Bi-material Beam Research Articles

Theoretical analysis of mode I fracture of adhesively bonded bi-material DCB joints

There has been a disputation about how to achieve a pure mode I fracture and thus measure its toughness for an adhesively bonded bi-material double cantilever beam (DCB) specimen. This paper therefore develops a theoretical methodology to calculate the normal and shear stress distributions in the adhesive layer and to perform data reduction and mode partitioning for generic bi-material DCB specimens. The theoretical model is validated using experimental data. The adhesive layer in a generic bi-material DCB joint is loaded both in normal and shear based on the predicted stress distributions in the adhesive layer, resulting in a mixed mode fracture behavior. The prediction results substantiate that a strain-based design principle eliminates the shear stresses and thus leads to pure mode I fracture in bi-material DCB specimens. This paper justifies that the established data reduction method is applicable for measuring mode I fracture toughness when a bi-material DCB is designed according to the strain-based design criterion.

Mechanical meta-sheets with independently tunable Poisson’s ratio and coefficient of thermal expansion

Mechanical metamaterials with tunable Poisson’s ratio (PR) and coefficient of thermal expansion (CTE) have received great attention due to their unique responsive behaviors under mechanical and thermal stimuli useful in a wide range of applications. But, in most designs, they are often coupled to each other, limiting the range of their values to be achieved and their independent controllability. Here, we present mechanical meta-sheets whose PR and CTE can be independently modulated by using rotating rectangular units consisting of alternated bi-material beams. Design principles for these mechanical meta-sheets are provided and verified through simulations and experiments. Both isotropic and anisotropic thermal deformations can be realized and diverse reconfigurations are possible under thermo-mechanical loads for sandwich panels utilizing these meta-sheets as skins. The proposed design principle would be powerful in a broad range of applications requiring shape morphing, stress mitigation, and smart actuation under a thermo-mechanical environment.

Modeling via peridynamics for crack propagation in laminated glass under fire

Despite laminated glass is widely used in building construction, its susceptibility to cracking and collapsing during an accidental fire can intensify fire propagation, leading to substantial casualties and economic losses. In this work, we developed a meshfree framework for modeling thermomechanical cracking of laminated glass via peridynamic (PD) theory. First, to overcome the incomplete input experimental data for implementing thermal boundary conditions, we proposed both the uniform and non-uniform temperature field strategies. Second, PD interface model and the stochastic distribution of defects were implemented within this framework. Third, the challenging problem of crack path selection in thermally loaded borosilicate/steel bi-material beams was studied to validate the feasibility of the proposed model. It is demonstrated that the proposed framework can capture the temperature history and crack features of laminated glass under fire, showing great consistency with experimental results. Our findings show that with the decrease of the glass strength or the increase of the defect content, more islands form in the central region of the glazing, implying higher risks of glass fall-out in fire conditions. This model constitutes a promising tool for predicting the fire performance of laminated glass and contributes to designing safer engineering structures.

3D piezoelectric composite honeycombs with alternating bi-material beam: An active control method for elastic properties

This paper introduces 3D piezoelectric composite honeycombs (3D PCH) that exhibit voltage-dependent elastic properties. The 3D PCH consists of alternating bi-material beams. Under the condition of applied stress and voltage, the deformation of the 3D PCH depends on the bi-material beam under the effect of electromechanical coupling. The coupling effect enables active elastic properties of the structure by controlling the voltage and stress. Only the bending deformation of the bi-material beam is considered and the axial deformation of the beam is ignored. The proposed analytical approach and the closed-form expression for the equivalent elastic properties are obtained based on the Euler-Bernoulli beam theory. The theoretical results are verified by finite element simulations. The results show that the elastic properties of the 3D PCH are related to the bi-material beam and the geometric parameters. The voltage-dependent Young's modulus and strain can be regulated by adjusting the applied voltage. This work provides an approach to studying the elastic properties of these active bi-material structures.

