Bi-material Beam Research Articles

Analytical solution for bi-material beam with graded intermediate layer

Abstract Using the Airy stress function, an analytical solution is obtained for a bi-material beam with a graded intermediate layer, which is subjected to a uniform load on the upper surface and has different boundary conditions at the two ends. Young’s modulus of the graded intermediate layer is assumed to be an arbitrary function of the thickness coordinate and its Poisson’s ratio is kept a constant. The solution can easily degenerate into the ones of the tri-material beam, the bi-material beam, the homogeneous beam, and the graded beam, and some of them coincide with the available solutions. The analytical and numerical (finite-element-based) results are in agreement with each other for several examples. The influence on the stress distribution for the cantilever beam is discussed when Young’s modulus of the graded intermediate layer takes different functions.

On the modal behavior of a three-dimensional functionally graded cantilever beam: Poisson’s ratio and material sampling effects

Modal behavior of a three-dimensional (3D) homogeneous and functionally graded (FG) cantilever beam is studied using the Rayleigh–Ritz (RR) method and the finite element method (FEM). The effect of Poisson’s ratio and material sampling point on the natural frequencies is further addressed using the FEM. The natural frequencies (first three) obtained using the RR method converge as the number of admissible shape functions increase. The natural frequencies (first 15) obtained using the FEM vary considerably with the material gradation , more so for the lower modes than for the higher modes. Poisson’s ratio significantly changes the torsional natural frequencies of the homogeneous and graded beams. Considerable change in the natural frequencies is seen for the linear graded beams whose material properties are sampled at the element centroid rather than at Gaussian integration points. This difference is easily observed for beams modeled using a coarse mesh rather than a fine mesh. The natural frequencies of the y direction hyperbolic tangent beam with material nonhomogeneity parameter β = 100 matches well with those of the y direction bi-material beam. The natural frequencies of the power-law graded 3D cantilever beam obtained using the FEM matches closely with the 2D reference (Qian and Ching, 2004 [1] ) solution obtained using the meshless local Petrov–Galerkin method.

Thermomechanical behavior and microstructural evolution of SiNx/Al bimaterial microcantilevers

Bimaterial microcantilevers are used in numerous applications in microelectromechanical systems (MEMS) for thermal, mechanical, optical, tribological and biological functionalities. Unfortunately, the residual stress-induced curvature and combined effects of creep and stress relaxation in the thin film significantly compromises the performance of these structures. To fully understand the themomechanical deformation and microstructural evolution of such microcantilevers, SiNx/Al bilayer cantilever beams were studied in this work. These microcantilevers were heated and subsequently cooled for five cycles between room temperature and 250 °C, with the peak temperature in each successive cycle increased in increments of 25 °C using a custom-built micro-heating stage. The in situ curvature change was monitored using an interferometric microscope. The general behavior of the bimaterial microcantilever beams can be characterized by linear thermoelastic regimes with (dκ/dT)ave = 0.079 mm−1 °C−1 and inelastic regimes. After thermal cycling with a maximum temperature of 225 °C, upon returning to room temperature, the bimaterial microcantilever beams were flattened and the curvature decreased by 99%. The thermoelastic deformation during thermal cycling was well described by the Kirchhoff plate theory. Deformation of bimaterial microcantilevers during long-term isothermal holding was studied at temperatures of 100 °C, 125 °C and 150 °C with a holding period of 70 h. The curvature of bimaterial microcantilever beams decreased more for higher holding temperatures. Finite element analysis (FEA) with power-law creep in Al was used to simulate the creep and stress relaxation and thus the curvature change of the bimaterial microcantilever beams. The microstructure evolutions due to isothermal holding in SiNx/Al microcantilevers were studied using an atomic force microscope (AFM). The grain growth in both the vertical and lateral directions was present due to isothermal holding. As the isothermal holding temperature increased, the surface roughness of the film increased with more prominent grain structures.

