Nonlocal Plasticity Research Articles

Our brains sense the future through a new quantum-like implicit learning mechanism

BackgroundImagine if our brains could unconsciously predict future events. This study explores this concept, presenting evidence for an inherent 'foreseeing' ability, termed anomalous cognition (AC). We introduce a new experimentally verifiable approach to explain anomalous information anticipation (AIA), a type of AC, based on an innovative, quantum-like model of implicit learning, grounded in Nonlocal Plasticity Theory (NPT). MethodsOur research involved 203 participants using methods such as continuous flash suppression, random dot motion, and advanced 3D EEG neuroimaging, along with IBM quantum random event generators for precise measurements across 144 trials. These trials tested contingencies between undetectable sensory stimuli and dot movements, focusing on participants' prediction abilities. The design conditions were strictly experimental, violating fundamental classical learning principles, particularly reflex conditioning. If these principles were immutable, their violation would prevent any systematic behavioral changes, resulting in random responses. This violation was implemented through two quantum physics concepts: the mathematical principle of nonlocality and entanglement. ResultsDespite the sensory stimulus being inaccessible, our results showed a significant prediction between the contingencies and an increase in AIA accuracy, with explained variances between 25 % and 48 %. EEG findings supported this, showing a positive link between brain activity in specific regions and AIA success. Electrochemical activations were detected in the posterior occipital cortex, the intraparietal sulcus, and the medial temporal gyri. AIA hits exceeded the threshold value corresponding to one standard deviation above the expected mean, showing moderate effect sizes in the experimental group (Cohen’s d = 0.461). Analyzing the learning curve using the derivation technique, we identified the acceleration point of the wave function, indicating systematic implicit learning. This result showed that from repetition 63 onwards, AIA hits increased significantly. ConclusionsThe results suggest that, despite violating fundamental classical learning principles, cognitive processes produced changes in participants' responses susceptible to neuromodulation, considering quantum physics principles of nonlocality and entanglement (both present in NPT). We discuss (a) why the priming effect does not explain the significant results; (b) the potential discovery of a new form of quantum-like implicit learning, which could scientifically resolve phenomena associated with anomalous cognitions (e.g., AIA); and (c) future research directions, including potential applications and clinical impact.

A mixed nonlocal finite element model for thermo‐poro‐elasto‐plastic simulation of porous media with multiphase fluid flow

SummaryA new mixed nonlocal finite element framework is developed for nonlinear thermo‐poro‐elasto‐plastic simulation of porous media with multiphase pore fluid flow and thermal coupling. The solid‐fluid interaction is accounted for using the mixture theory of Biot based on the volume fractions concept. Different sources of nolinearities arising from the multiphase fluid flow effects, advective‐diffusive heat transfer, inelastic deformation, fluid flux injection induced mechanical tractions, solid skeleton deformation permeability dependence, and temperature dependent viscosity are included in developing a robust numerical solver for the targeted coupled multiphysics problem. To address the effect of microstructure in inelastic localized deformation behaviour with dilational softening, a nonlocal plasticity model is proposed based on a characteristic length scale which rectifies the non‐physical pathological mesh dependence problem encountered in conventional plasticity. The accuracy and strength of the developed model is shown with comparing the obtained numerical results of a benchmark bilateral compression test with existing published data in the literature. To show the versatility and robustness of the developed computational framework in modelling the geomechanics of real‐case engineering practices, large scale thermo‐hydro‐mechanical (THM) subsurface stimulation processes with applications in enhanced oil recovery (EOR) are effectively simulated and the targeted enhanced recovery and performances are demonstrated. The current formulation does not include phase transformation modelling capability, and therefore, the developed models may not be applicable for simulation of the engineering processes that involve phase change behaviour (e.g., steam injection).

“Implicit” vs “Explicit” gradient plasticity models: Do they always remove mesh dependence in softening materials?

