In this investigation, the free in-plane vibration analysis of orthotropic rectangular plates with non-uniform boundary conditions and internal line supports is performed with a modified Fourier solution. The exact solution for the problem is obtained using improved Fourier series method, in which both two in-plane displacements of the orthotropic rectangular plates are represented by a double Fourier cosine series and four supplementary functions, in the form of the product of a polynomial function and a single cosine series expansion, introduced to remove the potential discontinuities associated with the original displacement functions along the edges when they are viewed as periodic functions defined over the entire x–y plane. The unknown expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh–Ritz procedure. The change of the boundary conditions can be easily achieved by only varying the stiffness of the two sets of the boundary springs at the all boundary of the orthotropic rectangular plates without the need of making any change to the solutions. The excellent accuracy of the current result is validated by comparison with those obtained from other analytical approach as well as the finite element method (FEM). Numerical results are presented to illustrate the current method that is applied not only to the homogeneous boundary conditions but also to other interesting and practically important boundary restraints on free in-plane vibrations of the orthotropic rectangular plates, including varying stiffness of boundary springs, point supported, partially supported boundary conditions and internal line supports. In addition to this, the effects of locations of line supports are also investigated and reported. New results for free vibration of moderately orthotropic rectangular plates with various edge conditions and internal line supports are presented, which may be used for benchmarking of researchers in the field.

Small-scale effects in nanobeams are effectively described by the Eringen model of nonlocal elasticity. The nonlocal elastostatic problem of Bernoulli–Euler nanobeams is here formulated in variational terms by recognizing that the nonlocality effect is equivalent to a bending curvature distortion prescribed on a corresponding local nanobeam, subjected to the same kinematic boundary constraints and applied loads. The conditions to be imposed for the kinematic integrability of the bending curvature field are also provided to evaluate the bending moment solution in statically indeterminate nonlocal nanobeams. Since the curvature distortion describing the nonlocality effect is kinematically integrable in statically determinate structures, bending moments do not exhibit small-scale effects in non-redundant nanobeams. The equivalence method illustrated in the present paper is resorted to for solving the nonlocal elastostatic problem of nanobeams under constant transversal load distributions.

In this paper, formulation of the thin cylindrical shell via the modified couple stress theory by taking account of shear deformation and rotary inertia is obtained. To do this, the study developed the first shear deformable cylindrical shell theory by considering the size effects via the couple stress theory and the equations of motion of shell with classical and non-classical boundary conditions were extracted through Hamilton's principle. In the end, as an example, free vibrations of the single-walled carbon nanotube (SWCNT) were investigated. Here, the SWCNT was modeled as a simply supported shell, and the Navier procedure was used to solve the vibration problem. The results of the new model were compared with those of the classical theory, pointing to the conclusion that the classical model is a special case of the modified couple stress theory. The findings also demonstrate that the rigidity of the nano-shell in the modified couple stress theory compared with that in the classical theory is greater, resulting in the increase in natural frequencies. In addition, the effect of the material length scale parameter on the vibrations of the nano-shell in different lengths and thickness was investigated.

An illustrative documentation of some standard experimental tests of electro-active VHB 4910 polymer under application of purely mechanical and electro-mechanically coupled loadings is presented. VHB 4910 is a very soft polymer that has potential applications as an electro-active polymer in the production of different types of actuators and sensors. The time-dependent viscoelastic phenomenon is ideal in polymers. Therefore, experiments with electro-mechanically coupled loads were conducted considering some standard tests that were usually used for a viscoelastic polymeric material characterization, i.e. loading-unloading tests, single-step relaxation tests, and multi-step relaxation tests. In all experimental cases, the polymer samples were pre-stretched up to several hundred per cent to make them thin enough initially so that the application of the electro-mechanically coupled load can show its effect to a larger extend. The pre-stretched samples were then subjected to various amounts of mechanical as well as coupled deformations at different strain rates. The data produced from several loading-unloading tests, single-step relaxation tests, and multi-step relaxation tests show that the electric loading has profound effect in the time-dependent behaviour of the electro-active VHB 4910 polymer. The data set either from single-step relaxation tests or multi-step relaxation tests can be used to identify electro-viscoelastic parameters for a suitable constitutive model that can capture electro-mechanically coupled behaviours of VHB 4910. For validation, loading-unloading cyclic tests data can be utilized.

