Nonlocal Formulation Research Articles

Displacement-driven approach to nonlocal elasticity

This study presents a physically-consistent displacement-driven reformulation of the concept of action-at-a-distance, which is at the foundation of nonlocal elasticity. In contrast to the class of existing approaches that adopts an integral stress–strain constitutive relation, the displacement-driven approach is predicated on an integral strain–displacement relation. One of the most remarkable consequence of this reformulation is that the (total) strain energy is guaranteed to be a convex and positive-definite function without imposing any constraint on the symmetry of the nonlocal kernel. This feature is critical to enable the application of nonlocal formulations to general continua exhibiting asymmetric interactions; ultimately a manifestation of material heterogeneity. Remarkably, the proposed approach also enables a strong (point-wise) satisfaction of the locality recovery condition and of the laws of thermodynamics, which are not foregone conclusions in most classical nonlocal elasticity theories. Additionally, the formulation is frame-invariant and the nonlocal operator remains physically consistent at material interfaces and domain boundaries. The study is complemented by a detailed analysis of the dynamic response of the nonlocal continuum and of its intrinsic dispersion, leading to the consideration that the choice of a nonlocal kernel should depend on the specific material. Examples of either exponential or power-law kernels are presented in order to demonstrate the applicability of the method to different classes of nonlocal media. The ability to admit generalized kernels reinforces the generalized nature of the displacement-driven approach over existing integral methodologies, which typically lead to simplified differential models based on exponential kernels. The theoretical formulation is also leveraged to perform numerical simulations of the linear static response of nonlocal beams and plates further illustrating the intrinsic consistency of the approach, which is free from unwanted and unrealistic boundary effects.

Dynamics of interfacial gravity–capillary waves in three-dimensional fluids of great depth

This paper proposes a new isotropic bidirectional model for weakly nonlinear gravity–capillary waves propagating between two incompressible, inviscid, and immiscible fluids of different densities. The newly developed equation is a generalization of the celebrated two-dimensional Benjamin equation. It is derived based on the nonlocal formulation of water wave (i.e., the Ablowitz–Fokas–Musslimani formulation) and computed with the modified exponential time-differencing method. It is found that horizontally two-dimensional, fully localized traveling waves (known as lumps) exist in the model equation, and plane solitary waves are unstable subject to long transverse perturbations, which evolve into groups of lumps in the long-term dynamics. When considering topographical disturbances on the rigid upper boundary, the nonlinear effect becomes important when a uniform flow passes beneath a locally confined topography with a near-critical speed, and the phenomenon of time-periodic shedding of lumps occurs. Unlike the near-critical situation, the subcritical flows usually generate steady elevation waves, while the supercritical ones produce a V-shaped pattern of wake lines. Furthermore, it is shown that a uniformly accelerated motion can also generate lumps provided that the flow stays in the transcritical regime for a considerably long time.

Numerical simulation of underground excavations in an indurated clay using non-local regularisation. Part 1: formulation and base case

The paper presents the numerical simulation of an underground excavation in the Callovo-Oxfordian argillaceous formation, performed in the Meuse/Haute-Marne underground research laboratory, within the context of deep geological nuclear waste disposal. The constitutive model adopted incorporates a number of characteristic features of the hydromechanical behaviour of the Callovo-Oxfordian claystone, and other indurated clays, including anisotropy and time-dependent deformations. Model parameters have been largely determined from available experimental results. Particular attention has been given to the incorporation of a non-local formulation in order to simulate localised deformations objectively. It is shown that the numerical analysis is able to provide a very satisfactory reproduction of the extent and configuration of the excavation-induced fractured zone, the development of rock displacements and the evolution of water pressures in the rock. The analysis is also able to provide a deeper insight into the mechanisms underlying the generation of water pressures inside the rock mass.

