Instytut Podstawowych Problemów Techniki

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A mathematically well-posed nonlocal model is formulated based on the variational approach and the transfer matrix method to investigate the size-dependent elastostatics of nonuniform miniaturized beams. The beams are composed of an arbitrary number of sub-beams with diverse material and geometrical properties, as well as small-scale size dependency. The model adopts a stress-driven nonlocal approach, a well-established framework in the Engineering Science community. The curvature of a sub-beam is defined through an integral convolution, considering the bending moments across all cross-sections of the sub-beam and a kernel function. The governing equations are solved and the deflections are derived in terms of some constants. The formulation uses local and interfacial transfer matrices, incorporating continuity conditions at cross-sections where sub-beams are joined, to define relations between constants in the solution of a generic sub-beam and those of the first sub-beam at the left end. The boundary conditions are then imposed to derive an explicit, closed-form solution for the deflection. The solution significantly simplifies the study of nonuniform beams with multiple sub-beams. The predictions of the model for two limiting cases, namely local nonuniform and nonlocal uniform beams, are in excellent agreement with the available literature data. The flexural behavior of nonuniform miniaturized beams, composed of two to five different sub-beams and subjected to different boundary conditions, is studied. The results are presented and discussed, emphasizing the effects of the material properties, nonlocalities, and lengths of the sub-beams on the deflection. It is demonstrated that the flexural response of nonlocal nonuniform beams is more complex than local counterparts. Unlike the local beams, dividing a nonlocal uniform beam into multiple sub-beams and then reconnecting them changes the overall stiffness of the beam. The study highlights the potential to design nonuniform miniaturized beams with specific configurations to control their flexural response effectively.

Słowa kluczowe:

Small-scale beam,Transfer matrix method,Multi-material,Size effect,MEMS,NEMS

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Structural schemes of applicative interests in Engineering Science frequently encounter the intricate phenomenon of discontinuity. The present study intends to address the discontinuity in the flexure of elastic nanobeam by adopting an abstract variational scheme. The mixture unified gradient theory of elasticity is invoked to realize the size-effects at the ultra-small scale. The consistent form of the interface conditions, stemming from the established stationary variational principle, is meticulously set forth. The boundary-value problem of equilibrium is properly closed and the analytical solution of the transverse displacement field of the elastic nanobeam is addressed. As an alternative approach, the eigenfunction expansion method is also utilized to scrutinize the efficacy of the presented variational formulation in tackling the flexure of elastic nanobeams with discontinuity. The flexural characteristic of mixture unified gradient beams with diverse kinematic constraints is numerically illustrated and thoroughly discussed. The anticipated nanoscopic features of the characteristic length-scale parameters are confirmed. The demonstrated numerical results can advantageously serve as a benchmark for the analysis and design of pioneering ultra-sensitive nano-sensors. The established variationally consistent size-dependent framework paves the way ahead in nanomechanics and inspires further research contributing to fracture mechanics of ultra-small scale elastic beams.

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Studying the dynamics of small-scale beams with attached particles is crucial for sensing applications in various fields, such as bioscience, material science, energy storage devices, and environmental monitoring. Here, a stress-driven nonlocal model is presented for the free transverse vibration of small-scale beams carrying multiple masses taking into account the eccentricity of the masses relative to the beam axis. The results show excellent agreement with the experimental and numerical data in the literature. New insights into the frequency shifts and mode shapes of the first four vibrational modes of stress-driven nonlocal beams with up to three attached particles are presented. The study investigates the inverse problem of detecting the location and mass of an attached particle based on natural frequency shifts. The knowledge acquired from the present study provides valuable guidance for the design and analysis of ultrasensitive mechanical mass sensors.

Słowa kluczowe:

Size effect, Mass sensor, Micro- and nanobeam, Nonlocal, Inverse problem

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The buckling instability of micro- and nanopillars can be an issue when designing intelligent miniaturized devices and characterizing composite materials reinforced with small-scale beam-like particles. Analytical modeling of the buckling of miniaturized pillars is especially important due to the difficulties in conducting experiments. Here, a well-posed stress driven nonlocal model is developed, which allows the calculation
of the critical loads and buckling configurations of the miniaturized pillars on an elastic foundation and with arbitrary numbers of edge cracks. The discontinuities in bending slopes and deflection at the damaged cross-sections due to the edge cracks are captured through the incorporation of both rotational and translational springs. A comprehensive analysis is conducted to investigate the instability of pillars containing a range of one to four cracks. This analysis reveals interesting effects regarding the influence of crack location, nonlocality, and elastic foundation on the initial and subsequent critical loads and associated buckling configurations. The main findings are: (i) the shielding and amplification effects related to a system of cracks become more significant as the dimensions of pillars reduce, (ii) the influence of the shear force at the damaged cross-section related to the translational spring must not be neglected when dealing with higher modes of buckling and long cracks, (iii) an elastic foundation decreases the effects of the cracks and size dependency on the buckling loads, and (iv) the effects of the edge cracks on the critical loads and buckling configurations of the miniaturized pillars are highly dependent on the boundary conditions.

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Multilayered metal-ceramic composites belong to the class of functionally graded materials with a step-wise gradient in material composition. These advanced structural materials can be tailored to meet design requirements. Aluminum-matrix composites are one of the most attractive metal-ceramic composites due to low specific weight, good thermal conductivity, enhanced specific strength, and low cost of the constituent materials. A comprehensive investigation of the fracture properties and mechanisms of layered aluminum-matrix composites is required to enhance their utilization in practical applications.
This chapter is focused on experiments and modeling of fracture in functionally graded AlSi12-Al2O3 composites. Three-layer bulk disks with 10, 20, and 30% volume fractions of Al2O3 are manufactured through powder metallurgy. Single-edge notched samples (SEVNB) are prepared from the bulk material and tested under four-point bending. The fracture tests are simulated using the phase-field modeling of brittle fracture. In the phase-field models, individual layers are considered homogeneous linear elastic isotropic materials with effective properties estimated by the rule of the mixture. The length scale parameter is calibrated by fitting the numerically determined fracture loads to the experimental data. The phase-field model is then used to investigate the impact of the stacking sequence on the load-displacement curves of the fracture specimens. It is revealed that the stacking sequence may significantly affect the load-displacement curves, including changes to the maximum load and post-peak response. The ability of the phase-field model to capture the crack arrestment, branching, and deflection in functionally graded layered materials is shown.

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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.

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The available experimental results in the literature on the quasi-static bending and free flexural vibration of microcantilevers and nanocantilevers are used to calibrate the length scale parameter of the stress-driven nonlocal elasticity model. The Bernoulli–Euler theory is used to define the kinematic field. The closed form solution derived for the bending problem is used to calibrate the length scale parameter by fitting the load–displacement curves to the experimental results. For the vibration problem, the calibration is done using the least-squares curve fitting method for the natural frequencies. The stress-driven nonlocal theory can adequately capture the size-dependent experimental results.

Słowa kluczowe:

nonlocal elasticity,stress-driven,experiment,length scale,calibration,MEMS,NEMS

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A unified approach is applied for determining both strain- and stress-driven differential formulations of Timoshenko nano-beams in presence of loading discontinuities. The consequent models can simulate small scale effects with different types of constitutive laws (such as pure nonlocal, mixture of local and nonlocal phases, and nonlocal gradient). A specific novel feature of the proposed models is the ability to consider loading discontinuities, i.e. points of discontinuities for generalized internal forces occurring in presence of external supports, forces, or couples concentrated at internal points of the nano-beam. To this end, novel constitutive continuity conditions (CCCs) are imposed at the beam interior points of loading discontinuities. CCCs contain integral convolutions of generalized forces or displacements over suitable parts of the nano-beam; they represent a valid alternative to Dirac delta function and are different from the well-known constitutive boundary conditions (CBCs) imposed at the end-points of the nano-beam. Finally, the proposed models are applied for finding closed-form solutions to cases of practical interest.

Słowa kluczowe:

nonlocal gradient elasticity, constitutive boundary conditions, constitutive continuity conditions, nano-beams, NEMS

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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.

