Constitutive Boundary Conditions Research Articles

A Gauss kernel non-local stress-driven plate theory

This study presents a novel stress-driven formulation, incorporating Kirchhoff’s plate assumptions and accounting for the non-local size-dependent feature in both plate geometric axes, which is practical, especially near the origin of the axis in the polar coordinates. The presented formulation does not impose any additional constraints on the elastic radial curvatures, which are known as constitutive boundary conditions. Therefore, taking into account the non-local behavior in both radial and circumferential directions does not lead to an overconstrained problem. The study focuses on the analysis of a circular plate with an outer radius (R), thickness (h), and the length scale parameter (L). The governing differential equations are derived in Cartesian coordinates and then transformed to the polar coordinates to analyze circular plates. When R tends to infinity, the governing equation is solved analytically. Using the Galerkin technique, a finite element solution is presented for the analysis of a bounded circular plate with a clamped boundary condition at the outer radius R. The study discusses the size effects and the gradient properties of functionally graded plates in stress-driven non-local theory. It reveals that while increasing the non-local parameter results in a reduction of transverse displacement, the moments remain unaffected.

Buckling and vibration analysis of axially functionally graded nanobeam based on local stress- and strain-driven two-phase local/nonlocal integral models

This paper presented an efficient finite element formulation for analysis of the buckling and free vibration of the axial functionally graded (AFG) Euler-Bernoulli nanobeam based on minimum total potential energy principle with linear constraints. The stress- and strain-driven two-phase local/nonlocal integral models are utilized to address small scale effects. The material properties of the AFG nanobeam vary exponentially along the axial direction. The nonlocal integral constitutive equation can be equivalently transformed into a differential equation with two constitutive boundary conditions. The governing equation and standard boundary conditions are derived via Hamilton's principle. The weak form associated with total potential energy is derived and the constitutive boundary conditions can be converted to external forces at the boundaries. By employing a numerical solution methodology on the basis of the finite element method (FEM) together with Lagrangian multiplier method (LMM), a unified finite element formulation based on stress- and strain-driven two-phase local/nonlocal elasticity is constructed to obtain the critical buckling load and vibration frequency. Meanwhile, the numerical results obtained through generalized differential quadrature method (GDQM) validate the efficiency and accuracy of the present results. The influence of gradient index, nonlocal parameter and local phase parameter on buckling and free vibration of AFG Euler-Bernoulli nanobeam is investigated in detail under different nonlocal models and boundary conditions. It is demonstrated that the gradient index has more effect on mode shapes than the buckling load and vibration frequency between the local stress- and strain-driven two-phase local/nonlocal models.

Nonlinear thermo-mechanical stress-driven modeling of nano arches augmented by higher order double-scaled kernel

Modeling the nonlocal behavior of small-scale structures is almost controversial. The recently developed approach, known as the stress-driven nonlocal model, in which the stress is the input of nonlocal integral, has been proved to pose as a remedy for the ill-posing behavior of strain-driven solutions. Though it is still in its early stages, stress-driven modeling of the nonlinear behavior of nano-scaled problems needs further thought. The coupled displacement field, in particular, makes nano-arches more challenging. The stress-driven formulation, enhanced by constitutive boundary conditions, reveals the thermo-mechanical buckling and post-buckling features of shallow nano-arches for the first time in this research. Furthermore, the single-scaled Helmholtz kernel is replaced with the more appropriate double-scaled kernel known as the bi-Helmholtz averaging nonlocal kernel. Higher-order continuity of the bi-Helmholtz kernel and, hence, constitutive boundary conditions and higher-order differential equations corresponding to the stress-driven scheme hold the key to efficiency. The arch is subjected to uniform temperature and radial pressure at the same time. To demonstrate the response of the nano-arch during buckling and post-buckling, the nonlinear equilibrium path, whether for mechanical or thermal loading, is given for the first time by prevailing on the challenges of the higher-order problems. Moreover, it would be beneficial if one could have the chance to recognize how and when the arch buckles. These valuable data are available in this study to give the limiting parameters at the moment of buckling, including limiting temperature, nonlocal parameter, and geometry.

