Since its original introduction in structural design, density-based topology optimization has been applied to a number of other fields such as microelectromechanical systems, photonics, acoustics and fluid mechanics. The methodology has been well accepted in industrial design processes where it can provide competitive designs in terms of cost, materials and functionality under a wide set of constraints. However, the optimized topologies are often considered as conceptual due to loosely defined topologies and the need of postprocessing. Subsequent amendments can affect the optimized design performance and in many cases can completely destroy the optimality of the solution. Therefore, the goal of this paper is to review recent advancements in obtaining manufacturable topology-optimized designs. The focus is on methods for imposing minimum and maximum length scales, and ensuring manufacturable, well-defined designs with robust performances. The overview discusses the limitations, the advantages and the associated computational costs. The review is completed with optimized designs for minimum compliance, mechanism design and heat transfer.

Crack initiation in brittle materials is not covered by classical fracture mechanics that deals only with the growth of pre-existing cracks. In order to overcome this deficiency, the Finite Fracture Mechanics concept assumes the instantaneous formation of cracks of finite size at initiation. Within this framework, a coupled criterion was proposed at the beginning of the 2000’s requiring two necessary conditions to be fulfilled simultaneously. The first one compares the tensile stress to the tensile strength, while the other uses an energy balance and the material toughness. The present analysis is restricted to the 2D case, and, through a wide list of references, it is shown that this criterion gives predictions in agreement with experiments in various cases of stress concentration, which can be classified in two categories: the singularities, i.e. indefinitely growing stresses at a point, and the non-singular stress raisers. It is applied to different materials and structures: notched specimens, laminates, adhesive joints or embedded inclusions. Of course, a lot of work remains to do in these domains but also in domains that are almost not explored such as fatigue loadings and dynamic loadings as well as a sound 3D extension. Some ideas in these directions are issued before concluding that FFM and the coupled criterion have filled a gap in fracture mechanics.L’approche classique de la mécanique de la rupture des matériaux fragiles n’aborde pas les problèmes d’initiation de nouvelles fissures, elle ne traite que la croissance de fissures préexistantes. Afin de surmonter cette déficience, l’approche connue sous la désignation anglo-saxonne de Finite Fracture Mechanics suppose la formation instantanée de fissures de taille finie à l’initiation. Développé dans ce cadre, le critère couplé requiert la vérification simultanée de deux conditions nécessaires : la première compare la contrainte de traction à la résistance en traction du matériau tandis que l’autre utilise une équation de conservation de l’énergie et fait appel à la ténacité. La présente analyse se limite au cas 2D et, à travers une longue liste de références, il est montré que le critère couplé donne, aux points de concentration de contraintes, des prédictions d’amorçage de fissures qui sont en accord avec les expériences. On peut distinguer deux catégories : les singularités, lorsque les contraintes croissent indéfiniment en s’approchant d’un point, et les simples concentrations de contraintes, lorsque celles-ci tout en étant élevées restent bornées. Le critère est appliqué à différents matériaux et structures: éprouvettes entaillées, composites stratifiés, joints adhésifs, inclusions. Bien sûr, beaucoup de travail reste à faire dans ces domaines, mais il existe aussi des sujets qui ne sont pratiquement pas explorés comme la fatigue, les chargements dynamiques ainsi qu’une généralisation aux situations tridimensionnelles. Quelques idées sont émises dans ces directions avant de conclure que la FFM et le critère couplé ont comblé une lacune en mécanique de la rupture.Die Initiierung von Rissen wird in der klassischen Bruchmechanik nicht umfasst, da sich diese auf die Beschreibung des Verhaltens vorhandener Risse beschränkt. In der Bruchmechanik finiter Risse wird diese Einschränkung durch die Betrachtung der instantanen Entstehung von Rissen endlicher Länge aufgehoben. Im Rahmen dieses Konzeptes wurde ein gekoppeltes Kriterium vorgeschlagen, das eine hinreichende Versagensbedingung in Form zweier gleichzeitig zu erfüllender notwendiger Bedingungen darstellt: eine Bedingung der Festigkeitsmechanik und eine Energiebedingung für den Bruchprozess. Die gegenwärtige Formulierung ist auf zweidimensionale Modelle beschränkt und ist, wie anhand zahlreicher Referenzen dargestellt, geeignet experimentelle Ergebnisse über Versagen an unterschiedlichsten Spannungskonzentratoren korrekt abzubilden. Diese lassen sich klassifizieren in singuläre Spannungskonzentratoren mit lokal unendlich hohen Spannungen und in nicht-singuläre Spannungskonzentratoren mit lokal stark erhöhten aber endlichen Spannungen. Das Kriterium wurde auf verschiedenste Struktursituationen und Materialen angewendet: gekerbte Bauteile, Laminate, Klebverbindungen oder auch Materialeinschlüsse. Jedoch verbleiben weitere unerschlossene Felder für weitere Untersuchungen wie etwa die Betrachtung von zyklischen und dynamischen Lasten oder eine gründliche Erweiterung auf dreidimensionale Risse. Mögliche Ansätze für solche Erweiterungen werden vorgestellt bevor der Schluss gezogen wird, dass die Bruchmechanik finiter Risse mit dem gekoppelten Kriterium eine Lücke in der Bruchmechanik geschlossen hat.

