Bi-layer Beam Research Articles

Interface crack between two interface deformable piezoelectric layers

Based on an interface deformable piezoelectric bi-layer beam model, a bonded piezoelectric bi-material beam with an interface crack perpendicular to the poling axis is analyzed within the framework of the theory of linear piezoelectricity. The layer-wise approximations of both the elastic displacements and electric potential are employed, and each sub-layer is modeled as a single linearly elastic Timoshenko beam perfectly bonded together through a deformable interface. Using the impermeable crack assumption, the closed form solutions for the energy release rate (ERR) and crack energy density (CED) are derived for the layered piezoelectric beam subjected to combined uniformly distributed electromechanical loading. Based on superposition principle, both the ERR and CED and their components are all reduced to the functions of the crack tip loading parameters. Loading dependence of the total CED with respect to the applied electric field is manifested with the analytical results, showing that there is a transformation from an even dependence to an odd dependence for the normalized CED when the applied mechanical loading increases. Compared with the commonly used equivalent single layer model, the proposed analysis augments the crack driving force by alleviating the stress concentration along the interface and thus increases the loading parameters at the crack tip. The proposed model provides improved solutions for fracture analysis of piezoelectric layered structures and sheds light on the loading dependence of the fracture parameters (i.e., the ERR and CED) with respect to the applied electromechanical loadings.

Elastic bending analysis of bilayered beams by an alternative two-variable method

Abstract An alternative two-variable method is used to reanalyze thermoelastic bending problems of bilayered beams subjected to external moments and internal stresses. The differences among zero-stress axis, zero-strain axis (i.e. neutral axis), bending axis, centroidal axis, and the parameter conditions for null/single/dual zero-stress axes are investigated analytically and numerically. Comparisons of thermoelastic stress predictions by the present model with Stoney's model and Hsueh's model are discussed in a representative case of GaAs top coat/Si substrate wafers. Results showed that the neutral axis does not coincide with the zero-stress axis in the general case, and the numbers and the locations of zero-strain or zero-stress axes depend on not only elastic modulus, thickness and/or thermal expansion coefficient ratios between the film and the substrate but also mechanical/thermal loading ratio.

Microbridge Tests on Bilayer Thin Films

In the present work, we report a microbridge testing method for a bilayer microbridge beam initially buckled by a residual compressive resultant force and a residual moment. A theoretical formula is derived in closed form with the consideration of substrate deformation. Measuring the profile of the initial buckling, one can evaluate the Young's modulus and residual stress of each layer of the bilayer beam. Alternatively, the Young's modulus and residual stress can be evaluated also from the load-deflection curve of the microbridge. Experimentally, Si 3 N 4 (200 nm)/SiO 2 (400 nm) microbridge samples were fabricated on a silicon wafer by the micromachining technique. The Young's modulus and residual stress were determined from the microbridge test to be 35 ± 2 GPa and -380 ± 20 MPa for the silicon oxide layer and 250 ± 7 GPa and 470 ±15 MPa for the silicon nitride layer.

Computational modeling of morphogenesis regulated by mechanical feedback

Mechanical forces cause changes in form during embryogenesis and likely play a role in regulating these changes. This paper explores the idea that changes in homeostatic tissue stress (target stress), possibly modulated by genes, drive some morphogenetic processes. Computational models are presented to illustrate how regional variations in target stress can cause a range of complex behaviors involving the bending of epithelia. These models include growth and cytoskeletal contraction regulated by stress-based mechanical feedback. All simulations were carried out using the commercial finite element code ABAQUS, with growth and contraction included by modifying the zero-stress state in the material constitutive relations. Results presented for bending of bilayered beams and invagination of cylindrical and spherical shells provide insight into some of the mechanical aspects that must be considered in studying morphogenetic mechanisms.

