Mechanics of deformation-triggered pattern transformations and superelastic behavior in periodic elastomeric structures

Recently, novel and uniform deformation-induced pattern transformations have been found in periodic elastomeric cellular solids upon reaching a critical value of applied load [Mullin, T., Deschanel, S., Bertoldi, K., Boyce, M.C., 2007. Pattern transformation triggered by deformation. Phys. Rev. Lett. 99, 084301; Boyce, M.C., Prange, S.M., Bertoldi, K., Deschanel, S., Mullin, T., 2008. Mechanics of periodic elastomeric structures. In: Boukamel, Laiarinandrasana, Meo, Verron (Eds.), Constitutive Models for Rubber, vol. V. Taylor & Francis Group, London, pp. 3–7]. Here, the mechanics of the deformation behavior of several periodically patterned two-dimensional elastomeric sheets are investigated experimentally and through numerical simulation. Square and oblique lattices of circular voids and rectangular lattices of elliptical voids are studied. The numerical results clearly show the mechanism of the pattern switch for each microstructure to be a form of local elastic instability, giving reversible and repeatable transformation events as confirmed by experiments. Post-deformation transformation is observed to accentuate the new pattern and is found to be elastic and to occur at nearly constant stress, resulting in a superelastic behavior. The deformation-induced transformations have been physically realized on structures constructed at the millimeter length scale. This behavior should also persist at the micro and nano length scales, providing opportunities for transformative photonic and phononic crystals which can switch in a controlled manner and also exploiting the phenomenon to imprint complex patterns.     

Peculiar features in the rheological behavior of electrorheological suspensions

Studies have been made of concentrated (up to 60%) diatomite suspensions in transformer oil, the structure and theological properties of which depend on an applied electric field. Studies have been conducted of steady-state and transient regimes of straining involving continuous and periodic shear. The structure in such suspensions is formed in the presence of an electric field of 10^–3 –10² duration. The suspensions under continuous stationary strain behave as non-Newtonian fluids with a yield stress dependent on electric intensity. Under periodic deformation conditions the test suspensions exhibit elasticity which abruptly diminishes with increasing deformation amplitude.     

Localized viscoelastoplastic strain in mesovolume of heterogeneous medium under different loading types

The localized viscoelastoplastic strain in the mesovolume of heterogeneous media under quasi-static and dynamic loading is investigated. The generalized Bingham–Shwedov model is used; it consists of a combination of Dragon–Mroz's for elastoplasticity and Maxwell's model of viscoelasticity. Any variational finite-difference scheme for solving the quasi-static problem of elastoplastic yielding of a heterogeneous solid can be taken into account. A modified Lagrange's variance equation for analyzing the stress–strain state can be described by the non-symmetric stress tensor. Approximation of spatial derivatives is made by using the twofold partition of spatial domain in tetrahedronal or three-angular (in two-dimensional space) unit cell of mesh-work. Finite difference for deformation is made use of in two or three space dimensions and time. Results for heterogeneous medium with complex form and large number of interior surfaces are obtained for quasi-static and dynamics problems.     

The Extended Malkus–Robbins Dynamo as a Perturbed Lorenz System

Recent investigations of some self-exciting Faraday-disk homopolar dynamos [Hide, R. and Moroz, I. M., Physica D 134, 1999, 387–301; Moroz, I. M. and Hide, R., International Journal of Bifurcation and Chaos 2000, 2701–2716; Moroz, I. M., International Journal of Bifurcation and Chaos 13, 2003, 147–161; Moroz, I. M., International Journal of Bifurcation and Chaos, to appear] have yielded the classic Lorenz equations as a special limit when one of the principal bifurcation parameters is zero. In this paper we focus upon one of those models [Moroz, I. M., International Journal of Bifurcation and Chaos 13, 2003, 147–161] and illustrate what happens to some of the lowest order unstable periodic orbits as this parameter is increased from zero.     