In this paper, an overview on a class of materials with high actual research interest will be given. Magnetic hybrid materials, i.e. liquid or elastic matrices filled with magnetic nano- or micro-particles provide the possibility to influence their mechanical behaviour by application of technically easily realizable magnetic fields. In particular, the viscous and elastic behaviour of magnetic hybrid materials can be influenced by magnetic fields. The physical reason for these changes are field-induced reconfigurations of the microstructure formed by the magnetic particles. Experimental techniques to observe these changes and their relation to changes in the mechanical behaviour of magnetic hybrid materials will be in focus of this review.

In view of the significant influences of multi-source uncertainties on structural safety, which generally exist in practical engineering (such as the dispersion of material, the uncertainty of external load and the error of processing technology), more academic research and engineering applications had paid attention to uncertainty in recent years. However, due to the complexity of the structural problems, there may be multiple uncertain parameters. Traditional methods of optimal design by single-source uncertainty model, particularly the one derived from probability theory, may no longer be feasible. This paper investigates a new formulation and numerical solution of reliability-based design optimization (RBDO) of structures exhibiting random and uncertain-but-bounded (interval and convex) mixed uncertainties. Combined with the non-probabilistic set-theory convex model and the classical probabilistic approach, the mathematical definition of hybrid reliability is firstly presented for a quantified measure of the safety margin. The reliability-based optimization incorporating such mixed reliability constraints is then formulated. The PSO algorithm is employed to improve the convergence and the stability in seeking the optimal global solution. Additionally, by introducing the general concept of the safety factor, the compatibility between the proposed hybrid RBDO technique and the safety factor-based model is further discussed. By virtue of the above two methods, two numerical examples of typical components (the cantilever structure and the truss structure) as well as one complex engineering example (the hypersonic wing structure) are performed, subjected to the strength or stiffness criteria. The accuracy and effectiveness of the present method are then demonstrated.

This paper verifies the existence of a single non-circular nano-inclusion with interface effect that achieves a uniform internal strain field in an elastic plane under uniform remote anti-plane shear loadings. The uniform strain field inside such a non-circular inclusion is prescribed via perturbations of the uniform strain field inside the analogous circular inclusion, and the unknown (non-circular) inclusion shape is characterized by a conformal mapping whose unknown coefficients are determined by a system of nonlinear equations. Numerical examples show various shapes of non-circular nano-inclusions with interface effects that achieve uniform internal strain fields. It is found that such a non-circular inclusion shape depends on the inclusion size and the specific uniform internal strain field. In particular, for given interface shear modulus and residual tension, it is shown that the minimum inclusion size required to guarantee the existence of such a non-circular inclusion usually increases as the shear modulus of the inclusion approaches that of the matrix. Moreover, the relationship between the shear traction jump across the interface and the curvature of the interface is discussed in detail for such non-circular inclusions.

