Multifield computational fluid dynamics model of particulate flow in curved circular tubes

Limitations of mass transfer resulting from non-optimized fluid mechanics can severely affect the performance of synthetic membrane filtration systems. To improve membrane efficiency, modern applications of this technology have extensively used curved membrane ducts that take advantage of Dean vortices (i.e., curvature-induced secondary flows) to minimize membrane fouling. This paper is concerned with a complete three-dimensional analysis of single-phase and two-phase particle/liquid flows around a curved membrane tube. The proposed multidimensional model was implemented in an advanced (next-generation) multiphase computational fluid dynamics (CFD) solver, NPHASE. The results of simulations have been validated against experimental data and compared against other findings available in the literature. The consistency and accuracy of the present approach have been demonstrated. The novel aspects of this work include: the demonstration that azimuthal vortices may bifurcate at Dean numbers lower than previously anticipated, the use of vorticity magnitude as a measure of vortex strength, and the explanation of the role that Dean vortices play to mitigate the effect of gravity on particle settling. The overall results have direct relevance to synthetic membrane fouling during filtration of particle suspensions.     

Whither compressional rheology?

The behaviour of compressible sediments and filter cakes is of significant fundamental interest and widespread practical importance; for example, in the area of solid–liquid separation engineering. In this paper, we review briefly recent progress made in the characterisation and modelling of the behaviour of compressible particulate gels using an approach which we shall for convenience call “compressional rheology”, following de Kretzer et al. [R.G. de Kretzer, D.V. Boger, P.J. Scales, Compressive Rheology: An Overview, Rheol. Rev., 1 2003, 125–166 (British Society of Rheology)] and others. The bulk of the paper will be concerned with one-dimensional consolidation under hydrostatic loading in one guise or another, modelled using extensions of an approach suggested by Philip and Smiles [J.R. Philip, D.E. Smiles, Macroscopic analysis of the behavior of colloidal suspensions, Adv. Colloid Interface Sci. 17 (1) (1982) 83–103] and Buscall and White [R. Buscall, L.R. White, The consolidation of concentrated suspensions. Part 1. The theory of sedimentation, J. Chem. Soc. Faraday Trans. I 83 (3) (1987) 873–891]. One-dimensional compressional rheology has been used to model processes such as filtration and sedimentation, and the field is being extended to centrifugation, drying and vacuum filtration currently. It is however the case that by no means all experiments and applications can be treated as one-dimensional, even approximately, and so towards the end the paper we anticipate and discuss the challenge of generalising to arbitrary loadings and higher-dimensional flows.     

Modelling large deformation behaviour under loading–unloading of semicrystalline polymers: Application to a high density polyethylene

In this work, the large deformation behaviour under monotonic loading and unloading of a high density polyethylene (HDPE) is studied. To analyze the nonlinear time-dependent response of the material, mechanical tests were conducted at room temperature under constant true strain rates and stress relaxation conditions. A physically-based inelastic model written under finite strain formulation is proposed to describe the mechanical behaviour of HDPE. In the model, the inelastic mechanisms involve two parallel elements: a visco-hyperelastic network resistance acting in parallel with a viscoelastic–viscoplastic intermolecular resistance where the amorphous and crystalline phases are explicitly taken into consideration. The semicrystalline polymer is considered as a two-phase composite. The influence of the crystallinity on the loading and unloading behaviour is investigated. Numerical results are compared to experimental data. It is shown that the model is able to accurately reproduce the experimental observations corresponding to monotonic loading, unloading and stress relaxation behaviours at different strain levels. Finally, the model capabilities to capture cyclic loading–unloading behaviour up to large strains are discussed. To demonstrate the improved modelling capabilities, simulations are also performed using the original model of Boyce et al. [Boyce, M.C., Socrate, S., Llana, P.G., 2000. Constitutive model for the finite deformation stress–strain behavior of poly(ethylene terephthalate) above the glass transition. Polymer 41, 2183–2201] modified by Ahzi et al. [Ahzi, S., Makradi, A., Gregory, R.V., Edie, D.D., 2003. Modeling of deformation behavior and strain-induced crystallization in poly(ethylene terephthalate) above the glass transition temperature. Mechanics of Materials 35, 1139–1148].     

