Nonlinear beam formulation incorporating surface energy and size effect: application in nano-bridges

A nonlinear beam formulation is presented based on the Gurtin-Murdoch surface elasticity and the modified couple stress theory. The developed model theoretically takes into account coupled effects of the energy of surface layer and microstructures sizedependency. The mid-plane stretching of a beam is incorporated using von-Karman nonlinear strains. Hamilton’s principle is used to determine the nonlinear governing equation of motion and the corresponding boundary conditions. As a case study, pull-in instability of an electromechanical nano-bridge structure is studied using the proposed formulation. The nonlinear governing equation is solved by the analytical reduced order method (ROM) as well as the numerical solution. Effects of various parameters including surface layer, size dependency, dispersion forces, and structural damping on the pullin parameters of the nano-bridges are discussed. Comparison of the results with the literature reveals capability of the present model in demonstrating the impact of nanoscale phenomena on the pull-in threshold of the nano-bridges.     

A non-classical Kirchhoff plate model incorporating microstructure,surface energy and foundation effects

A new non-classical Kirchhoff plate model is developed using a modified couple stress theory, a surface elasticity theory and a two-parameter elastic foundation model. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and the complete boundary conditions and provides a unified treatment of the microstructure, surface energy and foundation effects. The new plate model contains a material length scale parameter to account for the microstructure effect, three surface elastic constants to describe the surface energy effect, and two foundation moduli to represent the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the newly developed plate model includes the models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases and recovers the Bernoulli–Euler beam model incorporating the microstructure, surface energy and foundation effects. To illustrate the new model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection of the simply supported plate with or without the elastic foundation predicted by the current model is smaller than that predicted by the classical model. Also, it is observed that the difference in the deflection predicted by the new and classical plate models is very large when the plate thickness is sufficiently small, but it is diminishing with the increase of the plate thickness. For the free vibration problem, it is found that the natural frequency predicted by the new plate model with or without the elastic foundation is higher than that predicted by the classical plate model, and the difference is significant for very thin plates. These predicted trends of the size effect at the micron scale agree with those observed experimentally. In addition, it is shown both analytically and numerically that the presence of the elastic foundation reduces the plate deflection and increases the plate natural frequency, as expected.     

Bending of functionally graded nanobeams incorporating surface effects based on Timoshenko beam model

The bending responses of functionally graded(FG) nanobeams with simply supported edges are investigated based on Timoshenko beam theory in this article. The Gurtin-Murdoch surface elasticity theory is adopted to analyze the influences of surface stress on bending response of FG nanobeam. The material properties are assumed to vary along the thickness of FG nanobeam in power law. The bending governing equations are derived by using the minimum total potential energy principle and explicit formulas are derived for rotation angle and deflection of nanobeams with surface effects. Illustrative examples are implemented to give the bending deformation of FG nanobeam. The influences of the aspect ratio, gradient index, and surface stress on dimensionless deflection are discussed in detail.     

Free torsional vibration of cracked nanobeams incorporating surface energy effects

This paper investigates surface energy effects, including the surface shear modulus, the surface stress, and the surface density, on the free torsional vibration of nanobeams with a circumferential crack and various boundary conditions. To formulate the problem, the surface elasticity theory is used. The cracked nanobeam is modeled by dividing it into two parts connected by a torsional linear spring in which its stiffness is related to the crack severity. Governing equations and corresponding boundary conditions are derived with the aid of Hamilton's principle. Then, natural frequencies are obtained analytically, and the influence of the crack severity and position, the surface energy, the boundary conditions, the mode number, and the dimensions of nanobeam on the free torsional vibration of nanobeams is studied in detail. Results of the present study reveal that the surface energy has completely different effects on the free torsional vibration of cracked nanobeams compared with its effects on the free transverse vibration of cracked nanobeams.     

Analysis of nonlinear vibrations and stability of rotating asymmetrical nano-shafts incorporating surface energy effects

The objective of the present work is to investigate the nonlinear vibrations of the rotating asymmetrical nano-shafts by considering surface effect. In order to compute the surface stress tensor, the surface elasticity theory is used. The governing nonlinear equations of motion are obtained with the aid of variational approach. Bubnov–Galerkin is a very effective method for exploiting the reduced-order model of the equations of motion. The averaging method is employed to analyze the reduced-order model of the system. For this purpose, the well-known Van der Pol transformation in the complex form and angle-action transformation are utilized. The effect of surface stress on the forward and backward speeds, steady state responses of the system, fixed points, close orbits and stability of the solutions is examined. The preliminary results of the research show that the absolute values of forward and backward whirling speeds in the presence of surface effect with positive residual surface stress are higher than those of regarding the system without surface effect and in the presence of surface effect with negative residual surface stress. In addition, it is seen that the undamped rotating asymmetrical nano-shaft, for specified value of detuning parameter, in the absence or presence of surface effect has various number of stable and unstable periodic solutions. Besides, there is different number of separatrix (homoclinic orbit type). Furthermore, bifurcations, number of solutions and their stability for damped rotating asymmetrical nano-shaft are investigated. Also, the above results have been obtained for rotating symmetrical nano-shaft.     

