Nonlocal Continuum Research Articles

Dynamics of Stress-Driven Two-Phase Elastic Beams.

The dynamic behaviour of micro- and nano-beams is investigated by the nonlocal continuum mechanics, a computationally convenient approach with respect to atomistic strategies. Specifically, size effects are modelled by expressing elastic curvatures in terms of the integral mixture of stress-driven local and nonlocal phases, which leads to well-posed structural problems. Relevant nonlocal equations of the motion of slender beams are formulated and integrated by an analytical approach. The presented strategy is applied to simple case-problems of nanotechnological interest. Validation of the proposed nonlocal methodology is provided by comparing natural frequencies with the ones obtained by the classical strain gradient model of elasticity. The obtained outcomes can be useful for the design and optimisation of micro- and nano-electro-mechanical systems (M/NEMS).

NONLOCAL CONTINUUM ADAPTIVE ELASTIC BONE-COLUMN BUCKLING MODEL

It is well argued that stability-initiated failure dominates, especially in older bone, because of the instability of single trabeculae which is prone to inelastic buckling at stresses far less than expected for strength-based failure. It is also well known that when several horizontal struts have disappeared, trabecula fails due to compression-buckling load. In this contribution, our main goal is to improve, from theoretical point of view, the mechanistic understanding of bone buckling failure which is known to be at the core of important clinical problems. For that and with respect to previous works, an attempt is made in order to establish a simplified adaptive-beam buckling model, formulated within the context of the nonlocal adaptive continuum mechanics, from which numerical computations were performed in order to get a better knowledge about bone-column buckling mechanism affected by both bone density and bone density gradient distributions restricted to Euler–Bernoulli beam theory. An attempt is made to compare the experimental data with the response of our simplified model. For that, controlled buckling tests of single trabeculae were carried out from three medial tibia end sections (knee joint).

Size-dependent nonlinear harmonically soft excited oscillations of nonlocal strain gradient FGM composite truncated conical microshells with magnetostrictive facesheets

This paper investigates the size-dependent nonlinear primary resonant dynamics of truncated conical microshells (TSMs) made of magnetostrictive facesheets with a functionally graded material (FGM) under external magnetic field and mechanical harmonic soft excitation. To accomplish this purpose, a framework is established using a microstructure-dependent third-order shear deformation shell (TSDS) model based on nonlocal strain gradient continuum elasticity. By employing discretized differential operators achieved from generalized differential quadrature (GDQ) technique, a numerical solution is adopted to acquire nonlocal strain gradient amplitude- and frequency-responses for the primary resonant dynamics of FGM-TCMs. The jump phenomenon related to the loss of first stability branch and jumping down to lower stable branches could occur in the nonlinear primary resonances of FGM composite TCMs. It is illustrated that frequency-response bends to right by the hardening-type of nonlinearity. This pattern is enhanced by nonlocal size effect, which is weakened under strain gradient size dependency. Also, it can be observed that at higher material gradient index values, the hardening-type of nonlinearity enhances and nonlocal strain gradient frequency-response bends to right more considerably.

Isogeometric analysis of cracks with peridynamics

Isogeometric analysis (IGA) is an important mesh-free method that provides the technique for the integration of computer-aided design and analysis. However, its formulation is based on classical continuum mechanics and is not suitable for crack propagation problems. The peridynamics (PD) theory is based on the non-local integral equation, which avoids the discontinuous problem of classical continuum mechanics and can solve the crack problems. In this paper, a coupling model of IGA and bond-based PD (IGA-PD) is proposed to provide the solution to crack problem of model with exact geometry. We prove that in the IGA-PD coupling model, the PD nodes can be constructed by using the control net from IGA. The implicit and explicit formulas of the PD model are combined with IGA formulations based on the force balance principle. Through setting the isogeometric model on the domain near the boundaries, the surface effect of the PD model can be relieved. The IGA-PD model fully utilizes the advantages of non-local continuum theory and improves the precision and computational efficiency in calculation of crack problems. The coupling method is straightforward and effective, and can be integrated within existing IGA codes. Examples of static problems and crack propagations are given to prove the effectiveness of the proposed method.

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.