On the mechanism of cyclic flooding for oil extraction

A two-dimensional model is considered for the displacement of oil by water from a stratum that is nonuniform over its thickness when a periodic elastic flow regime is employed.Translated from Izvestiya Akademii Nauk SSSR, Meknanika Zhidkosti i Gaza, No. 3, pp. 58–66, May–June, 1980.     

Mechanics of granular materials: The discrete and the continuum descriptions juxtaposed

In recent years a discussion could be followed where the pros and cons of the applicability of the Cosserat continuum model to granular materials were debated [Bardet, J.P., Vardoulakis, I., 2001. The asymmetry of stress in granular media. Int. J. Solids Struct. 38, 353–367; Kruyt, N.P., 2003. Static and kinematics of discrete Cosserat-type granular materials. Int. J. Solids Struct. 40, 511–534; Bagi, K., 2003. Discussion on “The asymmetry of stress in granular media”. Int. J. Solids Struct. 40, 1329–1331; Bardet, J.P., Vardoulakis, I. 2003a. Reply to discussion by Dr. Katalin Bagi. Int. J. Solids Struct. 40, 1035; Kuhn, M., 2003. Discussion on “The asymmetry of stress in granular media”. Int. J. Solids Struct. 40, 1805–1807; Bardet, J.P., Vardoulakis, I., 2003b. Reply to Dr. Kuhn’s discussion. Int. J. Solids Struct. 40, 1809; Ehlers, W., Ramm, E., Diebels, S., D’Addetta, G.A., 2003. From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses. Int. J. Solids Struct. 40, 6681–6702; Chang, C.S., Kuhn, M.R., 2005. On virtual work and stress in granular media. Int. J. Solids Struct. 42, 3773–3793]. The authors follow closely this debate and try, with this paper, to provide a platform where the various viewpoints could find their position. We consider an ensemble of rigid, arbitrarily shaped grains as a set with structure. We establish a basic mathematical framework which allows to express the balance laws and the action–reaction laws for the discrete system in a “global” form, through the concepts of “part”, “granular surface”, “separately additive function” and “flux”. The independent variable in the balance laws is then the arbitrary part of the assembly rather than the single grain. A parallel framework is constructed for Cosserat continua, by applying the axiomatics established by [Noll, W., 1959. The foundation of classical mechanics in the light of recent advances in continuum mechanics. In: The axiomatic method, with special reference to Geometry and Physics, North-Holland Publishing Co., Amsterdam pp. 266–281, Gurtin, M.E., Williams, W.O., 1967. An axiomatic foundation of continuum thermodynamics. Arch. Rat. Mech. Anal. 26, 83–117, Gurtin, M.E., Martins, L.C., 1976. Cauchy’s theorem in classical physics. Arch. Rat. Mech. Anal. 60, 305–324]. The comparison between the two realisations suggests the microscopic interpretation for some features of Cosserat Mechanics, among which the asymmetry of the Cauchy-stress tensor and the couple-stress.     

Antiplane elastic wave propagation in pre-stressed periodic structures; tuning,band gap switching and invariance

The effect of nonlinear elastic pre-stress on antiplane elastic wave propagation in a two-dimensional periodic structure is investigated. The medium consists of cylindrical annuli embedded on a periodic square lattice in a uniform host material. An identical inhomogeneous deformation is imposed in each annulus and the theory of small-on-large is used to find the incremental wave equation governing subsequent small-amplitude antiplane waves. The plane-wave-expansion method is employed in order to determine the permissable eigenfrequencies. It is found that pre-stress significantly affects the band gap structure for Mooney–Rivlin and Fung type materials, allowing stop bands to be switched on and off. However, it is also shown that for a specific class of materials, their phononic properties remain invariant under nonlinear deformation, permitting some rather interesting behaviour and leading to the possibility of phononic cloaks.     

