This paper investigates the thermal vibration characteristics of double-walled carbon nanotubes (DWCNTs) embedded in an elastic medium. Unlike most existing studies, we employ both strain-driven (eD) and stress-driven (sD) two-phase local/nonlocal integral models (TPNIMs) to account for nonlocal effects in the beam deformation, elastic foundation response, and thermally induced stresses simultaneously. The governing equations and associated boundary conditions are rigorously derived through Hamilton's principle, establishing a complete thermomechanical formulation. The constitutive framework transforms integral relations between generalized strain and nonlocal stress fields into equivalent differential forms through systematic incorporation of constitutive boundary constraints. Notably, we provide closed-form solutions for nonlocal thermal stress components, capturing temperature-dependent effects. Numerical solutions for the vibration frequencies are obtained using the generalized differential quadrature method (GDQM). Our results demonstrate the significant influence of the nonlocal parameters, elastic foundation stiffness and thermal stress magnitude on the vibrational response. These effects are quantified for different boundary conditions, providing new insights into the thermo-mechanical behavior of DWCNTs.

In this work, we present a size-dependent study of the dynamics of axially functionally graded (AFG) rods with general constraints. General boundary conditions (GBCs) are modeled by setting elastic spring constraints with torsional stiffness at both ends of the rod, while the gradient variation of the rod characteristic parameters along the axial direction is described by a power-law model. The existence of cracks divides the beam into two parts connected by a rotational spring. In this case, by adopting the equivalently differential formulation derived from the stress-driven nonlocal integral theory-which is rigorously complemented by a complete set of constitutive boundary and continuity conditions-a mathematically well-posed model of the problem is established. All the variables in the formula of the differential problem are discretized, and the numerical solutions of the vibration frequencies of various bounded AFG rods are then established using the generalized orthogonal differential method (GDQM). After verifying the current formulas and results, the influence of various parameters, such as non-local scale parameters, FG index, elastic boundary constraint strength, and crack-related parameters, on the torsional frequency/formation of the structure is investigated in detail. It is expected that the outcomes are beneficial to the health monitoring and safety design of miniaturized components for micro/nano-technological applications.

This study offers an in-depth analysis of the thermal buckling and vibrational characteristics of double-walled carbon nanotubes (DWCNTs) embedded within a Winkler-type elastic medium. To address size-dependent effects, strain-driven (eD) and stress-driven (sD) two-phase nonlocal-local integral models (TPNIMs) are employed, considering Timoshenko beam deformation, foundation-structure interactions, and thermally induced stresses. The governing equations and corresponding boundary conditions are systematically derived using Hamilton's principle. The integral constitutive relations linking generalized strain fields to nonlocal stress tensors are reformulated into equivalent differential expressions, incorporating constitutive boundary conditions. A significant methodological contribution of this work lies in the derivation of closed-form analytical solutions for nonlocal thermal stress distributions. The generalized differential quadrature method (GDQM) is utilized to numerically determine critical buckling loads and natural vibration frequencies. Parametric studies are conducted to evaluate the interdependent influences of nonlocal scaling parameters, foundation stiffness, and temperature variation on the mechanical behavior of DWCNTs, revealing pronounced size effects that are dependent on the boundary conditions. These findings provide crucial insights into the stability-performance trade-offs inherent in thermo-mechanically coupled DWCNT systems, thereby establishing foundational design principles for the development of nanotube-based NEMS and MEMS devices operating under high-temperature conditions.

This study develops a novel finite element formulation for analyzing thermal buckling and free vibration behaviors of Timoshenko nanobeams, employing a unified strain- and stress-driven two-phase local/nonlocal integral model with bi-Helmholtz kernel. The governing equations are derived using Hamilton's principle, with nonlocal effects incorporated through equivalent differential formulations and constitutive boundary conditions. Key methodological innovations include explicit formulations of thermal-induced forces (axial, bending, and shear components), transformation of constitutive boundary conditions into equivalent external forces via the minimum potential energy principle, and implementation of a Lagrange multiplier-enhanced finite element approach to handle higher-order boundary variables. The model's validity is established through comparison with benchmark solutions, followed by systematic parametric studies examining nonlocal effects, thermal loading, and geometric influences. Numerical results demonstrate the model's effectiveness in capturing size-dependent behaviors across various boundary conditions, providing a robust computational tool for nanobeam analysis.