Integral nonlocal stress gradient elasticity of functionally graded porous Timoshenko nanobeam with symmetrical or anti‐symmetrical condition

AbstractUtilization of symmetrical or anti‐symmetrical condition could improve the calculation efficiency. In this paper, a mathematical formulation is proposed to deal with the symmetrical or anti‐symmetrical condition in an integral nonlocal stress gradient model (INSGM), which is transformed equivalently into differential form with constitutive boundary condition as well as constitutive symmetrical or anti‐symmetrical condition. Unlike general constitutive boundary conditions, an integral item is introduced to constitutive symmetrical and anti‐symmetrical conditions, and they are opposite to each other. Based on INSGM with symmetrical or anti‐symmetrical conditions, static bending of simply‐supported (SS) and clamped‐clamped (CC) functionally graded porous Timoshenko nanobeams is investigated for symmetrical loads, including uniformly distributed load (UDL) and middle point force, as well as anti‐symmetrical loads, including anti‐symmetrical UDL and middle point moment. The exact solutions are deduced and expressed in explicit form for different boundary and loading conditions. Calculation shows that, under UDL, bending deflections of half Timoshenko nanobeams based on current model agree well with those for whole Timoshenko nanobeams based on general INSGM for both SS and CC boundary conditions. Numerical study is performed to show the effectiveness of current model.

Size effect in ultrasensitive micro- and nanomechanical mass sensors

The size dependent free transverse vibration of the micro- and nanocantilever mass sensors is studied. The general case of a sensor with an arbitrary number of attached particles is considered. The domain is divided into different segments at the cross-sections where the particles are located, and the displacement fields are described based on the Bernoulli-Euler beam theory. The size effect is introduced into the formulation by assuming the constitutive equation of the stress-driven nonlocal theory of elasticity. The eigenvalue problem is generated by solving the equation of motion in each segment of the sensor and imposing the variationally consistent and higher-order constitutive boundary and continuity conditions. The natural frequencies and their sensitivity to the attachment of a small mass are analyzed analytically. It is shown that the frequency shifts resulting from attachment of a small mass can be explicitly defined as a function of the frequency and mode shape of the unloaded sensor. The model is used to numerically study the natural frequencies of sensors loaded by one to three particles. Comprehensive results are presented on the effect of the size dependency on the frequency shifts of the first four modes of vibration. It is revealed that neglecting the size effect may result in wrong detections of the masses of the attached particles.

Theoretical fracture analysis of double cantilever microbeam using two-phase local/nonlocal integral models with discontinuity

A mathematical approach is proposed to apply strain-driven (εD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIMs) with discontinuity of variable fields. The equivalently differential form together with constitutive boundary conditions and constitutive continuity conditions is derived explicitly. Compare to constitutive boundary conditions, constitutive continuity conditions have one more integral item, and the discontinuity plays no role on differential constitutive relation and constitutive boundary conditions. εD- and σD-TPNIMs are applied to formulate the Mode I and II fracture problem of double cantilever Euler-Bernoulli microbeams subjected symmetric and anti-symmetrical end moment and force loads. It is found that constitutive continuity conditions play no role on Mode I fracture problem since that the flexible deformation disappears for intact microbeam under symmetry loads. The bending deflections under different loads are solved analytically, and the energy release rate (ERR) and stress intensity factor (SIF) are derived explicitly. On the basis of the superposition principle, SIF for edge-cracked Euler-Bernoulli microbeams subjected generally end moment and force can be calculated. The influence of nonlocal parameters and the intact microbeam length on the size-dependent fracture behavior is investigated numerically. The obtained results can explain the superior fracture performance of nanomaterials.