A viscoplastic constitutive model incorporating dynamic strain aging effect during cyclic deformation conditions

In this paper, a viscoplastic constitutive model previously proposed by the authors was extended to apply to the cyclic deformation analysis of the modified 9Cr-1Mo steel. A series of cyclic deformation tests were conducted on modified 9Cr-1Mo steel at various temperatures, including those under anisothermal conditions. Furthermore, cyclic evolution of state variables used in the authors' constitutive model was experimentally measured. Based on the test results, cyclic softening behavior depending on the temperature and its history was introduced into the constitutive model. The extended model was applied to simulations of inelastic deformation behavior under monotonic tension, stress relaxation, creep, isothermal cyclic deformations including stress relaxation and anisothermal cyclic deformations. It was found that the present constitutive model has a capability of describing the inelastic deformation behavior of modified 9Cr-1Mo steel adequately at various loading conditions.     

Derivation of a well-posed and multidimensional drift-flux model for boiling flows

In this Note, we derive a multidimensional drift-flux model for boiling flows. Within this framework, the distribution parameter is no longer a scalar but a tensor that might account for the medium anisotropy and the flow regime. A new model for the drift-velocity vector is also derived. It intrinsically takes into account the effect of the friction pressure loss on the buoyancy force. On the other hand, we show that most drift-flux models might exhibit a singularity for large void fraction. In order to avoid this singularity, a remedy based on a simplified three field approach is proposed. To cite this article: O. Grégoire, M. Martin, C. R. Mecanique 333 (2005).     

A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

In this paper, a novel size-dependent functionally graded(FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton's principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.     

Validation of a particle impact breakage model incorporating impact number effect

This paper presents validation of a particle impact breakage model i.e. Vogel and Peukert model with a focus on the impact number. The Vogel and Peukert model developed based on mechanical and statistical foundation has been widely used in various fields such as mineral engineering and chemical engineering but is barely studied in the application of repeated impact. The selective breakage data in the literature is collected to provide the database for model validation. It has shown that the Vogel and Peukert model is generally applicable to all the breakage cases considering the impact number. The effect of impact number is further elaborated in the population balance model (PBM) whereas the particle dynamics are provided from Discrete Element Method (DEM) simulation of an impact pin mill. The global system analysis of impact number is carried out with the synergic effect from impact velocity. The successful validation of Vogel and Peukert model incorporating the effect of impact number demonstrates its versatility whilst other key parameters such as impact energy and particle size can be considered in parallel.     

A note on the effect of surface energy and void size to void growth

Recent studies revealed that rapid void growth is the dominant failure mechanism in an elasto-plastic solid under high mean tensile stress. This paper studies the effect of the surface energy and void size to the void growth. The models of a thick spherical shell and a thick spherical column in void growth are analyzed and numerically estimated. The main conclusion from this study is that, for typical metals, the surface energy effect is negligible for voids larger than 100 nm in size, but it may become significant when the void size is on the order of 10 nm.     

A nonlinear microbeam model based on strain gradient elasticity theory with surface energy

A microscale nonlinear Bernoulli–Euler beam model on the basis of strain gradient elasticity with surface energy is presented. The von Karman strain tensor is used to capture the effect of geometric nonlinearity. Governing equations of motion and boundary conditions are obtained using Hamilton’s principle. In particular, the developed beam model is applicable for the nonlinear vibration analysis of microbeams. By employing a global Galerkin procedure, the ordinary differential equation corresponding to the first mode of nonlinear vibration for a simply supported microbeam is obtained. Numerical investigations show that in a microbeam having a thickness comparable with its material length scale parameter, the strain gradient effect on increasing the beam natural frequency is higher than that of the geometric nonlinearity. By increasing the beam thickness, the strain gradient effect decreases or even diminishes. In this case, geometric nonlinearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for beams with some specific thickness-to-length parameter ratios, both geometric nonlinearity and size effect have significant role on increasing the frequency of nonlinear vibration.     

