Characteristic Length Scale Parameter Research Articles

Gurson-Cohesive modeling (GCM) for 3D ductile fracture simulation

A Gurson-Cohesive Model (GCM) is proposed to investigate 3D ductile fracture accounting for the void growth, coalescence, and complete failure phenomena. The proposed model idealizes the ductile fracture process as continuum damage evolution, cohesive crack initiation, nonlinear softening along the crack surface, and complete failure. The Gurson model is employed to describe continuum damage with void growth, while the cohesive zone model is utilized to introduce discontinuous cracks and represent nonlinear softening behavior. The transition from the continuum damage to discontinuous crack is taken into account systematically using a porosity-based crack initiation criterion considering stress triaxiality. The computational results successfully reproduce the experimental results of fracture tests with 15–5 PH steel and structural carbon steel 20 within the unified modeling framework. Furthermore, strong and stable convergence of both global and local responses (e.g., load-displacement curve, area reduction, crack tip location) are demonstrated under the mesh refinement without the aid of any ad hoc characteristic length scale parameter in the simulations.

Seismic wave propagation analysis in the framework of generalized continuum mechanics theory

For the problems of seismic wave propagation, scale effects will appear while considering the microstructures of medium. Scale effects mean that new components of wave fields appear in the seismograms due to the microstructure interactions in the medium and vary with the change of the characteristic length scale parameters. The generalized continuum mechanics (GCM) theory which introduces the intrinsic length scales of the microstructures is suitable for analyzing the scale effects of seismic wave propagation. We adopt the concept of multi-scale microstructure interactions, that is, the modified couple stress theory, and the one-parameter second strain gradient theory, as two branches of the GCM theory, have the same definition characteristic length scale parameters of medium, to describe the microstructure interactions at different scales. Through the quantitative relationship between the characteristic length scale parameters of the medium and the micropore, the layered model of the characteristic length scale parameters of the micropore is constructed for numerical modeling in a unified framework, and the scale effects of seismic wave propagation is analyzed. More specifically, the asymmetric elastic wave equations under a unified framework are derived, and the physical definition of the characteristic length scale parameters of medium is given. And the influence of the characteristic length scale parameter of the micropore on the seismic wave propagation is focused on. Finally, the scale effects of seismic waves excited by high-speed trains under the viaduct system are analyzed, and some conclusions are drawn.

A new method for free vibration analysis of nanobeams: Introduction of equivalent lattice stiffness method

Since the efficiency of non-classical continuum theories is strongly dependent on the recognition of the suitable values of small length scale parameters and there is still uncertainty about them, a novel approach, equivalent lattice stiffness method is developed here. This approach without the characteristic length scale parameter which arises in non-classical continuum theories, such as nonlocal theory and strain gradient theory, is capable to capture size effect more easily and accurately. This method is proposed based on the concept of lattice dynamics but a Taylor series expansion is involved to approximate the displacements of the continuous domain; accordingly, this approach is in accordance with lattice dynamics with tunable precision and is more simpler than continuum theories. Since the discrete phenomena in micro/nanostructures can be incorporated into the continuum formulation using this method, it can be competitive with other analytical methods like molecular mechanics approach. Moreover, this study can bridge the gap between the continuum theories and atomic models since it is capable to remedy the ongoing challenge on the accurate values of characteristic length scale parameters. To conform the accuracy of the current study, the dispersion curves of different theories are indicated and compared with lattice dynamics. Then, the longitudinal and transverse vibration of nanobeams subjected to different boundary conditions is investigated using equivalent lattice stiffness method.

Couple stress Rayleigh–Bénard convection in a square cavity

Consistent skew-symmetric couple stress theory, which introduces an intrinsic characteristic length scale parameter l into the formulation, is applied here to model the Rayleigh–Bénard convection problem. The focus is on the potential effect of size-dependency on the stability of Rayleigh–Bénard convection. We consider a two-dimensional flow and study the effects of couple-stresses by developing and then applying a stream function-vorticity-temperature computational fluid dynamics formulation. Details are provided on the governing equations for size-dependent flow, based on the Boussinesq approximation. Afterwards, the formulation is applied to the Rayleigh–Bénard convection problem in a square cavity to examine numerically the stability of the flow as a function of the length scale parameter l. The investigation covers a range of Rayleigh numbers, and includes an evaluation of the critical value beyond which convective instabilities start to appear, based upon an overall kinetic energy norm. The additional boundary conditions associated with consistent couple stress theory are found to play an important role in determining this critical Rayleigh number and also affect the flow patterns. Furthermore, the developments here may encourage the study of size-dependent multiphysics flow phenomena in general, which in turn may reveal new features and mechanisms of energy dissipation and storage at the smallest continuum scales.