Near critical phenomena in laminar boundary layers

Asymptotic analysis of boundary layer separation in the limit of large Reynolds number Re→∞ has shown that in a number of cases which are of importance from a practical point of view solutions of the resulting interaction equations describing two-dimensional (2-D) steady flows exist up to a limiting value Γ_c of the relevant controlling parameter Γ only while two branches of solutions exist in a regime Γ<Γ_c. The present study aims at a better understanding of near critical flows |Γ-Γ_c|→0 and in particular the changes of the flow behaviour associated with the passage of Γ through Γ_c.     

Shakedown in elastic contact problems with Coulomb friction

Elastic systems with frictional interfaces subjected to periodic loading are sometimes predicted to ‘shake down’ in the sense that frictional slip ceases after the first few loading cycles. The similarities in behaviour between such systems and monolithic bodies with elastic–plastic constitutive behaviour have prompted various authors to speculate that Melan’s theorem might apply to them – i.e., that the existence of a state of residual stress sufficient to prevent further slip is a sufficient condition for the system to shake down.In this paper, we prove this result for ‘complete’ contact problems in the discrete formulation (i) for systems with no coupling between relative tangential displacements at the interface and the corresponding normal contact tractions and (ii) for certain two-dimensional problems in which the friction coefficient at each node is less than a certain critical value. We also present counter-examples for all systems that do not fall into these categories, thus giving a definitive statement of the conditions under which Melan’s theorem can be used to predict whether such a system will shake down.     

Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity

In the context of single-crystal strain gradient plasticity, we focus on the simple shear of a constrained strip in order to study the effects of the material parameters possibly involved in the modelling. The model consists of a deformation theory suggested and left undeveloped by Bardella [(2007). Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scales involved. Int. J. Plasticity 23, 296–322] in which, for each glide, three dissipative length scales are considered; they enter the model through the definition of an effective slip which brings into the isotropic hardening function the relevant plastic strain gradients, averaged by means of a p-norm. By means of the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]), the model further involves an energetic material length scale. The application suggests that two dissipative length scales may be enough to qualitatively describe the size effect of metals at the microscale, and they are chosen in such a way that the higher-order state variables of the model be the dislocation densities. Moreover, we show that, depending on the crystallography, the size effect governed by the defect energy may be different from what expected (based on the findings of [Bardella, L., 2006. A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 54, 128–160] and [Gurtin et al. 2007. Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878]), leading mostly to some strengthening. In order to investigate the model capability, we also exploit a Γ-convergence technique to find closed-form solutions in the “isotropic limit”. Finally, we analytically show that in the “perfect plasticity” case, should the dissipative length scales be set to zero, the presence of the sole energetic length scale may lead, as in standard plasticity, to non-uniqueness of solutions.     

Influence of nonparallel flow on the development of disturbances in a supersonic boundary layer

The linear theory is used to solve the problem of the development of two-dimensional disturbances in the boundary layer of compressible fluid. In contrast to the stability theory of plane-parallel flows, the present paper takes into account the presence in the boundary layer of transverse (at right angles to the flow direction) motions, the dependence of the averaged flow parameters on the longitudinal coordinate, and also the deformation of the amplitude distribution profile of the disturbances as a function of the longitudinal coordinate. The calculations are made for Mach number M = 4.5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 26–31, March–April, 1980.     

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.     

Some exact solutions of the ferrohydrodynamics equations

The theoretical study of nonisothermal flows of magnetizable liquids presents serious matheical difficulties, which are intensified as compared to to the study of normal liquids by the necessity of simultaneous solution of both the hydrodynamics and Maxwell's equations, with corresponding boundary conditions for the magnetic field. Thus, in most cases problems of this type are solved by neglecting the effect of the liquid's nonisothermal state on the field distribution within the liquid, and also, as a rule, with use of an approximate solution for Maxwell's equations and fulfillment of the boundary conditions for the field [1–3]. The present study will present easily realizable practical formulations of the problem which permit exact one-dimensional solutions of the complete system of the equations of thermomechanic s of electrically nonconductive incompressible Newtonian magnetizable liquids with constant transfer coefficients. A common feature of the formulations is the presence of a longitudinal temperature gradient along the boundaries along which liquid motion is accomplished. Plane-parallel convective flows of this type in a conventional liquid and their stability were considered in [4–6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 126–133, May–June, 1979.     

