dr Hossein Darban 

Doktorat
20180510  Multiscale Modeling of Delamination Fracture in Multilayered Structures (University of Genoa)
 1472 
Ostatnie publikacje
1.  Darban H., Elastostatics of nonuniform miniaturized beams: Explicit solutions through a nonlocal transfer matrix formulation, International Journal of Engineering Science, ISSN: 00207225, DOI: 10.1016/j.ijengsci.2024.104054, Vol.198, No.104054, pp.118, 2024 Streszczenie: A mathematically wellposed nonlocal model is formulated based on the variational approach and the transfer matrix method to investigate the sizedependent elastostatics of nonuniform miniaturized beams. The beams are composed of an arbitrary number of subbeams with diverse material and geometrical properties, as well as smallscale size dependency. The model adopts a stressdriven nonlocal approach, a wellestablished framework in the Engineering Science community. The curvature of a subbeam is defined through an integral convolution, considering the bending moments across all crosssections of the subbeam 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 crosssections where subbeams are joined, to define relations between constants in the solution of a generic subbeam and those of the first subbeam at the left end. The boundary conditions are then imposed to derive an explicit, closedform solution for the deflection. The solution significantly simplifies the study of nonuniform beams with multiple subbeams. 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 subbeams 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 subbeams 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 subbeams 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: Smallscale beam,Transfer matrix method,Multimaterial,Size effect,MEMS,NEMS Afiliacje autorów:
 200p.  
2.  Darban H., Luciano R.^{♦}, Basista M. A., Modeling frequency shifts in smallscale beams with multiple eccentric masses, ThinWalled Structures, ISSN: 02638231, DOI: 10.1016/j.tws.2024.112005, Vol.201, No.Part A, pp.112005111200519, 2024 Streszczenie: Studying the dynamics of smallscale 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 stressdriven nonlocal model is presented for the free transverse vibration of smallscale 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 stressdriven 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 Afiliacje autorów:
 140p.  
3.  Darban H., Luciano R.^{♦}, Basista M.A., Effects of multiple edge cracks, shear force, elastic foundation, and boundary conditions on bucking of smallscale pillars, INTERNATIONAL JOURNAL OF DAMAGE MECHANICS, ISSN: 10567895, DOI: 10.1177/10567895231215558, Vol.33, No.4, pp.247268, 2024 Streszczenie: The buckling instability of micro and nanopillars can be an issue when designing intelligent miniaturized devices and characterizing composite materials reinforced with smallscale beamlike particles. Analytical modeling of the buckling of miniaturized pillars is especially important due to the difficulties in conducting experiments. Here, a wellposed stress driven nonlocal model is developed, which allows the calculation Afiliacje autorów:
 100p.  
4.  Darban H., Bochenek K., Węglewski W., Basista M.A., Experimental Evaluation and PhaseField Model of Fracture Behavior of AluminaAluminium Graded Composite, Advanced Structured Materials, ISSN: 18698433, DOI: 10.1007/9783031455544_4, Vol.199, pp.147166, 2024 Streszczenie: Multilayered metalceramic composites belong to the class of functionally graded materials with a stepwise gradient in material composition. These advanced structural materials can be tailored to meet design requirements. Aluminummatrix composites are one of the most attractive metalceramic 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 aluminummatrix composites is required to enhance their utilization in practical applications. Afiliacje autorów:
 
5.  Darban H., Size effect in ultrasensitive micro and nanomechanical mass sensors, MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 08883270, DOI: 10.1016/j.ymssp.2023.110576, Vol.200, pp.110576111057611, 2023  200p.  
6.  Darban H., Luciano R.^{♦}, Darban R.^{♦}, Buckling of cracked micro and nanocantilevers, ACTA MECHANICA, ISSN: 00015970, DOI: 10.1007/s0070702203417x, Vol.234, pp.693704 , 2023 Streszczenie: The sizedependent buckling problem of cracked micro and nanocantilevers, which have many applications as sensors and actuators, is studied by the stressdriven 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 largescale cracked cantilever beams when the nonlocal parameter vanishes. Afiliacje autorów:
 100p.  
7.  Darban H., Luciano R.^{♦}, Basista M.A., Calibration of the length scale parameter for the stressdriven nonlocal elasticity model from quasistatic and dynamic experiments, MECHANICS OF ADVANCED MATERIALS AND STRUCTURES, ISSN: 15376494, DOI: 10.1080/15376494.2022.2077488, Vol.30, No.17, pp. 3518 3524, 2023 Streszczenie: The available experimental results in the literature on the quasistatic bending and free flexural vibration of microcantilevers and nanocantilevers are used to calibrate the length scale parameter of the stressdriven 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 leastsquares curve fitting method for the natural frequencies. The stressdriven nonlocal theory can adequately capture the sizedependent experimental results. Słowa kluczowe: nonlocal elasticity,stressdriven,experiment,length scale,calibration,MEMS,NEMS Afiliacje autorów:
 70p.  