Finite element quasi-static solutions of rate-independent softening plasticity models are known to depend on the mesh size (i.e., are unreliable) due to loss of ellipticity of the governing partial differential equations, which allows for the development of non-smooth solutions, such as shear bands of zero thickness. A “regularization” scheme that has been proposed is the use of “non-local” (Bažant et al., 1984) or “strain-gradient” theories (Aifantis, 1984). We examine in detail the families of “explicit” and “implicit” gradient plasticity models and show that use of such theories does not always eliminate mesh dependence of finite element solutions in rate-independent softening materials. Depending on the value of the non-local hardening modulus, the mathematical problem may still lose ellipticity and allow for the development of solutions with discontinuous spatial velocity gradients, which are responsible for the mesh dependence of numerical solutions. As an example, a rate-independent “implicit” gradient version of the von Mises yield condition with the associated flow rule is implemented in ABAQUS, and the formation of shear bands in plane strain tension is analyzed numerically; it is shown that, unless the material parameters in the non-local model are such that ellipticity is retained, the finite element solutions depend on the mesh size. The example of isotropic porous metal plasticity with a hardening matrix material is considered (e.g., the Gurson (1977) model); it is shown that regularization is achieved if an implicit non-local model is used, in which the local porosity f is replaced by its non-local (neighborhood average) counterpart f̄ in the yield condition. In all non-local plasticity models, dimensional consistency requires the introduction of one or more “material lengths” ℓi, which multiply the additional terms with the higher-order spatial derivatives introduced by the non-local models. The boundary layers associated with the singular perturbation problem when the material lengths ℓi→0 are examined. A linear stability analysis is carried out to study the stability of homogeneous problems under small harmonic perturbations; it is shown that the stability range of the non-local models is always larger than that of the corresponding local models.

Nonlocal strain regularisation for critical state models with volumetric hardening

The finite element (FE) method may suffer from numerical instability and mesh dependency when modelling the formation of shear bands using strain-softening constitutive models. Nonlocal methods have been shown to be capable of avoiding these numerical issues effectively. This paper proposes a novel generic algorithm for incorporating existing nonlocal methods into the FE formulation of elasto-plastic constitutive models with volumetric hardening laws. Using the example of the modified Cam Clay (MCC) model this paper shows that instabilities are inevitable if the nonlocal plastic volumetric strains are directly used to update the hardening parameter. The paper further proposes a robust regularisation algorithm to overcome this issue. This algorithm has been implemented into a bespoke FE code and its capability in significantly reducing mesh dependency while maintaining numerical stability is verified through a series of numerical analyses of biaxial compression tests for highly overconsolidated clays. The performance of various existing nonlocal methods is also critically assessed.

Dynamic size effects across the scales: Localization in round notched bars

The strengthening and hardening of metals increase with diminishing size due to plastic strain gradient accompanied by Geometrically Necessary Dislocations (GNDs). In contrast, the inertia forces under severe dynamic loading conditions decrease. To shed the light on the interaction between the two mechanisms, the present work focuses on the localization in notched bars subject to high strain-rate tensile loading and first lays out a new computational model where the Fleck and Willis strain gradient plasticity theory is implemented into a dynamic finite element framework. The new framework allows accounting for strain gradient strengthening through the non-local plasticity material model and models the “smaller is stronger” mechanism (controlled by a built-in length parameter), while material inertia governs the “larger is more dynamic” effect. The work by Needleman (2018) [Effect of size on necking of dynamically loaded notched bars, Mech. Materials, 116:180-188] is taken as a starting point, and the present study first conveys the message that the scale-dependent dynamic effect observed in visco-plastic tensile bars also exists for rate-independent materials across scales (questioned by Needleman, 2018). Subsequently, the notched bar’s size and corresponding loading are scaled to the micrometer size range where plastic strain gradients influence the material response. Focusing attention on the shift between plastic localization in the notch and away from the notch (in the region subject to uni-axial tension) allows cementing this shift at the micron scale. The study shows that gradient strengthening owing the GNDs stabilizes the early localization in the notch and delays the shift between the localization mechanisms when the size of the notched bar diminishes while keeping the material length parameter constant. Moreover, severe gradient strengthening gives rise to a slow-growing neck-like (micron scale) localization mechanism where a neck-type deformation evolves near the notch — but not in the notch as oppose to larger bar sizes.