Viscoelastic nanobeams on viscoelastic foundations enable enhanced vibration suppression and novel functionalities in M/NEMS devices. However, accurately predicting their size-dependent, time-damped vibrations demands robust nonclassical theories. This study presents a precise theoretical framework for the free damping vibration characteristics of a curved viscoelastic Timoshenko nanobeam resting on a size-dependent viscoelastic foundation. We develop a mathematically well-posed viscoelastic integral nonlocal strain gradient theory (VINSGT) by integrating integral nonlocal strain gradient theory (INSGT with the Kelvin-Voigt model, while accounting for the size effect in the foundations’ reaction force. The integral constitutive equations are transformed into an equivalent differential form, incorporating essential constitutive boundary conditions (CBCs). The governing equations are discretized via the generalized differential quadrature method (GDQM), yielding a complex eigenvalue problem. A two-step numerical scheme resolves the vibration frequencies and establishes the relationship between damping and viscous coefficients. Numerical examples validate the VINSGT framework and systematically investigate size effects in the damped vibration behavior of viscoelastic curved Timoshenko nanobeams on size-dependent foundations. This work provides a reliable theoretical basis for designing and optimizing vibration control in advanced M/NEMS with viscoelastic nanobeam-foundation systems.

In this paper, the size-dependent bending and buckling behaviors of a piezoelectric semiconductor nanobeam are investigated using both strain-driven and stress-driven two-phase local/nonlocal integral models. The governing equations are derived based on a linearized one-dimensional phenomenological theory of piezoelectric semiconductors. The two-phase local/nonlocal integral formulation is implemented, and the integral equations are transformed into differential forms with corresponding constitutive constraints. Several dimensionless variables are introduced to simplify the formulations. The general differential quadrature method is employed to obtain numerical solutions. Based on the results, the influence of nonlocal parameters on the bending deflection, electric potential, and buckling loads of the piezoelectric semiconductor nanobeam is examined under various boundary and loading conditions. Critical comparisons between strain-driven and stress-driven modeling paradigms are highlighted to elucidate their distinct predictive capabilities in nanoscale electromechanical coupling phenomena.

In this paper, a novel isogeometric finite element formulation is developed to investigate buckling responses of curved Euler–Bernoulli nanobeams. The governing equations and standard boundary conditions are derived through the minimum total potential energy principle. The equivalent differential forms of the strain- and stress-driven two-phase local/nonlocal integral models, along with the corresponding constitutive boundary conditions, are considered in a unified form to account for the size effect phenomenon. The axial force and bending moment are explicitly obtained, and the weak form of the governing equation is derived accordingly. The isogeometric analysis (IGA)-based finite element method (FEM) is developed to obtain the novel isogeometric finite element formulation for the buckling behaviors of curved nanobeams under different boundary conditions, with the displacement field modeled by Non-uniform rational B-splines (NURBS) instead of traditional Lagrangian and Hermite cubic interpolation functions. The constitutive boundary conditions enable the flexible accommodation of higher-order variables. This two-phase isogeometric finite element model fulfills high-order boundary conditions, features straightforward shape functions, and exhibits favorable convergence behavior. The buckling analysis results are compared with other available results in the literature to demonstrate the efficiency and accuracy of the present IGA framework. Convergence and parameter sensitivity analyses conducted under various boundary conditions demonstrate the robustness of the proposed method. The numerical results show that the strain- and stress-driven two-phase local/nonlocal models of curved nanobeams exhibit consistent softening and stiffening effects on buckling responses, respectively. Additionally, the effects of the opening angle and length-to-height ratio of the curved nanobeam are investigated.