Słowa kluczowe:

cracked nanobeam, transverse vibration, nonlocal elasticity, size effect

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A novel approach is proposed to determine a physically meaningful length-scale parameter for the phase-field modeling of macroscale fracture in metal–ceramic composites on an example of chromium–alumina composite fabricated by powder metallurgy. The approach is based on the fractography analysis by the scanning electron microscopy (SEM) with the aim to measure the process zone size and use that value as the length-scale parameter in the phase-field modeling. Mode I and mixed-mode I/II fracture tests are conducted on Cr–Al2O3 composites at different reinforcement volume fractions and particle sizes using single-edge notched beams under four-point bending. The fracture surfaces are analyzed in detail by SEM to determine the size of the process zone where the microscale nonlinear fracture events occur. The model adequately approximates the experimentally measured fracture toughness and the fracture loads. It is shown that the model prediction of the crack initiation direction under the mixed-mode loading is in agreement with the experiments and the generalized maximum tangential stress criterion. These outcomes justify using the process zone size as the scale parameter in the phase-field modeling of macroscale fracture in chromium–alumina and similar metal–ceramic composites.

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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.

Słowa kluczowe:

nanobeams, nonlocal elasticity, stress gradient, buckling, Pasternak foundation

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Mode I fracture behavior of edge- and centrally-cracked nanobeams is analyzed by employing both stress-driven non-local theory of elasticity and Bernoulli–Euler beam theory. The present formulation implements the size-dependency experimentally observed at material micro- and nanoscale, by assuming a non-local constitutive law, that relates the strain to the stress in each material point of the body, through an integral convolution and a kernel. It is observed that the energy release rate decreases by increasing the nonlocality, showing the superior fracture performance of nanobeams with respect to large-scale beams.

Słowa kluczowe:

energy release rate, nanobeam, stress-driven, non-local integral model, stress intensity factor

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Injected anchors in masonry elements represent a widespread technique for improving the 'box behavior' of masonry structures, since they contribute to avoid or delay out-of-plane mechanisms under horizontal actions. Despite of their diffusion, no clear design indications for injected anchors are available in literature and codes. This paper is aimed to propose design formulations for the maximum pull-out force in injected anchors basing on wide numerical analyses realized through a 2D Finite Element (FE) model specifically tuned to simulate pull-out tests. Thanks to the variation of several parameters, the most significant ones influencing the maximum pull-out force are identified and introduced in the strength models and several coefficients are assessed through best fitting regression analyses carried out on the numerical results. Finally, based on a 'design by testing' approach, preliminar 5% percentile provisions for the maximum pull-out force are proposed too, and the reliability of the ‘design-oriented’ formulation is assessed by means of comparisons with experimental results of some pull-out tests available in literature.

Słowa kluczowe:

injected anchors, masonry pull-out force, bond, regression analysis, design formulations

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The size-dependent bending of perfectly/imperfectly bonded multilayered/stepwise functionally graded nanobeams, e.g. multiwalled carbon nanotubes with weak van der Waals forces, with any arbitrary numbers of layers, exhibiting different material, geometrical, and length-scale properties, is studied through a layerwise formulation of the stress-driven nonlocal theory of elasticity and the Bernoulli-Euler beam theory. The formulation is also valid for the continuously graded nanobeams, where the through-the-thickness material gradation with any arbitrary distribution is approximated in a stepwise manner through many layers. The size-dependency of each layer is accounted for through nonlocal constitutive relationships, which define the strains at each point as the output of integral convolutions in terms of the stresses in all the points of the layer and a kernel. Linear elastic uncoupled interfacial laws are implemented to model the mechanical response of the interfaces. The size-dependent system of equilibrium equations governing the deformations of the layers are derived and subjected to the variationally consistent edge boundary conditions and the constitutive boundary conditions associated with the stress-driven integral convolution. The formulation is applied to multilayered and sandwich nanobeams and the effects of the interfacial imperfections on the displacement fields and the interfacial displacement jumps are studied. It is found that the interfacial imperfections have greater impact on the field variables of multilayered nanobeams than that of the multilayered beams with the large-scale dimensions.

Słowa kluczowe:

layered nanobeam, discrete layer approach, size-effect, imperfect interface, nonlocal elasticity

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The instability of nanobeams rested on two-parameter elastic foundations is studied through the Bernoulli-Euler beam theory and the stress-driven nonlocal elasticity model. The size-dependency is incorporated into the formulation by defining the strain at each point as an integral convolution in terms of the stresses in all the points and a kernel. The nonlocal elasticity problem in a bounded domain is well-posed and inconsistencies within the Eringen nonlocal theory are overcome. Excellent agreement is found with the results in the literature, and new insightful results are presented for the buckling loads of nanobeams rested on the Winkler and Pasternak foundations.