The microstructure of dual-phase steels consisting of a ferrite matrix with embedded martensite inclusions is the main contributor to the mechanical properties such as high ultimate tensile strength, high work hardening rate, and good ductility. Due to the composite structure and the wide field of applications of this steel type, a wide interest exists in corresponding virtual computational experiments. For a reliable modeling, the microstructure should be included. For that reason, in this paper we follow a computational strategy based on the definition of a representative volume element (RVE). These RVEs will be constructed by a set of tomographic measurements and mechanical tests. In order to arrive at more efficient numerical schemes, we also construct statistically similar RVEs, which are characterized by a lower complexity compared with the real microstructure but which represent the overall material behavior accurately. In addition to the morphology of the microstructure, the austenite–martensite transformation during the steel production has a relevant influence on the mechanical properties and is considered in this contribution. This transformation induces a volume expansion of the martensite phase. A further effect is determined in nanoindentation test, where it turns out that the hardness in the ferrite phase increases exponentially when approaching the martensitic inclusion. To capture these gradient properties in the computational model, the volumetric expansion is applied to the martensite phase, and the arising equivalent plastic strain distribution in the ferrite phase serves as basis for a locally graded modification of the ferritic yield curve. Good accordance of the model considering the gradient yield behavior in the ferrite phase is observed in the numerical simulations with experimental data.

Based on Jeffcott rotor model, the coupled bending-torsional rotor-bearing system with rub-impact under electromagnetic excitation for hydraulic generating set is established and the corresponding equation is given. The influence of excitation current, mass eccentricity and electromagnetic torque in the system is investigated by taking use of numerical method. The simulation results show that eccentricity is the crucial factor causing the changes of system dynamic characteristics, compared to the effect of torsional degree of freedom. And the larger the eccentricity is, the more obviously it affects the system responses. While the complicated dynamic behavior of bending vibration response can be restrained, due to the introduction of torsion motion, which has the property to suppress complex dynamic response of system, particularly, the inhibited and improved effect is more evident, as the eccentricity turns to be smaller. In addition, the unstable jumping phenomena of torsional response can be motivated by electromagnetic torque, which has to be taken seriously during the process of model establishment and dynamic analysis of system. From the global view of electromechanical coupling characteristics, the coupled bending-torsional vibration model with rub-impact under electromagnetic excitation can supply a more comprehensive reflection about the dynamic characteristics of system and provide useful reference with system state recognition and fault diagnosis.

Based on the nonlinear von Kármán strain and the associated linear stress, the coupling nonlinear dynamic equations of a rotating, double-tapered, cantilever Timoshenko beam are derived using the Hamilton principle. The equation of motion is discretized via the Galerkin method in which the eigenfunctions of a clamped-free Euler–Bernoulli beam are utilized. The natural frequencies and the nonlinear responses of the rotating Timoshenko beam are investigated. Some interesting phenomena of frequency veering and mode shift are observed in the rotating tapered beam due to the coupling effect. The effects of the dimensionless parameters on the natural frequencies of a rotating double-tapered Timoshenko beam are studied through numerical examples.

With extending the Rayleigh–Ritz procedure to study the hub-plate system, the characteristics of global analytical modes are addressed for a typical rigid–flexible coupling dynamic system, i.e., a three-axis attitude stabilized spacecraft installed with a pair of solar arrays. The displacement field of the solar arrays is expressed as a series of admissible functions which is a set of characteristic orthogonal polynomials generated directly by employing Gram–Schmidt process. The rigid body motion of spacecraft is represented by the product of constant and generalized coordinate. Then, through Rayleigh–Ritz procedure, the eigenvalue equation of the three-axis attitude stabilized spacecraft installed with a pair of solar arrays is derived. Solving this eigenvalue equation, the frequencies and analytical expressions of global modes for the flexible spacecraft are obtained. To validate the present analysis, comparisons between the results of the present method and ANSYS software are performed and very good agreement is achieved. The convergence studies demonstrate the high accuracy, excellent convergence and high efficiency of the present approach. Finally, the method is applied to study the characteristics of global modes of the flexible spacecraft.

In this paper, a new damage indicator based on modal data, such as mode shapes and its derivatives, is presented for damage identification in plate-like structures. The proposed indicator is determined using modal analysis information extracted from a finite element code in MATLAB. After obtaining the mode shapes, the slope and curvature of the plate in each mode are calculated based on central finite difference methods. A numerical example with and without noise is considered to evaluate the exact location of different damage scenarios. In order to validate the proposed indicator for structural damage detection, the obtained results have been compared with another study which was based on experimental data. Moreover, in order to better assess the performance of the proposed indicator, a comparison has been made between the proposed indicator and two well-known indicators found in the literature. The results indicate that the proposed damaged indicator is able to detect precisely the location of single and multiple damage cases having different characteristics in plate-like structures.