Wavelet Method Identifying the Characteristic Length Scale Parameter of Fractal Rough Surface

Wavelet Method Identifying the Characteristic Length Scale Parameter of Fractal Rough Surface

Boundary element formulation for plane problems in size-dependent piezoelectricity

A new boundary element formulation is developed to analyze two-dimensional size-dependent piezoelectric response in isotropic dielectric materials. The model is based on the recently developed consistent couple stress theory, in which the couple-stress tensor is skew-symmetric. For isotropic materials, there is no classical piezoelectricity, and the size-dependent piezoelectricity or flexoelectricity effect is solely the result of coupling of polarization to the skew-symmetric mean curvature tensor. As a result, the size-dependent effect is specified by one characteristic length scale parameter l, and the electromechanical effect is specified by one flexoelectric coefficient f. Interestingly, in this size-dependent multi-physics model, the governing equations are decoupled. However, the problem is coupled, because of the existence of a flexoelectric effect in the boundary couple-traction and normal electric displacement. We discuss the boundary integral formulation and numerical implementation of this size-dependent piezoelectric boundary element method, which provides a boundary-only formulation involving displacements, rotations, force-tractions, couple-tractions, electric potential, and normal electric displacement as primary variables. Afterwards, we apply the resulting BEM formulation to several computational problems to confirm the validity of the numerical implementation and to explore the physics of the flexoelectric coupling. Copyright © 2016 John Wiley & Sons, Ltd.

Development of a precipitate size-dependent crystal plasticity constitutive model for two-phase materials and its implementation on a multi-scale computational framework

A new method to introduce size-dependence in crystal plasticity constitutive models, recently developed for single-phase polycrystal materials [1], is extended to two-phase single crystal materials. The precipitate size-dependent crystal plasticity constitutive model is developed by accounting for the resistance to dislocation nucleation and mobility at a material point in the matrix phase near the interface between the precipitate and matrix phases (referred to as precipitate-matrix interface influence region). Following the crystal plasticity constitutive modeling principle, changes in strength and straining-hardening characteristics in the precipitate-matrix interface influence region are captured by introducing a shear flow strain parameter equivalent to the resistance to dislocation nucleation. As a result, for the interface influence region with its thickness and the distribution of the equivalent shear flow strain remaining the same irrespective of the unit-cell size, the precipitate size-dependence is naturally evolved in the constitutive model. A simplified model that considers the precipitate-matrix interface effect on an average sense in the matrix phase is also developed under the general framework. Implementation of this general framework is demonstrated by considering the case of a power-law flow rule and a hyperbolic-secant hardening rule. Accordingly, a characteristic length-scale parameter that defines the effective precipitate size is introduced. Finally, the precipitate size-dependent constitutive model was implemented on a multi-scale computational framework developed by NASA Glenn Research Center. The elastic-plastic behavior of a full-scale Ni-based superalloy disk with variations in the precipitate size along the radius is analyzed as an example problem for the size-dependent multi-scale model.

Capturing tensile size-dependency in polymer nanofiber elasticity

As the name implies, tensile size-dependency refers to the size-dependent response under uniaxial tension. It defers markedly from bending size-dependency in terms of onset and magnitude of the size-dependent response; the former begins earlier but rises to a smaller value than the latter. Experimentally, tensile size-dependent behavior is much harder to capture than its bending counterpart. This is also true in the computational effort; bending size-dependency models are more prevalent and well-developed. Indeed, many have questioned the existence of tensile size-dependency. However, recent experiments seem to support the existence of this phenomenon. Current strain gradient elasticity theories can accurately predict bending size-dependency but are unable to track tensile size-dependency. To rectify this deficiency a higher-order strain gradient elasticity model is constructed by including the second gradient of the strain into the deformation energy. Tensile experiments involving 10wt% polycaprolactone nanofibers are performed to calibrate and verify our model. The results reveal that for the selected nanofibers, their size-dependency begins when their diameters reduce to 600nm and below. Further, their characteristic length-scale parameter is found to be 1095.8nm.

Numerical Evaluation of Size Effect in Piezoelectric Micro-Beam with Linear Micromorphic Electroelastic Theory

AbstractThe linear micromorphic electroelastic theory is proposed to solve bending problems of piezoelectric micro-beam in this paper. The basic governing equations with the boundary conditions are derived through the variational principle. Both the cantilever piezoelectric micro-beam subjected to a concentrated load at the free end and the simply supported micro-beam subjected to a distributed load are analyzed. It is found that the predictions from the micromorphic electroelastic theory are remarkably different from those from the classical theory when the micro-beam thickness is approximate or equal to the characteristic length scale parameter, but their difference is slight when the micro-beam thickness is much larger than the characteristic length scale parameter. As a result, it is concluded that the size effect is significant when the micro-beam thickness is comparable to the characteristic length scale parameter.