Numerical integration in G/XFEM analysis of 2-D fracture mechanics problems for physically nonlinear material and cohesive crack propagation

PurposeThe aim of this paper is to present a novel data transfer technique to simulate, by G/XFEM, a cohesive crack propagation coupled with a smeared damage model. The efficiency of this technique is evaluated in terms of processing time, number of Newton–Raphson iterations and accuracy of structural response.Design/methodology/approachThe cohesive crack is represented by the G/XFEM enrichment strategy. The elements crossed by the crack are divided into triangular cells. The smeared crack model is used to describe the material behavior. In the nonlinear solution of the problem, state variables associated with the original numerical integration points need to be transferred to new points created with the triangular subdivision. A nonlocal strategy is tailored to transfer the scalar and tensor variables of the constitutive model. The performance of this technique is numerically evaluated.FindingsWhen compared with standard Gauss quadrature integration scheme, the proposed strategy may deliver a slightly superior computational efficiency in terms of processing time. The weighting function parameter used in the nonlocal transfer strategy plays an important role. The equilibrium state in the interactive-incremental solution process is not severely penalized and is readily recovered. The advantages of such proposed technique tend to be even more pronounced in more complex and finer meshes.Originality/valueThis work presents a novel data transfer technique based on the ideas of the nonlocal formulation of the state variables and specially tailored to the simulation of cohesive crack propagation in materials governed by the smeared crack constitutive model.

Numerical simulation of underground excavations in an indurated clay using non-local regularisation. Part 2: sensitivity analysis

A sensitivity study is presented to evaluate the influence of different parameters on the simulation of an underground excavation in the Callovo-Oxfordian (COx) argillaceous formation performed in the Meuse/Haute-Marne underground research laboratory. An elasto-viscoplastic constitutive law representing the characteristic behaviour of indurated mudrocks and stiff clays has been employed. It incorporates anisotropy, strain-softening, creep deformations and dependence of permeability on damage. In addition, a non-local formulation, able to simulate localised deformations objectively, has been incorporated in the analyses. The following features affecting the excavation have been studied: initial stress, strength and stiffness anisotropy, strength parameters, hydraulic and hydromechanical parameters, and scale effects. A simulation reported in a companion paper provides the base case for benchmarking. The results are compared in terms of extent and configuration of the excavation fractured zone, vertical and horizontal tunnel convergences, and the development and evolution of pore pressures in the rock. From the comparisons, an enhanced understanding of the hydromechanical mechanisms associated with underground excavations in COx claystone, and other similar argillaceous materials, has been achieved.

A mechanism-based gradient damage model for metallic fracture

A new gradient-based formulation for predicting fracture in elastic–plastic solids is presented. Damage is captured by means of a phase field model that considers both the elastic and plastic works as driving forces for fracture. Material deformation is characterised by a mechanism-based strain gradient constitutive model. This non-local plastic-damage formulation is numerically implemented and used to simulate fracture in several paradigmatic boundary value problems. The case studies aim at shedding light into the role of the plastic and fracture length scales. It is found that the role of plastic strain gradients is twofold. When dealing with sharp defects like cracks, plastic strain gradients elevate local stresses and facilitate fracture. However, in the presence of non-sharp defects failure is driven by the localisation of plastic flow, which is delayed due to the additional work hardening introduced by plastic strain gradients.

A Generalized Interpolation Material Point Method for Shallow Ice Shelves. 2: Anisotropic Nonlocal Damage Mechanics and Rift Propagation.

Ice shelf fracture is responsible for roughly half of Antarctic ice mass loss in the form of calving and can weaken buttressing of upstream ice flow. Large uncertainties associated with the ice sheet response to climate variations are due to a poor understanding of these fracture processes and how to model them. Here, we address these problems by implementing an anisotropic, nonlocal integral formulation of creep damage within a large‐scale shallow‐shelf ice flow model. This model can be used to study the full evolution of fracture from initiation of crevassing to rifting that eventually causes tabular calving. While previous ice shelf fracture models have largely relied on simple expressions to estimate crevasse depths, our model parameterizes fracture as a progressive damage evolution process in three‐dimensions (3‐D). We also implement an efficient numerical framework based on the material point method, which avoids advection errors. Using an idealized marine ice sheet, we test the creep damage model and a crevasse‐depth based damage model, including a modified version of the latter that accounts for damage evolution due to necking and mass balance. We demonstrate that the creep damage model is best suited for capturing weakening and rifting over shorter (monthly/yearly) timescales, and that anisotropic damage reproduces typically observed fracture patterns better than isotropic damage. Because necking and mass balance can significantly influence damage on longer (decadal) timescales, we discuss the potential for a combined approach between models to best represent mechanical weakening and tabular calving within long‐term simulations.