The analysis of dynamic instability of a bimaterial beam with alternating magnetic fields and thermal loads

In this study, the dynamic instabilities and transient vibrations of a bimaterial beam with alternating magnetic fields and thermal loads are investigated. Materials are assumed isotropic, and the physical properties are assumed to have unique values in each layer. Based on the Hamilton's principle, the equation of motion is derived in which the damping factor, the electromagnetic force, the electromagnetic torque, and the thermal load are considered. The solution of thermal effect is obtained by superposing certain fundamental linear elastic stress states which are compatible with Euler Bernoulli beam theory. Using the Galerkin's method, the equation of motion is reduced to a time-dependent Mathieu equation. The numerical results of the regions of dynamic instability are determined by the incremental harmonic balance (IHB) method, and the transient vibratory behaviors are presented by the fourth-order Runge–Kutta method. The results show that the responses of the dynamic instability and transient vibrations of the system are influenced by the temperature increase, the magnetic field, the thickness ratio, the excitation frequency, and the dimensionless damping ratio. The effects of using different values of parameters are presented to display the instability and steady vibrations and reveal some interesting characteristics such as beats and resonance phenomenon.

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.

Generic non-linear modelling of a bi-material composite beam with partial shear interaction

Composite members composed of two materials joined by shear connection find widespread use in engineering infrastructure, in both traditional practice and innovative applications. Studies in the literature dating back nearly 60 years have elucidated the mechanics of the behaviour of these composite structural members in which the solution for the slip at the interface between the materials was determined by solving a linear differential equation. However, these solutions are based on a linear formulation of the strain–displacement relationship, and in some applications this relationship must be represented in non-linear form, so that the second order effects in the member can be quantified correctly. This paper presents such a study for a composite member with two materials, being typical of a steel–concrete composite beam in structural engineering. It quantifies the restraint of the member ends by longitudinal and rotational elastic springs, so that the axial tension developed is a function of the transverse loading, material properties, cross-sectional properties and the restraint stiffness. The problem is treated using minimisation of the total potential stored in the two members, the elastic shear connection at their interface, the restraints at the ends and the work done by the transverse forces, for which the differential equations for the deformations can be determined from routine variational calculus. The non-linear equation of equilibrium relating the external loading to the internal actions is stated in closed form by invoking the static and kinematic boundary conditions for the member. The solution is compared with closed form treatments derived elsewhere, and a representative member is analysed so that the influences of the non-linearity, end restraint stiffness and degree of partial shear interaction on its behaviour can be examined.

An improved adhesively bonded bi-material beam model for plated beams

An improved bi-material beam theory with adhesive interface is presented to study the interface behavior in a conventional material (e.g., wood or concrete) beam reinforced by an externally bonded (e.g., steel or FRP) plate. Both the adherend normal and tangential deformations induced by interface stresses are considered by introducing two interface compliances, from which an improved solution of interface stress distributions is obtained. In addition, an unrealistic restriction of the same curvatures in both the adherends commonly used in most of the existing studies is released in the present theoretical formulation. Closed-form solutions of interface stresses are derived, and favorable agreements between the present solutions and those of the literature are achieved, thus demonstrating the validity of the solutions. The effects of the adherend normal and tangential deformations on the interface stress distributions are studied, and the improved accuracy of the present flexible interface over the rigid interface is illustrated. The present improved bi-material beam theory provides a better prediction of the interface stress distributions, especially for adherend materials (i.e., beam and bonded plate) made of relatively low stiffness properties (e.g., wood material as the originally strengthened beam and thin FRP composite as the strengthening plate), and it can be used to analyze the interface debonding in the beams with externally adhesive bonded plates.

