Young’s modulus prediction of hexagonal nanosheets and nanotubes based on dimensional analysis and atomistic simulations

The present work investigates Young’s modulus of hexagonal nanosheets and nanotubes based on dimensional analysis and molecular mechanics. Using second derivatives of the strain energy density revealed from molecular dynamics simulations at 0 K (i.e., molecular mechanics) with harmonic potentials for various combinations of force constants, Young’s modulus have been computed for single-walled armchair and zigzag nanotubes of different radii. This parametric study with the aid of dimensional analysis allows explicitly establishing Young’s modulus of (n, n) armchair and (n, 0) zigzag nanotubes as functions of the force constants, bond length and chiral index n. Proposed formulae are applied to estimate Young’s modulus of graphene, boron nitride, silicon carbide sheets and their nanotubes. The accuracy of the proposed formulae are verified and discussed with available data in the literature for these three sheets and their nanotubes.     

Material behavior of steel – Modeling of complex phenomena and thermodynamic consistency

Steel has a complex material behavior. Stress- and strain-dependent phase transformations, transformation-induced plasticity (TRIP), and its interactions with plasticity are important phenomena of both theoretical and practical interest, as they may cause distortion of work-pieces. These phenomena continue to be intensively studied both experimentally and theoretically. In order to simulate real processes like heat treatment of work-pieces, one has to include the relevant phenomena in a suitable bulk model. It is the aim of the current paper to contribute to the formulation of such a model in the context of macroscopic continuum mechanics and to discuss the capabilities. Due to the possible interaction (coupling) of TRIP and plasticity, the usual approach in plasticity without phase transformations has to be modified substantially. We apply a general approach for non-linear hardening, allowing to model observable effects of interaction of plasticity and TRIP. Besides this, we prove the thermodynamic consistency under sufficient conditions.     

Internal loops in superelastic shape memory alloy wires under torsion – Experiments and simulations/predictions

Understanding torsional responses of shape memory alloy (SMA) specimens under partial or fully transformed cases with internal loops is of particular importance as the entire response might not be always utilized and only a portion of the entire response (internal loop) might be of significance to designers. In this work, we present experimental results of large complex loading and unloading torsional cycles which were conducted on superelastic SMA wires, under isothermal conditions with the purpose of elucidating the torsional internal loop response during loading and unloading. Such data hereto has not been available in open literature. Utilizing this data, we model the torsional response of superelastic SMA wires subjected to various loading and unloading situations that can result in different extents of transformation.A thermodynamically consistent Preisach model (Rao and Srinivasa, 2013) captures such complex internal loops with a high degree of precision by modeling driving force for phase transformation vs. volume fraction of martensite relationships. This approach is different from capturing purely phenomenological stress–strain or stress–temperature Preisach models. The thermodynamic approach utilized here has broader predictive capability. The model predictions indicate good agreement with the internal loop structures even though only the outer loop information was used for model calibration. The addition of a single inner loop information for model calibration greatly improves the predictions.     

Impact failure of granular materials – Non-equilibrium multiscale simulations and high-speed experiments

We present results of high-speed impact experiments on aluminum oxide (Al₂O₃) and silicon carbide (SiC) ceramics and propose a mesoscopic way to model the fracture behavior of these brittle materials based on discrete particles. The two-dimensional model used here has only three adjustable parameters, but is able of reproducing many salient features of the investigated ceramics under compressive, tensile and shock impact load. We discuss our particle model in detail and then consider strain and shear load simulations. In particular, we model explicitly the macroscopic experimental set-up of the edge-on impact experiment and show that the experimentally observed crack patterns can in principle be explained by the random distribution of particle overlaps and the thereby generated differences in the local strength of the material.     