Local–nonlocal stress-driven model for multi-cracked nanobeams

Exact closed-form solutions for multi-cracked Euler–Bernoulli nanobeams are provided by proposing two equivalent approaches. The considered multi-cracked nanobeams are modeled with a well-posed local–nonlocal stress-driven model and contain an arbitrary number n of isolated damaged cross-sections. In order to simulate the damaged sections, it is assumed that the nanobeam bending flexibility contains n Dirac delta functions, with a Dirac delta located at each damaged section: this representation of cracked sections, already used in local problem, is adopted for the first time here in the local–nonlocal stress-driven model. The first approach providing closed-form solutions is based on the integral definition of the local–nonlocal stress-driven model, where the bending curvature is given by the superposition of the local and nonlocal phases, and the nonlocal phase is an integral convolution of the bending moment. This approach provides an integro-differential equation, which is solved by taking firstly the Laplace transform and then the anti-transform. In the second approach, it is shown that the integral definition of the stress-driven multi-cracked nanobeam model is equivalent to a differential equation together with suitable constitutive boundary conditions. This equation is solved by making use of Laplace transform again. The two approaches provide the same solution, with the second one resulting computationally faster. Interesting results show that a jump of the rotation at the damaged cross-sections occurs in local beam, but does not occur in pure nonlocal nanobeams (i.e. local phase is absent): contrary to the local elasticity theory, the bending curvature of pure nonlocal nanobeams has no singularity at damaged sections. The typical stiffness increase, due to the increase in nonlocal fraction of the mixture or to the increase in the length-scale parameter, also appears in multi-cracked nanobeams and this stiffening nonlocal effect is more pronounced in more constrained nanobeams.

Size-dependent fracture analysis of Centrally-Cracked nanobeam using Stress-Driven Two-Phase Local/Nonlocal integral model with discontinuity and symmetrical conditions

A novel mathematical formulation is presented to deal with stress-driven two-phase local/nonlocal integral model (SD-TPNIM) with discontinuity and symmetrical conditions, which is transformed equivalently into the differential form together with four constitutive constraints. Compare with classic SD-TPNIM, two extra constitutive continuity conditions are introduced, and a few integral items are introduced to constitutive discontinuity conditions and constitutive boundary conditions at symmetrical point. SD-TPNIM with symmetrical or anti-symmetrical conditions is applied to formulate the size-dependent fracture behavior of simply-supported and clamped–clamped centrally-cracked nanobeams subjected to uniformly distributed load (UDL) or middle point force (MPF) along opposite (Mode I crack) or same (Mode II crack) directions. The bending deflections are deduced and expressed in explicit form for different boundary and loading conditions. Accordingly, one can calculate the work done by external loads and energy release rate for both mode I and II crack problems. In addition, the asymptotic solutions for energy release rate are deduced for small nonlocal length parameter. The influence of nonlocal parameters on the size-dependent fracture behaviors including external work and energy release rate as well as the accuracy of asymptotic solutions for energy release rate is investigated numerically. The size-dependent fracture behavior can explain the superior fracture performance of nanomaterial to some extent.

Buckling of cracked micro- and nanocantilevers

The size-dependent buckling problem of cracked micro- and nanocantilevers, which have many applications as sensors and actuators, is studied by the stress-driven nonlocal theory of elasticity and Bernoulli–Euler beam model. The presence of the crack is modeled by assuming that the sections at the left and right sides of the crack are connected by a rotational spring. The compliance of the spring, which relates the slope discontinuity and the bending moment at the cracked cross section, is related to the crack length using the method of energy consideration and the theory of fracture mechanics. The buckling equations of the left and right sections are solved separately, and the variationally consistent and constitutive boundary and continuity conditions are imposed to close the problem. Novel insightful results are presented about the effects of the crack length and location, and the nonlocality on the critical loads and mode shapes, also for higher modes of buckling. The results of the present model converge to those of the intact nanocantilevers when the crack length goes to zero and to those of the large-scale cracked cantilever beams when the nonlocal parameter vanishes.

On the dynamics of 3D nonlocal solids

The stress-driven nonlocal continuum theory is a well-established integral elasticity formulation. Modelling the elastic strain via a convolution integral between stress and an averaging kernel, the theory leads to an integro-differential structural problem for arbitrary 3D nonlocal solids. Integral elasticity solutions are mainly available for 1D solids by adopting a bi-exponential kernel and reverting the boundary-value integro-differential problem to an equivalent differential formulation involving both classical and constitutive boundary conditions. A general computational framework to implement the stress-driven theory for 3D nonlocal solids is still missing. This paper tackles this issue, solving the relevant integro-differential problem by a two-field finite-element approach involving displacements and stresses. The approach can handle the dynamics of complex small-size structures of arbitrary shape. Focusing on free vibration responses, two examples are presented: a 2D rectangular plate, serving as benchmark to demonstrate size effects captured by stress-driven nonlocal elasticity, and a 3D solid modelling a typical dual microcantilever resonator with overhang.