A new multi-scale modeling approach based on hygro-Cosserat theory for self-induced stress in hydrating cementitious mortars

The present paper focuses on the modeling of internal stresses induced by the restrained autogenous shrinkage of hydrating cementitious matrix in cement-based mortars. At very early age (0–48h), these self-induced stresses may be relatively high and even critical, especially for cementitious systems with low water-to-cement ratio, since the physico-chemical phenomena involved (hydration and self-desiccation) are particularly intense. To pursue the mentioned objective, an original multi-scale approach based on the application of hygro-Cosserat theory has been developed to model the self-induced stress variation in the cement paste surrounding the aggregates. In fact, the characteristic length scale parameter Lc in the Cosserat theory helps us to reduce the specimen size from macro-scale to micro-scale and even sub-micro-scale due to its explicit size effect features, which is not feasible in the classical theory, i.e. Cauchy–Bolzmann’s theory. The self-shrinkage phenomenon at early age has been observed and modeled via the experiments and a freshly defined Cosserat Size effect number (CS) based upon the Representative Volume Element (RVE) concept. The proposed method is capable of treating the internal stress and could be followed by cracks appearance investigation in the cementitious matrix surrounding the sand inclusions, which should occur inside of the RVE of mortar subjected to self-desiccation shrinkage during the hydration process at early age. The occurrence of these micro-cracking networks are confirmed by Scanning Electronic Microscopy (SEM) observations at the interface cement paste/aggregate performed on different mortars at early age. By taking advantage of the time-dependent Finite Element Analysis (FEA), the numerical outcomes are well agreed with the experimental observations coming from SEM. It concludes that the inclusion creates high hygro-stress concentration around the grains: when the number of inclusions increases, this hygro-stress could lead to a micro-crack network through the matrix.

New Approach of Multi Scale Modeling Based on the Hygro-Cosserat Theory for Self Desiccation Deformation of Cement Mortars at Early Age

In the present paper, we concentrate on the heterogeneous cement mortars and we treat them as Cosserat-based media. The autogenous shrinkage phenomenon at early age (from 1 up to 3 days after mixing) has been analyzed by means of Cosserat theory. The characteristic length scale parameter Lc in this theory helps us to change the size specimen from macro-scale to micro-scale using the theoretical size effect aspects. This methodology is also capable of treating cracks initiation and their appearance in the cementitious matrix surrounding the sand-inclusions, which should occurred inside of the Representative Volume Elementary (RVE) of mortar subjected to self-desiccation shrinkage during hydration at early age. By taking advantage of the Nonlinear Finite Element Analysis (NFEA), the numerical experiments have been performed. The numerical outcomes are well agreed with the experimental observations coming from Scanning Electronic Microscopy (SEM) images. It concludes that the inclusions create not only a hygro stress concentration around the grains but also the number of inclusions should influence the network in cementitous matrix.

Modeling the dynamic rupture propagation on heterogeneous faults with rate- and state-dependent friction

We investigate the effects of non-uniform distribution of constitutive parameters on the dynamic propagation of an earthquake rupture. We use a 2D finite difference numerical method and we assume that the dynamic rupture propagation is governed by a rate- and state-dependent constitutive law. We first discuss the results of several numerical experiments performed with different values of the constitutive parameters a (to account for the direct effect of friction), b (controlling the friction evolution) and L (the characteristic length-scale parameter) to simulate the dynamic rupture propagation on homogeneous faults. Spontaneous dynamic ruptures can be simulated on velocity weakening (a b) can arrest a dynamic rupture, but can be driven to an instability if suddenly loaded by the dynamic rupture front. Our simulations provide a picture of the complex interactions between fault patches having different frictional properties and illustrate how the traction and slip velocity evolutions are modified during the propagation on heterogeneous faults. These results involve interesting implications for slip duration and fracture energy.

A model of thermal contact conductance at high real contact area fractions

Thermal contact conductance (TCC) is studied in the whole range of real contact area fractions between zero and unity. For this purpose, a two-scale model is developed in which the effective (macroscopic) TCC coefficient is obtained from the solution of the heat conduction problem at the scale of asperities. Additional thermal resistance at the real contact spots is included in the model. The model is applied for several real 3D roughness topographies for which the effective TCC coefficient is determined as a function of the real contact area fraction and the local TCC coefficient at real contact spots. An analytical function is found which approximates this relationship in the whole range of parameters, and a characteristic length-scale parameter is introduced which characterizes the effective TCC properties of a rough surface.

A predictive model for permeability of correlated porous media

Starting from the analysis of correlated percolation lattices, we develop a new predictive model for the permeability of porous media. The model proposed introduces a new characteristic length scale parameter related to the pore-pore correlation function of the digitalized image of the porous structure. The model predicts with good accuracy the permeability of complex pore structures such as correlated percolation lattices, deterministic fractals, fractal percolation lattices and correlated porous media in general.