Dynamic stability analysis and DQM for beams with variable cross-section

The present paper deals with the dynamic behaviour of a clamped beam subjected to a sub-tangential follower force at the free end. The aim of this work is to obtain the frequency–axial load relationship for a beam with a variable circular cross-section. In this way, one can identify both divergence critical loads – where the frequency goes to zero – and the flutter critical load – in correspondence with two frequencies coalescence. The numerical approach adopted for solving the partial differential equation of motion is the differential quadrature method (henceforth DQM). This method was proposed by Bellmann and Casti [Bellmann, R.E., Casti, J., 1971. Differential quadrature and long-term integration. J. Math. Anal. 34, 235–238] and has been employed recently in the solution of solid mechanics problems by Bert and Malik [Bert, C.W., Malik, M., 1996. Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev., ASME, 49 (1), 1–28] and Chen et al. [Chen, W., Stritz, A.G., Bert, C.W., 1997. A new approach to the differential quadrature method for fourth-order equations. Int. J. Numer. Method Eng. 40, 1941–1956]. More precisely, a modified version of this method has been used, as proposed by De Rosa and Franciosi [De Rosa, M.A., Franciosi, C., 1998a. On natural boundary conditions and DQM. Mech. Res. Commun. 25 (3), 279–286; De Rosa, M.A., Franciosi, C., 1998b. Non classical boundary conditions and DQM. J. Sound Vibrat. 212(4), 743–748] to satisfy all the boundary conditions.Some frequencies–axial loads relationships are reported in order to show the influence of tapering on the critical loads.     

Heuristic search for a predictive strain-energy function in nonlinear elasticity

In this work, a new, quasi-structural model – bootstrapped eight-chain model – is proposed as a modification to the strain energy of eight-chain model [Arruda, E.M., Boyce, M.C., 1993. A three-dimensional constitutive model for the large stretch behaviour of rubber elastic materials. J. Mech. Phys. Solids 41, 389—412] that invokes the Langevin chain statistics. This development has been led to by our heuristic search into how the strain energy of eight-chain model may be adapted in order to account better for the mechanical behaviour of elastomeric materials in both linear and nonlinear elastic regimes [Treloar, L.R.G., 1944. Stress–strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc. 40, 59–70]. The eight-chain model appears to produce very similar results in predicting biaxial stress to those of a first stretch-invariant model that gives a good fit in uniaxial extension and, thus, it is shown that the former can not be significantly enhanced within the limitation of the latter. Evaluation of predictive capability for an additive invariant-separated form of strain energy shows that an explicit inclusion of a second stretch-invariant function would not work and that any thus added term ought to be dependent on both the first and second stretch-invariants of deformation tensor, and hints that an improvement is possibly needed at low strain. The composite and filament models [Miroshnychenko, D., Green, W.A., Turner, D.M., 2005. Composite and filament models for the mechanical behaviour of elastomeric materials. J. Mech. Phys. Solids 53 (4), 748–770] have their strain-energy functions in that suggested form and cope very well with predicting the experimental data of Treloar (1944). We use the form of strain energy for the filament model, that proved to be successful, to bootstrap the strain energy of eight-chain model in order to improve the performance of the latter at low strain. Thus, we derive a new model – bootstrapped eight-chain model – that requires only two material parameters – a rubber modulus and a limiting chain extensibility. The proposed model is quasi-structural due to bootstrapping and it retains the best traits and corrects the faults of the eight-chain model, conforming more closely to the classical experimental data of Treloar (1944).     

Non-Newtonian flow at lowest order,the role of the Reiner–Rivlin stress

In 1963 Giesekus [H. Giesekus, Die simultane Translations- und Rotationsbewegung einer Kugel in einer elastikoviskosen Flüssigkeit, Rheol. Acta 3 (1962) 59–71] showed that a Stokes velocity field also satisfies the equilibrium equation for the flow of a restricted form of the second order fluid. The same result was found by Tanner [R.I. Tanner, Plane creeping flows of incompressible second order fluids, Phys. Fluids 9 (1966) 1246–1247] in 1966 in the context of plane flow for which the restrictions on the second order fluid are not relevant. Tanner [R.I. Tanner, Some extended Giesekus-type theorems for non-Newtonian fluids, Rheol. Acta 28 (1989) 449–452] later showed that the velocity field for the inertialess, plane flow of the generalized Newtonian fluid is also the velocity field for the flow of a special form of the Criminale–Ericksen–Filbey (CEF) stress system [W.O. Criminale Jr., J.L. Ericksen, G.L. Filbey Jr., Steady flow of non-Newtonian fluids, Arch. Rat. Mech. 1 (1958) 410–417]. In this paper it will be shown that the results of Giesekus and Tanner are special cases of a more general theorem in which the velocity field, in any dimension, of the equilibrium Reiner–Rivlin problem also satisfies the corresponding problem for the materially steady stress system (a generalization of the CEF system) provided the coefficients of the Reiner–Rivlin stress [M. Reiner, A mathematical theory of dilatancy, Am. J. Math. 67 (1945) 350–362; R.S. Rivlin, The hydrodynamics of non-Newtonian fluids, Proc. R. Soc. Lond. 193 (1948) 260–281] are derivable from a strain-rate potential. As with the Giesekus–Tanner theorems the new theorem holds generally for velocity boundary conditions, but in some cases, such as the free jet, stress boundary conditions can be imposed.     