Using a lumped mass model of a single rub-impact rotor system considering the gyroscopic effect, the stability and steady-state response of the rotor system are investigated in this paper. The contact between the rotor and the stator is described by the simple Coulomb friction and piecewise linear spring models. An algorithm combining harmonic balance method with pseudo arc-length continuation is adopted to calculate the steady-state vibration response of a nonlinear system. Meanwhile, Hill’s method is used to analyze the stability of the system. The nonlinear dynamic characteristics of the system are investigated when the gap size, stator stiffness and unbalance are regarded as the control parameters. The results show that the gap size determines the location of the rub-impact; besides, the smaller gap can improve the stability of the system. The unsteady motion can be found as the stator stiffness increases. Moreover, the unbalance directly affects vibration amplitude, which becomes greater with the increasing imbalance.

This paper concerns the dynamic analysis of composite beams containing elastic and viscoelastic (VE) layers. A method for determination of the dynamic characteristics of multilayered beams with VE layers (i.e., natural frequencies, non-dimensional damping ratios and modes of vibration) is presented. The Euler–Bernoulli beam theory and the Timoshenko theory are used to describe the elastic and VE layers, respectively. To describe the mechanical properties of a VE layer, the four-parameter rheological model with fractional derivative is applied. A number of particular rheological models are particular cases of this general model. The virtual work principle and the finite element method together with the Laplace transform are used to derive the equation of motion in the frequency domain. The dynamic characteristics of a beam with VE layers are obtained as the solution to a properly defined nonlinear eigenvalue problem. The continuation method is adopted for solving the nonlinear eigenproblem. Several conclusions concerning the accuracy of the method and variability of results are presented on the basis of numerical studies.

The purposes of this article are to present new aspects of modeling multi-physically coupled fields, focusing particularly on the partitioned treatment of electro-thermo-mechanical problems. Coupled problems of this kind occur in many industrial applications, such as micro-electrical devices, field-assisted sintering processes or electrical fuses. In this paper, we restrict ourselves to the case of nonlinear thermo-elasticity at finite strains and a heat source resulting from the electrical field. Plasticity effects are not taken into consideration. The objective is to ascertain the coupling of the algorithm relating to the fields involved individually and to demonstrate a global partitioned solution strategy. We also introduce several methods that increase algorithmic stability and accelerate the iterative coupling process. This article aims to present an efficient partitioned coupling strategy for different coupling levels between the fields. To this end, we study the proposed algorithm with the help of several numerical examples ranging from linear to highly nonlinear problems, involving substantial geometric changes and finite strains.

In this paper, we revisit the limit equilibrium analysis of masonry arches. Firstly, the major contributions during the last three centuries associated with geometric and energy formulations are discussed, and subsequently, the paper explains that the problem of determining the minimum thickness of a masonry arch capable to support its own weight has multiple solutions. The infinite many neighboring solutions for the minimum thickness of a masonry arch result from the infinite many possible directions of rupturing that an arch with finite thickness may develop when becoming a mechanism. Given this infinite number of physically admissible rupturing directions, the energy approach expressed with the principle of stationary potential energy emerges as the most powerful tool to analyze masonry arches at their limit equilibrium state. The paper concludes that vertical rupturing is the most critical rupturing direction since it results to the largest value of the minimum thickness that an elliptical arch needs to support its own weight. For the common case where there is an intrados layer of voussoirs with physical joints perpendicular to the intrados, the initial rupture has to first follow the physical joint; therefore, the broken rupture pattern reported by Lamé and Clapeyron in 1823 corresponds to the larger value of the minimum allowable thickness.

In this paper, an algorithm for identifying rectangular notch parameters as damage in a plate using Lamb waves is presented. In this algorithm, a combination of pulse-echo and pitch-catch methods is used. The method is divided into two steps: notch localization and notch geometry detection. The bases for this algorithm are mode conversion and scattering phenomena because of interaction of Lamb wave modes with defects. The method is applied to some numerical examples, and the results show that it can successfully identify all of rectangular notch parameters, i.e., its location, depth, and width.