A Gradient-Enhanced Plasticity Based Phase-Field Model for Ductile Fracture Simulations

Ductile fracture is modeled by using a novel phase-field method of geometric type to avoid the use of the complicated discretization approaches for crack discontinuities. The plasticity model is defined by an over-nonlocal implicit gradient-enhanced framework, which is equivalent to the integral-type plasticity models and therefore strongly nonlocal. A modified phenomenological barrier function is used as the crack phase-field driving force by mainly considering the effects of the nonlocal plastic deformation under shear-dominated stress states. The ductile damage is assumed to solely affect the plastic energy stored capacity from the micro-mechanical perspective such that the proposed approach can be easily extended to more general loading conditions. The implementation of the proposed phase-field method is shown to be easily integrated into the commercial codes (e.g., ABAQUS) through the coupling use of several user interfaces. We present simulations of the shear band formation under axial compression and the ductile crack propagations in a single-edged notched plate, a slanted fracture specimen and a pure shear test specimen to elucidate the viability of the current nonlocal method. The numerical results adequately demonstrate that mesh dependency can be apparently alleviated if material softening occurs.

Effects of sand burial in integrated clonal systems of the invasive Carpobrotus edulis

ABSTRACT Sand burial represents one of the most common stresses for plant performance in coastal sand dunes. Clonal plants inhabiting sand dunes frequently experience situations in which the sand can bury part of the clone while other modules remain unburied. Here we present a field experiment in a coastal sand dune testing the effect of sand burial in the clonal Carpobrotus edulis. In the experiment, apical ramets were either or not subjected to burial in sand to a depth of 90% ramet height, and connected basal ramets were not subjected to sand burial. Sand burial significantly reduced survival and growth of apical ramets. Interestingly, we found a non-local plastic response in basal ramets that was dependent on the conditions experienced by their apical ramets. Thus, when apical ramets remained unburied a developmentally programmed division of labour was found. In this situation, basal ramets specialized in the acquisition of soil-based resources while apical ramets specialized in aboveground expansion. On the contrary, when apical ramets were subjected to sand burial, basal ramets increased the production of photosynthetic structures. This response could alleviate the light stress suffered by the buried apical ramets. We demonstrate that burial stress suffered by apical ramets of C. edulis triggered a compensatory response in their connected unburied ramets. This result highlights the importance of considering modular plasticity to understand plant responses to sand burial stress.

Nonlocal plastic softening algorithm and numerical investigation based on the Drucker–Prager criterion

Strain softening in geotechnical materials often leads to the ill condition of boundary value problems. That is, the regional governing equations or partial differential equations are no longer applicable under the original fixed solution conditions, and the over non-localization theory is used to solve the finite element equations to ensure the possibilities of the governing equations. Based on the Drucker–Prager (D–P) criterion, a nonlocal elastoplastic constitutive model and its numerical algorithm for geomaterials are established while considering the softening plasticity. Strain localization leads to high strain gradients and severe deformations in the zone. The introduction of characteristic lengths establishes the relationship between microstructure and macroscopic deformation. Accordingly, the surroundings will affect a certain point of a damaged rock material. When a Gaussian weight function is introduced, and the characterization is concerned on the damage point, the influence gradually decreases, and the characteristic length controls the influence range as the distance increases. When using the closest point projection method to calculate the stress regression of the D–P criterion, the plastic multiplier is weighted and averaged by the Gaussian weight function, and the local plastic multiplier is converted into a nonlocal plastic multiplier for numerical analysis. The results show that with the introduction of the weight function and the feature length, the localized zone width no longer depends on the grid size, that is, the grid sensitivity is overcome, and the result is more in line with the physical nature.

Characterization of size-affected non-uniform deformation of polycrystalline copper

In polycrystalline materials, with the increase in grain size relative to the macrostructure, the collective behavior of crystal grains affects their macroscopic deformation field. To investigate the effect of relative size on the mechanical behavior of polycrystalline materials, the interaction between the microstructure-induced non-uniform deformation and the specimen shape-induced non-uniform deformation was evaluated based on experimental and numerical studies of uniaxial tensile tests of polycrystalline copper specimens with a curved gauge section. The effects of the macroscopic stress gradient and grain size on the strain field of the specimen were evaluated using specimens with different curvature radii obtained from different thermal treatment conditions. The development of strain distribution was measured using the digital image correlation method. A high strain concentration was observed at the minimum cross-section region in the specimen with smaller grains, whereas such strain concentration was relaxed in the specimen with larger grains because a random strain distribution occurred owing to the polycrystalline structure. A full-scale crystalline plasticity finite element method simulation, under conditions similar to those in the experiment, was then performed. Deformation concentrated zone, in which the length and width depended on the grain size, occurred in the polycrystalline specimen. The cross-section of the specimen was locally reduced when the deformation concentrated zone reached the free surface, and the tensile force became smaller for the specimen with larger grains. To discuss the relative specimen size effect, the plastic strain was divided into local and nonlocal plastic strains. Both the experimental and simulation results clarified that the nonlocal plastic strain gradient evaluated in the finite volume region increased with the region-averaged stress. From these results, we proposed a constitutive equation for the plastic strain as a function of local stress and finite volume averaged stress. The nonlocal plastic work for the evaluation region, which is estimated using the nonlocal strain gradient, increased with the stress during the strain-hardening stage in both the simulation and experimental results.

