Micromechanical computational modeling of expansive porous media

A modified Terzaghi principle is proposed to describe the influence of locally coupled electro-chemo-mechanical processes in highly compacted swelling clays upon the form of the macroscopic modified effective stress principle. The two-scale model is derived using the homogenization procedure to upscale the microscopic behavior of a two-phase system composed of clay particles saturated by a completely dissociated electrolyte aqueous solution. Numerical experiments are performed to illustrate the results in a particular cell geometry. To cite this article: M.A. Murad, C. Moyne, C. R. Mecanique 330 (2002) 865–870.     

Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model

A simple model consisting of the Upper Convected Maxwell constitutive equation and a kinetic equation for destruction and construction of structure, first proposed by Fredrickson in 1970, is used here to reproduce the complex rheological behavior of viscoelastic systems that also exhibit thixotropy and rheopexy under shear flow. The model requires five parameters that have physical significance and that can be estimated from rheological measurements. Several steady and unsteady flow situations were analyzed with the model. The model predicts creep behavior, stress relaxation and the presence of thixotropic loops when the sample is subjected to transient stress cycles. Such behavior has been observed with surfactant-based solutions and dispersions. The role of the characteristic time for structure built up, λ, in the extent and shape of the thixotropic loops is demonstrated.     

Presentation of the creep curves of solid biological materials by a simplified mathematical version of the generalized Kelvin-Voigt model

Published creep equations, derived from the Kelvin-Voigt model with a discrete retardation time spectrum were used to generate data for linear presentation of the creep function. It is shown that 2–3 term creep functions, containing 4–6 constants, could be reduced to an algebraic expression having two constants only. Consequently, entire creep curves of a variety of biological and food materials could be described by a single type of equation in the form of:P (t) = k ₀ + k₁ t + t/(k₂ + k₃t) whereP (t) is the creep parameter (compliance, strain or deformation),k ₀ a constant representing the instantaneous compliance,k ₁ a constant representing the steady-state flow, andk ₂ andk ₃ the characteristic constants of the creep function (t) obtained by regression from the linear relationship oft/ (t) vst.     

Simulation of ratcheting strain to a high number of cycles under biaxial loading

The Ohno–Wang kinematic hardening rule is modified to incorporate the Burlet–Cailletaud radial evanescence term for an improved simulation of the ratcheting behavior. The Delobelle parameter δ^′ is implemented in the modified model to compromise shakedown of the Burlet–Cailletaud hardening rule and over-prediction of the Ohno–Wang model. An evolution equation is proposed for δ^′ to simulate the ratcheting strain over an extended domain of cycles. Ratcheting tests were conducted on S45C steel under four types of nonproportional axial–torsional loading. The new model is found to yield reasonably accurate predictions of ratcheting strain to a much higher number of cycles compared with other studies.     

An appraisal of rheological models as applied to polymer melt flow

Summary An attempt is made at giving an appraisal of some representative rheological models of both differential and integral type, using the standard rheological measurements of six polymer melts. Experimental data obtained were the steady shear viscosity and the first normal stress difference by means of aWeissenberg rheogoniometer over the range of shear rates: 10^–2 ~ 10 sec^–1, and by means of aHan slit/capillary rheometer over the range of shear rates: 10 ~ 10³ sec^–1. Also measured by means of theWeissenberg rheogoniometer were the dynamic viscosity and dynamic elastic modulus over the range of frequencies: 0.3 × 10^–2 ~ 3 × 10² sec^–1. Rheological models chosen for an appraisal are theSpriggs 4-constant model, theMeister model, and theBogue model.It is found that the capability of the three models considered is about the same in their prediction of the rheological behavior of polymer melts in simple shearing flow. It is pointed out however that, due to the ensuing mathematical complexities, the usefulness of these models is limited to the study of flow problems associated with simple flow situations. Therefore, in analysing the complex flow situations often encountered with various polymer processings, the authors suggest use of the empirical models of the power-law type for both the viscosity and normal stress functions.With 11 figures, 4 schemas and 1 table     

Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

Liquid–crystalline nematic polymers revisited

A new model for nematic polymers is proposed, based on the probability ψ(u,u,t) for a macromolecule to be oriented along direction u while embedded in a u environment created by its neighbours. The potential of the internal forces is written Φ(u,u) accordingly. The free energy contains a contribution ν Φ + k_BT ln ψ where the brackets mean an average over the probability distribution, while ν is the (uniform) polymer number density. An equation is derived for the time-evolution of the order parameter S = uu − I/3, together with an expression for the stress tensor. These two results offer a generalization of the Doi Model in so far as they include a distortional energy, analogue to the Frank elastic energy for low molecular mass nematics. Extending the Maier–Saupe variational procedure, we specify the way that the internal potential Φ(u,u) must be written for it to favour non-zero values of the order parameter, while giving a penalty to situations with gradients of the order parameter. The result is quite different from the potential proposed a decade ago by Marrucci and Greco (their Φ depends on u only), while it has a clear connection with the so-called Landau-de Gennes (LdG) tensor models, which are based on a free-energy depending on the order parameter and its gradients.     

Micro/macro-crack growth due to creep–fatigue dependency on time–temperature material behavior

The implicit character of micro-structural degradation is determined by specifying the time history of crack growth caused by creep–fatigue interaction at high temperature. A dual scale micro/macro-equivalent crack growth model is used to illustrate the underlying principle of multiscaling which can be applied equally well to nano/micro. A series of dual scale models can be connected to formulate triple or quadruple scale models. Temperature and time-dependent thermo-mechanical material properties are developed to dictate the design time history of creep–fatigue cracking that can serve as the master curve for health monitoring.In contrast to the conventional procedure of problem/solution approach by specifying the time- and temperature-dependent material properties as a priori, the desired solution is then defined for a class of anticipated loadings. A scheme for matching the loading history with the damage evolution is then obtained. The results depend on the initial crack size and the extent of creep in proportion to fatigue damage. The path dependent nature of damage is demonstrated by showing the range of the pertinent parameters that control the final destruction of the material. A possible scenario of 20 yr of life span for the 38Cr2Mo2VA ultra-high strength steel is used to develop the evolution of the micro-structural degradation. Three micro/macro-parameters μ^*, d^* and σ^* are used to exhibit the time-dependent variation of the material, geometry and load effects. They are necessary to reflect the scale transitory behavior of creep–fatigue damage. Once the algorithm is developed, the material can be tailor made to match the behavior. That is a different life span of the same material would alter the time behavior of μ^*, d^* and σ^* and hence the micro-structural degradation history. The one-to-one correspondence of the material micro-structure degradation history with that of damage by cracking is the essence of path dependency. Numerical results and graphs are obtained to demonstrate how the inherently implicit material micro-structure parameters can be evaluated from the uniaxial bulk material properties at the macroscopic scale.The combined behavior of creep and fatigue can be exhibited by specifying the parameter ξ with reference to the initial defect size a₀. Large ξ (0.90 and 0.85) gives critical crack size a_cr = 11–14 mm (at t < 20 yr) for a₀ about 1.3 mm. For small ξ (0.05 and 0.15), there results critical a_cr = 6–7 mm (at t < 20 yr) for a₀ about 0.7–0.8 mm. The initial crack is estimated to increase its length by an order of magnitude before triggering global to the instability. This also applies ξ ≈ 0.5 where creep interacts severely with fatigue. Fine tuning of a_cr and a₀ can be made to meet the condition oft = 20 yr.Trade off among load, material and geometric parameters are quantified such that the optimum conditions can be determined for the desired life qualified by the initial–final defect sizes. The scenario assumed in this work is indicative of the capability of the methodology. The initial–final defect sizes can be varied by re-designing the time–temperature material specifications. To reiterate, the uniqueness of solution requires the end result to match with the initial conditions for a given problem. This basic requirement has been accomplished by the dual scale micro/macro-crack growth model for creep and fatigue.     

A rheological investigation of solid soap

Summary The rheological behaviour of solid soaps has been investigated over several orders in time and shear by means of creep experiments, measurement of stressstrain curves at constant shear rate and dynamic experiments. The modulus at small deformations is almost constant over a frequency range of 10^–3–10³ Hz; a transition from linear to non-linear behaviour has been observed at a strain of about 10^–3. The results suggest that some kind of network structure is responsible for the rheological behaviour of soap.

