Nonlocal Integral Theory Research Articles

Nonlinear post‐buckling analysis of viscoelastic nano‐scaled beams by nonlocal integral finite element method

AbstractThe viscoelastic buckling and nonlinear post‐buckling behavior of nano‐scaled beams are analyzed using the nonlocal integral elasticity theory. Eringen's nonlocal theory is one of the well‐known and popular size‐dependent theories, which has been used by several researchers to study the mechanical behavior of, mostly, the elastic nanostructures. A finite element method is developed using Hamilton's principle based on the two‐phase nonlocal integral theory and taking into account the buckling related terms and viscoelastic effects. The corresponding formulations are derived by implementing the Euler–Bernoulli beam theory and Kelvin–Voigt viscoelastic model. Furthermore, for analyzing the post‐buckling and viscoelastic buckling behavior, the nonlinear strains, and initial displacement (imperfection) have been considered. By employing the variational relations, the governing equations are obtained and then solved numerically using the finite difference and Newton–Raphson methods. Because of the finite element nature of the current method, various boundary conditions can be properly implemented. The results of the current study are compared with those available in the literature and the effects of nonlocal parameter, viscoelastic parameter, axial compressive load and boundary conditions on the viscoelastic buckling, and postbuckling behavior have been investigated.

A bi-Helmholtz type of two-phase nonlocal integral model for buckling of Bernoulli-Euler beams under non-uniform temperature

It is well-acknowledged by the scientific community that Eringen’s nonlocal integral theory is not applicable to nanostructures of engineering interest due to conflict between equilibrium and constitutive requirements. In this paper, a well-posed two-phase nonlocal integral elasticity with the bi-Helmholtz kernel is developed to study the size-dependent buckling response of Bernoulli-Euler beams under non-uniform temperatures. The governing equation is derived by invoking the variational principle of virtual work, and the temperature effect is equivalent to the thermal load along the axial direction, which is determined by nonlocal heat conduction. The two-phase nonlocal integral constitutive equation is transformed into a differential one equipped with four constitutive boundary conditions, and then exact solutions for the buckling loads of the beam with various boundary edges are obtained. Numerical results are validated by comparing them with those from local elasticity. Moreover, the effects of parameters related to the two-phase nonlocal elastic model and the nonlocal heat conductive model are investigated.

Nonlinear flexure mechanics of beams: stress gradient and nonlocal integral theory

In order to study the intrinsic size-effects, the stress gradient theory is implemented to a nano-scale beam model in nonlinear flexure. The nonlocal integral elasticity model is considered as a suitable counterpart to examine the softening behavior of nano-beams. Reissner variational principle is extended consistent with the stress gradient theory and applied to establish the differential, constitutive and boundary conditions of a nano-sized beam in nonlinear flexure. The nonlinear integro-differential and boundary conditions of inflected beams in the framework of the nonlocal integral elasticity are determined utilizing the total elastic strain energy formulation. A practical series solution approach in terms of Chebyshev polynomials is introduced to appropriately estimate the kinematic and kinetic field variables. A softening structural behavior is observed in the flexure of the stress gradient and the nonlocal beam in terms of the characteristic parameter and the smaller-is-softer phenomenon is, therefore, confirmed. The flexural response associated with the stress gradient theory is demonstrated to be in excellent agreement with the counterpart results of the nonlocal elasticity model equipped with the Helmholtz kernel function. The nonlocal elasticity theory endowed with the Error kernel function is illustrated to underestimate the flexural results of the stress gradient beam model. Detected numerical benchmark can be efficiently exploited for structural design and optimization of pioneering nano-engineering devices broadly utilized in advanced nano-electro-mechanical systems.

Free vibration characteristics of nonlocal viscoelastic nano‐scaled plates with rectangular cutout and surface effects

AbstractIn the present study, the viscoelastic free vibration behavior of nano‐scaled plates is studied by employing a finite element method based on the two‐phase nonlocal integral theory. Various boundary conditions, surface effects and cutouts have been assumed. The principle of total potential energy is used for developing the nonlocal finite element method, and the classical plate theory is assumed to derive the formulations. By numerically solving the eigenvalue problem, which has been obtained by the variational principle, the complex eigenvalues of free vibration of the viscoelastic nano‐scaled plates are acquired. The current results are compared with those available in the literature and those obtained by commercial finite element software, and the influences of the nonlocal parameter, viscoelastic parameter, geometrical parameters (e.g. cutout and size), surface effects and different boundary conditions on the complex eigenvalues are studied. It is noted that the current method is able to handle quite versatile boundary conditions and geometries like cutouts, which are rather difficult (or impossible) to be tackled by employing other methods available in most researches.