Bending and buckling analysis of functionally graded Timoshenko nanobeam using Two-Phase Local/Nonlocal piezoelectric integral model

Previous studies indicate that nonlocal piezoelectric differential model would lead to an inconsistent bending response of Euler-Bernoulli nanobeam. In this paper, static bending and elastic buckling of functionally graded piezoelectric (FGP) Timoshenko nanobeam are studied through two-phase local/nonlocal piezoelectric integral model. The differential governing equations and standard boundary conditions are derived on the basis of the principle of minimum potential energy. The relations between general strain and nonlocal stress components are expressed as integral forms and then transformed equivalently into differential forms with constitutive boundary conditions. Several nominal variables are introduced to simplify the mathematical formulation. It shows that Timoshenko nanobeam based on purely nonlocal piezoelectric integral model would lead to ill-posed mathematical model and inconsistent size-dependent response with and without considering constitutive boundary conditions, respectively. The general differential quadrature method is applied to obtain the numerical results for bending deflections and buckling loads. The influence of nonlocal parameters, boundary conditions, functionally graded index and buckling order on the bending deflections and buckling loads is investigated numerically. A consistent softening response is obtained.

Bearing behavior of high-performance concrete-filled high-strength steel tube composite columns subjected to eccentrical load

In order to study the bearing capacity of high-performance concrete-filled high-strength steel tube (HCHST) composite stub columns subjected to eccentrical load, 22 HCHST composite stub columns, 4 concrete-filled steel tube (CFST) composite stub columns and 8 high-performance concrete-filled steel tube (HCST) composite stub columns were designed with the cubic compressive strength of concrete (fcu), the yield strength of steel tube (fy), the thickness of tube wall (t), the eccentricity (e) and slenderness ratio (λ) as the main parameters. Considering the nonlinear constitutive model of concrete and simplified constitutive model of steel, the finite element (FE) model of HCHST composite stub columns was established by ABAQUS software. By comparison with the existing test results, the rationality of the constitutive model and boundary conditions was verified. The variation of ultimate bearing capacity and the typical failure modes of HCHST composite stub columns under different parameters was analyzed. The results show that the specimens exhibit obvious bulge outward at the end of the steel tube and shear failure at the end of concrete. High-performance concrete (HPC) can significantly improve the ultimate eccentrical compression bearing capacity of composite columns, and high-strength steel tubes have better restraint effect on HPC. With the increasing of t, the ultimate eccentrical compression bearing capacity and the load-holding capacity of HCHST columns increases gradually, while the ultimate eccentrical compression bearing capacity decreases gradually with the increasing of λ and e. By introducing the reduction coefficient of eccentricity (φ1) and the reduction coefficient of slenderness ratio (φ2), the calculation formula of the eccentric bearing capacity of HCHST columns is proposed by statistical regression, which can lay a foundation for the application of HCHST columns in practical engineering.

Stress-driven nonlinear behavior of curved nanobeams

A nonlocal stress-driven model for geometrically nonlinear behavior of a shallow arch under radial pressure is presented. The analytical nonlinear equilibrium equation and also buckling equations by variational principles and virtual work are found. As it has been proven before, the strain-driven models encountered some contradictions with local equilibrium conditions, while stress-driven formulation has solved these difficulties. The simultaneous participation of the constitutive boundary conditions has led to this improvement. To obtain nonlocal strains, local nonlinear strains are incorporated in the nonlocal stress-based equations. In this way, nonlocal loads corresponding to the limit-load and bifurcation instabilities are exactly calculated by considering the effect of nanoscale and geometry parameters. In this article, the geometric limits which determine when and how the arch bifurcates, are found. The results in specific cases are compared with those available in the literature. These comparisons show the accuracy of the work. Finally, the 3D view of the equilibrium nonlinear paths, associated with the pinned curved nanobeam, are presented for different nanoscale and also geometry ratio parameters.