Size-dependent nonlinear bending behavior of porous FGM quasi-3D microplates with a central cutout based on nonlocal strain gradient isogeometric finite element modelling

With the aid of the non-uniform rational B-spline (NURBS)-based isogeometric technique, for the first time, the size-dependent geometrically nonlinear bending characteristics of microplates made of porous functionally graded materials (FGMs) having a central cutout with different shapes are studied. The nonlocal strain gradient continuum elasticity within the framework a hybrid higher-order quasi-3D plate theory is adopted to describe the kinematic relations via only four unknowns. To capture the effective material properties, a porosity-dependent rule of mixture is employed. The nonlocal strain gradient nonlinear load–deflection responses are obtained corresponding to various geometrical and material parameters as well as different boundary conditions. It is revealed that the significance of both the nonlocal and strain gradient reduces. This prediction is the same for all values of the material property gradient index as well as the porosity index. Also, it is demonstrated that a central cutout leads to change the trend of load–deflection response, and this change occurs at a specific value for the applied distributed load which depends on several parameters such as the cutout geometry and boundary conditions. In addition, it is displayed that corresponding to different maximum deflections, the significance of the strain gradient size effect in the absence of nonlocality on the nonlinear flexural stiffness of a porous FGM microplate is more than that of the nonlocal size effect in the absence of the strain gradient size dependency.

On the theoretical and molecular dynamic methods for natural frequencies of multilayer graphene nanosheets incorporating nonlocality and interlayer shear effects

In this paper, a multiplate nonlocal shear model and molecular dynamic simulations are presented to investigate the effects of interlayer shear and nonlocality on the natural frequencies of multilayer graphene sheets (MLGSs). From one aspect in the optimal design of such structures, the interaction between graphene layers, which can significantly vary the static and dynamic behavior due to lack of solidity of layers stack, should be considered. On the other hand, it is requied that the nonlocality phenomenon which has an effective role in the mechanical analysis of nanostructures is taken into account. To this aim, the equation of motion along with corresponding boundary conditions is derived by using the Kirchhoff plate model and nonlocal continuum theory to capture these effects. For a case study, the free vibration problem of simply supported MLGSs is studied by presenting the closed-form Navier’s solution of natural frequencies and carrying out Molecular dynamics (MD) simulations to estimate the value of nonlocal parameter. Finally, the influences of interlayer shear modulus, number of layers, foundation stiffness, and nonlocal parameter on the natural frequencies of such MLGSs are examined in detail. Findings show that natural frequencies including nonlocality effect become smaller than ones of the classical continuum theory and also have good agreements with MD simulations. Moreover, it is seen that the sensitivity of natural frequency to the nonlocal parameter becomes more significant when the number of layers or interlayer shear modulus increases as well as the aspect ratio or width of sheets decreases.

Discrete Kernel Functions for fcc Crystals Within Eringen’s Nonlocal Theory of Elasticity

The dilemma with the deficiencies of the nonlocal kernel functions as the building blocks of the Eringen’s nonlocal theory has been of concern. The aim of the current work is to provide a remedy for the calculation of the components of the nonlocal moduli tensor pertinent to face center cubic (fcc) crystals accounting for their true symmetry group. To this end, three new distinct nonlocal kernel functions which are the discrete atomistic Green’s functions in the stress space are obtained through the nonlocal dispersion relations associated with the longitudinal and shear waves in fcc crystals combined with the corresponding ones calculated via ab initio based on density functional perturbation theory (DFPT). In other words, based on nonlocal continuum and quantum mechanical considerations different kernel functions pertinent to each component of the elastic moduli tensor are calculated for the first time. Moreover, the nonzero components of the classical elastic moduli as well as the equilibrium lattice parameter associated with Al, Cu, Ni, Pd, and Ag fcc crystals have been computed using ab initio density functional theory (DFT) and compared with the available experimental data. In order to show the importance of the anisotropy of the kernel functions, the nonlocal stress field of a screw dislocation inside an infinitely extended fcc medium has been calculated; the solutions for two cases where the dislocation line is along the $\left [ 0\ 0\ 1 \right ]$ and $\left [ 1\ 0\ 1 \right ]$ directions are compared. Within this theory the unrealistic singular behavior of the classical stresses at the dislocation core is eliminated.