Modèles discrets de structures tissées : Analyse de stabilité et de drapéDiscrete models of woven structures: stability and draping analysis

A discrete model of a woven fabric structure is established, whereby nodes endowed with a mass and a rotational rigidity are connected by rigid bars to form a two-dimensional truss. The set of four bars that delineate a quadrilateral area is further endowed with a torsion deformation mode. The kinematics of the truss reproduces the large rotations and displacements encountered for real tissues. The equilibrium shape of such a structure is obtained as the minimum of its total potential energy versus the whole set of kinematic translational and rotational variables, accounting for eventual kinematic constraints due to contact with a rigid surface by the Lagrange multipliers method. A stability analysis is conducted, and the potentiality of the model is illustrated by fabric draping simulations. To cite this article: B. Ben Boubaker et al., C. R. Mecanique 330 (2002) 871–877.     

Parallelization of pseudo-particle modeling and its application in simulating gas–solid fluidization

Pseudo-Particle Modeling (PPM) is a particle method proposed by Ge and Li in 1996 [Ge, W., & Li, J. (1996). Pseudo-particle approach to hydrodynamics of particle–fluid systems. In M. Kwauk & J. Li (Eds.), Proceedings of the 5th international conference on circulating fluidized bed (pp. 260–265). Beijing: Science Press] and has been used to explore the microscopic mechanism in complex particle–fluid systems. But as a particle method, high computational cost remains a main obstacle for its large-scale application; therefore, parallel implementation of this method is highly desirable. Parallelization of two-dimensional PPM was carried out by spatial decomposition in this paper. The time costs of the major functions in the program were analyzed and the program was then optimized for higher efficiency by dynamic load balancing and resetting of particle arrays. Finally, simulation on a gas–solid fluidized bed with 102,400 solid particles and 1.8 × 10⁷ pseudo-particles was performed successfully with this code, indicating its scalability in future applications.     

Implicit finite element formulations for multi-phase transformation in high carbon steel

An anomalous plastic deformation observed during the phase transformation of steels was implemented into the finite element modeling. The constitutive equations for the transformation plasticity originally proposed by Greenwood and Johnson [Greenwood, G.W., Johnson, R.H., 1965. The deformation of metals under small stresses during phase transformation. Proc. Roy. Soc. A 283, 403] and further extended by Leblond et al. [Leblond, J.B., Mottet, G., Devaux, J.C., 1986a. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, I. Derivation of general relations. J. Mech. Phys. Solids 34, 395–409; Leblond, J.B., Mottet, G., Devaux, J.C., 1986b. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, II. Study of classical plasticity for ideal-plastic phases. J. Mech. Phys. Solids 34, 411–432; Leblond, J.B., Devaux, J., Devaux, J.C., 1989a. Mathematical modeling of transformation plasticity in steels, I: case of ideal-plastic phases. Int. J. Plasticity 5, 511–572; Leblond, J.B., 1989b. Mathematical modeling of transformation plasticity in steels, II: coupling with strain hardening phenomena. Int. J. Plasticity 5, 573–591] were modified to consider the thermo-mechanical response of generalized multi-phase steel during phase transformations from austenite at high temperature. An implicit numerical solution procedure to calculate the plastic deformation of each constituent phase was newly proposed and implemented into the general purpose implicit finite element program via user material subroutine. The new algorithms include efficient calculation of consistent tangent modulus for the transformation plasticity and application of general anisotropic yield functions without limitation to the isotropic yield function. Besides the thermo-elastic–plastic constitutive equations, non-isothermal transformation kinetics was characterized by the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation and additivity relationship for the diffusional transformation, while the model proposed by Koistinen and Marburger was used for the diffusionless transformation. Numerical verifications for the continuous cooling experiments under various loading conditions were conducted to demonstrate the applicability of the developed numerical algorithms to the high carbon steel SK5.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part-I: A very low work hardening aluminum alloy (Al6061-T6511)