Deformation and stress build-up in bi-material beam specimens with a curing FM300 adhesive interlayer

A dynamic mechanical analyzer was used to measure warpage of bi-material beam specimens with a curing adhesive interlayer during the cure cycle. Specimens consisting of a steel substrate, a curing FM300 adhesive layer, and a pre-cured composite substrate were subjected to a variety of cure cycles to determine the effect of cure time, cure temperature, and ramp rate in single- and multi-hold cycles on the development of warpage and residual stress. Experiments and a developed mathematical model showed that warpage and residual stress can be reduced if cure cycles are carefully chosen with respect to the cure kinetics and property evolution of the adhesive.

Correlation between fracture and damage for quasi-brittle bi-material interface cracks

Fracture at a bi-material interface is essentially mixed-mode, even when the geometry is symmetric with respect to the crack and loading is of pure Mode I, due to the differences in the elastic properties across an interface which disrupts the symmetry. The linear elastic solutions of the crack tip stress and displacement fields show an oscillatory type of singularity. This poses numerical difficulties while modeling discrete interface cracks. Alternatively, the discrete cracks may be modeled using a distributed band of micro-cracks or damage such that energy equivalence is maintained between the two systems. In this work, an approach is developed to correlate fracture and damage mechanics through energy equivalence concepts and to predict the damage scenario in quasi-brittle bi-material interface beams. The study is aimed at large size structures made of quasi-brittle materials failing at concrete–concrete interfaces. The objective is to smoothly move from fracture mechanics theory to damage mechanics theory or vice versa in order to characterize damage. It is concluded, that through the energy approach a discrete crack may be modeled as an equivalent damage zone, wherein both correspond to the same energy loss. Finally, it is shown that by knowing the critical damage zone dimension, the critical fracture property such as the fracture energy can be obtained.

Determination of nonlinear properties of thermal sprayed ceramic coatings via inverse analysis

A robust procedure to estimate nonlinear mechanical properties of thermally sprayed (TS) ceramic coatings is introduced. Such a method is valuable particularly for TS coatings since each may possess unique properties depending upon its processing condition. Traditionally, their responses have been assumed as linear elastic (isotropic or anisotropic) except in very high temperature environments where softening may occur. However, recent inspections revealed their properties to be more accurately characterized as nonlinear elastic. It appears their distinctive morphology consisting of cracks and interfaces are responsible for such a response. In this work, a versatile procedure to identify nonlinear properties of thermally sprayed coatings is developed. First, a suitable stress–strain model is proposed and then a nonlinear bi-material beam solution is derived. Afterward, an inverse analysis technique is utilized to process the measured curvature–temperature to extract the unknown parameters. Prior to implementing in actual specimens, a detailed simulation study is performed to verify the method’s accuracy as well as robustness. This computational analysis closely replicates deposition processes of TS coatings. With the successful verification, actual curvature measurement of TS yttria-stabilized zirconia (YSZ) coating is used to determine the nonlinear properties. The estimated results clearly reveal a significant nonlinearity of the TS coating. The main advantage of this procedure is that it requires essentially no specimen preparation and allows continuous measurements after TS deposition. Furthermore, the data interpretation does not require complex computational models and calculations. This streamlined process makes the present method attractive when evaluations of many specimens are required. The present procedure can be also extended to identify nonlinear properties of other coatings/films on substrates.

Free vibration analysis of delaminated bimaterial beams

Non-dimensional parameters, namely, axial stiffness ratio and bending stiffness ratio, are introduced to study the vibration of bimaterial beams with a single delamination. The use of these parameters provide a better understanding on the influence of the axial stiffness and bending stiffness of the delaminated layers on the vibration of a delaminated beam. Both the ‘free mode’ and ‘constrained mode’ models in the study of the vibration of delaminated beams are used in the present analysis. A relative slenderness ratio specific for delamination vibration is further introduced, which is shown to dominate the vibration characteristics of the beam. For lower slenderness ratios, the influence of the delamination decreases and a global vibration mode dominates. For higher slenderness ratios, the influence of the delamination increases and a local vibration mode prevails.

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.