Existing research has shown that nonlocal piezoelectric differential models often yield inconsistent dynamic responses for nanostructures. To address this issue, the two-phase local-nonlocal integral formulation has been proposed and has garnered increasing scholarly attention as an effective alternative. This study presents the first implementation of this theoretically consistent and paradox-free framework to investigate the size-dependent dynamic stability and free vibration behavior in piezoelectric Timoshenko nanobeams. The generalized boundary conditions are simulated through elastic constraints incorporating both translational and rotational springs at both beam ends. Departing from conventional approaches, the present formulation simultaneously accounts for size effects in both bending deformation and axial deformation caused by external voltages via the derivation of an equivalent differential representation of the well-posed local-nonlocal integral piezoelectric model. This formulation is rigorously complemented by a complete set of constitutive constraint conditions, ensuring mathematical well-posedness throughout the analytical framework. The generalized differential quadrature method (GDQM) is used to discretize the governing differential equations, enabling numerical determination of dynamic instability regions (DIRs) for various boundary configurations. Following comprehensive validation through comparative analyses, we systematically examine the influence of nonlocal parameters, static force factors, and boundary stiffness characteristics on the DIRs of the beams. Furthermore, this investigation underscores the significance of incorporating nonlocal effects into voltage-induced axial loading, addressing a critical gap in the current understanding of electromechanical coupling at nanoscale dimensions.

Mixed-mode behaviors are critical in both quasi-static and dynamic fracture. To investigate dynamic mixed-mode fracture in quasi-brittle materials, we developed a rate-independent Phase-Field Cohesive-Zone Model (PF-CZM). This model incorporates damage-induced anisotropy through a directional decomposition scheme. A modified G-criterion determines the fracture plane's orientation, and a multiscale framework is introduced to stabilize crack orientation once formed. We solved the governing equations using an implicit time integration scheme implemented in Julia, and adaptive mesh refinement (AMR) accelerated computations. Numerical examples confirm the model's length-insensitivity and its flexibility in capturing failure mode transitions. Notably, our work reveals that Y-joint crack structures in dynamic Brazilian split test form from the intersection of a central mode-I crack and corner-generated mode-II cracks. This study marks the first application of a directional decomposition scheme to dynamic fracture, offering novel into-brittle mixed-mode fracture.

In this study, a novel isogeometric finite element approach is proposed to analyze the free vibration of general curved nanobeams with small initial curvatures, within the framework of strain- and stress-driven two-phase local/nonlocal integral theory. An isogeometric analysis (IGA) framework, combined with the finite element method (FEM), is employed to compute the vibrational response of curved nanobeams with variable curvatures. The NURBS basis functions enable flexible curve modeling, ensuring accurate representation of complex geometric microstructures. This model features low computational cost, high accuracy, good convergence, and strong applicability for curved beam structures. Parabolic nanobeams are examined as case studies, and convergence and parameter sensitivity analyses are conducted to demonstrate the robustness of the proposed method. Results indicate that small initial curvature exhibits a significant influence on the vibration characteristics of curved nanobeams and cannot be ignored. This analysis provides valuable insights for researching vibrations in slightly curved beam structures with variable curvatures.

Nanolattice metamaterials, known for their exceptional mechanical performance, exhibit size-dependent properties that are crucial for their design and application. This work introduces an isogeometric analysis framework to investigate these materials' mechanical responses. The framework integrates the Euler-Bernoulli beam model to capture the kinematics of individual beams with the stress-driven two-phase local/nonlocal integral model (sigma D-TPNIM) to account for nanoscale size-dependent strengthening effects. The framework's accuracy is validated through comparisons with established results for both a single beam and a re-entrant hexagonal lattice. We then analyze the static and dynamic responses of various nanolattices, including re-entrant hexagonal, Kagome, and triangular geometries. Our analysis covers static tension, steady-state vibration, and incident wave propagation, demonstrating the framework's versatility in predicting complex mechanical behaviors. This research provides a robust and promising tool for the advanced modeling of diverse nanolattice metamaterials.