Nonlocal Thermodynamics: Mathematical Model of Two-Dimensional Thermal Conductivity

Nonlocal models of thermodynamics are becoming more and more popular in the modern world. Such models make it possible to describe materials with a complex structure and unique strength and temperature properties. Models of nonlocal thermodynamics of a continuous medium establish a relationship between micro and macro characteristics of materials. A mathematical model of thermal conductivity in nonlocal media is considered. The model is based on the theory of nonlocal continuum by A.K. Eringen. The interaction of material particles is described using local and nonlocal terms in the law of heat conduction. The nonlocal term describes the effect of decreasing the influence of the surrounding elements of the material structure with increasing distance. The effect of nonlocal influence is described using the standard non-locality function based on the Gaussian distribution. The nonlocality function depends on the distance between the elements of the material structure. The mathematical model of heat conduction in a nonlocal medium consists of an integro-differential heat conduction equation with initial and boundary conditions. A numerical solution to the problem of heat conduction in a nonlocal plate is obtained. The numerical solution of a two-dimensional problem based on the finite element method is described. The influence of nonlocal effects and material parameters on the thermal conductivity in a plate under highintensity surface heating is analyzed. The importance of nonlocal characteristics in modelling the thermodynamic behaviour of materials with a complex structure is demonstrated.

On Ductile Damage Modelling of Heterogeneous Material Using Second-Order Homogenization Approach

The paper deals with the numerical modelling of ductile damage responses in heterogeneous materials using the classical second-order homogenization approach. The scale transition methodology in the multiscale framework is described. The structure at the macrolevel is discretized by the triangular C1 finite elements obeying nonlocal continuum theory, while the discretization of microstructural volume element at the microscale is conducted by means of the mixed type quadrilateral finite element with the nonlocal equivalent plastic strain as an additional nodal variable. The ductile damage evolution at the microlevel is modelled by using the gradient enhanced elastoplasticity. The macrolevel softening is governed by two criterions expressed by the increase in homogenized damage variable and the threshold of the local equivalent strain. The softening at each material point at the macrolevel is detected by the critical value of the homogenized damage, where homogenization of the damage variable is performed only within softening area. Due to the nonlocal continuum theory applied, a realistic softening behaviour is demonstrated after the damage initiation, compared to the widely used first-order homogenization approach. All algorithms derived have been embedded into the finite element code ABAQUS by means of the user subroutines and verified on the standard benchmark problems. The damage evolution at both microlevel and macrolevel has been demonstrated.

Nonlinear vibration of nonlocal strain gradient functionally graded beam on nonlinear compliant substrate

Beams supported on compliant substrate are used in nanostructures. Due to atomistic dimensions, mechanics of nanostructures are governed by molecular interactions, modeled by the Nonlocal (NL) and Strain Gradient (SG) continuum model. Due to their slender nature (owing to significant strength/stiffness of nano-materials), nanostructures are liable to geometric nonlinearity. Additionally, the supporting substrate may be nonlinear. However, nonlinear forced vibration of nanostructures is not researched well. This study investigates the geometrically nonlinear vibration of NL-SG beams on nonlinear substrate with shear interactions. The higher-order curvature, von Karman nonlinearity are included along with nonlinear Pasternak model for the substrate. The depth-wise functional gradation of material properties is also included. The governing equations of motion are derived following the variational method. A two-step perturbation is employed to obtain a closed-form solution. The free vibration nonlinear frequency, response time history, and forced vibration transmissibility characteristics are presented. The influence of important parameters is illustrated. Specifically, the dominant role of nonlinear bending and substrate stiffness is noted. The effect of NL and SG interactions is shown to significantly influence the vibration behavior. However, the effect of functional gradation of material is shown to be minor.

Nonlinear coupling vibrations of graphene composite laminated sheets impacted by particles

In this work, the graphene composite sheets coated with metal were equivalent to composite laminated sheets. The nonlinear coupling vibrations of graphene composite laminated sheets (GCLSs) under particle impact were investigated. The size-dependent effect and viscoelasticity were incorporated to model GCLSs in the nonlocal continuum theory and Kelvin-Voigt model framework. Particle impact was regarded as stochastic excitation on the GCLS. According to the von Karman strain-displacement relationship for the geometric nonlinearity and Reddy's third-order shear deformation plate theory, the nonlinear governing partial differential equations were derived. To examine the validity of this model, a comparison with published results was carried out. With the purpose of ensuring the convergence and reliability of the results, a large number of mode truncations were performed. The emphasis of numerical simulations was placed on studying how the various factors, including the size-dependent effect, forcing amplitude, and viscoelasticity, influencing the nonlinear responses of GCLS. The motivation of this study is to present a novel model providing both practical and theoretical guidance for devices involving graphene composites subjected to particle impacts.

