Power Flow Response Based Dynamic Topology Optimization of Bi-material Plate Structures

Work on dynamic topology optimization of engineering structures for vibration suppression has mainly addressed the maximization of eigenfrequencies and gaps between consecutive eigenfrequencies of free vibration, minimization of the dynamic compliance subject to forced vibration, and minimization of the structural frequency response. A dynamic topology optimization method of bi-material plate structures is presented based on power flow analysis. Topology optimization problems formulated directly with the design objective of minimizing the power flow response are dealt with. In comparison to the displacement or velocity response, the power flow response takes not only the amplitude of force and velocity into account, but also the phase relationship of the two vector quantities. The complex expression of power flow response is derived based on time-harmonic external mechanical loading and Rayleigh damping. The mathematical formulation of topology optimization is established based on power flow response and bi-material solid isotropic material with penalization(SIMP) model. Computational optimization procedure is developed by using adjoint design sensitivity analysis and the method of moving asymptotes(MMA). Several numerical examples are presented for bi-material plate structures with different loading frequencies, which verify the feasibility and effectiveness of this method. Additionally, optimum results between topological design of minimum power flow response and minimum dynamic compliance are compared, showing that the present method has strong adaptability for structural dynamic topology optimization problems. The proposed research provides a more accurate and effective approach for dynamic topology optimization of vibrating structures.

METHOD TO CALCULATE STRESS INTENSITY FACTOR OF V-NOTCH IN BI-MATERIALS

Based on Zak＇s stress function, the eigen-equation of stress singularity ofbi-materials with a V-notch was obtained. A new definition of stress intensity factor for a perpendicular interfacial V-notch of bi-material was put forward. The effects of shear modulus and Poisson＇s ratio of the matrix material and attaching material on eigen-values were analyzed. A generalized expression for calculating/（i of the perpendicular V-notch of bi-materials was obtained by means of stress extrapolation. Effects of notch depth, notch angle and Poisson＇s ratio of materials on the singular stress field near the tip of the V-notch were analyzed systematically with numerical simulations. As an example, a finite plate with double edge notches under uniaxial uniform tension was calculated by the method presented and the influence of the notch angle and Poisson＇s ratio on the stress singularity near the tip of notch was obtained.

Evaluation of stress intensity factors for bi-material interface cracks using displacement jump methods

The aim of the present work is to investigate the numerical modeling of interfacial cracks that may appear at the interface between two isotropic elastic materials. The extended finite element method is employed to analyze brittle and bi-material interfacial fatigue crack growth by computing the mixed mode stress intensity factors(SIF). Three different approaches are introduced to compute the SIFs. In the first one, mixed mode SIF is deduced from the computation of the contour integral as per the classical J-integral method,whereas a displacement method is used to evaluate the SIF by using either one or two displacement jumps located along the crack path in the second and third approaches. The displacement jump method is rather classical for mono-materials,but has to our knowledge not been used up to now for a bimaterial. Hence, use of displacement jump for characterizing bi-material cracks constitutes the main contribution of the present study. Several benchmark tests including parametric studies are performed to show the effectiveness of these computational methodologies for SIF considering static and fatigue problems of bi-material structures. It is found that results based on the displacement jump methods are in a very good agreement with those of exact solutions, such as for the J-integral method, but with a larger domain of applicability and a better numerical efficiency(less time consuming and less spurious boundary effect).

Analysis of stress intensity factor in orthotropic bi-material mixed interface crack

Adopting the complex function approach, the paper studies the stress intensity factor in orthotropic bi-material interface cracks under mixed loads. With con- sideration of the boundary conditions, a new stress function is introduced to transform the problem of bi-material interface crack into a boundary value problem of partial dif- ferential equations. Two sets of non-homogeneous linear equations with 16 unknowns are constructed. By solving the equations, the expressions for the real bi-material elastic constant εt and the real stress singularity exponents λt are obtained with the bi-material engineering parameters satisfying certain conditions. By the uniqueness theorem of limit, undetermined coefficients are determined, and thus the bi-material stress intensity factor in mixed cracks is obtained. The bi-material stress intensity factor characterizes features of mixed cracks. When orthotropic bi-materials are of the same material, the degenerate solution to the stress intensity factor in mixed bi-material interface cracks is in complete agreement with the present classic conclusion. The relationship between the bi-material stress intensity factor and the ratio of bi-material shear modulus and the relationship be- tween the bi-material stress intensity factor and the ratio of bi-material Young＇s modulus are given in the numerical analysis.