Modeling of buckling of nanobeams embedded in elastic medium by local-nonlocal stress-driven gradient elasticity theory

A novel buckling model is formulated for the Bernoulli-Euler nanobeam resting on the Pasternak elastic foundation. The formulation is based on the local-nonlocal stress-driven gradient elasticity theory. In order to incorporate the size-dependency, the strain at each point is defined as the integral convolutions in terms of the stresses and their first-order gradients in all the points, accounting also for the local contribution. The differential form of the nonlocal constitutive equation, together with a set of constitutive boundary conditions, are used to define the buckling equation in terms of transverse displacement, which is solved in closed form. Both variationally consistent and the constitutive boundary conditions are imposed to calculate the buckling loads and the corresponding mode shapes. The predictions of the present model are in agreement with the results available in the literature for the carbon nanotubes based on the molecular dynamics simulations. Insightful results are presented for the first three buckling modes of local-nonlocal nanobeams considering the gradient effects. The distinctive feature of the present model is its capability to capture both stiffening and softening behaviors at the small-scales, which result in, respectively, higher and lower buckling loads of the nanobeams with respect to those of the large-scale beams.

A new finite element method framework for axially functionally-graded nanobeam with stress-driven two-phase nonlocal integral model

Nano-structures always show size effects. In the paper, a new FEM framework is developed to analyze the mechanical responses of nanobeams made of axially functionally-graded material (FGM) under different boundary conditions. The stress-driven two-phase nonlocal integral model is utilized to introduce the size effect into the model. The present work is based on the nonlocal constitution in a differential form and the corresponding constitutive boundary conditions are converted to external loads equivalently. A series of numerical examples, including static bending, buckling, and free vibration are carried out for the beams with different boundary conditions and various nonlocal and FGM parameters. The present framework shows a promising way for its flexibility in handling complex FGM distribution patterns, boundary conditions, and loads.

Free transverse vibrations of nanobeams with multiple cracks

A nonlocal model is formulated to study the size-dependent free transverse vibrations of nanobeams with arbitrary numbers of cracks. The effect of the crack is modeled by introducing discontinuities in the slope and transverse displacement at the cracked cross-section, proportional to the bending moment and the shear force transmitted through it. The local compliance of each crack is related to its stress intensity factors assuming that the crack tip stress field is undisturbed (non-interacting cracks).The kinematic field is defined based on the Bernoulli-Euler beam theory, and the small-scale size effect is taken into account by employing the constitutive equation of the stress-driven nonlocal theory of elasticity. In this manner, the curvature at each cross-section is defined as an integral convolution in terms of the bending moments at all the cross-sections and a kernel function which depends on a material characteristic length parameter. The integral form of the nonlocal constitutive equation is elaborated and converted into a differential equation subjected to a set of mathematically consistent boundary and continuity conditions at the nanobeam's ends and the cracked cross-sections. The equation of motion in each segment of the nanobeam between cracks is solved separately and the variationally consistent and constitutive boundary and continuity conditions are imposed to determine the natural frequencies. The model is applied to nanobeams with different boundary conditions and the natural frequencies and the mode shapes are presented at the presence of one to four cracks. The results of the model converge to the experimental results available in the literature for the local cracked beams and to the solutions of the intact nanobeams when the crack length goes to zero. The effects of the crack location, crack length, and nonlocality on the natural frequencies are investigated, also for the higher modes of vibrations. Novel findings including the amplification and shielding effects of the cracks on the natural frequencies are presented and discussed.

Nonlocal gradient integral models with a bi-Helmholtz averaging kernel for functionally graded beams

In this paper, we develop the bi-Helmholtz averaging kernel-based variationally nonlocal strain and stress gradient (NStrainG and NStressG) integral models to study the size-dependent bending of functionally graded (FG) beams. The integral constitutive relation is transformed into a higher-order differential form while four non-standard constitutive boundary conditions are derived in detail. The Laplace transform technique is utilized to derive closed-form solutions. Interestingly, we find that, unlike the Helmholtz kernel case, the local stress contribution must be included in the bi-Helmholtz kernel-based NSrainG model to avoid the conflict between equilibrium and constitutive requirements. In the numerical simulations, a series of examples are presented to study the size-effects on the bending deformation of the beams with various boundary types. Numerical results show that the bi-Helmholtz kernel-based nonlocal gradient theories are capable of describing both softening and stiffening size-effects in small structures by suitably tuning the nonlocal and gradient relevant parameters involved in nonlocal gradient models. Moreover, the effect of FG parameter is also presented.