The Strain Energy Release Rate of a Bi-Material Beam with Interfacial Crack

Multi-layer structures are common in electronic package especially for the micro devices manufactured via the semi-conductor processes or MEMS processes. Interfacial crack due to the delamination significantly weakens the multi-layer structure. It is desired to understand the interfacial fracture properties of the electronic packaging materials. In this research, three specimens named Doubled Cantilever Beam (DCB), End-Notched Flexure (ENF), and Four-Point-Bending are proposed to investigate the fracture toughness associated with mode I, mode II and mixed mode. Basing on the Euler-Bernoulli beam theory, the strain energy in a bi-layer beam is derived. The strain energy before and after the propagation of the interfacial crack are calculated, lead to the determination of the strain energy release rate. The analytical results of strain energy release rate derived in this investigation are compared with the numerical results obtained from finite element method. The effects of material properties and thickness between the adjacent layers of interfacial crack are examined through the parametric study.

Fracture Analysis of Shear Deformable Bi-Material Interface

Based on a novel split bi-layer shear deformable beam model capable of capturing the local deformation at the crack tip, the explicit closed-form solutions of bi-material interface fracture are presented in this paper. A recently developed novel shear deformable bi-layer beam theory is briefly reviewed, from which the deformation at the crack tip is explicitly derived. A new expression for the energy release rate is then obtained using the J integral, in which several new terms associated with the transverse shear force are present; this represents an improved solution compared to the one from the classical beam model. By exploiting the two concentrated crack tip forces, the general loadings acting at the crack tip are decomposed into two groups which produce only the mode I and mode II energy release rates, respectively; the total energy release rate is thus decomposed into the mode I and II components in a global sense. The stress intensity factor referred to as local decomposition is also obtained including the transverse shear effect. The difference between the global and local mode decompositions is clarified, and a simple relationship between them is provided. The effect of the existence of a thin layer of adhesive on the stress intensity factor is further studied by an asymptotic analysis. A simple and improved expression for the T stress, the nonsingular term of stress at the crack tip, is also given. The fracture parameters of several commonly used interface fracture specimens are summarized. The present fracture analysis including the transverse shear effect is in better agreement with finite element analyses and shows advantages and improved accuracy over the available classical solutions.

Novel joint deformation models and their application to delamination fracture analysis

Split beam type of fracture specimens is commonly used in the delamination evaluation of laminated composites. The compliance and energy release rate (ERR) of the specimens are examined in this study in the new light of the crack tip deformation. Three joint deformation models (i.e., the rigid, semi-rigid, and flexible joint models) describing the different degrees of crack tip deformation are obtained based on three corresponding bi-layer beam theories (i.e., the conventional composite beam, shear deformable bi-layer beam, and interface deformable bi-layer beam). Due to different considerations of the interface displacement compatibility in each bi-layer beam theory, these joint models, among which the semi-rigid and flexible joint ones are new, show three distinct levels of accuracy in predicting the crack tip deformation. By using these two novel joint models, the new terms, which are “missing” in the rigid joint model, are recovered for the compliances and ERRs of six common delamination specimens. As a validation, an asymmetric DCB specimen is examined by three joint models, as well as the numerical finite element analysis, and three distinct accuracy levels of the solutions are revealed again in comparison with finite element analysis. It is shown that the rigid joint model is the most approximate due to neglecting the crack tip deformation, the semi-rigid model provides better solutions due to partially inclusion of the joint deformation, and the flexible joint model is the most accurate because of the fully consideration of the crack tip deformation. The distinction on accuracy also indicates the significant effect of joint (crack tip) deformation on delamination specimen analysis. The novel semi-rigid and flexible joint deformation models presented in this study provide explicit closed-form solutions of fracture parameters which can be easily adopted in practice.