Nonlocal wrinkling instabilities in bilayered systems using peridynamics

Wrinkling instabilities occur when a stiff thin film bonded to an elastic substrate undergoes compression. Regardless of the nature of compression, this phenomenon has been extensively studied through local models based on classical continuum mechanics. However, the experimental behavior is not yet fully understood and the influence of nonlocal effects remains largely unexplored. The objective of this paper is to fill this gap from a computational perspective by investigating nonlocal wrinkling instabilities in a bilayered system. Peridynamics (PD), a nonlocal continuum formulation, serves as a tool to model nonlocal material behavior. This manuscript presents a methodology to precisely predict the critical conditions by employing an eigenvalue analysis. Our results approach the local solution when the nonlocality parameter, the horizon size, approaches zero. An experimentally observed influence of the boundaries on the wave pattern is reproduced with PD simulations which suggests nonlocal material behavior as a physical origin. The results suggest that the level of nonlocality of a material model has quantitative influence on the main wrinkling characteristics, while most trends qualitatively coincide with predictions from the local analytical solution. However, a relation between the film thickness and the critical compression is revealed that is not existent in the local theory. Moreover, an approach to determine the peridynamic material parameters across a material interface is established by introducing an interface weighting factor. This paper, for the first time, shows that adding a nonlocal perspective to the analysis of bilayer wrinkling by using PD can significantly advance our understanding of the phenomenon.

Modelling of nonlocal damage and failure in ductile steel sheets under multiaxial loading

A continuum damage model is proposed capturing the influence of damage on the stress tensor for ductile steel sheets under multiaxial loading conditions. The model consists of a critical strain, representing damage initiation, as well as a failure strain. Both are Hosford-Coulomb type functions of the triaxiality and the Lode parameter. In order to prevent the pathological mesh-sensitivity for continuum damage models, the approach is extended towards nonlocal damage evolution. Thereby, two different established methods are compared: the integral based nonlocal formulation and the gradient enhanced damage method. A critical assessment of both methodologies is carried out by comparing the simulation results with test data of various tensile, shear and biaxial specimens, made of the microalloyed steel HX340LAD. The outcome demonstrate the superiority of the gradient enhanced approach for industrial applications, since nearly the same prediction accuracy is obtained as with the integral nonlocal approach, but the simulation time is significantly shorter.

A thermodynamically nonlocal damage model using a surface-residual-based nonlocal stress

ABSTRACT In this research, a surface-residual-based nonlocal stress was introduced into nonlocal damage theory to describe the long-range actions among microstructures that were excluded in the definition of Cauchy stress. By using the surface-residual-based nonlocal stress tensor, a thermodynamically consistent nonlocal integral damage model was established to simulate the strain localization behavior for elastic-brittle damage problems. In this model, both the strain and the damage were taken as nonlocal variables in the free energy function, and the integral-type damage constitutive relationships and the evolution equation were derived via thermodynamic laws in order to ensure the self-consistency within the thermodynamic framework. Based on the nonlocal damage formulations using a real nonlocal stress concept, we simulated the strain localization phenomenon in an elastic bar subjected to uniaxial tension. The results showed clear localizing and softening features of strain in the damage zone, and the boundary effects arising from the nonlocal surface residual were illuminated. Furthermore, the strain localization behaviors for different internal characteristic lengths were simulated, through which we found that the characteristic length was comparable to the size of the strain localization zone.