Contribution of nonlocality to surface elasticity

The underlying mechanisms of surface phenomena are very complex and still not entirely clear. The aim of this work attempts to reveal the important contribution of the nonlocal integral theory of elasticity to surface elasticity, which is of fundamental scientific interest. By considering a uniform and isotropic half-space medium subjected to an arbitrary uniform strain, it is shown that the bulk is homogeneous, however, the whole medium is heterogeneous due to the bond loss near the free surface. The nonlocal effect cannot be observed at all in the homogeneous bulk, but the bond loss characterized by the nonlocal integral theory can show an important contribution to the surface elasticity. This in turn allows to propose a simplified surface model to replace the complex modeling of nonlocal integral theory. The thickness of surface zone can be evaluated from the intrinsic characteristic length used in the nonlocal integral theory. According to the intrinsic correlation and using the molecular dynamics simulations, the thickness of the surface layer of silicon films is 2.6911 Å. Furthermore, with application of the simplified surface model to a thin film in tension, it has been noted that the geometric dimension of size-dependence is generally not that of traditional mechanics. For instance, in most situations, the main contribution to size-dependence has actually come from the thickness (or radius) direction of a rod-type structure, rather than its axial direction which is intuitively- and widely-used in the current practice in open literature.

Nonlocal Elasticity Response of Doubly-Curved Nanoshells

In this paper, we focus on the bending behavior of isotropic doubly-curved nanoshells based on a high-order shear deformation theory, whose shape functions are selected as an accurate combination of exponential and trigonometric functions instead of the classical polynomial functions. The small-scale effect of the nanostructure is modeled according to the differential law consequent, but is not equivalent to the strain-driven nonlocal integral theory of elasticity equipped with Helmholtz’s averaging kernel. The governing equations of the problem are obtained from the Hamilton’s principle, whereas the Navier’s series are proposed for a closed form solution of the structural problem involving simply-supported nanostructures. The work provides a unified framework for the bending study of both thin and thick symmetric doubly-curved shallow and deep nanoshells, while investigating spherical and cylindrical panels subjected to a point or a sinusoidal loading condition. The effect of several parameters, such as the nonlocal parameter, as well as the mechanical and geometrical properties, is investigated on the bending deflection of isotropic doubly-curved shallow and deep nanoshells. The numerical results from our investigation could be considered as valid benchmarks in the literature for possible further analyses of doubly-curved applications in nanotechnology.

Semi-Implicit Numerical Strategy for Implementing a Nonlocal Finite-Deformation Gurson Model for Glassy Polymer Using Finite Element Analysis

Post-peak strain softening for some thermoplastic resins often leads to localized deformation with plastic flow instability, spurious energy dissipation, and numerical solutions with mesh dependency using finite element analysis (FEA) due to the lack of a length scale. This paper introduces the nonlocal integral theory with a length scale into the Gurson model to capture objective void evolution and shear bands for glassy polymer under quasi-static loading. Nonlocal numerical technique using the finite-deformation Gurson model is developed, where the nonlocal implicit stress return algorithm and the nonlocal consistent tangent stiffness are presented in the corotational configuration by performing nonlocal averaging on the void volume fraction in a semi-implicit way. The numerical strategy is implemented by combining the finite element software ABAQUS-UMAT, USDFLD, and UEXTERNALDB subroutines. For plane tension problem, the developed nonlocal numerical technique circumvents mesh dependency well by studying the void growth, the evolution of shear bands, and the load responses.

Stress-strains state calculation of a rod at uniaxial tension with non-local effects

A numerical calculation of a uniaxial extension of a rod is carried out in terms of the nonlocal integral theory of elasticity using the finite element method for various parameters of the material, taking into account its microstructure, and contributions of nonlocal effects. The dependencies of the strain energy are presented with increasing number of finite elements, which confirm the convergence and correctness of the mathematical model and method.