High temperature strength and failure of the Ni-base superalloy PM 3030

The strength, fatigue life and fracture behavior of the oxide dispersion strengthened (ODS) nickel-base superalloy PM 3030 are investigated. The high Al content in PM 3030 leads to the formation of coherent γ′ particles and, thus, to additional precipitation strengthening. A coarse and elongated grain structure (R34) and two isotropic batches with mean grain sizes of 1 μm (R90) and 17 μm (R901315) are considered. Compressive constant strain rate tests and high cycle fatigue (HCF) tests are performed. Optical, scanning and transmission electron microscopy (OM, SEM and TEM) are carried out. The properties are compared with those of the solely oxide dispersion strengthened Ni-base alloy PM 1000 [Estrin, Y., Heilmaier, M., Drew, G., 1999. Creep properties of an oxide dispersion strengthened nickel-base alloy: the effect of grain orientation and grain aspect ratio. Mater. Sci. Eng. A 272(1), 163–173]. It is found that additional γ′ hardening provides an increase in quasi-static strength by about a factor 2 and in HCF life by about a factor 10²–10³ at temperatures up to 850 °C. When fatigue life is compared at a fixed ratio of stress amplitude-to-yield or ultimate compressive strength, R34 shows a fatigue life similar to that of PM 1000 at lower temperature (e.g. 600 °C) indicating that the quasi-static strength advantage is proportionally translated into improved fatigue performance; for higher temperatures (850 °C) however, R34 shows a shorter fatigue life as compared to PM 1000. Grain size reduction, as exemplified with the fine grain R90 batch, also provides an increase in strength up to the equicohesion temperature (T_E) [Dieter, G.E., 1986. Mechanical Metallurgy. SI Metric ed. McGraw-Hill Book Company, London]. Above T_E, faster diffusion and grain boundary sliding [Raj, R., Ashby, M.F., 1971. On the grain boundary sliding and diffusional creep. Metall. Trans. 2, 1113–1127; Spingarn, J.R., Nix, W.D., 1978. Diffusional creep and diffusionally accommodated grain rearrangement. Acta Metall. 26, 1389–1398] lead to a drastic drop in strength for the R90 material. In contrast, the batch with intermediate grain size (R901315) shows strength comparable to that of R34 up to 850 °C. Furthermore, R901315 shows improved crack tolerance compared to its coarse grain counterpart R34. Due to premature crack initiating coarse oxide particles however, R901315 does not show any improvement in elongation to failure during tensile tests. Eliminating those coarse particles is expected to improve the ductility and toughness of this isotropic batch.     

Turbulent pipe flow downstream a 90° pipe bend with and without superimposed swirl

In the present work, the turbulent flow downstream a 90° pipe bend is investigated by means of stereoscopic particle image velocimetry. In particular, the three dimensional flow field at the exit of the curved pipe is documented for non-swirling and swirling flow conditions, with the latter being generated through a unique axially rotating pipe flow facility. The non-swirling flow was examined through snapshot proper orthogonal decomposition (POD) with the aim to reveal the unsteady behaviour of the Dean vortices under turbulent flow conditions, the so-called “swirl-switching” phenomenon. In respect to the swirling turbulent pipe flow, covering a wide range of swirl strengths, POD has been employed to study the effect of varying strength of swirl on the Dean vortices as well as the interplay of swirling motion and Dean cells. Furthermore, the visualised large scale structures in turbulent swirling flows through the bend are found to incline and tear up with increasing swirl intensity. The present time-resolved, three component, experimental velocity field data will provide a unique and useful database for future studies; in particular for the CFD community.     