Finite element analysis of deformation and failure mechanisms in nanoscale hexagonal cellular structures of metallic glasses

Cellular metallic glasses (MGs) have been found to be a potential candidate for structural and functional applications due to their attractive properties such as high strength to weight ratio, excellent energy absorption and enhanced plastic deformation. The experiments have shown transition in deformation mode from global failure caused by localization in a shear band to the local failure by damage confined to few cells with reduction in relative density of specimen from a large to moderate value. The mode of deformation again changes over to the collective buckling of ligaments through row by row collapse when the relative density is decreased to a sufficiently lower level. The atomistic simulations on nanoscale cellular MGs have also reported transition from localized but confined to few cells to almost homogeneous deformation with increasing cell size. They have also shown strain localization in a dominant shear band for cell spacing along diagonal direction above a threshold value which was correlated to shear bandwidth in monolithic MG. However, it is not clear as to why and how the shear band thickness in monolithic MG controls the threshold cell-spacing. Therefore, 2D plane strain finite element (FE) simulations of compressive loading are performed on nanoscale cellular MGs using thermodynamically consistent finite strain non-local plasticity model. The present FE simulations successfully predict the two transitions in deformation mode as observed in MD simulations and experiments. It is found that the interaction stress associated with the flow defects such as shear transformation zones (STZs) plays an important role in the deformation response of cellular MGs. Results show that the transition in deformation behavior is governed by the ratio of cell-wall thickness to the intrinsic material length associated with interaction stress. Also, the moderate change in sample size has marginal effect on the deformation response of MG cellular structure. The present work may provide guidelines for designing cellular MG structures capable of showing enhanced plastic deformation for practical engineering applications.

Parametric Study on an Integral-Type Nonlocal Elastoplasticity Model Regularized with Tikhonov-Phillips Method

An integral-type nonlocal elastoplasticity model is proposed and formulated in a one-dimensional scenario to address the strain-softening problem. This nonlocal model includes not only the nonlocal plasticity but also the nonlocal elasticity. The resulting elastic constitutive equation and the plastic consistency condition are ill-posed Fredholm integral equations of the first kind whose solutions are instable without appropriate regularization. In this paper, the Tikhonov-Phillips regularization method is used to reformulate the integral-differential consistency condition to obtain an approximate solution of the local internal variable, which is stable and smooth. A detailed parametric study is carried out to examine the influences of the regularization parameter, the internal length scale, the plastic modulus ratio, and the tolerance of the nonlocal yield condition on the regularized solutions. A numerical example shows that the solutions from the proposed model and the regularization method are mesh-independent. A comparison of the results from the proposed model with those from the nonlocal plasticity model and from the gradient-dependent plasticity model shows good agreement.

A non-local plasticity model for porous metals with deformation-induced anisotropy: Mathematical and computational issues