Identification of non-local continua for lattice-like materials

The paper is focused on the dynamic homogenization of lattice-like materials with lumped mass at the nodes to obtain energetically consistent models providing accurate descriptions of the acoustic behavior of the discrete system. The equation of motion of the Lagrangian one-dimensional lattice is transformed according to a unitary approach aimed to identify equivalent non-local continuum models of integral-differential and gradient type, the latter obtained through standard or enhanced continualization. The bilateral Z-transform of the difference equation of motion of the lattice, mapped on the unit circle, is matched to the governing integral-differential equation of the equivalent continuum in the transformed Fourier space, which has the same frequency band structure as the Lagrangian one. Firstly, it is shown that the approximation of the kernels via Taylor polynomials leads to the differential field equations of higher order continua endowed with non-local constitutive terms. The field equations derived from such approach corresponds to the ones obtained through the so called standard continualization. However, the differential problem turns out to be ill-posed because the non-positive definiteness of the potential energy density of the higher order continuum. Energetically consistent equivalent continua have been identified through a proper mapping correlating the transformed macro-displacements in the Fourier space with a new auxiliary regularizing continuum macro-displacement field in the same space. Specifically, the mapping here introduced has zeros at the edge of the first Brillouin zone. The integral-differential governing equation and the corresponding differential one has been reformulated through an enhanced continualization, that is characterized by energetically consistent differential equations with inertial and constitutive non-localities. Here, the constitutive and inertial kernels of the integral-differential equation exhibit polar singularities at the edge of the first Brillouin zone. The proposed approach is generalized in a consistent way to two-dimensional lattices by using multidimensional Z- and Fourier transforms, a procedure that may be easily extended to three-dimensional lattices. Finally, two examples of lattice-like systems consisting of periodic pre-stressed cable-nets of point mass at the nodes are analyzed. The resulting gradient continuum models provide dispersion curves who are in excellent agreement with those of the Lagrangian systems.

Preface to the special issue “Recent advances in nonlocal and non-classical continuum mechanics: in honor of Professor J. N. Reddy” on the occasion of his 75th Birthday

Preface to the special issue “Recent advances in nonlocal and non-classical continuum mechanics: in honor of Professor J. N. Reddy” on the occasion of his 75th Birthday

Grain size dependence of polycrystalline plasticity modeling in cylindrical indentation

Grain boundary strengthening effect for polycrystalline copper is studied numerically using crystal plasticity in conjunction with cylindrical indentation simulations under the plane strain condition. In order to compare with an isotropic, heterogeneous continuum model a new constitutive relation is developed. This new nonlocal continuum model that encompasses the heterogeneity in yield strength based on the exponentiated Weibull function can predict the plastic properties of materials in the micron length scale. The spatial description of the deformation gradient two-point tensor is utilized to capture the intrinsic size effect in line with the subsequent deformation measures. Moreover, the total geometrically necessary dislocation density is obtained from the non-zero components of Nye dislocation density tensor. From the simulation, the relationship between the effective Green–Lagrange strain and effective stress measures is explained using the persistent long-range order and intermittent short-range order. The observation derived from the analogy between the cylindrical indentation and the progress in cylindrical voids describes how different average grain sizes of polycrystalline materials are compared with the behavior of isotropic materials. The trajectories of directions of both principal stretch and maximum shear strain explain that the internal stresses induced by cylindrical indentation either hinder or reinforce the dislocation flow, forming strain localization sporadically. The grain size dependence of polycrystalline modeling incorporates the Hall–Petch strengthening as well as the homogenization of anisotropic polycrystalline metal into the isotropic effective medium. This is a physically-based model that is used to model failure characterization in metals.