A computationally efficient non-linear beam model

Aerospace structures with large aspect ratio, such as airplane wings, rotorcraft blades, wind turbine blades, and jet engine fan and compressor blades, are particularly susceptible to aeroelastic phenomena. Finite element analysis provides an effective and generalized method to model these structures; however, it is computationally expensive. Fortunately, the large aspect ratio of these structures is exploitable as these potential aeroelastically unstable structures can be modeled as cantilevered beams, drastically reducing computational time.In this paper, the non-linear equations of motion are derived for an inextensional, non-uniform cantilevered beam with a straight elastic axis. Along the elastic axis, the cross-sectional center of mass can be offset in both dimensions, and the principal bending and centroidal axes can each be rotated uniquely. The Galerkin method is used, permitting arbitrary and abrupt variations along the length that require no knowledge of the spatial derivatives of the beam properties. Additionally, these equations consistently retain all third-order non-linearities that account for flexural-flexural-torsional coupling and extend the validity of the equations for large deformations.Furthermore, linearly independent shape functions are substituted into these equations, providing an efficient method to determine the natural frequencies and mode shapes of the beam and to solve for time-varying deformation.This method is validated using finite element analysis and is extended to swept wings. Finally, the importance of retaining cubic terms, in addition to quadratic terms, for non-linear analysis is demonstrated for several examples.     

The effect of the beam shapes on the doubly-clamped piezoelectric energy harvester

For a piezoelectric energy harvester composed of a doubly-clamped beam with arbitrary width shapes and a proof mass, the influence of beam shapes and electrode arrangements on different electric outputs is analyzed. The output performances of piezoelectric energy harvesters with rectangular shape, concave trapezoidal shape, and concave parabolic shape are compared, and an optimization way is given. The experimental results validate the effectiveness of the methods.     

A plasticity model for powder compaction processes incorporating particle deformation and rearrangement

This paper develops a mechanistic model of granular materials that can be used with a commercial finite element package (ABAQUS). The model draws on the ideas of critical state soil mechanics and combines them with the theory of envelopes to develop an elasto-plastic model with a non-associated flow rule. The model incorporates both local deformation at the granule contacts, and rearrangement of the granules so that jointly they account for any bulk deformation. The mechanics of the model closely reflect the physicality of the material behaviour and the model parameters are closely linked (although not simplistically identical) to the characteristics of the granules. This not only gives an insight into the material behaviour, but also enables the model to be used to facilitate design of the material, its processing properties and, hence, component development. The model is used to simulate drained triaxial tests, settlement of a powder in a bin, and some examples of die pressing. Simulations are compared with experimental data and with predictions obtained using other models.     

A well-posed problem for the exterior Stokes equations in two and three dimensions

This paper treats the Stokes problem in exterior Lipschitz-continuous domains of ² and ³. Using the weighted Sobolev spaces of Hanouzet (in ³) and Giroire (in ²), we establish the inf-sup condition between the velocity and pressure spaces. This fundamental result shows that the variational Stokes problem is well-posed in those spaces. In the last paragraph, we obtain additional regularity of the solution when the data are smoother.     

A biaxial hysteretic model for a structural system incorporating strength deterioration and pinching phenomena

A biaxial hysteretic model is developed to take into account the commonly observed hysteretic characteristics of strength and stiffness degradation, pinching and biaxial interaction. The concept of zero force point is proposed to develop the biaxial hysteretic model. Two non-linear inelastic springs and one linear elastic spring are connected in parallel to define the cyclic behavior of the proposed biaxial hysteretic model. The proposed biaxial hysteretic model is rate-independent and capable of simulating not only global (such as story shear force versus story drift response of the whole structure) but also the local hysteretic behavior of structural member (such as moment versus curvature or plastic rotation response of a structural element). The biaxial cyclic loading test data of six reinforced concrete columns, which are designed for flexural failure and shear failure, are used to validate the proposed biaxial hysteretic model.     

A micromechanical model of surface steps

The surface of an epitaxial thin film typically consists of terraces separated by steps of atomic height and it evolves largely by the motion of steps. Steps are sources of stress that interact with other residual stress fields, and these interactions have a profound effect on surface evolution. A model of the elastic field arising from a two-dimensional step is presented as a departure from the commonly used half-plane point-multipole model. The field is calculated asymptotically for small step height up to second order in terms of ‘structural’ parameters that can be determined from empirical data or atomistic calculations. Effects of a lattice mismatch and surface stress are included. The model is shown to be in remarkable agreement with atomistic predictions. It is demonstrated that second-order terms are necessary for understanding non-trivial step-step interactions, and that these second-order fields cannot be described by point sources on a half-plane.     

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.