In the present study, the initial and subsequent yield surfaces in Al 6061-T6511, based on 10 με deviation from linearity definition of yield, are presented. The subsequent yield surfaces are determined during tension, free end torsion, and combined tension–torsion proportional loading paths after reaching different levels of strains. The yield surfaces are also obtained after linear, bi-linear and non-linear unloading paths after finite plastic deformation. The initial yield surface is very close to the von-Mises yield surface and the subsequent yield surfaces undergo translation and distortion. In the case of this low work hardening material, the size of the yield surfaces is smaller and negative cross-effect is observed with finite plastic deformation. The subsequent yield have a usual “nose” in the loading direction and flattened shape in the reverse loading direction; the observed nose is more dominant in the case of tension and combined tension–torsion loading than in torsional loading. The size of the yield surfaces after unloading is smaller than the initial yield surface but larger than subsequent yield surfaces obtained during prior loading, show much smaller cross-effect, and the shape of these yield surfaces depends strongly on the loading and unloading paths. Elastic constants (Young’s and shear moduli) are also measured within each subsequent yield surfaces. Evolution of these constants with finite deformation is also presented. The decrease of the two moduli is found to be much smaller than reported earlier in tension by Cleveland and Ghosh [Cleveland, R.M., Ghosh, A.K., 2002. Inelastic effects on springback in metals. Int. J. Plast. 18, 769–785]. Part-II and III [(Khan et al., 2009a) and (Khan et al., 2009b)] of the papers will include experimental results on annealed 1100 Al (a very high work hardening material) and on both Al alloys (Al6061-T6511 and annealed 1100 Al) in tension- tension stress space, respectively. The results for both cases are quite different than the ones that are presented in this paper.     

Nonstationary waves in the continuously stratified flow of a fluid of finite depth

A theoretical investigation is made of the development of linear two-dimensional waves in a continuously stratified flow of an ideal incompressible fluid. The waves are generated by pressures that are independent of time and that are applied at time t=0 to a bounded region on the free surface of an initially undisturbed flow. The stationary internal waves generated by such a disturbance have been investigated in [1–3]. The nonstationary waves in a continuously stratified fluid that are generated by initial disturbances or periodic surface pressures applied to the entire free surface have been studied in [4–7] in the absence of a slow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 87–93, November–December, 1976.     

Relationship between plastic and intrinsic dissipations

In this paper, the relationship between the plastic and intrinsic dissipations is addressed within the normality structure of [Rice, J.R., 1971. Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455; Rice, J.R., 1975. Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Argon, A.S. (Ed.), Constitutive Equations in Plasticity. MIT Press, Cambridge, MA, pp. 23–79.] It is shown that the plastic dissipation is generally not equal to the intrinsic dissipation. Within the normality structure, the microscale and macroscale thermodynamic fluxes and forces are related by the conditions of energy and dissipation equivalence. If the plastic dissipation is required to be equal to the intrinsic dissipation, J₂ potential and the Levy–Mises equation are recovered from the condition of dissipation equivalence for incompressible plastic flows.     

Role of two-dimensional acoustic wave cutoff in the instability of subsonic MHD flows

One of the important factors affecting the structure of the natural vibrations and the conditions under which they build up in an inhomogeneous subsonic flow may be the cutoff of non-one-dimensional sound waves expressed in the strong reflection of such waves from the critical sections (caustics). In this study the case of natural two-dimensional acoustic perturbations in an inhomogeneous subsonic conducting gas flow in the presence of critical sections is subjected to an asymptotic analysis. Special attention is paid to the conditions of growth of the two-dimensional acoustic perturbations in the internal resonator formed by two critical sections and the walls of an MHD channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 26–36, March–April, 1988.The authors are grateful to seminar participants L. M. Biberman and G. A. Lyubimov for useful discussions.