The correct capture and understanding of the bearing- induced rotor vibrations is nowadays a rather compulsory task, which should accompany the modeling and simulation work flow of high-speed rotor systems, such as turbochargers. The oil-film concentrated in the rotor's journal bearings is the root cause of the system's occurring nonlinear effects known as sub-synchronous vibrations, the behavior of which depends on both the system's geometric and dynamic configuration. In this paper, a methodology is applied for the case of a turbocharger with full-floating ring bearings that allows the quantification of the sub-synchronous vibrations during run-up simulations. It is conducted by considering both the wheel shaft and shaft-bearing geometry as a set of input parameters, the variation of which contributes in quantifying their influence upon the sub-synchronous evolution with respect to amplitude and duration criteria. Motivated by linear multivariate regression algorithms and data mining techniques, i.e., correlation coefficients and global sensitivity methods, the influence of each design parameter on the sub-synchronous formation is analyzed. Furthermore, with the help of the non-supervised neural network methods, design configurations are indicated that could be set as a compromise in terms of feasibility and low-cost production.

Aiming at the oil-film instability of the sliding bearings at high speeds, this paper systematically investigates oil-film instability laws of an overhung rotor system with parallel and angular misalignments in the run-up and run-down processes. A finite element (FE) model of the overhung rotor system considering the gyroscopic effect is established, and the sliding bearings are simulated by a nonlinear oil-film force model based on the assumption of short-length bearings. Moreover, the effectiveness of the FE model is also verified by comparing our simulation results with the experimental results in the published literature. In the run-up and run-down processes with constant angular acceleration, the effects of parallel misalignment (PM) and angular misalignment (AM) on oil-film instability laws are simulated. The results show that under the perfectly aligned condition, the onsets of the first and second vibration mode instability in the run-down process are less than those in the run-up process due to the hysteresis effect. Under the misalignment conditions, the misalignment of the coupling can delay the onset of the first vibration mode instability and decrease its vibration amplitude. In comparison with the PM, the amplitudes of multiple frequency components are more obvious under the given AM conditions. Moreover, in the run-up and run-down processes with different misalignment conditions, the variation of the dominant vibration energy was observed according to the rotating frequency $$f_{\mathrm{r}}$$ f r , the first-mode whirl/whip frequency $$f_{\mathrm{n}1}$$ f n 1 , the second-mode whirl/whip frequency $$f_{\mathrm{n}2}$$ f n 2 , or the their combinations, such as $$f_{\mathrm{r}}$$ f r – $$2f_{\mathrm{n}2}$$ 2 f n 2 .

Helical structures are among the most universal building blocks in nature and engineering. In this work, I performed three-dimensional finite element simulations to study the transitions of shapes and multi-stability in the mechanically self-assembled helical structures driven by anisotropic misfit strains. The shape transition between a purely twisted ribbon, or a helicoid, and a general helical ribbon can be achieved by tuning a few relevant geometric and mechanical parameters, including the misfit strains, the geometric misorientation angle, the dimensions, and the mechanical properties of the composite layers. The results of our work show good agreement with the recent theoretical works and will serve as a powerful tool to facilitate on-demand designs of spontaneously curved structures at both macroscopic and microscopic scales, for a number of engineering applications including nanoelecromechanical systems, drug delivery, sensors, drug delivery, active materials, optoelectronics, and microrobotics.

The mechanical behavior of automotive dual-phase steel (DP) is modeled by two different approaches: with a full-field representative volume element (RVE) and with a mean-field model. In the first part of this work, the full-field RVE is constituted by a crystal plasticity-based ferrite matrix with von Mises-type martensite inclusions. To isolate the martensite influence, the full-field DP results were compared to a full-field comparison RVE. In the comparison RVE, all martensite inclusions were replaced by a phase that exhibits the average ferrite behavior. A higher relative martensite grain boundary coverage facilitates an increased average dislocation density after quenching. However, for uniaxial deformations above ∼10%, the grain size-dependent relation reverses and exhibits slowed-down hardening. In the second part, we incorporate the main findings from the full-field simulations into a nonlinear mean-field model of Hashin–Shtrikman type. The dislocation density production parameter and the saturated dislocation density are modeled based on grain size and martensite coverage. The comparison of both approaches shows good agreement for both the overall and constituent averaged behavior.