Theory of early onset of turbulence in a dispersion

The effects of phase interaction are considered for the hydrodynamic instability of a flowing dispersion, and a model is proposed for premature onset of turbulence.A disperse phase can stabilize the laminar flow of a suspension or aerosol by impeding the growth of neutral perturbations and the onset of turbulent conditions. This stabilizing action is seen also in the shift of the curves for neutral stability towards higher Reynolds number R [1,2] and in the occurrence of additional dissipation of pulsation energy as a resutt of phase interaction [3,4].These conclusions from the theory are closely confirmed by tests on the motion of finely divided or neatly equal-density systems, where turbulence starts at R. higher than those for the homogeneous dispersion medium, However, systems with coarser particles and substantial density differences show the converse effect (lower R,). This has been observed [5] for a liquid containing sand in vertical pipes, where R. was only t60-200 instead of 2300. Similar results have been obtained with oil in sands [6]. Loss of laminar flow at low R. is characteristic also of emulsions having disperse material less dense than the dispersion medium, e.g., water-oil emulsions of very low concentration [7]. The quantity R. is not an invariant characteristic of the type of motion but is dependent on the physical parameters of the phases and the concentration.This indicates that a different mechanism accompanies the usual mechanism for toss of hydrodynamic stability (from inertial effects, undamped shear waves, and secondary flows); energy is transferred to pulsations and causes premature onset of turbulence. This energy transfer must be due to interaction between phases, which produces differences between the local speeds of liquid and particles; it does not require shear in order to arise. A preliminary analysis of early turbulence thus can be based on the stability of a one-dimensional homogeneous two-phase flow.     

A criterion for detection of the onset of Dean instability in Newtonian fluids

Dean instability for Newtonian fluids in laminar secondary flow in 180° curved channels was studied experimentally and numerically. The numerical study used Fluent CFD code to solve the Navier–Stokes equations, focusing on flow development conditions and the parameters influencing Dean instability. An accurate criterion based on the radial gradient of the axial velocity was defined that allows detection of the instability threshold, and this criterion is used to optimize the grid geometry. The effects on Dean instability of the curvature ratio (from 5.5 to 20) and aspect ratio (from 0.5 to 12) are studied. In particular, we show that the critical value of the Dean number decreases with the increasing duct curvature ratio. The variation of the critical Dean number with duct aspect ratio is less regular.In the experimental study, flows were visualized in several tangential positions of a 180° curved channel with aspect ratio 8 and curvature ratio 10. The flow is hydrodynamically developed at the entrance to the curved channel. The critical Dean number is detected and the development of secondary flow vortices by additional counter-rotating vortex pairs is observed. A diagram of different critical Dean numbers is established.     

Direct numerical simulation of particle interaction with ejections in turbulent channel flows

Direct numerical simulations (DNS) of incompressible turbulent channel flows coupled with Lagrangian particle tracking are performed to study the characteristics of ejections that surround solid particles. The behavior of particles in dilute turbulent channel flows, without particle collisions and without feedback of particles on the carrier fluid, is studied using high Reynolds number DNS (Re = 12,500). The results show that particles moving away from the wall are surrounded by ejections, confirming previous studies on this issue. A threshold value separating ejections with only upward moving particles is established. When normalized by the square root of the Stokes number and the square of the friction velocity, the threshold profiles follow the same qualitative trends, for all the parameters tested in this study, in the range of the experiments. When compared to suspension thresholds proposed by other studies in the Shields diagram, our simulations predict a much larger value because of the measure used to characterize the fluid and the criterion chosen to decide whether particles are influenced by the surrounding fluid. However, for intermediate particle Reynolds numbers, the threshold proposed here is in fair agreement with the theoretical criterion proposed by Bagnold (1966) [Bagnold, R., 1966. Geological Survey Professional Paper, vol. 422-1]. Nevertheless, further studies will be conducted to understand the normalization of the threshold.     

An analytical wall-function for recirculating and impinging turbulent heat transfer

The performance of the analytical wall-function (AWF) of Craft et al. [Craft, T.J., Gerasimov, A.V., Iacovides, H., Launder, B.E., 2002, Progress in the generalisation of wall-function treatments. Int. J. Heat Fluid Flow 23, 148–160.] is improved for predicting turbulent heat transfer in recirculating and impinging flows. Since constant parameters of the eddy viscosity formula were used to derive the AWF, the prediction accuracy of the original AWF tends to deteriorate in complex flows where those parameters need changing according to the local turbulence. To overcome such shortcomings, the present study introduces a functional behaviour on the strain parameter into the coefficient of the eddy viscosity of the AWF. The presently modified version of the AWF is validated in turbulent heat transfer of pipe flows, channel flows, back-step flows, pipe flows with abrupt expansion and plane impinging slot jets. The results confirm that the present modification successfully improves the performance of the original AWF for all the flows and heat transfer tested.     