Dynamic anti-plane analysis for two symmetrically interfacial cracks near circular cavity in piezoelectric bi-materials

The present paper is exposed theoretically to the influence on the dynamic stress intensity factor （DSIF） in the piezoelectric bi-materials model with two symmet- rically permeable interracial cracks near the edges of a circular cavity, subjected to the dynamic incident anti-plane shearing wave （SH-wave）. An available theoretical method to dynamic analysis in the related research field is provided. The formulations are based on Green＇s function method. The DSIFs at the inner and outer tips of the left crack are obtained by solving the boundary value problems with the conjunction and crack- simulation technique. The numerical results are obtained by the FORTRAN language program and plotted to show the influence of the variations of the physical parameters, the structural geometry, and the wave frequencies of incident wave on the dimensionless DSIFs. Comparisons with previous work and between the inner and outer tips are con- cluded.

Calculation of thermal neutron albedo for mono-material and bi-material reflectors

Thermal neutron albedo has been investigated for different thicknesses of mono-material and bi-material reflectors. An equation has been obtained for a bi-material reflector by considering the neutron diffusion equation. The bi-material reflector consists of binary combinations of water, graphite, lead, and polyethylene. An experimental measurement of thermal neutron albedo has also been conducted for mono-material and bi-material reflectors by using a^(241) Am–Be(5.2 Ci) neutron source and a BF3 detector. The maximum value of thermal neutron albedo was obtained for a polyethylene–water combination(0.95 ± 0.02).

Dispersion of Axisymmetric Longitudinal Waves in A Bi-Material Compound Solid Cylinder Made of Viscoelastic Materials

The paper studies the dispersion of axisymmetric longitudinal waves in the bi-material compound circular cylinder made of linear viscoelastic materials.The investigations are carried out within the scope of the piecewise homogeneous body model by utilizing the exact equations of linear viscoelasto-dynamics.The corresponding dispersion equation is derived for an arbitrary type of hereditary operator and the algorithm is developed for its numerical solution.Concrete numerical results are obtained for the case where the relations of the constituents of the cylinder are described through fractional exponential operators.The influence of the viscosity of the materials of the compound cylinder on the wave dispersion is studied through the rheological parameters which indicate the characteristic creep time and long-term values of the elastic constants of these materials.Dispersion curves are presented for certain selected dispersive and non-dispersive attenuation cases under various values of the problem parameters and the influence of the aforementioned rheological parameters on these curves is discussed.As a result of the numerical investigations,in particular,it is established that in the case where the rheological parameters of the components of the compound cylinder are the same,the viscosity of the layers’materials causes the axisymmetric wave propagation velocity to decrease.

Temperature dependence of the photothermal laser cooling efficiency for a micro-cantilever

The relationship between the photothermal cooling efficiency of a micro-cantilever＇s mechanical mode and the en- vironmental temperature is studied. The micro-cantilever and a polished fiber end form a low finesse Fabry-Perot （FP） cavity. Experimental results in a temperature range from 77 K to 298 K show that temperature has an obvious influence on photothermal cooling efficiency. The photothermal cooling efficiency, ηph, at 100 K is 10 times that at 298 K. This accords well with the theoretical analysis that the high photothermal cooling efficiency can be achieved when photothermal response time, τph, and mechanical resonant frequency, ω0, are close to the optimal photothermal cooling condition ω0τph = 1. Our study provides an important approach for high effective photothermal cooling and high-sensitivity measurement for force microscopy.