Dynamic Void Growth and Localization Behaviors of Glassy Polymer Using Nonlocal Explicit Finite Element Analysis

This work introduces the nonlocal integral theory into the finite-deformation viscoplastic Gurson-type model for glassy polymer with plastic softening under strain rate loading. In order to improve computational efficiency and numerical robustness, nonlocal averaging on the void volume fraction is coupled with the update of stress and strain using ABAQUS dynamic subtoutines. From two numerical examples using nonlocal FEA on the unnotched square plate and the dumbbell-shaped specimen under tension respectively, the introduced length scale is demonstrated to regularize the dynamic initial-value problem well that remains hyperbolic by predicting the nonlocal void growth, the objective load responses, and the contours of localization bands accurately. In addition, the relationship between the width of localization band and the length scale is also studied.

Viscoelastic free vibration behavior of nano-scaled beams via finite element nonlocal integral elasticity approach

In this paper, the free-vibration behavior of viscoelastic nano-scaled beams is studied via the finite element (FE) method by implementing the principle of total potential energy and nonlocal integral theory. The formulations are derived based on the Kelvin–Voigt viscoelastic model and Euler–Bernoulli beam theory considering the nonlocal integral theory. The eigenvalue problem of the free vibration is extracted by employing the variational relations. To the best of the authors knowledge it is the first time that the viscoelastic characteristics are implemented in the nonlocal integral FE method to study mechanical behavior of nano-scaled beams. Various boundary conditions can be properly modeled by the current method. Numerical results are compared with literature in order to validate the proposed approach. Then, the effects of nonlocal parameter, viscoelastic parameter, geometrical parameters and different boundary conditions on the complex natural frequencies of the nano-scaled Euler– Bernoulli beams are studied.

Finite Element Analysis of Progressive Failure and Strain Localization of Carbon Fiber/Epoxy Composite Laminates by ABAQUS

Interaction mechanism between the intralaminar damage and interlaminar delamination of composite laminates is always a challenging issue. It is important to consider the progressive failure and strain softening behaviors simultaneously during the damage modeling and numerical simulation of composites using FEA. This paper performs three-dimensional finite element analysis of the progressive failure and strain localization of composites using FEA. An intralaminar progressive failure model based on the strain components is proposed and the nonlinear cohesive model is used to predict the delamination growth. In particular, the nonlocal integral theory which introduces a length scale into the governing equations is used to regularize the strain localization problems of composite structures. Special finite element codes are developed using ABAQUS to predict the intralaminar and interlaminar damage evolution of composites simultaneously. The carbon fiber/epoxy composite laminates with a central hole demonstrates the developed theoretical models and numerical algorithm by discussing the effects of the mesh sizes and layups patterns. It is shown the strain localization problem can be well solved in the progressive failure analysis of composites when the energy dissipation due to the damage of the fiber, matrix and interface occurs at a relatively wide area.

Wave propagation in 1D elastic solids in presence of long-range central interactions

In this paper wave propagation in non-local elastic solids is examined in the framework of the mechanically based non-local elasticity theory established by the author in previous papers. It is shown that such a model coincides with the well-known Kröner–Eringen integral model of non-local elasticity in unbounded domains. The appeal of the proposed model is that the mechanical boundary conditions may easily be imposed because the applied pressure at the boundaries of the solid must be equilibrated by the Cauchy stress. In fact, the long-range forces between different volume elements are modelled, in the body domain, as central body forces applied to the interacting elements. It is shown that the shape change of travelling disturbances coalesces with those predicted by the non-local integral theory of elasticity in unbounded domains, but several differences arise in the case of bounded domains. The wave propagation problem has been formulated by means of the Hamiltonian functional of the proposed mechanically based model of non-local elasticity, introducing an additional term to the elastic potential energy that accounts for elastic long-range interactions. In this way, the wave equation may be obtained in a weak formulation and be further used to provide approximate analytical solutions to the governing equation in the context of standing wave analysis. An equivalent discrete point-spring model, similar to lattice-type networks, has also been introduced to show the mechanical equivalence of the non-local elastic model as well as to provide a mechanical scheme suitable for the numerical treatment of pressure waves travelling in non-local bounded domains.