Mechanical behaviour of carbon nanotubes under combined twisting–bending

The main objective of this paper is to investigate the mechanical behaviour (strength and stiffness) of carbon nanotubes (CNTs) under combinations of bending and twisting. In order to achieve this goal, molecular dynamics (MD) simulations of bended and twisted CNTs are performed. The LAMMPS code is used, the AIREBO potential is considered for CC bonds, the temperature is kept at 300 K and incremental bending and twisting rotations are imposed to the CNT. Two types of CNTs are analyzed, including zig-zag (8,0) and armchair (5,5) CNTs with similar radius and length. The CNTs are also analyzed for pure bending and pure twisting. The main results are shown in the form of diagrams of energy and moment against imposed rotations. Some relevant conclusions are drawn concerning the influence of loading (bending and twisting) on the stiffness, strength and failure of CNTs: namely, it is concluded that armchair CNTs possess higher strength and fracture toughness under twisting–bending loading than zigzag CNTs; additionally, it is found that both CNTs (armchair and zigzag) still support moderate-to-high bending levels without failure after being extremely twisted and torsionally buckled, even for twisting angles four times those corresponding to torsional buckling; finally, the results prove that CNTs, mostly armchair ones, exhibit very high twisting–bending stiffness and strength and can be used with confidence as torsional spring elements in nanoelectromechanical systems (NEMS).     

Nonlinear behavior of a rotor-AMB system under multi-parametric excitations

A rotor-active magnetic bearing (AMB) system subjected to a periodically time-varying stiffness with quadratic and cubic nonlinearities under multi-parametric excitations is studied and solved. The method of multiple scales is applied to analyze the response of two modes of a rotor-AMB system with multi-parametric excitations and time-varying stiffness near the simultaneous primary and internal resonance. The stability of the steady state solution for that resonance is determined and studied using Runge-Kutta method of fourth order. It is shown that the system exhibits many typical non-linear behaviors including multiple-valued solutions, jump phenomenon, hardening and softening non-linearities and chaos in the second mode of the system. The effects of the different parameters on the steady state solutions are investigated and discussed also. A comparison to published work is reported.     

Mechanical and microstructural investigations of an austenitic stainless steel under non-proportional loadings in tension–torsion-internal and external pressure

This paper is concerned with the experimental behaviour of a 316 austenitic stainless steel at room temperature and under non-proportional cyclic and ratchet strainings in tension–torsion-internal and external pressures. The main investigations deal with the over-strengthening due to the multiaxiality of the loadings. A classification of the different kinds of cyclic tests can be established with respect to the increasing maximum over-strengthening. Concerning the ratchetting effect, from tests performed under in or out-of-phase cyclic tension–torsion plus a static stress due to internal pressure, it is shown that the rate of the diametrical ratchetting is an increasing function of the phase lag between the cyclic components. Dislocation substructures resulting from cyclic and ratchetting tests are investigated and various kinds of microstructures are reported. An analysis of these microstructures shows that the over-strengthening is not solely related to the slip multiplicity but also to the development of heterogeneous substructures. It has been also possible to evaluate the intra- and inter-granular back stresses and the effective stress as a function of the strengthening.     

Quasi-soft opto-mechanical behavior of photochromic liquid crystal elastomer: Linearized stress–strain relations and finite element simulations

Based on the neo-classical elastic energy of liquid crystal elastomers, the opto-mechanical behavior is modeled by considering the effect of photoisomerization on the nematic-isotropic transition of liquid crystal phase. Linearized stress–strain relation is derived for infinitesimal deformations with a very unusual shear stress that does not vanish identically as in the case of the soft behavior but is proportional to the rotation of directors. In other words, the shear stress depends on both the shear strain and the skew symmetric part of the displacement gradient with the shear modulus induced by the effect of photoisomerization. Finite element implementation for plane stress problems is obtained through a self-defined material subroutine in ABAQUS FEA tool. Numerical simulations show that the light induced deformations of two dimensional specimens consist of contractions, expansions and bending in different directions. The stress distributions indicate that the driving force for the light induced bending is produced by the bending moment of the normal stress along the director, while the other stress components are much smaller for two dimensional beam shaped specimens. However, the shear stress of the soft LCE is generally nonzero under light illumination due to the inhomogeneity of the opto-mechanical effect. It can be concluded from the strain distributions that the transversal plane cross section could remain plane after deformation if the light intensity or the decay distance is not too small and the sample is in the deep nematic phase. However, the shear strain and in plane rotation are of the same order as the other strain components, and thus should not be neglected. This indicates that the classical simple bending assumptions such as the Euler–Bernoulli beam theory should not be directly applied to model the light induced bending of neo-classical liquid crystal elastomers due to the soft behavior of the materials.     