This paper extends the one-dimensional (1D) nonlocal strain gradient integral model (NStraGIM) to the two-dimensional (2D) Kirchhoff axisymmetric nanoplates, based on nonlocal strain gradient integral relations formulated along both the radial and circumferential directions. By transforming the proposed integral constitutive equations into the equivalent differential forms, complemented by the corresponding constitutive boundary conditions (CBCs), a well-posed mathematical formulation is established for analyzing the axisymmetric bending and buckling of annular/circular functionally graded (FG) sandwich nanoplates. The boundary conditions at the inner edge of a solid nanoplate are derived by L’Hôspital’s rule. The numerical solution is obtained by the generalized differential quadrature method (GDQM). The accuracy of the proposed model is validated through comparison with the data from the existing literature. A parameter study is conducted to demonstrate the effects of FG sandwich parameters, size parameters, and nonlocal gradient parameters.

Under complex stress-states, mixed-mode fracture is critical to the crack propagation. Additionally, in quasi-brittle materials, the toughness and strength can differ across fracture modes. Therefore, to analyze mixed-mode fracture behaviors under different stress conditions, we developed a new phase-field cohesive zone model (PF-CZM). A directional strain energy decomposition scheme with anisotropic constitution is applied to better describe the mechanical behaviors of damaged materials. A mixed-mode ratio is introduced to describe the relative contribution of mode I and mode II fracture to the crack propagation. Thus, the phase-field governing equation can be still derived by taking variation to the potential energy with respect to the phase-field. The crack orientation for propagation is assumed to be the direction that results in the maximum increase in crack area, which is demonstrated to be consistent with the modified G-criterion. The mode II crack orientation is determined using a deformation gradient- assistant approach. We also propose a new numerical frozen mechanism to take into account the interaction between the existing and incremental crack. Several numerical examples are provided to validate the current PF-CZM. The current study addresses when and how a crack will propagate in complex scenarios and significantly broadens the PFM's applicability range for mixed-mode fracture, making it suitable for usage with a variety of materials.

The elastic buckling of Double-walled carbon nanotubes (DWCNTs) embedded in an elastic foundation is investigated in this article. In contrast to the majority of the literature on this subject, the behaviors of both the beam and elastic foundation are considered as nonlocal by applying the strain-driven (eD) and stress-driven (SD) two-phase local/nonlocal integral models (TPNIMs). The differential governing equations and standard boundary conditions are derived by the Principle of Minimum Potential Energy. The relations between the general strain and general nonlocal stresses are expressed as integral equations, which are further transformed unified into equivalent differential forms with constitutive boundary conditions. The general differential quadrature method (GDQM) is applied to obtain the numerical results of buckling loads of DWCNTs. The numerical simulation results present the size-effect of nonlocal parameter and elastic foundation on the critical buckling loads of DWCNTs with various boundary types.

In the present work, a novel nonlocal finite element model is presented for functionally graded (FG) Timoshenko nanobeams resting on a size-dependent elastic foundation. In contrast to the previous major studies, the size-dependent effects of both nanobeam and elastic foundation are taken into account simultaneously and modeled with the equivalent stress-driven two-phase local/nonlocal differential model equipped with two constitutive boundary conditions. The weak form of governing equations is derived and the higher-order variables in the additional external forces are eliminated with the aid of the constitutive boundary conditions. A finite element formulation based on the differential nonlocal constitutive relations is developed for buckling and free vibration analysis of FG nanobeams. Several comparative studies are conducted to verify the efficiency and accuracy of the proposed nonlocal finite element method (FEM). Considering the nonlocality of the elastic foundation, the effects of two-phase local/nonlocal elasticity on critical buckling load and vibration frequency of FG Timoshenko nanobeam are investigated in detail with different gradient index, nonlocal parameter, local volume fraction and buckling as well as vibration orders under different boundary conditions.

In this paper, a fractional-order kinematic model is utilized to capture the size-dependent static bending and free vibration responses of piezoelectric nanobeams. The general nonlocal strains in the Euler-Bernoulli piezoelectric beam are defined by a frame-invariant and dimensionally consistent Riesz-Caputo fractional-order derivatives. The strain energy, the work done by external loads, and the kinetic energy based on the fractional-order kinematic model are derived and expressed in explicit forms. The boundary conditions for the nonlocal Euler-Bernoulli beam are derived through variational principles. Furthermore, a finite element model for the fractional-order system is developed in order to obtain the numerical solutions to the integro-differential equations. The effects of the fractional order and the vibration order on the static bending and vibration responses of the Euler-Bernoulli piezoelectric beams are investigated numerically. The results from the present model are validated against the existing results in the literature, and it is demonstrated that they are theoretically consistent. Although this fractional finite element method (FEM) is presented in the context of a one-dimensional (1D) beam, it can be extended to higher dimensional fractional-order boundary value problems.

