In this study, the effective out-of-plane rigidities of several 2D lattices, consisting of Euler beams, with different common unit cell topologies are investigated. The effective out-of-plane rigidities per weight density of these lattices, normalized by those of the full solid plates with the same material and thickness, are determined. The effective out-of-plane rigidities are computed by the homogenization method based on equivalent strain energy and the Kirchhoff plate theory. Particularly, the homogenization-related equations, including the equivalent strain energy equation itself, are not taken from their corresponding equations for 3D solids, but are derived directly using the Kirchhoff plate equations. Moreover, the exact forms, having some dimensionless factors, of the effective material constants for 2D-lattice plates are analytically derived. By using exact curve fitting, these exact forms, in most cases, yield the closed forms of the effective material constants. Finally, the efficiency of the considered unit cell topologies, in terms of the normalized effective rigidities per weight density of their resulting lattices, is discussed.
This article investigates the nonlinear vibrations of asymmetric vertically supported Jeffcott rotor system. Asymmetry in both linear and nonlinear stiffness coefficients of the rotating shaft is considered. The disk eccentricity and its orientation angle are included in the system model. Asymptotic analysis is sought to obtain an analytical approximate solution for the considered system model in the primary resonance case. Bifurcation diagrams for the different system parameters are obtained to explore the system steady-state lateral vibrations. The main acquired results revealed that (1) the symmetric system can oscillate by one of three stable forward whirling amplitudes at the same rotational speed depending on the initial position of the rotating disk. (2) Asymmetry in the linear stiffness coefficient does not affect the symmetry of the whirling motion, but it may change the system natural frequency. (3) Asymmetry in the nonlinear stiffness coefficient is responsible for both asymmetrical and backward whirling motions. All obtained analytical results have been verified via solving the system original equations numerically, where the analytical and numerical results are in excellent agreement.
This paper revisits the fundamental structural dynamic systems with regard to the effect of gravity, and thus self-weight, on their dynamic characteristics and response. Far from being a purely theoretical exercise, as would have been the case in the past, this study is a first step in structural dynamics inspired by—and anticipating—the potential of building under extraterrestrial conditions. More specifically, five basic structural models are considered: (a) the simple pendulum (SP), (b) the rigid inverted pendulum (RIP), (c) the flexural inverted pendulum (FIP), (d) the rigid rocking block (RRB), and (e) the flexural rocking block (FRB). The focus is to identify patterns and regions where low gravity can have a beneficial or detrimental role on the structural response. The paper initially presents the effect of low gravity on the dynamic characteristics of each system and then proceeds with highlighting their self-similar response, along with the differences in response due to low gravity. It is proved that low gravity is detrimental for the SP, while it is beneficial for the RIP and FIP models. Nevertheless, the effect can be both beneficial and detrimental for the RRB and FRB, depending on their parameters as revealed from this investigation. Finally, the main dynamic characteristics of the five cases studied, factorized by the gravitational multiplier ( $$\alpha )$$ α ) , are quantified and summarized in the form of a representative table.
In this study, we obtained analytical solutions for functionally graded curved beams with different properties in tension and compression, in which the moduli of elasticity in tension and compression are assumed as two different exponential functions. First, by determining the unknown neutral layer, we established a simplified mechanical model concerning tension and compression subzone and derived the one-dimensional solution (i.e., the solution in the scope of mechanics of materials). Given that the one-dimensional solution is a relatively simplified one, thus a comprehensive understanding of this problem is still needed. For this purpose, we established the consistency equation expressed in terms of stress function under two-dimensional theory of elasticity. Combining boundary conditions of inner and outer edges with continuity conditions of the neutral layer, we applied power series method for the solution of stress components under pure bending. The variations of radial and circumferential stresses in different cases of bimodular functionally graded parameters are comprehensively analyzed with numerical examples. Results indicate that the position of the neutral layer is generally related to the elastic modulus and the functionally graded coefficients of the materials. Moreover, the maximum tensile or compressive bending stress may not take place at the outer or inner edges of the curved beam but inside the beam, which should be given more attention in the analysis and design of functionally graded curved beams with different properties in tension and compression.