Analysis of beam-type fracture specimens with crack-tip deformation

A novel bi-layer beam model is developed to account for local effects at the crack tip of a bimaterial interface by modeling a bi-layer composite beam as two separate shear deformable beams. The effect of interface stresses on the deformations of sub-layers, which is referred to as the elastic foundation effect in the literature, is considered in this model by introducing two interface compliance coefficients; thus a flexible joint condition at the crack tip is considered in contrast to the “rigid joint” condition used in the conventional bi-layer model. An elastic crack tip deformable model is presented, and the closed-form solutions of local deformation at the crack tip are then obtained. By applying this novel crack tip deformation model, the new terms due to the local deformations at the crack tip, which are missing in the conventional composite beam solutions of compliance and energy release rate (ERR) of beam-type fracture specimens, are recovered. Several commonly used beam-type fracture specimens are examined under the new light of the present model, and the improved solutions for ERR and mode mixity are thus obtained. A remarkable agreement achieved between the present and available solutions illustrates the validity of the present study. The significance of local deformation at the crack tip is demonstrated, and the improved solutions developed in this study provide highly accurate predictions of fracture properties which can actually substitute the full continuum elasticity analysis such as the finite element analysis. The new and improved formulas derived for several specimens provide better prediction of ERR and mode mixity of beam-type fracture experiments.

Mechanics of Bimaterial Interface: Shear Deformable Split Bilayer Beam Theory and Fracture

A novel split beam model is introduced to account for the local effects at the crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable beams bonded perfectly along their interface. In comparisons with analytical two-dimensional continuum solutions and finite element analysis, better agreements are achieved for the present model, which is capable of capturing the local deformation at the crack tip in contrast to the conventional composite beam theory. New solutions of two important issues of cracked beams, i.e., local buckling and interface fracture, are then presented based on the proposed split bi-layer shear deformable beam model. Local buckling load of a delaminated beam considering the root rotation at the delamination tip is first obtained. By considering the root rotation at the crack tip, the buckling load is lower than the existing solution neglecting the local deformation at the delamination tip. New expressions of energy release rate and stress intensity factor considering the transverse shear effect are obtained by the solution of local deformation based on the novel split beam model, of which several new terms associated with the transverse shear force are present, and they represent an improved solution compared to the one from the classical beam model. Two specimens are analyzed with the present model, and the corresponding refined fracture parameters are provided, which are in better agreement with finite element analysis compared to the available classical solutions.

Mechanics and fracture of crack tip deformable bi-material interface

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).

Analysis of thin film piezoelectric microaccelerometer using analytical and finite element modeling

Microaccelerometers using piezoelectric lead zirconate titanate (PZT) thin films have attracted much interest due to their simple structure and potentially high sensitivity. In this paper, we present a theoretical model for a microaccelerometer with four suspended flexural PZT-on-silicon beams and a central proof mass configuration. The model takes into account the effect of device geometry and elastic properties of the piezoelectric film, and agrees well with the results obtained by the finite element analysis. This study shows that the accelerometer sensitivity decreases with increases in the beam width, in the thickness of the bilayer beams, and in the elastic modulus of the mechanical microstructure. Increases in the beam length increase sensitivity. For a fixed beam thickness, a maximum sensitivity exists for appropriate PZT/Si thickness ratio. In addition, it is found that with appropriate geometrical dimensions, both high sensitivity and broad frequency bandwidth can be achieved. The calculation of the stress distribution in the suspended PZT/Si beam structure when the device is subjected to large vibrational acceleration indicates that the thin film microaccelerometer can stand extremely large g conditions with very good mechanical reliability. In the dynamic analysis, it is found that both analytical model and finite element modeling give very close results of the resonance frequency of the device. The results of this study can be readily applied to on-chip piezoelectric microaccelerometer design and its structural optimization.

Thermomechanical response of bare and Al2O3-nanocoated Au/Si bilayer beams for microelectromechanical systems