Derivation of nonlocal FEM formulation for thermo-elastic Timoshenko beams on elastic matrix

Free thermal vibration analysis of nanobeams surrounded by an elastic matrix is examined via nonlocal elasticity and Timoshenko beam theories in this article. Elastic matrix is formulated with a two-parameter elastic foundation modelled by combining Winkler and Pasternak assumptions. The equation of motion for free vibration is reached via Hamilton’s principle and solved by analytical method (separation of variable). However, since the separation of variable cannot be applied for boundary conditions other than simply supported nanobeams, a weighted residue-based finite element formulation is developed. Within the scope of numerical results, firstly, it is understood from the comparisons given for nondimensional frequencies that the nonlocal finite element formulation has a high accuracy and then, using this formulation, nondimensional frequencies of nanobeams with different boundary conditions are computed under different parameters. Additionally, the detailed discussions of the numerical results are presented. Finally, by giving some general results, the effect of size dependency and environmental factors on the dynamic behavior of nanobeams is expressed.

A stabilized computational nonlocal poromechanics model for dynamic analysis of saturated porous media

AbstractIn this article we formulate a stable computational nonlocal poromechanics model for dynamic analysis of saturated porous media. As a novelty, the stabilization formulation eliminates zero‐energy modes associated with the original multiphase correspondence constitutive models in the coupled nonlocal poromechanics model. The two‐phase stabilization scheme is formulated based on an energy method that incorporates inhomogeneous solid deformation and fluid flow. In this method, the nonlocal formulations of skeleton strain energy and fluid flow dissipation energy equate to their local formulations. The stable coupled nonlocal poromechanics model is solved for dynamic analysis by an implicit time integration scheme. As a new contribution, we validate the coupled stabilization formulation by comparing numerical results with analytical and finite element solutions for one‐dimensional and two‐dimensional dynamic problems in saturated porous media. Numerical examples of dynamic strain localization in saturated porous media are presented to demonstrate the efficacy of the stable coupled poromechanics framework for localized failure under dynamic loads.

Compaction localization and anisotropic permeability evolution

The oedometric compaction of materials with collapsible structure is investigated from a numerical point of view. Such materials form planar zones of localized volumetric deformation that may or may not be accompanied by grain crushing. The purpose of the present work is to investigate the evolution of the permeability and its anisotropy as the result of the formation of compaction bands. The finite element method is used for the simulations with the commercial software ABAQUS. The mechanical response is calibrated on experimental results from the literature. Due to the localization of the deformation, the problem is not numerically stable and shows mesh dependence when a simple material model is used. To resolve the issue a nonlocal formulation is implemented. Drained oedometric tests are simulated. For the permeability different formulations are used, either ignoring the anisotropic evolution of the pore network or accounting for it. The results show that permeability anisotropy at the microscale needs to be considered, because otherwise the predicted reduction in permeability is qualitatively not in agreement with experimental results. On the whole, it is concluded that the permeability reduction following the formation of compaction bands can be highly anisotropic both at the micro and the macro scale.

A heat conduction equation for heterogeneous media and its connections to various known equations

We derive the governing equation of the macroscopically average temperature field in a heterogeneous medium. Although the constituents of the heterogeneous medium obey Fourier's law, the average temperature field is governed by a spatiotemporally nonlocal integro-differential formulation. Then, we find that the average fluctuation of the temperature in the matrix region of the heterogeneous medium plays a key role in determining the various degenerate forms of the governing equation. Depending on the value of this average fluctuation, the governing equation can degenerate into the well-known equations in the literature, including the Jeffreys-type equation, Nunziato equation, Gurtin and Pipkin equation, peridynamic formulation, and dual-phase-lag (DPL) equation. These connections establish the relations between the coefficients in these equations with the properties of the constituents of a heterogeneous medium, and provide guidelines for their applications to the prediction of heat conduction in heterogeneous media.