Compact and cancellous bone behave as a “bimaterial beam model” of constant strength

A mechanical model of bone should describe stress and strain distributions according to minimum/maximum laws. At the macrolevel the shape of the bone enables the support of significant loads if these are applied in physiological directions. With this concept in mind we have examined the geometric distributions of compact and cancellous bone and compared these to variations of V in the shear diagram and M in the bending moment diagram respectively. Two types of analytical methods have been used: (i) basic mechanical models of cylindrical and fusiform tube shapes with different thicknesses of tube plates, all compared with bone and presented as FEM models ; (ii) FEM models of human long bones obtained from CT data. Preprocessing was carried out using CATIA V5R14, and ABAQUS/Standard was used for subsequent analysis. In this analysis FEM simulation of different bending tests was performed in all samples. It was observed that there were: (a) variations of V in the shear diagram and M in the bending moment diagram according to the shape, (b) variations of the cortical thickness of different cross sections of bone shafts according to the polar moment of inertia and (c) variations of the CT densities of bone tissue and Young's modulus of elasticity distribution according to previously published data. The results from these models indicate that the local compact bone volume mainly correlates with the bending moment stresses and cancellous bone volume correlates with the shear stresses. Bone avoids higher local stresses by distributing the internal forces and bone volume according to a minimum/maximum law which continuously optimises the bone shape to one with constant strength.

An experimental investigation of residual stresses in an epoxy–steel laminate

This paper describes an investigation of residual stresses in epoxy–steel systems using a bi-material beam. Residual stresses due to curing shrinkage, differential thermal contraction and hygroscopic expansion were investigated. Curvature in the bi-material beams was used as a measure of residual stress and simple beam theory was used to relate curvature to stresses and strains in the beam. Good agreement was achieved between predicted and experimental curvature of the beam. It was shown that in the system studied, the shrinkage stresses associated with the initial cure reaction were small, but considerable stresses developed on cooling due to differential thermal contraction of the two materials. However, on repeat thermal cycling, the residual stresses increased, which was interpreted as evidence of a post-cure effect. Samples exposed to moisture demonstrated significant stress relaxation, which was proportional to the weight of water absorbed. This was attributed mainly to the hygroscopic expansion of the adhesive.

Design and fabrication of a novel bimorph microoptomechanical sensor

We have designed a so-called flip-over bimaterial (FOB) beam to increase the sensitivity of micromechanical structures for sensing temperature and surface stress changes. The FOB beam has a configuration such that a material layer coats the top and bottom of the second material at different regions along the beam length. By multiple interconnections of FOB beams, the deflection or sensitivity can be amplified, and the out-of-plane motion of a sensing structure can be achieved. The FOB beam has 53% higher thermomechanical sensitivity than a conventional one. Using the FOB beam design, we have developed a microoptomechanical sensor having a symmetric structure such that beam deflection is converted into a linear displacement of a reflecting surface, which is used for optical interferometry. The designed sensor has been fabricated by surface micromachining techniques using a transparent quartz substrate for optical measurement. Within a sensor area of 100 /spl mu/m/spl times/100 /spl mu/m, the thermomechanical sensitivity S/sub T/=180 nm/K was experimentally obtained.

On solution schemes for time-independent thermomechanical analysis for structures containing polymeric materials

Contradicting ideas about implementing temperature-dependent Young’s modulus in a time-independent quasi-static thermomechanical analysis can be found regularly in the literature. The incremental (quasi-static evolution) and the non-incremental (Hookean) solution schemes represent simplifications of the viscoelastic behavior of polymeric materials according to different theoretical disciplines. These two schemes lead to completely different solutions when Young’s modulus is a function of the temperature. In this paper we review the ideas about implementing temperature-dependent Young’s modulus in a time-independent quasi-static thermomechanical analysis. Differences of the ideologies are highlighted using bimaterial beam solutions. Thermomechanical deformations of a bimaterial structure, which resembles a plastic ball grid array package assembly, at different temperatures are measured using shadow Moiré interferometry. Numerical solutions from different schemes are compared with measurement results.