Estimation of deformation behavior of TRIP steels — smooth/ringed-notched specimens under monotonic and cyclic loading

Uniaxial tension tests were performed under a constant strain rate and various environmental temperatures from 77 to 373 K to identify the concrete form of the constitutive equation for TRIP steels. To elucidate the dependence of the martensitic transformation on the nonuniform deformation and to validate the proposed constitutive equation, the volume fraction of the martensite phase is predicted and measured using computational simulation and experimental procedures, respectively, for the uniaxial tension of bars with a ringed notch. The good correspondence between the local volume fraction of the martensite phase around the notch, obtained by both methods, verifies the validity of the proposed constitutive equation for the nonuniform deformation behavior. Subsequently, the computational simulations were performed to elucidate the deformation behavior of ringed-notched bars under compression and that of smooth/ringed-notched bars under cyclic loading.     

A second strain gradient elasticity theory with second velocity gradient inertia – Part I: Constitutive equations and quasi-static behavior

A multi-cell homogenization procedure with four geometrically different groups of cell elements (respectively for the bulk, the boundary surface, the edge lines and the corner points of a body) is envisioned, which is able not only to extract the effective constitutive properties of a material, but also to assess the “surface effects” produced by the boundary surface on the near bulk material. Applied to an unbounded material in combination with the thermodynamics energy balance principles, this procedure leads to an equivalent continuum constitutively characterized by (ordinary, double and triple) generalized stresses and momenta. Also, applying this procedure to a (finite) body suitably modelled as a simple material cell system, in association with the principle of the virtual power (PVP) for quasi-static actions, an equivalent structural system is derived, featured by a (macro-scale) PVP having the typical format as for a second strain gradient material model. Due to the surface effects, the latter model does work as a combination of two subsystems, i.e. the bulk material behaving as a Cauchy continuum, and the boundary surface operating as a membrane-like boundary layer, each subsystem being in (local and global) equilibrium by its own. Further, the applied (ordinary) boundary traction splits into two (response-dependent) parts, i.e. the “Cauchy traction” transmitted to the bulk material and the “Gurtin–Murdoch traction” acting, together with all other boundary tractions, upon the boundary layer. The role of the boundary layer as a two-dimensional manifold enclosing a Cauchy continuum is elucidated, also with the aid of a discrete model. A strain gradient elasticity theory is proposed which includes a minimum total potential energy principle featuring the relevant boundary-value problem for quasi-static loads and its (unique) solution. A simple application is presented. Two appendices are included, one reports the proof of the global equilibrium of the boundary layer, the other is concerned with double and triple stresses. The paper is complemented by a companion Part II one on dynamics. Previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137] are improved and extended.     

Finite element simulations of dynamics of multivariant martensitic phase transitions based on Ginzburg–Landau theory

A finite element approach is suggested for the modeling of multivariant stress-induced martensitic phase transitions (PTs) in elastic materials at the nanoscale for the 2-D and 3-D cases, for quasi-static and dynamic formulations. The approach is based on the phase-field theory, which includes the Ginzburg–Landau equations with an advanced thermodynamic potential that captures the main features of macroscopic stress–strain curves. The model consists of a coupled system of the Ginzburg–Landau equations and the static or dynamic elasticity equations, and it describes evolution of distributions of austenite and different martensitic variants in terms of corresponding order parameters. The suggested explicit finite element algorithm allows decoupling of the Ginzburg–Landau and elasticity equations for small time increments. Based on the developed phase-field approach, the simulation of the microstructure evolution for cubic-tetragonal martensitic PT in a NiAl alloy is presented for quasi-statics (i.e., without inertial forces) and dynamic formulations in the 2-D and 3-D cases. The numerical results show the significant influence of inertial effects on microstructure evolution in single- and polycrystalline samples, even for the traditional problem of relaxation of initial perturbations to stationary microstructure.     