In order to reduce structural vibrations in narrow frequency bands, tuned mass absorbers can be an appropriate measure. A quite similar approach which makes use of applied piezoelectric elements, instead of additional oscillating masses, are the well-known resonant shunts, consisting of resistances, inductances, and possibly negative capacitances connected to the piezoelectric element. This paper presents a combined approach, which is based on a conventional tuned mass absorber, but whose characteristics can be strongly influenced by applying shunted piezoceramics. Simulations and experimental analyses are shown to be very effective in predicting the behavior of such electromechanical systems. The vibration level of the absorber can be strongly attenuated by applying different combinations of resistant, resonant, and negative capacitance shunt circuits. The damping characteristics of the absorber can be changed by applying a purely resistive or resonant resistant shunt. Additionally, the tuning frequency of the absorber can be adapted to the excitation frequency, using a negative capacitance shunt circuit, which requires only the energy to supply the electric components.

Piezoelectric inertia motors use the inertia of a body to drive it by means of a friction contact in a series of small steps. It has been shown previously in theoretical investigations that higher velocities and smoother movements can be obtained if these steps do not contain phases of stiction (“stick-slip” operation), but use sliding friction only (“slip-slip” operation). One very promising driving option for such motors is the superposition of multiple sinusoidal signals or harmonics. In this contribution, the theoretical results are validated experimentally. In this context, a quick and reliable identification process for parameters describing the friction contact is proposed. Additionally, the force generation potential of inertia motors is investigated theoretically and experimentally. The experimental results confirm the theoretical result that for a given maximum frequency, a signal with a high fundamental frequency and consisting of two superposed sine waves leads to the highest velocity and the smoothest motion, while the maximum motor force is obtained with signals containing more harmonics. These results are of fundamental importance for the further development of high-velocity piezoelectric inertia motors.

Torsion of a non-deformable circular punch attached to an elastic half-space with functionally graded or multilayered coating is considered. The coating is represented by a set of alternating soft and hard (in terms of shear modulus values) layers. The boundaries of the layers can be sharp (piecewise-constant variation of the shear modulus) or smooth (continuously inhomogeneous or functionally graded coatings). An approximate analytical solution of high accuracy applicable for any value of coating thickness is constructed. The influence of number of layers and type of the boundaries between them on the kernel transform of the integral equation and on the contact stresses under the punch is investigated. It is shown that for certain parameter values of the problem substantial differences between the results for the continuous and piecewise-constant variation of the shear modulus with depth are obtained.

This paper investigates the free vibration of a homogeneous Euler–Bernoulli beam with multiple transverse cracks under non-symmetric boundary conditions. The differential equation is formulated by introducing Dirac’s delta function into the uniform flexural stiffness, and the close-form solution of mode shapes is then derived by applying the Laplace transform technique. The proposed method is validated against existing experimental method for damaged cantilever beams. With the validated method, a parametric study is performed to study the effect of crack numbers, damage parameters and crack locations on the natural frequencies and mode shapes for three non-symmetric boundary conditions (pinned–clamped, clamped–free shear and pinned–free shear).

Small-scale effects in carbon nanotubes are effectively assessed by resorting to the methods of nonlocal continuum mechanics. The crucial point of this approach consists in defining suitable constitutive laws which lead to reliable results. A nonlocal elastic law, diffusely adopted in literature, is that proposed by Eringen. According to this theory, the elastic equilibrium problem of a nonlocal nanostructure is equivalent to that of a corresponding local nanostructure subjected to suitable distortions simulating the nonlocality effect. Accordingly, transverse displacements and bending moments of a Bernoulli–Euler nonlocal nanobeam can be obtained by solving a corresponding linearly elastic (local) nanobeam, subjected to the same loading and kinematic constraint conditions of the nonlocal nanobeam, but with the prescription of suitable inelastic bending curvature fields. This observation leads naturally to the definition of a higher-order Eringen version for Bernoulli–Euler nanobeams, in which the elastic energy is assumed to be dependent on the total and inelastic bending curvatures and on their derivatives. Weak and strong formulations of elastic equilibrium of first-order gradient nanobeams are provided by a consistent thermodynamic approach. Exact solutions of fully clamped and cantilever nanobeams are given and compared with those of literature.