We present results on the thermomechanical behavior of bare and nanocoated gold/polysilicon (Au/Si) bilayer cantilever beams for microelectromechanical system applications. The cantilever beams have comparable thicknesses of the Au and Si layers and thus experience significant out-of-plane curvature due to a temperature change. The experiments focus on the inelastic behavior of the bilayer beams due to thermal holding and thermal cycling. In uncoated Au/Si beams, thermal holding directly after release or thermal cycling both lead to a curvature decrease as a function of time or cycle number, respectively. The drop in curvature during thermal cycling or thermal holding in uncoated beams was not accompanied by a change in the slope of the thermoelastic curvature–temperature relationship. The absolute change in curvature depends on the temperature and the holding time. When holding or cycling to a temperature of 175 °C, the curvature change in uncoated beams is minimal for hold times up to 4500 min or 15,000 cycles. When holding or cycling to temperatures of 200 or 225 °C, the curvature in uncoated beams drops by a factor of three for hold times up to 4500 min or 15,000 cycles. The surface structure induced by long-term holding of uncoated beams shows grooving at the grain boundaries while the surface structure induced by cycling of uncoated beams shows consolidation of the grain boundaries. The Au/Si beams with a conformal 40-nm atomic layer deposition Al2O3 coating show a considerably different response compared to identical Au/Si bare beams subjected to the same thermal histories. The coating completely suppresses decreases in curvature when the beams are held at 225 °C for 4500 min. On the contrary, the coating does not always suppress thermal ratcheting when the beam is cycled from a low temperature to 225 °C. In the coated beams, the drop in curvature due to thermal cycling was accompanied by a change in the thermoelastic slope of the curvature–temperature relationship. Negligible microstructural changes were detected on the Al2O3-coated Au surface after holding or cycling. The results are discussed in light of potential deformation mechanisms and a simple analysis linking the mismatch strain between the layers to the curvature in the beams.

Diffusion-induced beam bending in hydrogen sensors

The diffusion-induced bending of both single-layer and bilayer beam structure is analyzed by using linear elastic beam theory and the Moutier theorem. A closed form solution of the radius of curvature due to diffusion is obtained. For the single-layer beam structure, the radius of curvature is inversely proportional to the bending moment created by nonuniform concentration distribution. For the bilayer beam structure, the curvature is a linear function of the mismatch strain between the two layers and the bending moment introduced by diffusion. The mismatch strain depends on the concentration and the partial molar volume of the diffusing component in both layers. Application to microelectromechanical systems hydrogen sensors with a layer of Pd is shown.

Fabrication of soft X-ray beam splitters for use in the wavelength region around 13 nm

Two kinds of soft X-ray beam splitters were fabricated for use at wavelengths around 13 nm: a Mo/Si multilayer beam splitter for near normal incidence and a Ru/SiN bilayer beam splitter for grazing incidence. In the wavelength region around 13 run, measurements showed the Mo/Si multilayer beam splitter to have a reflectivity of 28% and a transmittance of 32°to at a grazing incidenct angle of 80°, and the Ru/SiN bilayer beam splitter to have a reflectivity and transmittance of 11% at a grazing incidenct angle of 17.5°.

Dependence of the resonance frequency of thermally excited microcantilever resonators on temperature

The dependence of the resonance frequency of thermally excited microcantilever resonators on temperature has been reported by several papers. All these papers ignored the film grown or deposited on the surface of a cantilever and regarded the beam as a single layer one. As a result, there was difference between the theoretical model and experimental results. The paper proposes a bilayer beam model to calculate the dependence of resonance frequency on temperature. This novel model regard the film as another mechanical layer and takes the induced mass by the surrounding gas into account. The calculated results are in good agreement with experimental results.

Elastic flexure of bilayered beams subject to strain differentials

The residual stresses present in a thin film and the curvature formed at its substrate during deposition have been a great concern to electrochemists and process engineers. Here a new hybrid analytical method is presented to reanalyze the flexural problem subjected to a strain differential in the general case. It was shown that the present solutions for ultrathin films agree with Stoney's equation. Moreover, single or dual neutral axes resulted, depending on materials and thickness ratios between the film and the substrate. Quantitative differences with others in the solutions of deformed curvature and residual stress are discussed in a representative case of GaAs top coat/Si substrate wafers.