8.  Caporale A.^{♦}, Darban H., Luciano R.^{♦}, Nonlocal strain and stress gradient elasticity of Timoshenko nanobeams with loading discontinuities, International Journal of Engineering Science, ISSN: 00207225, DOI: 10.1016/j.ijengsci.2021.103620, pp.103620117, 2022 Streszczenie: A unified approach is applied for determining both strain and stressdriven differential formulations of Timoshenko nanobeams 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 nanobeam. 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 nanobeam; they represent a valid alternative to Dirac delta function and are different from the wellknown constitutive boundary conditions (CBCs) imposed at the endpoints of the nanobeam. Finally, the proposed models are applied for finding closedform solutions to cases of practical interest. Słowa kluczowe: nonlocal gradient elasticity, constitutive boundary conditions, constitutive continuity conditions, nanobeams, NEMS Afiliacje autorów:
 200p.  
9.  Darban H., Luciano R.^{♦}, Basista M., Free transverse vibrations of nanobeams with multiple cracks, International Journal of Engineering Science, ISSN: 00207225, DOI: 10.1016/j.ijengsci.2022.103703, Vol.177, pp.103703120, 2022 Streszczenie: A nonlocal model is formulated to study the sizedependent 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 crosssection, 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 (noninteracting cracks).The kinematic field is defined based on the BernoulliEuler beam theory, and the smallscale size effect is taken into account by employing the constitutive equation of the stressdriven nonlocal theory of elasticity. In this manner, the curvature at each crosssection is defined as an integral convolution in terms of the bending moments at all the crosssections 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 crosssections. 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 Afiliacje autorów:
 200p.  
10.  Darban H., Bochenek K., Węglewski W., Basista M., Experimental Determination of the LengthScale Parameter for the PhaseField Modeling of Macroscale Fracture in Cr–Al2O3 Composites Fabricated by Powder Metallurgy, METALLURGICAL AND MATERIALS TRANSACTIONS APHYSICAL METALLURGY AND MATERIALS SCIENCE, ISSN: 10735623, DOI: 10.1007/s11661022066773, pp.123, 2022 Streszczenie: A novel approach is proposed to determine a physically meaningful lengthscale parameter for the phasefield 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 lengthscale parameter in the phasefield modeling. Mode I and mixedmode I/II fracture tests are conducted on Cr–Al2O3 composites at different reinforcement volume fractions and particle sizes using singleedge notched beams under fourpoint 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 mixedmode 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 phasefield modeling of macroscale fracture in chromium–alumina and similar metal–ceramic composites. Afiliacje autorów:
 200p.  
11.  Darban H., Luciano R.^{♦}, Caporale A.^{♦}, Basista M., Modeling of buckling of nanobeams embedded in elastic medium by localnonlocal stressdriven gradient elasticity theory, COMPOSITE STRUCTURES, ISSN: 02638223, DOI: 10.1016/j.compstruct.2022.115907, Vol.297, pp.115907111, 2022 Streszczenie: A novel buckling model is formulated for the BernoulliEuler nanobeam resting on the Pasternak elastic foundation. The formulation is based on the localnonlocal stressdriven gradient elasticity theory. In order to incorporate the sizedependency, the strain at each point is defined as the integral convolutions in terms of the stresses and their firstorder 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 localnonlocal nanobeams considering the gradient effects. The distinctive feature of the present model is its capability to capture both stiffening and softening behaviors at the smallscales, which result in, respectively, higher and lower buckling loads of the nanobeams with respect to those of the largescale beams. Słowa kluczowe: nanobeams, nonlocal elasticity, stress gradient, buckling, Pasternak foundation Afiliacje autorów:
 140p.  
12.  Vantadori S.^{♦}, Luciano R.^{♦}, Scorza D.^{♦}, Darban H., Fracture analysis of nanobeams based on the stressdriven nonlocal theory of elasticity, MECHANICS OF ADVANCED MATERIALS AND STRUCTURES, ISSN: 15376494, DOI: 10.1080/15376494.2020.1846231, pp.110, 2022 Streszczenie: Mode I fracture behavior of edge and centrallycracked nanobeams is analyzed by employing both stressdriven nonlocal theory of elasticity and Bernoulli–Euler beam theory. The present formulation implements the sizedependency experimentally observed at material micro and nanoscale, by assuming a nonlocal 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 largescale beams. Słowa kluczowe: energy release rate, nanobeam, stressdriven, nonlocal integral model, stress intensity factor Afiliacje autorów:
 70p.  