INTERFACIAL CRACK ANALYSIS IN THREE-DIMENSIONAL TRANSVERSELY ISOTROPIC BI-MATERIALS BY BOUNDARY INTEGRAL EQUATION METHOD

The integral-differential equations for three-dimensional planar interfacial cracks of arbitrary shape in transversely isotropic bimaterials were derived by virtue of the Somigliana identity and the fundamental solutions, in which the displacement discontinuities across the crack faces are the unknowns to be determined. The interface is parallel to both the planes of isotropy. The singular behaviors of displacement and stress near the crack border were analyzed and the stress singularity indexes were obtained by integral equation method. The stress intensity factors were expressed in terms of the displacement discontinuities. In the non-oscillatory case, the hyper-singular boundary integral-differential equations were reduced to hyper-singular boundary integral equations similar to those of homogeneously isotropic materials.

Crack-Tip Stress Analysis at a Bi-Material Interface by Photoelastic, Isopachic and FEA

The paper investigates the stress state at the bi-material interface crack-tip by the Photoelastic and Isopachic methods and the Finite Element Analysis (FEA). The principal stresses at the bi-material interface crack-tip are theoretically determined using the combination photoelastic and isopachic fringes. The size and the shape of crack-tip isochro-matic and isopachic fringes, at a bi-material interface under static load, are studied. When the crack-tip, which is perpendicular to interface, is placed at the interface of the bi-material, the isochromatic and the isopachic fringes depend on the properties of the two materials. Thus, the isochromatic and the isopachic fringes are divided into two branches, which present a jump of values at the interface. The size of the two branches mainly depends on the elastic modulus and the Poisson’s ratio of the two materials. From the combination of the isochromatic and the isopachic fringes, the principal stresses σ1 and σ2 can be estimated and the contour curves around the crack-tip can be plotted. For the FEA analysis, the program ANSYS 11.0 was used. The bi-material cracked plates were made from Lexan (BCBA) and Plexiglas (PMMA).

Precracking and interfacial delamination in a bi-material structure:Static and dynamic loadings

The behavior of a precracked bi-material structure interface under given static and dynamic axial loading is an interest object in the present paper.Firstly,it is shown that the shear-lag model is a proper tool to analyze a delamination process in a precracked bi-material structure undergoing static loading.Secondly,the＂shear-lag model＂is applied to the structure under dynamic loading.To solve the problem for an interface delamination of the structure and to determine the debond length along the interface,our own 2D boundary element method（BEM）code is proposed in the case of static loading,and the shear-lag model together with the Laplace transforms and half-analytical calculations are used in the case of dynamic loading.The interface layer is assumed as a very thin plate compared with the other two.The parametric（geometric and elastic）analysis of the debond length and interface shear stress is done. The results from the 2D BEM code proved the validity of analytical solutions to the shear-lag model.In the dynamic case,the influence of loading characteristics,i.e.,frequencies and amplitude fluctuations on the shear stress and the value of debond length for an interval of time,is discussed. The analysis of the obtained results is illustrated by an example of the modern ceramic-metal composite,namely cermet, and depicted in figures.

Analysis and Design of Thermally Actuated Micro-Cantilevers for High Frequency Vibrations Using Finite Element Method