The design of mechanical microstructures having auxetic behaviour is proposed in this paper using techniques of topology optimization for compliant mechanisms. A robust hybrid algorithm based on evolutionary algorithms and local search steps is used. The result may need verification in order to accommodate needs not taken into account in the topology optimization. Therefore, a numerical homogenization scheme is used in order to show that the final design still has the wished negative Poisson’s property.

Based on Mindlin’s strain gradient elasticity theory capturing microscale effects, a new extended Timoshenko beam element is proposed to study the postbuckling behavior of microbeams. So as to develop the size-dependent finite element formulation, the higher-order tensors of energy pairs in the energy functional are vectorized and represented in the quadratic form. In comparison with the standard Timoshenko beam element, the present one needs two further nodal degrees of freedom including derivatives of lateral translation and rotation. The Hermite polynomials are also implemented as shape functions. The developed model is general so that its formulation can be used for modified couple stress, modified strain gradient and classical elasticity theories. In the numerical results, the influences of the small-scale factor, geometrical parameters and boundary conditions on the bifurcation diagrams of microbeams are examined.

The authors have proposed a new integration method for structural dynamics by utilizing uniform quintic B-spline polynomial interpolation. In this way, with two adjustable parameters, the proposed method is successfully formulated for solving of the differential equation of motion governing a SDOF system and later generalized for a MDOF system. In the proposed method, the straightforward recurrence formulas were derived based on quintic B-spline interpolation approximation and collocation method, and the calculation process for MDOF systems was also provided. Stability analysis shows that the proposed method can attain both conditional and unconditional stability. The validity of the proposed method is verified with three numerical simulations. Compared with the latest Bathe and Noh–Bathe methods, the proposed method not only has higher computation efficiency, but also possesses better numerical dissipation characteristics.

A theoretical model is proposed to predict the interfacial debonding length and fiber pull-out length in fiber-reinforced polymer-matrix composites. The stress and displacement fields of fiber and matrix are derived considering the dual phase region model, and the relation between the pull-out length and debonding length of fiber is obtained. The interface debonding criterion is given based on the energy release rate relation in an interface debonding process. The formulas are applied in glass fiber-reinforced epoxy composites to demonstrate the newly theoretical model. The theoretical predictions of present model agree well with the experimental results. Several parameters studies are performed to analyze the debonding length and the pull-out length of fiber in glass fiber-reinforced epoxy composites.

A dimensionless governing equation of clamped square plates was obtained by introducing dimensional analysis method to the basic governing equations of plates. The dimensionless governing equation includes three different influence aspects on structural dynamic response: the geometry of the structure, the dynamic resistance ability of material and the ratio of dynamic loads to the resistance ability of material. The dimensionless governing equation was applied to the dynamic response study of plates under blast loading, and a new dimensionless number was suggested for clamped square plates under explosion loads. The suggested dimensionless number has clear physical meaning, and the parameters included in the dimensionless number are easy to get. The dimensionless number suggested in this paper was applied to analysis the experimental data of clamped square plates under blast loading. Meanwhile, comparative analysis was proceeded which indicated the dimensionless number suggested in this paper can be effectively used to predict the dynamic response. The results showed the deflection–thickness ratio is in direct proportion to the dimensionless damage number. An empirical expression was obtained which can be used to the prediction of dynamic response of similar clamped plate under different blast loading condition.

The half-power bandwidth method is commonly used to evaluate the system damping by using frequency response curves and assuming a small damping ratio. Preceding derivations obtain the third-order corrections to the classical formula but still show large errors when the damping ratio is high, especially for the acceleration case. In this note, new approximate formulas for damping ratio in terms of the half-power bandwidth are established using displacement and acceleration frequency response functions, respectively. The proposed formulas, third-order correction and classical approximations are compared by using relative errors of calculated damping ratios for both displacement and acceleration cases. The new formulas for damping ratio are brief and show excellent accuracy for small as well as high damping ratios.

In this paper, the application of non-local fractional continuum model for plane strain and plane stress elasticity is presented. The kinematics and stress concepts are discussed, and governing equations in terms of displacements for both plane problems are defined. The numerical implementation utilising generalised finite difference method is shown in detail. Three cases are solved to indicate the role of order of fractional continua and length scale: biaxial tension, pure shear and complex state. Classical (local) solution is obtained as a special case.