13.  Ceroni F.^{♦}, Darban H., Caterino N.^{♦}, Luciano R.^{♦}, Efficiency of injected anchors in masonry elements: evaluation of pullout strength, CONSTRUCTION AND BUILDING MATERIALS, ISSN: 09500618, DOI: 10.1016/j.conbuildmat.2020.121707, Vol.267, pp.121707112170714, 2021 Streszczenie: Injected anchors in masonry elements represent a widespread technique for improving the 'box behavior' of masonry structures, since they contribute to avoid or delay outofplane 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 pullout force in injected anchors basing on wide numerical analyses realized through a 2D Finite Element (FE) model specifically tuned to simulate pullout tests. Thanks to the variation of several parameters, the most significant ones influencing the maximum pullout 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 pullout force are proposed too, and the reliability of the ‘designoriented’ formulation is assessed by means of comparisons with experimental results of some pullout tests available in literature. Słowa kluczowe: injected anchors, masonry pullout force, bond, regression analysis, design formulations Afiliacje autorów:
 140p.  
14.  Darban H., Caporale A.^{♦}, Luciano R.^{♦}, Nonlocal layerwise formulation for bending of multilayered/functionally graded nanobeams featuring weak bonding, EUROPEAN JOURNAL OF MECHANICS ASOLIDS, ISSN: 09977538, DOI: 10.1016/j.euromechsol.2020.104193, Vol.86, pp.104193112, 2021 Streszczenie: The sizedependent 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 lengthscale properties, is studied through a layerwise formulation of the stressdriven nonlocal theory of elasticity and the BernoulliEuler beam theory. The formulation is also valid for the continuously graded nanobeams, where the throughthethickness material gradation with any arbitrary distribution is approximated in a stepwise manner through many layers. The sizedependency 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 sizedependent 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 stressdriven 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 largescale dimensions. Słowa kluczowe: layered nanobeam, discrete layer approach, sizeeffect, imperfect interface, nonlocal elasticity Afiliacje autorów:
 100p.  
15.  Darban H., Fabbrocino F.^{♦}, Feo L.^{♦}, Luciano R.^{♦}, Sizedependent buckling analysis of nanobeams resting on twoparameter elastic foundation through stressdriven nonlocal elasticity model, MECHANICS OF ADVANCED MATERIALS AND STRUCTURES, ISSN: 15376494, DOI: 10.1080/15376494.2020.1739357, pp.19, 2021 Streszczenie: The instability of nanobeams rested on twoparameter elastic foundations is studied through the BernoulliEuler beam theory and the stressdriven nonlocal elasticity model. The sizedependency 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 wellposed 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, stressdriven Afiliacje autorów:
 70p.  
16.  Darban H.^{♦}, Luciano R.^{♦}, Caporale A.^{♦}, Fabbrocino F.^{♦}, Higher modes of buckling in shear deformable nanobeams, International Journal of Engineering Science, ISSN: 00207225, DOI: 10.1016/j.ijengsci.2020.103338, Vol.154, pp.103338118, 2020 Streszczenie: The sizedependent buckling instability of shear deformable nanobeams rested on a twoparameter elastic foundation is studied through the stressdriven nonlocal theory of elasticity and the kinematic assumptions of the Timoshenko beam theory. The smallscale 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 wellposed and does not include inconsistencies usually arising using other nonlocal models. The sizedependent governing differential equations in terms of the transverse displacement and the crosssectional 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 Afiliacje autorów:
 200p.  
17.  Darban H., Fabbrocino F.^{♦}, Luciano R.^{♦}, Sizedependent linear elastic fracture of nanobeams, International Journal of Engineering Science, ISSN: 00207225, DOI: 10.1016/j.ijengsci.2020.103381, Vol.157, pp.103381113, 2020 Streszczenie: 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 stressdriven nonlocal theory of elasticity to account for the sizedependency in the constitutive equations, and the BernoulliEuler 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 loaddisplacement 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 nonlinear postpeak response and the unstable crack propagation, the snapback instability, which is more intense for nanospecimens with higher nonlocality. Słowa kluczowe: cracked nanobeams, nonlocal fracture, energy release rate, cohesive, crack propagation Afiliacje autorów:
 200p.  
18.  Ceroni F.^{♦}, Darban H.^{♦}, Luciano R.^{♦}, Analysis of bond behavior of injected anchors in masonry elements by means of finite element modeling, COMPOSITE STRUCTURES, ISSN: 02638223, DOI: 10.1016/j.compstruct.2020.112099, Vol.241, pp.112099118, 2020 Streszczenie: Injected anchors made of steel bars embedded in masonry elements by means of cementbased grout represented in the past a wide solution for avoiding outofplane 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 bargrout and the groutmasonry 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 pullout tests available in literature and, then, is used for investigating the effects of several parameters on both local and global behaviour. Loaddisplacement 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 bargrout and the groutmasonry 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, pullout test, FE model Afiliacje autorów:
 140p.  