A non-local (gradient) plasticity model for porous metals that accounts for deformation-induced anisotropy is presented. The model is based on the work of Ponte Castañeda and co-workers on porous materials containing randomly distributed ellipsoidal voids. It takes into account the evolution of porosity and the evolution/development of anisotropy due to changes in the shape and the orientation of the voids during plastic deformation. A “material length” ℓ is introduced and a “non-local” porosity is defined from the solution of a modified Helmholtz equation with appropriate boundary conditions, as proposed by Geers et al. (2001); Peerlings et al. (2001). At a material point located at x, the non-local porosity f(x) can be identified with the average value of the “local” porosity floc(x) over a sphere of radius R≃3ℓ centered at x. The same approach is used to formulate a non-local version of the Gurson isotropic model. The mathematical character of the resulting incremental elastoplastic partial differential equations of the non-local model is analyzed. It is shown that the hardening modulus of the non-local model is always larger than the corresponding hardening modulus of the local model; as a consequence, the non-local incremental problem retains its elliptic character and the possibility of discontinuous solutions is eliminated. A rate-dependent version of the non-local model is also developed. An algorithm for the numerical integration of the non-local constitutive equations is developed, and the numerical implementation of the boundary value problem in a finite element environment is discussed. An analytical method for the required calculation of the eigenvectors of symmetric second-order tensors is presented. The non-local model is implemented in ABAQUS via a material “user subroutine” (UMAT or VUMAT) and the coupled thermo-mechanical solution procedure, in which temperature is identified with the non-local porosity. Several example problems are solved numerically and the effects of the non-local formulation on the solution are discussed. In particular, the problems of plastic flow localization in plane strain tension, the plane strain mode-I blunt crack tip under small-scale-yielding conditions, the cup-and-cone fracture of a round bar, and the Charpy V-notch test specimen are analyzed.

Theoretical Estimation of The Influence of Plastic Deformation on Average Coefficient of Friction in the Process of Nanostructuring Burnishing of Metal Samples

The problem of burnishing metal samples with a smooth surface using a spherical indenter is considered. The nonlocal plastic strains in the burnished sample are taken into account. The dimensionless parameters are found that determine the effective friction coefficient under the influence of external oscillations. Numerical modeling is used to obtain the dependencies of the effective friction coefficient on the dimensionless parameters.

Finite element analysis of tensile deformation of nanoglass-metallic glass laminate composites

Nanolaminate composites consisting of alternate layers of Nanoglasses (NGs) and metallic glasses (MGs) have shown enhanced tensile ductility without great penalty on strength. Recent atomistic simulations on such NG-MG nanolaminate composites have revealed that peak stress attained in these materials do not follow rule of mixture. Also, a transition in deformation behaviour takes place when MG layer thickness is reduced below a threshold level which is correlated with average size of glassy grain in NG layer. However, the mechanistic reasons for this correlation is not well understood. Therefore, continuum simulations of tensile loading on NG-MG laminate composites are performed using thermodynamic consistent non-local plasticity model. Results show that interaction stress associated with flow defects such shear transformation zones (STZs) plays a pivotal role in the deformation response of laminate composites. Also, shear band width in these materials, scales with intrinsic material length associated with the interaction stress. Further, the material length with respect to MG layer thickness governs the transition in deformation behaviour. The present work may provide guidelines in developing highly ductile NG-MG laminate composites for practical engineering applications.

Dynamic Model of Elastoplastic Normal Collision of a Spherical Particle with a Half-Space with Allowance for Adhesion Interaction in a Contact Zone

The problem of a collision of a spherical homogeneous particle with a half-space is solved numerically. The coefficient of the normal velocity restitution is calculated as a function of the initial velocity in the presence of the mechanical energy dissipation due to plastic deformations and adhesion interaction between the surfaces of contacting bodies. The strain hardening effect is taken into account using the criterion of nonlocal plasticity introduced before. The study is carried out in dimensionless parameters, which makes it universal. The analytical expression is found for the critical initial velocity above which a plastic deformation starts. The simulation results agree well with the available experimental data.

Size effect in concrete beams under bending – influence of the boundary layer and the numerical description of cracks

In the paper the size effect phenomenon in concrete is analysed. The results of numerical simulations of using FEM on geometrically similar un-notched and notched concrete beams under bending are presented. Concrete beams of four different sizes and five different notch heights under three-point bending test were simulated. In total 18 beams were analysed. Two approaches were used to describe cracks in concrete. First, eXtended Finite Element Method (XFEM) describing cracks as discrete cohesive ones with bilinear softening was chosen. Alternatively, an elasto-plastic constitutive law with Rankine criterion, associated flow rule and bilinear softening was defined. In order to ensure mesh-independent FE results, a non-local theory in an integral format as a regularisation technique was applied in the softening regime. In both approaches the influence of the decrease of the material parameters (mainly fracture energy) in the boundary layer on obtained maximum loads was studied. Additionally the influence of the averaging method in non-local plasticity was also examined. Obtained results were compared with experimental outcomes available in literature.

