirchhoff constitutive theories for stress in solids

We investigate the possibility of linear elasticity as an infinitesimal theory based on a genuinely linear response function which retains its validity even for finite deformations. Careful consideration of the domain of definition of the stress response function, the definition of linearity and the notion of material frame-indifference leads to our main result that an exact linear constitutive theory for elastic solids is impossible. We then generalize our result to viscoelasticity theory where the stress response is dependent on deformation gradient histories.

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Zusammenfassung Verf. betrachten die Möglichkeit linearer Elastizität als infinitesimale Theorie begründet auf einer echt linearen Reaktionsfunktion die ihre Gültigkeit sogar für endliche Deformationen behält. Genaue Betrachtung des Definitionsbereiches der Spannungsreaktionsfunktion, der Definition von Linearität, und des Objektivitätsbegriffes führen zum Hauptresultat dass eine echt lineare Theorie für elastische Körper unmöglich ist. Das Resultat wird dann auf viskoelastische Theorie verallgemeinert, wobei die Spannungsreaktion von der Vorgeschichte des Deformationsgradienten abhängt.

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Department of Aerospace Engineering and Mechanics

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Department of Mathematics     

A stochastic approach to the internal damage in a structure solid: Part 1: Theoretical model

The material system is considered as heterogenous medium of actual microstructural elements. These elements exhibit random geometric and physical characteristics and are further disturbed by a latitude of randomly oriented, second phase particles. A stochastic model is presented for the occurring damage process due to the nucleation and growth of microvoids under external loading. From a micromechanical point of view, the nucleation of a void at a partile-matrix interface is considered to be associated with the cut-off of the interfacial binding potential. The growth of an elemental void is seen, then, to follow a random walk of the discrete Markov type. The latter is associated with the build-up of strain in front of the tip of the advancing void and the redistribution of local stress. As the void reaches the boundary between neighbouring elements, a dicrete inter-elemental fracture process is examined in relation to the intensities of transformation within the elemental boundary.