A quasi-zero stiffness (QZS) isolator is devised to acquire the feature of high-static-low-dynamic stiffness. Cam–roller–nonlinear spring mechanisms, where two horizontal dampers are installed symmetrically, are employed as a negative stiffness provider to connect in parallel with a vertical spring. From the static analysis, the piecewise restoring force in the vertical direction of the system is inferred considering possible separation between the cam and roller. The stiffness characteristics and parameters for offering zero stiffness at the equilibrium position are then determined. The dynamic equation is established and used for the deduction of the amplitude–frequency equation by means of the Harmonic Balance Method. The definitions of force and displacement transmissibility are introduced, and their expressions are derived for subsequent investigations of the effects of horizontal spring’s nonlinearity, excitation amplitude, horizontal damping, and vertical damping on the transmissibility performance. The comparative study is implemented on the isolation performance afforded by the QZS isolator and an equivalent linear counterpart, whose static bearing stiffness is same as the QZS isolator. Results indicate that the system with softening nonlinear horizontal spring can exhibit better performance than that with opposite stiffness spring. With the increase in horizontal damping ratio, the force transmissibility is further suppressed in resonance frequency range but increased in a small segment of higher frequencies and tends to unite in high frequency range. However, the horizontal damper deteriorates the ability to isolate the displacement excitation to a certain extent. Besides, the isolation capability of the QZS system depends on the magnitude of excitation amplitude. The quasi-zero stiffness system possesses lower initial isolation frequency and better isolation ability around resonance frequency compared with the linear system. Therefore, the quasi-zero stiffness isolator has superior low-frequency ability in isolating vibration over its linear counterpart.
The problem of determining the steady-state dynamic response of a granular elastic half-plane to a load moving on its surface is solved analytically. The granular material is modeled as a gradient elastic solid with one material constant with length dimensions in addition to the two classical elastic moduli. The load is uniformly distributed of constant magnitude and moves with constant speed. The resulting two partial differential equations of motion are of the sixth order with respect to the horizontal x and vertical y coordinates and of the second order with respect to time t. These equations are solved with the aid of complex Fourier series involving x and t, which reduce them to a system of two ordinary differential equations, which can be easily solved. The so-obtained solution is used to easily assess the microstructural effect on the various response quantities through parametric studies.
A first endeavor is made in this paper to explore new analytic buckling solutions of moderately thick rectangular plates by a straightforward double finite integral transform method, with focus on typical non-Lévy-type fully clamped plates that are not easy to solve in a rigorous way by the other analytic methods. Solving the governing higher-order partial differential equations with prescribed boundary conditions is elegantly reduced to processing four sets of simultaneous linear equations, the existence of nonzero solutions of which determines the buckling loads and associated mode shapes. Both numerical and graphical results confirm the validity and accuracy of the developed method and solutions by favorable comparison with the literature and finite element analysis. The succinct but effective technique presented in this study can provide an easy-to-implement theoretical tool to seek more analytic solutions of complex boundary value problems.
This work proposes a new constitutive model for shape memory polymers and shape memory polymeric composites under the use of level-set functions as additional variables of state. The model regards shape memory polymers as inhomogeneous bodies consisting of two different phases, an active (rubbery elastic) phase and a frozen (glassy elastic) phase. Introducing the level-set function provides the potential to describe the interface separating the two distinct phases in an implicit manner. Under proper thermodynamical arguments, an appropriate evolution equation is derived that provides the tool to describe phase transformations in shape memory polymers. Furthermore, an extended model version is developed that applies in shape memory polymeric composites by introducing two (or more) level-set functions so as to represent implicitly three (or more) material phases capturing the behavior of multi-shape memory polymers. The level-set constitutive equations are formulated in three dimensions, although the one-dimensional case is adopted and analyzed thoroughly. The reproduction of the shape memory thermomechanical cycles of polymers and polymeric composites provides validity and credibility to the current model.
This article is intended to present an overview of various mechanical analyses of rectangular nanobeams and single-, double-, and multi-walled (SW-, DW-, and MW-) carbon nanotubes (CNTs) with combinations of simply supported, free, and clamped edge conditions embedded or non-embedded in an elastic medium, including bending, free vibration, buckling, coupled thermo-elastic and hygro-thermo-elastic, dynamic instability, wave propagation, geometric nonlinear bending, and large amplitude vibration analyses. This review introduces the development of various nonlocal beam and shell theories incorporating Eringen’s nonlocal elasticity theory and the application of strong- and weak-form-based formulations to the current issue. Based on the principle of virtual displacements and Reissner’s mixed variational theorem, the corresponding strong- and weak-form formulations of the local Timoshenko beam theory are reformulated for the free vibration analysis of rectangular nanobeams and SW-, DW-, and MW-CNTs, and presented for illustrative purposes. A comparative study of the results obtained using assorted nonlocal beam and shell theories in combination with the analytical and numerical methods is carried out.