Experimentally determined properties of softening functions in pseudo-elastic models of the Mullins effect

This paper aims to identify the true source of limitations of pseudo-elastic models for describing the stress-softening phenomenon in elastomers which were recently proposed in the literature [Ogden, R.W., Roxburgh, D.G., 1999. A pseudo-elastic model for the Mullins effect in filled rubber. Proceedings of the Royal Society of London A 455 (1988), 2861–2877; Elías-Zúñiga, A., Beatty, M.F., 2002. A new phenomenological model for stress-softening in elastomers. Zeitschrift für angewandte Mathematik und Physik (ZAMP) 53 (5), 794–814]. These models as well as their modified versions [Mars, W.V., 2004. Evaluation of pseudo-elastic model for the Mullins effect. Tire Science and Technology, TSTCA 32 (3), 120–145; Elías-Zúñiga, A., 2005. A phenomenological energy-based model to characterize stress-softening effect in elastomers. Polymer 46 (10), 3496–3506] fail to give fully satisfactory coincidence of experimental data and theoretical predictions. In this paper a suitable analysis of experimental data published in the open literature is presented. This analysis shows several interesting features regarding the nature of the stress-softening phenomenon (widely known as the Mullins effect). In particular, it is shown that the magnitude of stress softening varies with strain in a non-monotonous manner and this non-monotonous character of the stress-softening phenomenon strongly depends on magnitude of the pre-strain. This experimental fact is in contradiction with the basic assumption used in pseudo-elastic models that the stress softening is described by a monotonously increasing function of strain. The common theoretical basis of pseudo-elastic models of stress softening and the source of this conflict are clarified.     

Constitutive modeling of shape memory polymer based self-healing syntactic foam

In a previous study, it was found that the shape memory functionality of a shape memory polymer based syntactic foam can be utilized to self-seal impact damage repeatedly, efficiently, and almost autonomously [Li G., John M., 2008. A self-healing smart syntactic foam under multiple impacts. Comp. Sci. Technol. 68(15–16), 3337–3343]. The purpose of this study is to develop a thermodynamics based constitutive model to predict the thermomechanical behavior of the smart foam. First, based on DMA tests and FTIR tests, the foam is perceived as a three-phase composite with interfacial transition zone (interphase) coated microballoons dispersed in the shape memory polymer (SMP) matrix; for simplicity, it is assumed to be an equivalent two-phase composite by dispersing elastic microballoons into an equivalent SMP matrix. Second, the equivalent SMP matrix is phenomenologically assumed to consist of an active (rubbery) phase and a frozen (glassy) phase following Liu et al. [Liu, Y., Gall, K., Dunn, M.L., Greenberg, A.R., Diani J., 2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling. Int. J. Plasticity 22, 279–313]. The phase transition between these two phases is through the change of the volume fraction of each phase and it captures the thermomechanical behavior of the foam. The time rate effect is also considered by using rheological models. With some parameters determined by additional experimental testing, the prediction by this model is in good agreement with the 1D test result found in the literature. Parametric studies are also conducted using the constitutive model, which provide guidance for future design of this novel self-healing syntactic foam and a class of light-weight composite sandwich structures.     

Study of spatial mixed flows in nozzles with asymmetrical inlets

The results of some investigations into the behavior of the spatial mixed flows of a nonviscous and nonheat-conducting gas in nozzles deviating from axisymmetrical shape in the subsonic region are presented. The investigation is based on the numerical integration of the transient gasdynamic equations using the Godunov scheme [1, 2] generalized to the spatial case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–169, March–April, 1975.The author wishes to thank M. Ya. Ivanov, A. N., Kraiko, and G. G. Chernyi for help and discussion of the results.     