The nonlinear vibration of the rotor with unbalanced magnetic pull (UMP) has been extensively investigated in the literature. Most of them focused on the calculation of UMP considering only dynamic eccentricity, while the static eccentricity was generally ignored in current studies. Static eccentricity is actually inevitable due to many reasons. This paper aimed to study the nonlinear responses of a rotor with dynamic and static eccentricities (DSE). The air-gap length unified formula considering DSE is established, and the UMP of the rotor system is obtained by numerical method. The vibration characteristics of a UMP excited rotor with only dynamic eccentricity and DSE are, respectively, discussed in detail for comparison. The effects of rotating frequency and initial static eccentricity on rotor shaft orbit and displacement spectra are acquired. Results illustrate that dynamic behaviors of the UMP excited rotor with DSE are different from those of the rotor with only dynamic eccentricity. The rotor shaft orbit is axis symmetry rather than central symmetry, and the rotor is more likely to have contact with the stator in the direction of static eccentricity. Zero and double supply frequency always exists in the frequency components of the displacement. Not only two and four times of the rotating frequency harmonic appear, but also double supply frequency plus one and two times of rotating frequency harmonic are discovered.

In this study, a new type of variable stiffness laminate combining the curving path planning with the potential flow theory was experimentally investigated. By a reasonably discrete pair of mutual conjugate functions, contour curves in the flow function and potential fields were generated individually. According to these, the narrowband fiber tracks on the two variable stiffness plies were individually defined. On each intersection of the two plies, the tracks were mutually orthogonal and played a key role in improving the properties of the laminate. The tensile tests of the laminate with central holes show that the ultimate tensile load and elastic modulus of the variable stiffness laminate increased by $${\sim }129$$ ∼ 129 and 110 %, respectively. The curving paths were effective in delivering stress from the area around the hole to the edge regions of the specimen. The path planning method was based on the flow function, and all the tracks on the entire ply were defined as a whole and thus may significantly benefit the subsequent studies such as parameter optimization of scalar function and the structure of the variable stiffness laminate. The analysis of the experimental data and fracture morphology comparison indicates that the tensile property of the laminates is significantly enhanced.

The nonlinear equations of motion governing the overturning (rocking) instability of a freely standing three-rigid block assembly under ground horizontal excitation are analytically derived. Under certain conditions regarding the slenderness of each rigid block, the aforementioned nonlinear equations of motion can be linearized and then, after integration, lead to closed-form solutions. Assuming that the friction between consecutive blocks as well as the lower block and the ground is sufficiently large to prevent sliding, attention is focused on determining the minimum amplitude of ground excitation, which leads to overturning (rocking) instability with or without impact either between blocks or between the lower block and the ground. To this end, all possible configuration patterns that may lead to rocking instability (with or without impact) through an escaped motion are properly discussed.

In this paper, static and dynamic behavior of bi-stable composite laminates with $$[0-90]_{T}$$ [ 0 - 90 ] T stacking sequence and piezoelectric layers is studied. The governing equations of system were obtained using Rayleigh–Ritz method and Hamilton’s principle. In order to improve the accuracy of results, a set of higher-order shape functions were employed. The dynamic response of the system under various electrical fields applied to the piezoelectric actuators was studied, and the effects of the presented shape functions on short-circuit natural frequency and the lowest electrical field required for snap-through were analyzed. The results obtained from the developed analysis have been compared with the conventional and FEM models. Good correlation was observed between the proposed model and the finite element method.

In this paper, the dynamics of piezo-actuated stick–slip micro-drives are studied experimentally and theoretically. First, the stick–slip-based force-generating test stand is introduced, and experimental results are presented. Then, a numerical model is formulated which explicitly includes the dynamics of normal and tangential properties of the contact areas in the frictional driving elements of the drive. The contact forces are simulated using the method of dimensionality reduction. We show that the experimentally observed behavior can be described without using any fitting parameters or assuming any generalized laws of friction if the explicit contact mechanics of the frictional contacts is taken into account. Furthermore, an even simpler model of the drive is developed to get a qualitative understanding of the system. It is employed to gain a new actuation method, which reduces the vibrations of the drive’s runner and therefore enhances its performance.