Segregation of particulate solids＂ Experiments and DEM simulations

Segregation of granular materials is a complex phenomenon, difficult to measure quantitatively and to predict. Discrete element method （DEM） can be a useful tool to predict segregation effects and to support the industrial design. In this context, a very challenging idea is the characterization of the granular solids to provide the key parameters needed for a successful DEM simulation of segregation processes. Rolling friction, sliding friction and the coefficient of restitution are the critical parameters to be studied. These microscopic simulation parameters are calibrated by comparing the macroscopic behavior of granular matter in standard bulk experiments, which have the advantage of being highly repeatable and reliable. An experimental method is presented to characterize free surface segregation. The effects of different particle properties, particularly, shape and size, on segregation of cohesionless materials were investi- gated. From the experiments, particle size demonstrated a stronger effect on segregation than particle shape. Finally, the corresponding DEM simulations of the segregation experiments were presented. The parameters obtained by calibration were validated by the comparison of the modeled segregation behav- ior with the experimental results. Thus, calibrated DEM simulations are capable of predicting segregation effects.     

Anisotropic hardening – numerical application of a cubic yield theory and consideration of variable r-values for sheet metal

The first part of this paper contains a polynomial yield condition of third order connected with evolution equations for material tensors of higher orders. They are formulated by formal generalisation of an approach by Danilov. The second part presents a possibility of taking into account the rotation of the yield surface as a result of a variable planar anisotropy (r-value) in sheet metal. This is done by an extension of the evolution equations, based on a quadratic yield function. The corresponding deformation law and the set of evolution equations are numerically integrated for selected loading paths in the subspaces σ₁,σ₂ and σ,τ. Some of the experimentally observed effects, such as the increasing curvature of the yield locus curve in the loading direction or the specific rotation of the yield surface, are correctly reproduced.     

A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior

This paper is the sequel of a companion Part I paper devoted to the constitutive equations and to the quasi-static behavior of a second strain gradient material model with second velocity gradient inertia. In the present Part II paper, a multi-cell homogenization procedure (developed in the Part I paper) is applied to a nonhomogeneous body modelled as a simple material cell system, in conjunction with the principle of virtual work (PVW) for inertial actions (i.e. momenta and inertia forces), which at the macro-scale level takes on the typical format as for a second velocity gradient inertia material model. The latter (macro-scale) PVW is used to determine the equilibrium equations relating the (ordinary, double and triple) generalized momenta to the inertia forces. As a consequence of the surface effects, the latter inertia forces include (ordinary) inertia body forces within the bulk material, as well as (ordinary and double) inertia surface tractions on the boundary layer and (ordinary) inertia line tractions on the edge line rod; they all depend on the acceleration in a nonstandard way, but the classical laws are recovered in the case of no higher order inertia. The classical linear and angular momentum theorems are extended to the present context of second velocity gradient inertia, showing that the extended theorems—used in conjunction with the Cauchy traction theorem—lead to the local force and moment (stress symmetry) motion equations, just like for a classical continuum. A gradient elasticity theory is proposed, whereby the dynamic evolution problem for assigned initial and boundary conditions is shown to admit a Hamilton-type variational principle; the uniqueness of the solution is also discussed. A few simple applications to wave propagation and dispersion problems are presented. The paper indicates the correct way to describe the inertia forces in the presence of higher order inertia; it extends and improves previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137]. Overall conclusions are drawn at the end of the paper.     

Effects of polydispersity on the micro–macro behavior of granular assemblies under different deformation paths

The micromechanical and macromechanical behavior of idealized granular assemblies, made by linearly elastic, frictionless, polydisperse spheres, are studied in a periodic, triaxial box geometry, using the discrete element method. Emphasis is put on the effect of polydispersity under purely isotropic loading and unloading, deviatoric(volume conserving), and uniaxial compression paths.We show that scaled pressure, coordination number and fraction of rattlers behave in a very similar fashion as functions of volume fraction, irrespective of the deformation path applied. Interestingly, they show a systematic dependence on the deformation mode and polydispersity via the respective jamming volume fraction. This confirms that the concept of a single jamming point has to be rephrased to a range of variable jamming points, dependent on microstructure and history of the sample, making the jamming volume fraction a state-variable.This behavior is confirmed when a simplified constitutive model involving structural anisotropy is calibrated using the purely isotropic and deviatoric simulations. The basic model parameters are found to depend on the polydispersity of the sample through the different jamming volume fractions. The predictive power of the calibrated model is checked by comparison with an independent test, namely uniaxial compression. The important features of the uniaxial experiment are captured and a qualitative prediction for the evolution of stress and fabric is shown involving a "softening" regime in both stress and fabric – stronger for the latter – that was not prescribed into the model a priori.     