Słowa kluczowe:

buckling, closed form solution, nanobeam, nonlocal elasticity, Pasternak foundation, stress-driven

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The size-dependent buckling instability of shear deformable nanobeams rested on a two-parameter elastic foundation is studied through the stress-driven nonlocal theory of elasticity and the kinematic assumptions of the Timoshenko beam theory. The small-scale size effects are taken into account by nonlocal constitutive relationships, which define the strains at each point as integral convolutions in terms of the stresses in all the points and a kernel. In this manner, the nonlocal elasticity formulation is well-posed and does not include inconsistencies usually arising using other nonlocal models. The size-dependent governing differential equations in terms of the transverse displacement and the cross-sectional rotation are decoupled, and closed form solutions are presented for the displacement functions. Proper boundary conditions are imposed and the buckling problem is reduced to finding roots of a determinant of a matrix, whose elements are given explicitly for different classical edge conditions. The closed form treatment of the problem avoids the numerical instabilities usually occurring within numerical techniques, and allows to find also higher buckling loads and shape modes. Several nanobeams rested on the Winkler or Pasternak elastic foundations and characterized by different boundary conditions, shear deformability, and nonlocality are considered and the critical loads and shape modes are presented, including those for the higher modes of buckling. Excellent agreements are found with the available approximate numerical results in the literature and novel insightful findings are presented and discussed, which are in accordance with experimental observations.

Słowa kluczowe:

nanobeam, buckling, elastic foundation, closed form solution, nonlocal elasticity, size effect

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A nonlocal linear elastic fracture formulation is presented based on a discrete layer approach and an interface model to study cracked nanobeams. The formulation uses the stress-driven nonlocal theory of elasticity to account for the size-dependency in the constitutive equations, and the Bernoulli-Euler beam theory to define the kinematic field. Two fundamental mode I and mode II fracture nanospecimens with applications in Engineering Science are studied to reveal principal characteristics of the linear elastic fracture of beams at nanoscale. The domains are discretized both through the transverse and longitudinal directions and the field variables are derived by solving systems of the nonlocal equilibrium equations subjected to the variationally consistent and constitutive boundary and continuity conditions. The energy release rates of the fracture nanospecimens are calculated both from the global energy consideration and from the localized fields at the tip of the crack, i.e. the cohesive forces and the displacement jumps. The results are shown to be the same, proving the capability of the interface model to predict localized fields at the crack tip which are important for the cohesive fracture problems. It is found that the nanospecimens with higher nonlocality have higher fracture resistance and load bearing capacity due to higher energy absorptions and lower energy release rates. The crack propagation in the nanospecimens are also studied and load-displacement curves are presented. The nonlocality considerably increases the stiffness of the initial linear response of the nanospecimens. The fracture model is also able to capture the non-linear post-peak response and the unstable crack propagation, the snap-back instability, which is more intense for nanospecimens with higher nonlocality.

Słowa kluczowe:

cracked nanobeams, nonlocal fracture, energy release rate, cohesive, crack propagation

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Injected anchors made of steel bars embedded in masonry elements by means of cement-based grout represented in the past a wide solution for avoiding out-of-plane mechanisms. Corrosion phenomena in steel bars reduced the effectiveness of such type of intervention over time. Innovative materials, as the Fiber Reinforced Plastic ones, can represent a suitable alternative to increase durability and performance of injected anchors. Since the effectiveness of injected anchors is strictly related to bond behaviour along both the bar-grout and the grout-masonry interfaces, a detailed analysis by means of a Finite Element model was developed for different types of bars embedded in masonry elements. The numerical model was firstly calibrated on some experimental results of pull-out tests available in literature and, then, is used for investigating the effects of several parameters on both local and global behaviour. Load-displacement curves and local distributions of shear stresses are examined in detail. The numerical analyses evidenced that the maximum tensile force in the anchor mainly depends on the shear strength of the bar-grout and the grout-masonry interfaces and on the embedded length, but for very long embedded length, it can be limited by the tensile failure in the anchor or in the masonry.