Development of a multiphysics model of synergistic effects between environmental exposure and damage in woven glass fiber reinforced polymeric composites

Extending the application of fiber-reinforced polymer (FRP) composite materials into the automotive, aerospace and military manufacturing is a significant advance in recent years. However, few extending have been achieved in civil infrastructure and building construction industry. This gap is partly due to (i) the lack of multiphysics and multiscale models unifying degradation-deformation damage phenomena to assess the safety and durability of newly adopted or proposed material and structural systems, and (ii) no available computational models that are succinctly fundamental and directly interactable with other digital files from architecture designer or manufacturer. To begin addressing this gap, this paper presents a structure level multiphysics model that uses the UV and moisture deterioration variables homogenized from a micromechanics model to integrate a nonlocal continuum damage model in a curvilinear coordinate system. This model links the application of analysis to the architecture or manufacturing digital model by enabling more accurate predictions of mechanical performance. An example of the model is presented for analysis of a group of glass fiber-reinforced polymer (GFRP) composite plates in SFMOMA Façade System, the first and largest architectural application of fiber reinforced polymer (FRP) in the United States.

A dynamic hybrid local/nonlocal continuum model for wave propagation

In this work, we develop a dynamic hybrid local/nonlocal continuum model to study wave propagations in a linear elastic solid. The developed hybrid model couples, in the dynamic regime, a classical continuum mechanics model with a bond-based peridynamic model using the Morphing coupling method that introduced in a previous study (Lubineau et al., J Mech Phys Solids 60(6):1088–1102, 2012). The classical continuum mechanical model is known as a local continuum model, while the peridynamic model is known as a nonlocal continuum model. This dynamic hybrid model aims to introduce the nonlocal model into the key structural domain, in which the dispersions or crack nucleations may occur due to flaws, while applying the local model to the rest of the structural domain. Both the local and nonlocal continuum domains are overlapped in the coupled subdomain. We study the speeds and angular frequencies of the plane waves, with small and large wavenumbers obtained by the hybrid model and compare them to purely local and purely nonlocal solutions. The error of the hybrid model is discussed by analyzing the ghost forces, and the work done by the ghost forces is considered equivalent to the energy of spurious reflections. One- and two-dimensional numerical examples illustrate the validity and accuracy of the proposed approach. We show that this dynamic hybrid local/nonlocal continuum model can be successfully applied to simulate wave propagations and crack nucleations induced by waves.

State-of-the-Art Review on the Progressive Failure Characteristics of Geomaterials in Peridynamic Theory

AbstractPeridynamic (PD) theory is an integral-type nonlocal continuum mechanics theory that reformulates the equation of motion in local continuum mechanics as an integrodifferential equation. PD ...

Formulation, analysis and computation of an optimization-based local-to-nonlocal coupling method

We present an optimization-based coupling method for local and nonlocal continuum models. Our approach couches the coupling of the models into a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the local and nonlocal problem domains, and the virtual controls are the nonlocal volume constraint and the local boundary condition. We present the method in the context of Local-to-Nonlocal diffusion coupling. Numerical examples illustrate the theoretical properties of the approach.

Virtual characterization of nonlocal continuum damage model parameters using a high fidelity finite element model

A virtual finite element framework is presented to provide the transition from a high fidelity model to a more computationally efficient continuum model for simulating fibre-dominated damage behavior in composite laminates subjected to tensile loadings. The high fidelity method in LS-DYNA is based on the use of cohesive interface elements to simulate both intra-laminar matrix cracks and delamination. Applied to IM7/8552 carbon fibre reinforced plastic laminates in over-height compact tension (OCT) tests, this modeling strategy is used to determine the effective damage parameters for the nonlocal continuum damage model, CODAM2, implemented as MAT219 in LS-DYNA. The prediction of CODAM2 is thoroughly assessed against quantitative and qualitative results obtained from the high fidelity model. Furthermore, the characterized CODAM2 model is applied to simulations of large-scale OCT, open–hole tension (OHT) and center-notched tension (CNT) specimens and results are compared to corresponding experimental data. Due to its nonlocal feature to track the crack trajectory without the need to align the mesh with fibre orientation, the continuum damage model CODAM2 is able to accurately predict the structural response in all large-scale OCT, OHT and CNT test cases with significant computational efficiency.