Thermo-physical properties of plasma electrolytic oxide coatings on aluminium

Plasma electrolytic oxide coatings appear to offer attractive combinations of hardness, wear resistance, corrosion resistance and interfacial adhesion. In order to optimise such characteristics, however, more basic thermo-physical property data are required, together with an understanding of how they are affected by processing conditions and microstructure. In the present study, coatings were produced on 6082 aluminium and characterised using profilometry, scanning electron microscopy, X-ray diffraction and nanoindentation. The in-plane thermal expansivity of detached coatings was measured by dilatometry to be about 8 microstrain K −1. There is thus a rather substantial mismatch between the expansivities of coating and substrate, amounting to about 15 microstrain K −1. The global in-plane Young's modulus was estimated using cantilever bending of sandwich coated substrates and also by measuring the curvature generated in a bi-material beam on cooling to low temperature. It was found to lie in the approximate range of 10–40 GPa. Values of this order, which are low compared with the figure of around 370 GPa expected for fully dense polycrystalline alumina, are thought to be associated with the presence of a network of microcracks and voids. A low value is expected to be beneficial in terms of conferring good strain tolerance, and hence resistance to spallation driven by differential thermal expansion.

Design and simulation of an uncooled double-cantilever microbolometer with the potential for ∼mK NETD

A new concept for uncooled infrared (IR) imaging with a double-cantilever microbolometer structure has been proposed. The approach has the potential of reaching a noise equivalent temperature difference (NETD) in the millikelvin range. The proposed microbolometer architecture comprises a thermally sensitive bimaterial element that controls the position of two capacitor plates coupled to the input of a low noise MOS amplifier. The top and the bottom plates of the sensing capacitor are composed of two overlapped free-standing bimaterial cantilever beams. The bimaterial element with maximum difference in thermal expansion coefficients will convert heat into mechanical movement. The orientation of the bimaterials for the top and the bottom cantilever beams is such that the metal layer on the bottom cantilever beam faces the metal layer on the top cantilever beam. IR irradiation will result in a deflection of the double-cantilever beams in opposite directions, enabling a large change in the sensing capacitor. Residual stress and curvature in each layer can be modified by an ion implantation technique. The advantages of the pixel design include extremely high sensitivity and low noise. In addition, the absorption of incident infrared irradiation can be enhanced dramatically by the aid of a resonant cavity configuration between the top and bottom plates. The design and manufacture of double-cantilever microbolometers are very flexible since the resonant cavity and the sensing capacitor gap are realized using two separate sacrificial layers. The effects of sensing capacitor gap distance, leg length, and bimaterial thickness on the microbolometer performance have also been investigated.

Peeling in Bimaterial Beams: The Peeling Moment and Its Relation to the Differential Rigidity

Two-layer beams (i.e., bimaterials) under thermomechanical stress are prone to peeling or delamination at the free edges. The self-equilibrating peeling stress gives rise to a peeling moment Mp close to the free ends. A simple and exact formula for Mp is presented in which the sign of Mp indicates whether the beam is prone or resistant to delamination. The effect on both the sign and magnitude of Mp of the flexural rigidities of both layers is examined. As the stiffness of one layer becomes dominant, the magnitude of the differential rigidity converges to one-half the thickness of the opposite layer.

The Peeling Moment–A Key Rule for Delamination Resistance in I.C. Assemblies

The peeling stress near the free edge of a bimaterial beam under uniform temperature change creates a moment which causes both layers to have identical curvatures at the interface. A new formula from beam theory is given for this Peeling Moment. A beam with a negative Peeling Moment resists delamination at the free edge. A physical explanation for the moment is developed; the sign of the Peeling Moment is also the determinant of the location of the equivalent centroid of the bimaterial beam. This provides a valuable new rule for designing resistance to thermomechanical delamination of a bimaterial structure.