Dynamics of viscoelastic liquid filaments: Low capillary number flows

Many applications of viscoelastic free surface flows requiring formation of drops from small nozzles, e.g., ink-jet printing, micro-arraying, and atomization, involve predominantly extensional deformations of liquid filaments. The capillary number, which represents the ratio of viscous to surface tension forces, is small in such processes when drops of water-like liquids are formed. The dynamics of extensional deformations of viscoelastic liquids that are weakly strain hardening, i.e., liquids for which the growth in the extensional viscosity is small and bounded, are here modeled by the Giesekus, FENE-P, and FENE-CR constitutive relations and studied at low capillary numbers using full 2D numerical computations. A new computational algorithm using the general conformation tensor based constitutive equation [M. Pasquali, L.E. Scriven, Theoretical modeling of microstructured liquids: a simple thermodynamic approach, J. Non-Newtonian Fluid Mech. 120 (2004) 101–135] to compute the time dependent viscoelastic free surface flows is presented. DEVSS-TG/SUPG mixed finite element method [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409] is used for the spatial discretization and a fully implicit second-order predictor–corrector scheme is used for the time integration. Inertia, capillarity, and viscoelasticity are incorporated in the computations and the free surface shapes are computed along with all the other field variables in a fully coupled way. Among the three models, Giesekus filaments show the most drastic thinning in the low capillary number regime. The dependence of the transient Trouton ratio on the capillary number in the Giesekus model is demonstrated. The elastic unloading near the end plates is investigated using both kinematic [M. Yao, G.H. McKinley, B. Debbaut, Extensional deformation, stress relaxation and necking failure of viscoelastic filaments, J. Non-Newtonian Fluid Mech. 79 (1998) 469–501] and energy analyses. The magnitude of elastic unloading, which increases with growing elasticity, is shown to be the largest for Giesekus filaments, thereby suggesting that necking and elastic unloading are related.     

An interpretation of the yield behaviour of polymers in terms of correlated motion

An interpretation of yield behaviour in polymers is given in terms of the stress-activated flow of structural units over an energy barrier together with an additional correlated component to the motion. This correlated contribution takes account, in the simplest way possible, of the state of other conformations, and it is shown to lead to an effective activation-energy barrier which depends in part on the strain present at any time in the material. In this manner, the present work relates to previous work by M.G. Brereton, S.G. Croll, R.A. Duckett and I.M. Ward (1974), who, on purely phenomenological grounds, proposed a relation between stress and strain which had the form of a feedback equation. Specifically, the strain resulting from an applied stress was assumed to modify the material in a way which reduced its resistance to stress. The basic equation obtained here is non-linear and shows a yield-like behaviour resulting from a dynamical (as opposed to a geometrical) instability. Furthermore, it indicates a consistent relation between yield in creep tests and in constant strain-rate tests.     

Similarity analysis and exact solutions for a general discrete two-velocity model of Boltzmann equation

Summary This paper concerns with the similarity analysis for a general discrete two-velocity model of the Boltzmann equation introduced by Illner [12]. We find the general groups of invariance and we get some exact solutions, recovering general results which contain either solutions extensively described in the literature or undiscovered ones.

---

Sommario In questa nota si applica l'analisi dei gruppi infinitesimi di trasformazione ad un modello generale discreto a due velocità dell'equazione di Boltzmann introdotto da Illner [12]. Si trovano i più generali gruppi di invarianza e si ottengono alcune soluzioni esatte, ritrovando risultati generali che contengono sia soluzioni ampiamente descritte in letteratura che nuove soluzioni.

---

---

Work supported by the C.N.R. through the G.N.F.M.     

A symmetric over-nonlocal microplane model M4 for fracture in concrete

This paper analyzes the effectiveness of a nonlocal integral-type formulation of a constitutive law such as microplane model M4 in which the yield limits soften as a function of the total strain for prediction of fracture propagation. For a correct regularization of the mathematical problems caused by the softening behavior, an “over-nonlocal” generalization of the type proposed by Vermeer and Brinkgreve [Vermeer, P.A., Brinkgreve, R.B.J., 1994. A new effective non-local strain measure for softening plasticity. In: Chambon, R., Desrues, J., Vardoulakis, I. (Eds.), Localization and Bifurcation Theory for Soil and Rocks, Balkema, Rotterdam, pp. 89–100.] is adopted. Moreover, the symmetric weight function, proposed by Borino et al. [Borino, G. Failla, B., Parrinello, F., 2003. A symmetric nonlocal damage theory. International Journal of Solids and Structure 40, 3621–3645.] for damage mechanics, is introduced for the calculation of the nonlocal averaging of the total strain upon which the yield limits depend. The capability of the proposed model for reproducing the stress and strain fields in the vicinity of a notch is also investigated. Finally, the symmetric over-nonlocal generalization of microplane model M4 has been applied for the simulation of a mixed-mode fracture test such as the four-point-shear test and the test of axial tension at constant shear force [Nooru-Mohamend, M.B., 1992. Mixed-mode fracture of concrete: an experimental approach. Doctoral Thesis Delft University of Thechnology, Delft, The Netherlands.]