Динамическая модель упруго-пластического нормального столкновения сферической частицы с полупространством с учетом адгезионного взаимодействия в зоне контакта

AbstractThe problem of a collision of a spherical homogeneous particle with a half-space is solved numerically. The coefficient of the normal velocity restitution is calculated as a function of the initial velocity in the presence of the mechanical energy dissipation due to plastic deformations and adhesion interaction between the surfaces of contacting bodies. The strain hardening effect is taken into account using the criterion of nonlocal plasticity introduced before. The study is carried out in dimensionless parameters, which makes it universal. The analytical expression is found for the critical initial velocity above which a plastic deformation starts. The simulation results agree well with the available experimental data.

A non-local plasticity model of stimulated volume evolution during hydraulic fracturing

Hydraulic fracturing in naturally fractured rocks often leads to the creation of a stimulated zone in which the target rock formation is deformed and fractured by the reactivation and shear dilation of natural fractures and the plastic deformation, damaging, and fracturing of the bulk. In this paper, we present a novel mathematical model with the goal of simulating the evolution of the stimulated volume during hydraulic fracturing. This was achieved by introducing an equivalent continuum non-local poro-elastic-plastic zone of enhanced permeability for the stimulated region, characterized by an internal length scale. The non-local plastic constitutive behavior of the rock, combined with the classical Biot’s poroelastic theory, was implemented using a new implicit C0 non-local finite element method. A predictor-corrector return algorithm for the non-local plasticity model was formulated as an extension of the classical plasticity algorithm. To improve the performance of the iterative solution scheme, a consistent algorithmic stiffness tangent modulus was developed. First, the elastic-plastic constitutive behavior of the proposed methodology is verified using the standard non-porous biaxial compression test with strain softening behavior. Next, it is verified that the poro-elastic-plastic model correctly simulates the evolution of the stimulated zone and the subsequent change in the flow and fluid pressure for several hydraulic fracturing examples under various far-field in-situ stress conditions. Lastly, the non-local poro-elastic-plastic model is shown to be mesh-independent and capable of capturing a wide range of complex fracturing behavior.

Nonlocal plasticity modelling of strain localisation in stiff clays

The paper addresses the numerical simulation of strain localisation in stiff clays that exhibit softening behaviour. An elastoplastic constitutive model developed to incorporate key features of stiff clay behaviour is described first. A non-local formulation is then introduced for the regularisation of the analysis of localisation. A series of analyses were conducted to explore relevant aspects of the numerical simulation of localisation. A 3D analysis was also performed to assess the suitability of the approach presented for 3D applications. Finally, application to the simulation of a laboratory test on Beaucaire marl results in an excellent reproduction of experimental observations.

Characterizing the Stimulated Reservoir Volume During Hydraulic Fracturing-Connecting the Pressure Fall-Off Phase to the Geomechanics of Fracturing

Microseismic imaging of the hydraulic fracturing operation in the naturally fractured rocks confirms the existence of a stimulated volume (SV) of enhanced permeability. The simulation and characterization of the SV evolution is uniquely challenging given the uncertainty in the nature of the rock mass fabrics as well as the complex fracturing behavior of shear and tensile nature, irreversible plastic deformation and damage. In this paper, the simulation of the SV evolution is achieved using a nonlocal poromechanical plasticity model. Effects of the natural fracture network are incorporated via a nonlocal plasticity characteristic length, ℓ. A nonlocal Drucker–Prager failure model is implemented in the framework of Biot's theory using a new implicit C0 finite element method. First, the behavior of the SV for a two-dimensional (2D) geomechanical injection problem is simulated and the resulting SV is assessed. It is shown that breakdown pressure and stable fracturing pressure are the natural outcomes of the model and both depend upon ℓ. Next, the post-shut-in behavior of the SV is analyzed using the pressure and pressure derivative plots. A bilinear flow regime is observed and it is used to estimate the flow capacity of the SV. The results show that the flow capacity of the SV increases as ℓ decreases (i.e., as the SV behaves more like a single hydraulic fracture); however, for 0.1m≤ℓ≤1m, the calculated flow capacity indicates that the conductivity of the SV is finite. Finally, it is observed that as ℓ tends to zero, the flow capacity of the SV tends to infinity and the SV behaves like a single infinitely conducting fracture.