Słowa kluczowe:

masonry, FRP bars, injected anchors, bond, pull-out test, FE model

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In this work, variational formulations are proposed for solving numerically the problem of bending and buckling of Timoshenko nano-beams. The present work belongs to research branch in which the non-local theory of elasticity has been used for analysis of beam-like elements in smart materials, micro-electro-mechanical (MEMS) or nano-electro-mechanical systems (NEMS). In fact, the local beam theory is not adequate to describe the behavior of beam-like elements of smart materials at the nano-scale, so that different non-local models have been proposed in last decades for nano-beams. The nano-beam model considered in this work is a convex combination (mixture) of local and non-local phases. In the non-local phase, the kinematic entities in a point of the nano-beam are expressed as integral convolutions between internal forces and an exponential kernel. The aim is to construct a functional whose stationary condition provides the solution of the problem. Two different functionals are defined: one for the pure non-local model, where the local fraction of the mixture is absent, and the other for the mixture with both local and non-local phases. The Euler equations of the two functionals are derived; then, attention focuses on the mixture model. The functional of the mixture depends on unknown Lagrange multipliers and the Euler equations of the functional provide not only the governing equations of the problem but also the relationships between these Lagrange multipliers and the other variables on which the functional depends. In fact, approximations of the variables of the functional can not be chosen arbitrarily in numerical analyzes but have to satisfy suitable conditions. The Euler equations involving the Lagrange multipliers are essential in the numerical analyzes and suggest the correct approximations that have to be adopted for Lagrange multipliers and the other unknown variables of the functional. The proposed method is verified by comparing numerical solutions with exact solutions in bending problem. Finally, the method is used to determine the buckling load of Timoshenko nano-beams with mixture of phases.

Słowa kluczowe:

non-local elasticity, variational methods, Timoshenko beam, buckling load, smart materials

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Free flexural vibrations of nanobeams with non-rigid edge supports are studied by means of the stress-driven nonlocal elasticity model and Euler-Bernoulli kinematics. The elastic deformations of the supports are modelled by transversal and flexural springs, so that, in the limit conditions when the springs stiffnesses tend to zero or infinity, the classical free, pinned, and clamped boundary conditions may be recovered. An analytical procedure is used to derive the closed form solution of the spatial differential equation. The problem of finding the natural frequencies is then reduced to find the roots of the determinant of a matrix, whose elements are explicitly given. The proposed technique, then, avoids the numerical instabilities usually arising when the numerical techniques are used to obtain the solution. The effects of both non-rigid supports elastic deformations and nonlocal parameter on the natural frequencies are studied also for higher vibrations modes. The comparison between the solutions of the proposed model and those available in the literature shows an excellent agreement, and new insightful results and discussions are presented.

Słowa kluczowe:

elastically constrained beam, nanostructures, natural frequency, size effects, well-posed nonlocal formulation

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Interfacial tractions generated at the interface in two-layered nanobeams are studied through the stress-driven nonlocal theory of elasticity and an interface model. The model uses a layerwise description of the problem and satisfies the continuity conditions at the interface. The size-dependency are incorporated into formulation through a nonlocal constitutive law which defines the strain at each point as an integral convolution in terms of the stresses in all the points and a kernel. The Bernoulli-Euler beam theory is used separately for each layer to describe kinematic field, and to derive size-dependent system of coupled governing equations. The displacement components within the layers are derived and the interfacial tractions are obtained through the interfacial constitutive relations. Results are presented for the interfacial shear and normal tractions, exhibiting a different behavior at the nano-scale compared to those of the layered beams with large-scale dimensions including different maximum interfacial tractions and the location where maxima occur. A superior resistance of nanobeams against debondings and delaminations due to the interfacial normal tractions compared to that of the beams with large-scale dimensions is observed. The formulation and the understandings presented here are expected to stimulate further researches on multilayered nanobeams, including their interfacial fracture mechanics.

Słowa kluczowe:

multilayered nanobeams, weak bonding, interfacial tractions, delamination, nonlocal elasticity

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Streszczenie:

A novel mathematical formulation is presented for the applications of the stress-driven nonlocal theory of elasticity to engineering nano-scale problems requiring longitudinal discretization. Specifically, a differential formulation accompanied with novel constitutive continuity conditions is provided for determining exact closed-form solutions of nonlocal Euler-Bernoulli beams with loading discontinuities, i.e. points of discontinuity for external loads and internal forces. Constitutive continuity conditions have to be satisfied in interior points where a loading discontinuity occurs and contain integral convolutions of the stress over suitable parts of the nonlocal beam. Several results show the effectiveness of the proposed method.

Słowa kluczowe:

closed-form solutions, discretization, Euler-Bernoulli beams, nanobeams

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