Interfacial debonding, a critical failure mechanism in heterogeneous materials, is often characterized by mixed-mode fracture. This study develops a numerical framework to simulate bulk and interfacial fractures in composite materials. A phase-field cohesive zone model, incorporating a directional energy decomposition scheme and a modified toughness method, is employed to capture complex fracture behaviors. A level-set method explicitly defines interface positions, while an adaptive mesh refinement strategy enhances computational efficiency. Numerical examples validate the model’s accuracy and efficiency in predicting mixed-mode crack propagation and interfacial debonding. This work provides a robust and efficient approach to simulate complex fracture phenomena in heterogeneous materials, especially for the mixed-mode fracture.

In this study, the size-dependent static tensile and axial vibrational responses of piezoelectric semiconductor nanobars are investigated using strain-driven and stress-driven dual-phase local/nonlocal integral constitutive models. A linearized one-dimensional phenomenological framework for piezoelectric semiconductors is employed to establish the governing equations. The two-phase local/nonlocal integral formulation is implemented, which is subsequently transformed into differential formulations with corresponding constitutive constraints. A few dimensionless variables are introduced to streamline mathematical derivations. The general differential quadrature method is utilized to obtain the numerical solutions. The influence of nonlocal parameters on the static extension displacement, electric potential, and undamped natural frequencies of piezoelectric semiconductor nanobar are investigated under different boundary and loading conditions. Critical comparisons between strain-driven and stress-driven modeling paradigms are highlighted to elucidate their distinct predictive capabilities in nanoscale electromechanical coupling phenomena.

Current studies on carbon nanotube (CNT) size effects predominantly employ Eringen’s differential nonlocal model, which is widely recognized as ill-suited for bounded domains. This paper investigates the free vibration of multi-walled CNTs (MWCNTs) with mathematically well-posed two-phase strain-driven and stress-driven nonlocal integral models incorporating the bi-Helmholtz kernel. The van der Waals (vdW) forces coupling MWCNT layers are similarly modeled as size-dependent via the bi-Helmholtz two-phase nonlocal integral framework. Critically, conventional pure strain-driven or stress-driven formulations become over-constrained when nonlocal vdW interactions are considered. The two-phase strategy resolves this limitation by enabling consistent coupling. Each bi-Helmholtz integral constitutive equation is equivalently transformed into a differential form requiring four additional constitutive boundary conditions (CBCs). The numerical solutions are obtained with the generalized differential quadrature method (GDQM) for these coupled higher-order equations. The parametric studies on double-walled CNTs (DWCNTs) and triple-walled CNTs (TWCNTs) elucidate the nonlocal effects predicted by both formulations. Additionally, the influence of nonlocal parameters within vdW forces is systematically evaluated to comprehensively characterize the size effects in MWCNTs.

We present a formulation for the size-affected vibration study of multi-cracked non-uniform Timoshenko beams based on the well-posed stress-driven nonlocal elastic theory with discontinuities. The beam ends are assumed to be constrained by elastic springs with translational and rotational stiffness to simulate general boundary conditions. The presence of cracks divides the beam into segments connected by translational and rotational springs, and compatibility conditions are established to address the geometric discontinuities introduced by these cracks. The stress-driven constitutive equations are integrated into an equivalent differential form, equipped with a set of constitutive boundary conditions at the two ends of the entire structure and multi-sets of constitutive continuity conditions at the junctions of the sub-structures. To solve the equations of motion, the constraint conditions and the integrals involved, we employ the differential quadrature method (DQM) alongside an interpolation quadrature formula, which allows us to efficiently compute the frequencies of the cracked beams across various boundary types. After validating our approach against results in the existing literature, we present numerical studies that examine the effects of the nonlocal parameter, the slope of the beam’s thickness variation, crack location, severity, number, and the stiffness of the springs on the vibrational behavior of the beams.