The dynamic power transmission characteristics of a finite stiffened Mindlin plate subject to different boundary conditions are analytically studied. The stiffened plate is modeled as a coupled structure comprising a plate and stiffeners. Dynamic responses calculated by the analytical solutions are verified through comparison of the results with those generated using the finite element method. The computed results show that Mindlin plate and Timoshenko beam theory is more suitable for studying dynamic power of the stiffened plate over a broad frequency range than classical plate and beam theory. The stiffness and inertia characteristics of the Mindlin plate can be enhanced using the stiffeners, which can significantly affect the dynamic response, especially in low-frequency range. It can be also noticed that the stop band in the low-frequency range can become wider by increasing the number and dimension (height and width) of the stiffeners, so vibratory power of the stiffened Mindlin plate in the low-frequency range can be greatly reduced.
In this paper, we analyze a general quasi-static two-dimensional multiple contact problem between two elastically similar half-planes under the constant normal (including applied moments) and oscillatory tangential loading utilizing the classical singular integral equations approach. Boundary conditions at nonsingular edges of discrete contact zones are applied and new side conditions are extracted and named “the consistency conditions” for multiple contacts. These conditions are mandatory for determination of the positions of the nonsingular edges of the contact and stick zones, when the number of them exceeds the number of the discrete contact and stick zones, respectively. Consequently, a mathematical relation between the pressure and shear distribution functions and between the extent of the contact and stick zones is obtained for the mentioned problem that shows all of the contact zones reach the full slip state simultaneously. Moreover, we show that for the weak normal loading, the approximated extent of the contact zones in multiple contacts with nonsingular edges may be estimated conveniently by assuming that the extent of the contact zones is the same as the overlapped extent in the free interpenetration figure.
Inerter, which is defined as a two-terminal mechanical element, has the characteristic that the force generated at its two terminals is proportional to the relative acceleration of the two ends. In this paper, a vibration isolator with lateral inerters is proposed and the effect of this geometrical nonlinear inerter on its dynamic performance is investigated. The force of the inerters in the moving direction of the mass and the acceleration term in the dynamic equation are nonlinear. The dynamic response is obtained using the averaging method and further checked by the numerical results, the stability analysis is also considered. The critical surface of the structural parameters which leads to no jump phenomenon and the jump frequencies when jump phenomenon occurs are determined by the Sylvester resultant method. The isolation performance of the lateral inerter-based vibration isolator is evaluated using four performance indexes: maximum dynamic displacement, maximum transmissibility, isolation frequency band and transmissibility in the higher isolation frequency band, and is compared with the parallel and series-connected inerter-based vibration isolators, as well as the linear vibration isolator. The results show that when the force amplitude is small, compared with the linear vibration isolator, the lateral inerter-based vibration isolator proposed in this paper can have a smaller maximum force transmissibility and larger isolation frequency band; the force transmissibility in the higher isolation frequency band is the same, which has the corresponding advantages of the parallel and series-connected inerter-based vibration isolators, respectively.
This article is intended to present an overview of various mechanical analyses of rectangular nanobeams and single-, double-, and multi-walled (SW-, DW-, and MW-) carbon nanotubes (CNTs) with combinations of simply supported, free, and clamped edge conditions embedded or non-embedded in an elastic medium, including bending, free vibration, buckling, coupled thermo-elastic and hygro-thermo-elastic, dynamic instability, wave propagation, geometric nonlinear bending, and large amplitude vibration analyses. This review introduces the development of various nonlocal beam and shell theories incorporating Eringen's nonlocal elasticity theory and the application of strong- and weak-form-based formulations to the current issue. Based on the principle of virtual displacements and Reissner's mixed variational theorem, the corresponding strong- and weak-form formulations of the local Timoshenko beam theory are reformulated for the free vibration analysis of rectangular nanobeams and SW-, DW-, and MW-CNTs, and presented for illustrative purposes. A comparative study of the results obtained using assorted nonlocal beam and shell theories in combination with the analytical and numerical methods is carried out.
In multi-layered composite laminates, Lamb wave equations are obtained using the transfer matrix method and global matrix method. These methods have numerical issues (missing roots or spurious roots) while solving the Lamb wave equations especially at high frequencies and for the laminates with a large number of layers. In the present work, an effective stiffness matrix method (ESM) is presented to solve the Lamb wave equations without numerical issues. The proposed ESM method offers a simple and mathematically straightforward formulation as it considers the multi-layered laminate as a single homogenous layer with effective stiffness properties. The Lamb wave equations of a single monoclinic layer are first derived by considering the displacement field in three directions and solved for obtaining dispersion curves. The proposed ESM method is then applied to various laminate configurations to test the effectiveness of the method. The different laminate configurations include quasi-isotropic, cross-ply, generally anisotropic and orthotropic laminates. The efficacy of the proposed method is established in these cases. In addition, the directional dependency of Lamb wave propagation characteristic (wave velocity) with laminate configurations is evaluated and analysed.