19.  Luciano R.^{♦}, Caporale A.^{♦}, Darban H.^{♦}, Bartolomeo C.^{♦}, Variational approaches for bending and buckling of nonlocal stressdriven Timoshenko nanobeams for smart materials, Mechanics Research Communications, ISSN: 00936413, DOI: 10.1016/j.mechrescom.2019.103470, Vol.103, pp.10347017, 2020 Streszczenie: In this work, variational formulations are proposed for solving numerically the problem of bending and buckling of Timoshenko nanobeams. The present work belongs to research branch in which the nonlocal theory of elasticity has been used for analysis of beamlike elements in smart materials, microelectromechanical (MEMS) or nanoelectromechanical systems (NEMS). In fact, the local beam theory is not adequate to describe the behavior of beamlike elements of smart materials at the nanoscale, so that different nonlocal models have been proposed in last decades for nanobeams. The nanobeam model considered in this work is a convex combination (mixture) of local and nonlocal phases. In the nonlocal phase, the kinematic entities in a point of the nanobeam 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 nonlocal model, where the local fraction of the mixture is absent, and the other for the mixture with both local and nonlocal 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 nanobeams with mixture of phases. Słowa kluczowe: nonlocal elasticity, variational methods, Timoshenko beam, buckling load, smart materials Afiliacje autorów:
 70p.  
20.  Luciano R.^{♦}, Darban H.^{♦}, Bartolomeo C.^{♦}, Fabbrocino F.^{♦}, Scorza D.^{♦}, Free flexural vibrations of nanobeams with nonclassical boundary conditions using stressdriven nonlocal model, Mechanics Research Communications, ISSN: 00936413, DOI: 10.1016/j.mechrescom.2020.103536, Vol.107, pp.10353615, 2020 Streszczenie: Free flexural vibrations of nanobeams with nonrigid edge supports are studied by means of the stressdriven nonlocal elasticity model and EulerBernoulli 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 nonrigid 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, wellposed nonlocal formulation Afiliacje autorów:
 70p.  
21.  Fabbrocino F.^{♦}, Darban H., Luciano R.^{♦}, Nonlocal layerwise formulation for interfacial tractions in layered nanobeams, Mechanics Research Communications, ISSN: 00936413, DOI: 10.1016/j.mechrescom.2020.103595, Vol.109, pp.10359515, 2020 Streszczenie: Interfacial tractions generated at the interface in twolayered nanobeams are studied through the stressdriven 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 sizedependency 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 BernoulliEuler beam theory is used separately for each layer to describe kinematic field, and to derive sizedependent 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 nanoscale compared to those of the layered beams with largescale 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 largescale 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 Afiliacje autorów:
 70p.  
22.  Caporale R.^{♦}, Darban H.^{♦}, Luciano R.^{♦}, Exact closedform solutions for nonlocal beams with loading discontinuities, MECHANICS OF ADVANCED MATERIALS AND STRUCTURES, ISSN: 15376494, DOI: 10.1080/15376494.2020.1787565, pp.111, 2020 Streszczenie: A novel mathematical formulation is presented for the applications of the stressdriven nonlocal theory of elasticity to engineering nanoscale problems requiring longitudinal discretization. Specifically, a differential formulation accompanied with novel constitutive continuity conditions is provided for determining exact closedform solutions of nonlocal EulerBernoulli 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: closedform solutions, discretization, EulerBernoulli beams, nanobeams Afiliacje autorów:
 70p. 
Abstrakty konferencyjne
1.  Darban H., Basista M., Modeling size effect in miniaturized mass sensors, THERMEC 2023, International Conference on PROCESSING & MANUFACTURING OF ADVANCED MATERIALS Processing, Fabrication, Properties, Applications, 20230702/0707, Wiedeń (AT), No.249, pp.170170, 2023  
2.  Darban H., Bochenek K., Węglewski W., Basista M., PHASEFIELD LENGTH SCALE MEASUREMENT BASED ON THE FRACTOGRAPHY: A CASE STUDY OF CRAL2O3 COMPOSITES, CMMSolMech 2022, 24th International Conference on Computer Methods in Mechanics; 42nd Solid Mechanics Conference, 20220905/0908, Świnoujście (PL), pp.1, 2022  
3.  Darban H., Bochenek K., Węglewski W., Basista M., Phasefield modeling of fracture in CrAl2O3 metal–ceramic composites with experimental verification, ICEAF VI, 6th International Virtual Conference of Engineering Against Failure, 20210623/0625, Spetses Island (GR), No.207, pp.12, 2021 