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Zusammenfassung Das Theologische Verhalten von Hartseifen wurde für mehrere Größenordnungen von Zeit und Schubgeschwindigkeit untersucht mittels Kriechversuche, Messung der Spannungs-Deformationskurven bei konstanter Schubgeschwindigkeit und dynamischer Experimente. Der Modul bei geringen Deformationen ist fast konstant im Frequenzbereich von 10^–3–10³ Hz; ein Übergang von linear zu nicht-linear wurde bei einer Deformation von etwa 10^–3 beobachtet. Die Ergebnisse lassen auf eine netzartige Struktur schließen, die für das Theologische Verhalten der Seife verantwortlich ist.

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The authors are indebted to Mr.C. J. Nederveen of the Central Laboratory T. N. O., Delft for the dynamic measurements, and thank Mr.D. W. Brinkman of the Unilever Research Laboratory, Vlaardingen for experimental assistance.     

Investigations into cyclic creep of materials (review)

Conclusions In this review, we analyzed the studies known to the author on cyclic creep of structural materials published since 1936. Some of the studies could not be examined but references to these studies and a brief analysis of the results can be found in [3, 54–56, 115]. The classification of these studies and interpretation of the results presented by the author and also the terminology used do not pretend to be unambiguous and universal. Slghtly different approaches to solving this problem were proposed in, for example, [7, 55, 66].It is important to continue investigations in the cyclic creep area in the following directions:generalization of the unidimensional models of cyclic creep for the multiaxial stress states;experimental examination and construction of the theory of cyclic creep for nonstationary loading conditions;examination of the development of cyclic creep in heterogeneous and anisotropic materials;development of the method of calculating the stress-strain state and endurance of structural members in the creep conditions in cyclic loading.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev, Translated from Prikladnaya Mekhanika, Vol. 23, No. 12, pp. 3–19, December, 1987.     

HYDRODYNAMIC DAMPING OF A CYLINDER AT β≈10

This paper describes delicate, but large-scale, experiments aimed at measuring the hydrodynamic damping of a circular cylinder oscillating in still water and transversely in a current. Attention is concentrated on the regime of very small Keulegan–Carpenter numbers, in which the drag coefficient is inversely proportional to the Keulegan–Carpenter number. Measurements in still water at β=650 000 and 1250 000 point to drag coefficients about twice those appropriate to two-dimensional laminar flow, in common with earlier measurements at β≈10⁵. In the presence of a slowly varying transverse current (generated by placing the cylinder at the node of standing waves of long period), the damping increased with the reduced velocity of the ambient flow at a rate that increased with the Reynolds number.     

The creep deformation of vibrating structures

In a recent paper [1] it was shown that the evaluation of certain bounding solutions for a structure subjected to cyclic loading was equivalent to assuming that the cycle time Δt was short compared with a stress redistribution time. Comparisons between values which are likely to occur in creep design situations indicated that Δt may often be assumed to be small and the bounding solution may be expected to closely approximate the actual stress history. In this paper the solution for the limiting case when Δt → 0 is evaluated for a class of constitutive relationships which may be expressed in terms of a finite number of state variables. Strain-hardening viscous, visco-elastic and Bailey-Orowan equations are discussed and particular solutions for which the residual stresses remain constant in time are derived. The solution for a non-linear visco-elastic model indicates that, for the stationary cyclic state, the constitutive equation need only predict the creep strain over a discrete number of cycles and need not predict the strains during a cycle. This observation should considerably simplify creep analysis.The solution of a simple example demonstrates the similarity between the predicting of the various constitutive relationships for isothermal problems. In fact they provide virtually identical solutions when expressed in terms of reference stress histories. The finite element solution of a plate containing a hole and subjected to variable edge loading is also presented for a viscous material. The solutions show behaviour which is similar to that of the two bar structure.     

A class of linear viscoelastic models based on Bessel functions

In this paper we investigate a general class of linear viscoelastic models whose creep and relaxation memory functions are expressed in Laplace domain by suitable ratios of modified Bessel functions of contiguous order. In time domain these functions are shown to be expressed by Dirichlet series (that is infinite Prony series). It follows that the corresponding creep compliance and relaxation modulus turn out to be characterized by infinite discrete spectra of retardation and relaxation time respectively. As a matter of fact, we get a class of viscoelastic models depending on a real parameter \(\nu > -1\). Such models exhibit rheological properties akin to those of a fractional Maxwell model (of order 1/2) for short times and of a standard Maxwell model for long times.     