Stress and internal friction associated with light-induced structural changes of a-Si:H deposited on crystalline silicon microcantilevers

The study of light-induced changes of the mechanical properties of a-Si:H should be valuable in the search for the microscopic mechanisms behind the Staebler–Wronski (SW) effect. We have developed a sensitive technique for studying such changes by depositing a-Si:H films onto commercial scanning probe microscope Si microcantilevers. The detection system of the microscope provides for measurements of beam bending, oscillation resonant frequency, and in-resonance damping factor. The internal friction of an a-Si:H film is much larger than that of crystalline Si and is the largest damping factor of the bilayer beam. We observed an increase in relative volume, Δ V/ V, with photocarrier generation rate, G, and exposure time, t, following ΔV/ V∝ G 0.7 t 0.45 in intrinsic as well as in 1 ppm PH 3/SiH 4 doped a-Si:H. The volume changes could be reversed by annealing and were the same for CW and pulsed light exposures using 400 μs long square pulses at a rate of 200 s −1. Based on the magnitude of Δ V/ V and the fact that it does not saturate we suggest that the structural changes causing Δ V/ V permeate the whole film and are not limited to defect sites.

Microbridge Testing of Silicon Oxide/Silicon Nitride Bilayer Films

ABSTRACTThe present work further develops the microbridge testing method to characterize mechanical properties of bilayer thin films. A closed-form formula for deflection versus load under small deflection is derived with consideration of the substrate deformation and residual stress in each layer. The analysis shows that the solution for bending a bilayer beam is equivalent to that for bending a single-layer beam with an equivalent bending stiffness, an equivalent residual force and a residual moment. One can estimate the Young's modulus and residual stress in a layer if the corresponding values in the other layer are known. The analytic results are confirmed by finite element calculations. The microbridge tests are conducted on low-temperature-silicon oxide (LTO)/silicon nitride bilayer films as well as on silicon nitride single-layer films. All microbridge specimens are prepared by the microfabricating technique. The tests on the single-layer films provide the material properties of the silicon nitride films. Then, applying the proposed method for bilayer films under small deflection yields the Young's modulus of 37 GPa and the residual stress of -148 MPa for LTO films.

Measurement of Thin Film Mechanical Properties by Microbeam Bending

AbstractAn improved microbeam bending technique has been developed for the study of mechanical properties of thin films on substrates. This testing method utilizes a triangular beam geometry and improved micromachining techniques compared to previously used methods. The technique permits the stress-strain law for a metal film on a substrate to be determined. Single crystal Si beams and bi-layer Si/Al beams of lengths 25–100 pgm have been fabricated and tested. The beams are deflected with a nanoindenter, which accurately imposes a load on the beam and measures the corresponding displacement. For the bi-layer beams, a simple numerical model utilizing a Ramburg-Osgood constitutive law the film has been developed to determine the stress-strain behavior of the Al film.

Mode I Fracture Toughness of Fiber Reinforced Composite-Wood Bonded Interface

An experimental characterization of the opening-mode (Mode I) fracture toughness of bonded interfaces for hybrid laminates is presented. A bi-layer Contoured Double Cantilever Beam (CDCB) specimen, which is designed by the Rayleigh-Ritz method, is effectively used for the fracture toughness tests of the bonded interfaces. The bi-layer specimen consists of constant thickness adherends bonded to straight tapered sections of an easily machinable material, and it is contoured to achieve a constant rate of compliance change with respect to crack length. Using linear slope CDCB specimens, Mode I fracture tests for wood-wood and Fiber Reinforced Plastic (FRP)-wood bonded interfaces are performed to determine the critical loads for crack initiation and crack arrest, and from the critical loads measured, the critical strain energy release rates (GI,) are easily evaluated by making use of the experimentally verified constant compliance rate change over defined crack lengths. Based on finite element analyses, the predicted GI, values by the Jacobian Derivative Method correlate closely with experimental results. The FRP-wood bonded interface considered in this study exhibits a stable crack propagation behavior, and the Resorcinol Formaldehyde adhesive used offers a good potential for bonding FRP to wood sub-strates. The efficient CDCB specimen and experimental/analytical program presented in this paper can be used to evaluate the Mode I fracture toughness (Gkn) for hybrid material interface bonds, such as FRP-wood.