A micromechanics-based strain gradient damage model for fracture prediction of brittle materials – Part II: Damage modeling and numerical simulations

In this paper, we established a strain-gradient damage model based on microcrack analysis for brittle materials. In order to construct a damage-evolution law including the strain-gradient effect, we proposed a resistance curve for microcrack growth before damage localization. By introducing this resistance curve into the strain-gradient constitutive law established in the first part of this work (Li, 2011), we obtained an energy potential that is capable to describe the evolution of damage during the loading. This damage model was furthermore implemented into a finite element code. By using this numerical tool, we carried out detailed numerical simulations on different specimens in order to assess the fracture process in brittle materials. The numerical results were compared with previous experimental results. From these studies, we can conclude that the strain gradient plays an important role in predicting fractures due to singular or non-singular stress concentrations and in assessing the size effect observed in experimental studies. Moreover, the self-regularization characteristic of the present damage model makes the numerical simulations insensitive to finite-element meshing. We believe that it can be utilized in fracture predictions for brittle or quasi-brittle materials in engineering applications.     

Controlled preparation and characterization of nano-sized hexagonal Mg（OH）2 flame retardant

Nano-sized hexagonal magnesium hydroxide （Mg（OH）2） with good dispersibility was synthesized by a double injection-hydrothermal method, utilizing polyvinylpyrrolidone （PVP） as an additive and with optimized processing parameters. SEM and BET analysis showed that the mean particle size and specific surface area of the Mg（OH）2 particles were 174 nm and 50.77 m^2/g, respectively. The FT-IR spectra and the XRD patterns showed that PVP was adsorbed on the surface of the Mg（OH）2 crystal, thus effectively limiting particle agglomeration and hindering crystal growth along the （1 01 ） plane. TGA showed a decrease in the decomposition temperature and an increase in the weight loss of the Mg（OH）2 particles due to addition of PV/.     

Multiple time scales of fluvial processes — theory and applications

Fluvial processes comprise water flow, sediment transport and bed evolution, which normally feature distinct time scales. The time scales of sediment transport and bed deformation relative to the flow essentially measure how fast sediment transport adapts to capacity region in line with local flow scenario and the bed deforms in comparison with the flow, which literally dictates if a capacity based and/or decoupled model is justified. This paper synthesizes the recently developed multiscale theory for sediment-laden flows over erodible bed, with bed load and suspended load transport, respectively. It is unravelled that bed load transport can adapt to capacity sufficiently rapidly even under highly unsteady flows and thus a capacity model is mostly applicable, whereas a non-capacity model is critical for suspended sediment because of the lower rate of adaptation to capacity. Physically coupled modelling is critical for fluvial processes characterized by rapid bed variation. Applications are outlined on very active bed load sediment transported by flash floods and landslide dam break floods.     

Transition from progressive buckling to global bending of circular shells under axial impact––Part I: Experimental and numerical observations

The influence of impact velocity and material characteristics on the dynamic buckling response of circular shells subjected to axial impact loads is studied. It is shown experimentally that the critical buckling length, which marks the transition between progressive and global buckling of aluminium alloy circular tubes, is significantly influenced by the axial impact velocity. A finite element analysis is undertaken to further explore the effects of material yield stress, strain hardening and strain rate sensitivity on the transition phenomenon. It is observed that circular tubes made of ductile alloys with a high yield stress and low strain hardening characteristics have a better performance as energy absorbers than tubes made of alloys with a low yield stress and high strain hardening characteristics. Theoretical analysis of some particular features of the dynamic buckling transition is presented in Part II [International Journal of Solids and Structures (2004)].