Vibrational behavior of thermally actuated cantilever micro-beams and their mechanical response at moderately high frequency under a non-harmonic periodic loading is studied in this paper. Two different configurations are considered: 1) a straight beam with two actuation layers on top and bottom which utilizes the bimorph effect to induce bending;2) a uniform beam with base excitation, where the beam is mounted on an actuator which moves it periodically at its base perpendicular to its axis. Generally, vibrating micro-cantilevers are required to oscillate at a specified frequency. In order to increase the efficiency of the system, and achieve deflections with low power consumption, geometrical features of the beams can be quantified so that the required vibrating frequency matches the natural frequencies of the beam. A parametric modal analysis is conducted on two configurations of micro-cantilever and the first natural frequency of the cantilevers as a function of geometrical parameters is extracted. To evaluate vibrational behavior and thermo-mechanical efficiency of micro-cantilevers as a function of their geometrical parameters and input power, a case study with a specified vibrating frequency is considered. Due to significant complexities in the loading conditions and thermo-mechanical behavior, this task can only be tackled via numerical methods. Selecting the geometrical parameters in order to induce resonance at the nominal frequency, non-linear time-history (transient) thermo-mechanical finite element analysis (using ANSYS) is run on each configuration to study its response to the periodic heating input. Approaches to improve the effectiveness of actuators in each configuration based on their implementation are investigated.

Theoretical optimization of micropillar arrays for structurally stable bioinspired dry adhesives

Inspired by the excellent adhesion performances of setae structure from organisms,micro/nano-pillar array has become one of the paradigms for adhesive surfaces.The micropillar arrays are composed of the resin pillars for adhesion and the substrate with different elastic modulus for supporting.The stress singularity at the bi-material corner between the pillars and the substrate can induce the failure of the micropillar-substrate corner and further hinder the fabrication and application of micropillar arrays,yet the design for the stability of the micropillar array lacks systematical and quantitative guides.In this work,we develop a semi-analytical method to provide the full expressions for the stress distribution within the bi-material corner combining analytical derivations and numerical calculations.The predictions for the stress within the singularity field can be obtained based on the full expressions of the stress.The good agreement between the predictions and the FEM results demonstrates the high reliability of our method.By adopting the strain energy density factor approach,the stability of the pillar-substrate corner is assessed by predicting the failure at the corner.For the elastic mismatch between the pillar and substrate given in this paper,the stability can be improved by increasing the ratio of the shear modulus of the substrate to that of the micropillar.Our study provides accurate predictions for the stress distribution at the bi-material corner and can guide the optimization of material combinations of the pillars and the substrate for more stable bioinspired dry adhesives.

STATIC STUDY OF CANTILEVER BEAM STICTION UNDER ELECTROSTATIC FORCE INFLUENCE

The model and analysis of the cantilever beam adhesion problem under the action of electrostatic force are given. Owing to the nonlinearity of electrostatic force, the analytical solution for this kind of problem is not available. In this paper, a systematic method of generating polynomials which are the exact beam solutions of the loads with di?erent distributions is provided. The polynomials are used to approximate the beam displacement due to electrostatic force. The equilibrium equation o?ers an answer to how the beam deforms but no information about the unstuck length. The derivative of the functional with respect to the unstuck length o?ers such information. But to compute the functional it is necessary to know the beam deformation. So the problem is iteratively solved until the results are converged. Galerkin and Newton-Raphson methods are used to solve this nonlinear problem. The e?ects of dielectric layer thickness and electrostatic voltage on the cantilever beam stiction are studied. The method provided in this paper exhibits good convergence. For the adhesion problem of cantilever beam without electrostatic voltage, the analytical solution is available and is also exactly matched by the computational results given by the method presented in this paper.

Interface microstructure evolution and bonding strength of TC11/c-TiA l bi-materials fabricated by laser powder deposition

Laser powder deposition was applied to fabricate the Ti-6.5Al-3.5Mo-1.5Zr-0.3Si （wt%）/Ti-47Al- 2Cr-2Nb-0.2W-0.15B （at%） bi-material system. The asdeposited TC11 alloy shows a basket-wave-like morphology while the as-deposited y-TiAl alloy consists of fully α2/γ lamellar microstructures. Regarding the thermal mismatch between TC11 and γ-TiAl during processing, the interface microstmcture evolution was concerned. The transformation pathway was illustrated. It is found that the content changes of Al elements and β-stabilizers Mo, Cr, and Nb are responsible for the evolution of microstructures at the interface. The fracture surfaces are located at the y-TiAl side. The bi-material shows a brittle-fracture manner, with the ultimate tensile strength of 560 MPa.