Stress-driven local/nonlocal mixture model for buckling and free vibration of FG sandwich Timoshenko beams resting on a nonlocal elastic foundation

We study the buckling and free vibration of functionally graded (FG) sandwich Timoshenko beams resting on an elastic foundation. 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 stress-driven strategy equipped with a bi-Helmholtz kernel. We find that in the presence of a nonlocal elastic foundation, the local/nonlocal mixture should be adopted in order to obtain a well-posed formulation of the problem at hand. The equations of motion and the standard boundary conditions are obtained by invoking Hamilton's principle. Each integro-differential constitutive law is transformed into an equivalently differential form equipped with four non-standard constitutive boundary conditions. The generalized differential quadrature method (GDQM) is then utilized to solve the corresponding eigenvalue problems numerically. Several comparative studies are conducted to validate the effectiveness of our solution. The numerical simulation results present the size-effect on the critical buckling loads and natural frequency of the beams with various boundary types, providing a new benchmark for further study of modeling small-scale beam structures using the bi-Helmholtz kernel-based stress-driven mixture model. Moreover, the influence of considering the size-dependency of the elastic foundation is also investigated.

Size dependent effects of two phase viscoelastic medium on damping vibrations of smart nanobeams: an efficient implementation of GDQM

This paper studies the dynamics of nonlocal piezo-magnetic nanobeams (PMNBs) embedded in the local/nonlocal viscoelastic medium through the consistent and paradox-free model of the nonlocal theory. Besides, to perform the dynamic analysis, an exact solution and an efficient approach of generalized differential quadrature method (GDQM) are introduced. Since the size-dependency of the uniform loads is wrongly neglected by the nonlocal elasticity in differential form, the size-dependency of piezo-magnetic load is applied through the two-phase theory. Also, size dependency of the viscoelastic medium is accurately applied and examined through the solutions presented employing the differential two-phase theory and satisfying the constitutive boundary conditions. In this regard, the two-phase resultant equations of motions together with boundary conditions including the constitutive ones related to two-phase PMNB and the two-phase medium are attained. To confirm the credibility and efficiency of the extracted equations as well as presented solution procedures, several analogical studies are accomplished, and it is shown that the results obtained from the differential relations are reliable and consistence with those extracted from the integral nonlocal relations. It is shown that the present approach of the GDQM simplifies the solution procedures of the nonlocal problems and improves the precisions in the cases close to the pure nonlocal state. The presented results emphasize that the size-dependency of viscoelastic medium, external electric, and magnetic loads play significant roles on the vibration characteristics, and therefore it must be considered based on two-phase theory. The available results can be helpful to achieve an excellent design of smart nanobeams embedded in viscoelastic medium.

A mixed two-phase stress/strain driven elasticity: In applications on static bending, vibration analysis and wave propagation

In recent years, there are numerous papers that deal with two-phase strain-driven model, including softening effect, and two-phase stress-driven model, with stiffening effect, and also the nonlocal strain gradient model, with both the softening and stiffening effects, to seek the impacts of length scales on the mechanics of structures in small scales. The current paper is a novel well-posed mixture model to cover all previous theories through employing various proposed small scales. Concurrent accounting of stiffening and softening effects is performed through employing a combination of the two-phase strain-driven elasticity with the two-phase stress-driven elasticity, with two additional local phase fraction factors and two various type of nonlocal parameters. Accordingly, in this model the total strain or stress at a certain point is regarded as a function of the stress or strain of all neighboring points in addition to the point of interest. Compatibility between integral and differential relations without any conflict of restrictions is doable through considering constitutive boundary conditions as key point of this match. To demonstrate its application values, the proposed mixture model as an efficient theoretical tool is hired for static bending, vibration and wave propagation in a mixed two-phase stress/strain driven elasticity system and the new essential relations are developed through examples for static bending, free vibration and wave propagating in Euler-Bernoulli nanobeams. The results are obtained by proposing an efficient exact solution, and the integrity as well as reliability of the present constitutive differential relations are evaluated through some validation studies. The output results in the framework of new suggested mixture model address some points about static bending, free vibration and wave propagation that is more comprehensive than the previous results of the contemporary continuum theories. Thus, based on the observations, this mixed two-phase stress/strain driven elasticity could be cover a widespread range of dynamic response in the nano-systems that the contemporary continuum theories can be only investigated some aspects of those problems.