A refined nonlocal isogeometric model for multilayer functionally graded graphene platelet-reinforced composite nanoplates

This article presents an effective and simple approach based on the four-variable refined plate theory (RPT) and isogeometric analysis (IGA) for bending and free vibration analyses of multilayer functionally graded graphene platelet-reinforced composite (FG GPLRC) nanoplates. The Eringen’s nonlocal elasticity theory is employed to take account of the size-dependent effect of nanoplates. Different distribution patterns of graphene platelets (GPLs) including uniform and non-uniform in the polymer matrix are considered. Governing equations of FG GPLRC nanoplates are then deduced from the principle of virtual work and resolved utilizing NURBS basis functions in the IGA framework. Accordingly, the requirement of third-order derivatives of approximate variables in the nonlocal formulation is fulfilled. Obtained outcomes indicate that the GPLs reinforcement can dramatically improve the stiffness of nanoplates, and GPLs rich at the bottom and the top of the nanoplate can be considered as the best reinforcing effect. Several new results are also considered as benchmark solutions for further studies on the FG GPLRC nanoplates.

Nonlinear thermoelastic fractional-order model of nonlocal plates: Application to postbuckling and bending response

This study presents a comprehensive frame-invariant fractional-order framework for the geometrically nonlinear bending and postbuckling analysis of nonlocal plates subject to combined thermal and mechanical loads. The fractional-order kinematic framework, unlike classical nonlocal formulations based on integer-order mechanics, is positive-definite and thermodynamically consistent. The positive-definite nature enables the application of variational principles to derive well-posed thermomechanical governing equations. A direct advantage of these theoretical advancements is reflected in the ability of the fractional-order approach to enable an energy-based asymptotic method obtained by extending the classical (local) Koiter’s approach. This method allows the analytical assessment of the impact of nonlocal interactions on the nature of the bifurcation and post-critical response of slender structures when subject to compressive loads. Further, the positive-definite nature of the fractional-order approach allows the development of an accurate fractional-order finite element method enabling the simulation of nonlocal structures subject to any combination of thermomechanical loads and boundary conditions. The above stated advantages are particularly important for a comprehensive and consistent analysis of nonlocal structures in the postbuckling regime. The theoretical framework is used to provide analytical and numerical insights on the effect of combined thermal and mechanical loads on the nonlinear response of nonlocal plates. Numerical results also demonstrate the high-level consistency of the framework when applied to the nonlinear thermoelastic analysis of nonlocal plates under different loading and boundary conditions.

Coupling Peridynamics with the Classical Methods for Modeling Hydraulic Fracture Growth in Heterogeneous Reservoirs

Summary Hydraulic fracture (HF) modeling is a multiscale and multiphysics problem. It should capture various effects, including those of in-situ stresses, poroelasticity, and reservoir heterogeneities at different length scales. A peridynamics (PD)-based hydraulic fracturing simulator has been demonstrated to reproduce this physics accurately. However, accounting for such details leads to a reduction in computational speed. In this paper, we present a novel coupling of the PD-based simulator with numerically efficient finite element methods (FEMs) and finite volume methods (FVMs) to achieve a significant improvement in computational performance. Unlike classical methods, such as FEM and FVM that solve differential equations, PD uses an integral formulation to circumvent the undefined spatial derivatives at crack tips. We implemented four novel coupling schemes of our PD-based simulator with FEM and FVM: static PD region scheme, dynamic PD region scheme, adaptive mesh refinement scheme, and dynamic mesh coarsening scheme. PD equations are solved using a refined mesh close to the fracture, whereas FE/FV equations are solved using a progressively coarser mesh away from the fracture. As the fracture grows, a dynamic conversion of FE/FV cells to PD nodes and adaptive mesh refinement are incorporated. To improve the performance further, the dynamic mesh coarsening scheme additionally converts the fine PD nodes back to coarse FE/FV cells as the HF grows in length. The coupling schemes are verified against the Kristianovich-Geertsma-de Klerk (KGD) fracture propagation problem. No spurious behavior is observed near the transition between PD and FE/FV regions. In the first three coupling schemes, the computational runtime for single fracture propagation is reduced by up to 10, 20, and 50 times, respectively, compared to a pure PD model. Laboratory experiments on the interaction of an HF with a natural fracture (NF) are revisited. The model captures complex fracture behavior, such as turning in the case of low stress contrast and low angle of interaction, kinking for higher stress contrast or higher angle of interaction, and fracture crossing for near-orthogonal NFs. Moreover, several previously reported phenomena, including fracture propagation at an angle to the principal stress directions, competing fracture growth from multiple closely spaced clusters, and interaction with layers of varying mechanical properties are successfully modeled. Thus, the coupling of PD with FEM and FVM offers an innovative and fundamentally comprehensive solution to alleviate the high computational costs typically associated with the pure PD-based hydraulic fracturing simulations. At the same time, these coupling schemes retain the versatility of the nonlocal PD formulation at modeling the evolution of arbitrary material damage, commonly observed during HF propagation in complex heterogeneous reservoirs.