Development of X-FEM methodology and study on mixed-mode crack propagation

The extended finite element method（X-FEM） is a novel numerical methodology with a great potential for using in multi-scale computation and multi-phase coupling problems. The algorithm is discussed and a program is developed based on X-FEM for simulating mixed-mode crack propagation. The maximum circumferential stress criterion and interaction integral are deduced. Some numerical results are compared with the experimental data to prove the capability and efficiency of the algorithm and the program. Numerical analyses of sub-interfacial crack growth in bi-materials give a clear description of the effiect on fracture made by interface and loading condition.

Manipulating a micro-cantilever between its optomechanical bistable states in a lever-based Fabry-Pérot cavity

In this paper, we demonstrate experimentally switching a cantilever between its optomechanical bistable states in a low finesse optical cavity. Our experiment shows that the deformation of cantilever can be manipulated by tuning the cavity resonance. When the laser power increases across the threshold value of 110 ?W, optomechanical bistability is induced by strong static photothermal backaction at room temperature. Numerical calculation revealed that the bistable effect originates from the multi-well potential created via the optomechanical interaction. Switching of the cantilever between the bistable states was achieved by tuning the cavity to the corresponding boundaries of the bistable region, where the barrier between the bistable states vanishes.

Transient Response in a Bi-material Cylinder of Soft Ferromagnetic Material Subjected to Magnetic Shock

The transient response in a bi-material cylinder of soft ferromagnetic material under magnetic shock is investigated in this study.The analytical solutions for displacement and stress have been derived using the finite Hankel transform and the Laplace transform.The numerical examples show that the displacement and stress fields respond dynamically in the bi-material cylinder under magnetic shock.The derived displacement at the center and radial stress on the surface of the cylinder satisfy the boundary conditions,showing the correctness of calculation.The displacement and stress waves propagate from the surface to the center of the cylinder when the magnetic field is loaded.The stress fields increase from the center to the surface of the cylinder and are much larger than the quasi-static state since the waves reflect,collide and concentrate in the body of the cylinder.The method of this paper can be used in the design of soft ferromagnetic structures.

Topology optimization of piezoelectric bi-material actuators with velocity feedback control

In recent years,the new technologies and discoveries on manufacturing materials have encouraged researchers to investigate the appearance of material properties that are not naturally available.Materials featuring a specific stiffness,or structures that combine non-structural and stmctural functions are applied in the aerospace,electronics and medical industry fields.Particularly,structures designed for dynamic actuation with reduced vibration response are the focus of this work.The bi-material and multifunctional concepts are considered for the design of a controlled piezoelectric actuator with vibration suppression by means of the topology optimization method(TOM).The bi-material piezoelectric actuator(BPEA)has its metallic host layer designed by the TOM,which defines the structural function,and the electric function is given by two piezo-ceramic layers that act as a sensor and an actuator coupled with a constant gain active velocity feedback control(AVFC).The AVFC,provided by the piezoelectric layers,affects the structural damping of the system through the velocity state variables readings in time domain.The dynamic equation analyzed throughout the optimization procedure is fully elaborated and implemented.The dynamic response for the rectangular fbur-noded finite element analysis is obtained by the Newmark's time-integration method,which is applied to the physical and the adjoint systems,given that the adjoint formulation is needed for the sensitivity analysis.A gradient-based optimization method is applied to minimize the displacement energy output measured at a predefined degree-of-freedom of the BPEA when a transient mechanical load is applied.Results are obtained for different control gain values to evaluate their influence on the final topology.

A Novel Sensor for Detecting the Vibration

The micro-cantilever beam with a twin long period optic fiber grating sensitive to the strain and the vibration is designed to use as the sensor head.The micro-displacement of wavelength caused by strain or vibration is amplified in the system.Special cladding material is used to eliminate the interference brougth about the temperature.The designing structure is enabled to detect the micro-information.