Ferrogels are soft elastic materials into which magnetic particles are embedded. The resulting interplay between elastic and magnetic interactions and the materials’ response to external fields makes them promising candidates for applications such as actuation and drug delivery. In this article, after providing a very brief introduction to particle-based simulation methods, we give an overview on how they can be applied to magnetic gels. We focus on the different mechanisms by which ferrogels can deform in an external magnetic field. Based on examples from our previous work, we show how these mechanisms can be captured by particle-based simulations. Lastly, we provide some links to simulation techniques on larger length scales.
In this article, the transient response of a cylinder, with a piezoelectric coating, weakened by multiple radial cracks is investigated. The problem is under torsional transient loading. First, the solution of the problem, weakened by a Volterra-type screw dislocation, is achieved by using Laplace and the finite Fourier sine transform. The solution is obtained for displacement and stress fields in the bar with a piezoelectric layer. At the next step, the dislocation solution is used to derive a set of Cauchy singular integral equations for analysis of bars with a circular cross section containing some radial cracks. The solution of the singular integral equations is used to determine the torsional rigidity of the cross section and also the stress intensity factors of the crack tips. In addition, several examples are presented to show the effect of the piezoelectric coating and torsional transient loading on the stress intensity factors and torsional rigidity of the system.
We consider a coated rigid inclusion inserted into an elastic matrix subjected to uniform remote anti-plane shear stresses and examine whether the inclusion can be made neutral (meaning that its introduction will not disturb the original uniform stress field in the surrounding uncut matrix) despite the presence of partial debonding along the inclusion–coating interface. Our analysis involves the introduction of a conformal mapping function (expressed in terms of a Laurent series) for the (thick) coating, a Laurent series expansion for the corresponding Plemelj function and simple matrix algebra. Our method demonstrates that coated neutral inclusions continue to be available under these challenging yet more realistic physical conditions. Numerical results are presented to demonstrate the feasibility of the solution method.
The present paper proposes an interphase model for the simulation of damage propagation in masonry walls in the framework of a mesoscopic approach. The model is thermodynamically consistent, with constitutive relations derived from a Helmholtz free potential energy. With respect to classic interface elements, the internal stress contribute is added to the contact stresses. It is considered that damage, in the form of loss of adhesion or cohesion, can potentially take place at each of the two blocks–mortar physical interfaces. Flow rules are obtained in the framework of the Theory of Plasticity, considering bilinear domains of ‘Coulomb with tension cut-off’ type. The model aims to be a first research step to solve the inverse problem of damage propagation in masonry generated by vertical ground movements, in order to ex-post identify the cause of a visible damage. The constitutive model is written in a discrete form for its implementation in a research-oriented finite element program. The response at the quadrature point is analyzed first. Then, the model is validated through comparisons with experimental results and finally employed to simulate the failure occurred in a wall of an ancient masonry building, where an arched collapse took place due to a lowering of the ground level under part of its foundation.
The main objective of the present paper is to study the temperature and thermal stress analysis of a functionally graded rectangular plate with temperature-dependent thermophysical characteristics of materials under convective heating. The nonlinear heat conduction equation is reduced to linear form using Kirchhoff’s variable transformation. Analytic solution of the heat conduction equation is obtained in the transform domain by developing an integral transform technique for convective-type boundary conditions. Goodier’s displacement function and Boussinesq harmonic functions are used to obtain the displacement profile and its associated thermal stresses. A mathematical model is prepared for functionally graded ceramic–metal-based material. The results are illustrated numerically and depicted graphically for both thermosensitive and nonthermosensitive functionally graded plate. During this study, one observed that notable variations are seen in the temperature and stress profile, due to the variation in the material parameters.
The problem of multiple adhesive contact is considered for an elastic substrate modeled as a transversely isotropic elastic half-space. It is assumed that a large number of the Kendall-type microcontacts are formed between the substrate and circular rigid (i.e., nondeformable) and frictionless micropads, which are interconnected between themselves, thereby establishing a load sharing. The effect of microcontacts interaction is accounted for in the formulation of the detachment criterion for each individual microcontact. A number of different asymptotic models are presented for the case of dilute clusters of microcontacts with their accuracy tested against a special case of two-spot contact, for which an analytical solution is available. The pull-off force has been estimated and the effects of the array size and the microcontact spacing are studied. It is shown that the flexibility of the micropads fixation, which is similar to that observed in mushroom-shaped fibrils, significantly increases the pull-off force. The novelty of the presented approach is its ability to separate different effects in the multi-scale contact problem, which allows one to distinguish between different mathematical models developed for bioinspired fibrillar adhesives.