A new two-point deformation tensor and its relation to the classical kinematical framework and the stress concept

Starting from the issue of what is the correct form for a Legendre transformation of the strain energy in terms of Eulerian and two-point tensor variables we introduce a new two-point deformation tensor, namely H=(F−F^−T)/2, as a possible deformation measure involving points in two distinct configurations. The Lie derivative of H is work conjugate to the first Piola–Kirchhoff stress tensor P. The deformation measure H leads to straightforward manipulations within a two-point setting such as the derivation of the virtual work equation and its linearization required for finite element implementation. The manipulations are analogous to those used for the Lagrangian and Eulerian frameworks. It is also shown that the Legendre transformation in terms of two-point tensors and spatial tensors require Lie derivatives. As an illustrative example we propose a simple Saint Venant–Kirchhoff type of a strain-energy function in terms of H. The constitutive model leads to physically meaningful results also for the large compressive strain domain, which is not the case for the classical Saint Venant–Kirchhoff material.     

Evidence and modelling of physical aging in green wood

Physical aging typical of semi-crystalline polymers was evidenced in green (never dried) wood from three tropical hardwood species. The assumption of uniform aging rate was verified by the construction of master curves from series of tensile creep tests in the fibre direction, performed at increasing time elapsed after a quench following heating above the glassy transition. The rheological response during periods of creep small enough to neglect the progress of aging was described by a model made of a spring in series with a parabolic dash-pot where only the characteristic time depends on the aging time. The model was able to describe results obtained by a previous author on softwood loaded transversally to the fibres. The possible role of a transient adsorption process consecutive to the quench is discussed.

Gelation of a radiation crosslinked model polyethylene

A radiation crosslinked model linear low-density polyethylene (LLDPE) exhibits power-law relaxation,G(t) =St ^–n at its gel point (GP). The relaxation exponent has a value of about 0.46. The relaxation behavior is dominated by power laws, not only directly at GP, but in a very broad vicinity of GP and in a frequency window, which narrows with distance from the gel point. The power law exponent decreases with increasing radiation dose (increasing extent of crosslinking). Independent measurements of the gel fraction and the molecular-weight distribution of the radiated samples' soluble fraction support the rheological observations.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=F^eF^p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F^e contributes to a standard elastic free-energy ψ^(e), while , the energetic part of F^p, contributes to a defect energy ψ^(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F^e=FF^p-1 and , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F^p and – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.     

A torsional creep apparatus

Summary This paper describes a torsional creep apparatus for the determination of the creep compliance in shear as a function of time. The instrument is suitable for the measurement of compliances below 5·10^–8 m²/N, in the time range between 1 and 10⁵ seconds. Within five minutes and with an accuracy of 1 C, the temperature of the specimen can be adjusted to any value between –175 and –200 C. From the torsional creep measurements can be calculated the dynamic shear modulus and the corresponding damping. The lowest damping, determined in this way, is several times 10^–3.

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Zusammenfassung Ein Apparat zur Messung des Torsions-Kriechverhaltens wird beschrieben. Das Instrument ist zur Bestimmung von Nachgiebigkeiten kleiner als 5·10^–5 m²/N im Zeitbereich zwischen 1 und 10⁵ Sekunden geeignet. Die Temperatur der Probe läßt sich innerhalb 5 Minuten und mit einer Genauigkeit von 1 C auf jeden Wert zwischen –175 und –200 C einstellen. Aus der gemessenen Kriechfunktion können der dynamische Schubmodul und die dynamische Dämpfung berechnet werden. Eine untere Grenze für die Dämpfung, die man auf diese Weise noch bestimmen kann, ist einige Male 10^–3.

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The work was partly sponsored by the Office of Naval Research, Washington, D. C., under contract number N 62558-3884.     

Stress relaxation,creep, failure and hysteresis in a linear elastic material with voids

I study several specific boundary-value problems in a theory recently proposed to model linear elastic materials with voids. I show that, in addition to the known fact that the model exhibits stress relaxation, it also exhibits creep, hysteresis, and a phenomenon which can be interpreted as failure. In order to maintain plausible physical behavior, I suggest an a priori inequality not contained in the original theory.