Coupling of non‐ordinary state‐based peridynamics and finite element method for fracture propagation in saturated porous media

AbstractIn the present manuscript fracture propagation in a saturated porous medium is modeled based on the classical Biot theory, where solid skeleton and fluid flow are represented by separate two layers. The non‐ordinary state‐based peridynamics (NOSBPD) layer is employed to capture deformation including fracturing of the solid skeleton, while the fluid flow is controlled by the finite element method (FEM) layer. The interaction between the layers is realized by considering the effects of pore pressure from the FEM layer on the NOSBPD layer and, vice versa, the effect of the volumetric strain, porosity, and permeability variations from the NOSBPD layer on the FEM layer. The coupling terms retain their parent characteristics, that is, the interaction term in the momentum balance equation is approximated by the local FEM formulation whereas the interaction term in the mass balance equation is approximated by the nonlocal NOSBPD formulation. By doing so, the model retains the flexibility of coupling two independent discretizations. The coupled system is solved by a fully implicit solution scheme. The accuracy of the proposed method has been verified against available closed‐form solutions and published numerical approaches for the pressure‐ and fluid‐driven facture propagation problems.

Modelling the behaviour of sensitive clays experiencing large deformations using non-local regularisation techniques

This paper presents a methodology to simulate the strain-softening-hardening response of sensitive clays when subjected to extensive remoulding. A non-local strain-softening technique is implemented into the Abaqus Finite Element Analysis (FEA) software through its user defined material subroutine UMAT, in the form of a critical state based strain-softening constitutive model. For the comparison of conventional and non-local formulations, a series of analyses have been performed investigating mesh dependency issues and the validity of a simple softening-scaling rule for practical applications. The non-local model significantly reduces solution mesh-dependency for small-strain analyses of biaxial shear and updated Lagrangian analyses of buried pipe uplift. Finally, the non-local model is applied to simulate undrained cycles of penetration and the consolidated-undrained capacity of a T-bar in kaolin clay. This analysis highlights the limitations of the use of standard element tests (e.g. triaxial tests) to calibrate the model input parameters that control the strain-softening aspect of the constitutive law, and the need for a constitutive law that captures the partial recovery of sensitivity during consolidation.

A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type—part II: boundary-value problems in the one-dimensional case

This paper is the second in a series of two that deal with a generalized theory of nonlocal elasticity of n-Helmholtz type. This terminology is motivated by the fact that the attenuation function (kernel) of the integral type nonlocal constitutive equation is the Green function associated with a generalized Helmholtz differential operator of order n. In the first paper, the governing equations have been derived and supported by suitable thermodynamic arguments. In this second paper, the proposed nonlocal model is specialized for the one-dimensional case to solve boundary-value problems. First, the relevant higher-order nonstandard boundary conditions in the differential (or, more precisely, integro-differential) version of the theory are derived. These boundary conditions are consistent with the particular family of attenuation functions adopted in the integral formulation. Then, some simple applications to statics and dynamics problems are presented. In particular, the theory is used to capture the static response and to perform free vibration analysis of a discrete lattice model with periodic microstructure (mass-and-spring chain) featured by nearest neighbor and next nearest neighbor particle interactions. In the latter case, boundary effects arise at the two lattice ends that are well captured by the proposed nonlocal continuum formulation. The nonlocal material parameters are identified a priori by matching the dispersion curve of the discrete lattice model, and a comparison in terms of attenuation function is also presented.