Embedding Fiber Bragg Grating Sensors in Carbon Composite Structures for Accurate Strain Measurement

Fiber Bragg Grating (FBG) sensors written by femtosecond laser pulses in polyamide-coated low bending loss optical fibers are succesfully embedded in carbon composite structures, following laminating and light resin moulding processes which optimize the size of each ply to address aesthetic, drapability and structural requirements of the final components. The sensors are interrogated by a tunable laser operating at around 1.55 μm and their response to temperature and strain variations is characterized in a thermally controlled chamber and by bending tests using suspended calibrated loads and a laser scanning system. Experimental results are in good agreement with simulations, confirming that the embedding process effectively overcomes potential issues related to FBG spectral distortion, birefringence and losses. In particular the effects of the composite material non homogeneity and FBG birefringence are investigated to evaluate their impact on the monitoring capabilities. A bi-material mechanical beam model is proposed to characterize the orthotropic laminates, pointing out better accuracy in estimating the applied load with respect to the classical homogeneous beam model. A comparative analysis, performed on different instrumented carbon composite samples and supported by theory, points out the repeatability of the FBG sensors embedding process and the effectiveness of the technology for real time accurate strain measurement. Based on such measurements damages and/or changes in local stiffness can be effectively detected, allowing for structural health monitoring of composite structures for applications in specific industrial fields like automotive and aerospace.

Bi-material sinusoidal beam-based temperature responsive multistable metamaterials

The ability of materials to adapt to changing environments is attractive for various applications, ranging from deployable structures to soft robots. In this work, we develop a multistable metamaterial that allows for autonomous recovery and programmable deformation in response to varying temperatures. The tunable multi-stability is achieved by incorporating a bi-material sinusoidal beam as the constituent element of the metamaterial. The mechanical response of the bi-material beam is explored theoretically. A phase diagram of its temperature induced bistability-to-monostability transition is obtained, validated by experiments and finite element simulations. With the phase diagram, the bi-material beam based building blocks of the metamaterial can be designed to transit from one stable state to another one via snap-through after compression, and return to its first stable configuration when the temperature rises above a critical temperature. Numerical and experimental results show clearly from the force–displacement-temperature path that the developed metamaterial possesses temperature-induced recoverability and programmable deformation. Construction of bi-material sinusoidal beams with more triggering temperatures for stability transition is also discussed.

Cohesive-zone based fracture mechanics model of an edge delamination in bimaterial beam under mixed-mode bending test

The problem of an edge-split bilayer beam consisted of bonded dissimilar materials under mixed-mode bending (MMB) is investigated by using a structural mechanics procedure. In this analytical procedure, the bilayer beam is modeled as two Timoshenko beams joined by using interface tensile and shear springs, and the fracture mechanics parameters including the strain energy release rate and phase angle are estimated from the spring responses. Numerical examples of bilayer beams consisted of isotropic or orthotropic bimaterials and with symmetric or asymmetric dimensions are presented and compared to continuum based finite-element solutions with either cohesive-zone or perfectly-bonded interface models. It is shown that the strain energy release rates obtained from these models are in good agreement. The phase angles obtained from the analytical model and the cohesive-zone-interface based continuum model are also consistent with each other, and agree to the crack-tip deformation. On the other hand, the phase angles obtained from the oscillating crack-tip fields related to the perfectly-bonded interface do not match to the crack-tip deformation as well as the cohesive-zone interface models. Consequently, it is recommended to characterize the mode-mixity dependent interfacial adhesion and fatigue responses by using the fracture mechanics parameters obtained by using the analytical solution for the MMB fracture problem.

Design of hourglass-lattice metastructure with near-zero thermal expansion using structural optimization method

Zero thermal expansion (ZTE) is strongly demanded for critical components of spacecraft structures exposed to extreme temperature variations in space. In this paper, a lattice metastructure with near-ZTE is numerically achieved by structural optimization method. A sandwich panel with one single layer of hourglass-type lattice cells with variable cross-sectional and bi-material beams are assembled to produce either positive or negative thermal expansion (PTE or NTE). Then, a multilayered construction of metastructures is proposed by the combination of both PTE and NTE layers to realize the overall behaviors of near-zero deformation. The structural optimization with the gradient-based algorithm is utilized to obtain an extreme low value of coefficient of thermal expansion (∼10-9 /K). The shape constraint is adopted to regulate the thermal deformation of contact surfaces of each layer. Furthermore, bi-material prototypes with nylon and short carbon fiber reinforced composites are fabricated and tested, and the results demonstrate a NTE value of −6.32 × 10-5 /K, PTE value of + 1.28 × 10-4 /K, and ZTE value of 2.53 × 10-6 /K. This agrees well with the optimization result. Overall, the proposed hourglass-lattice metastructure with a high thermal stability is a promising solution to enable large-scale space structures such as space telescope, antenna, and radar, etc.

Elastic mechanics solution of thermal expansion of bi-material curved beam and its application to negative thermal expansion metamaterials

Thermal stress impacts various engineering fields significantly, such as aerospace and precision instruments. This adverse effect can be greatly reduced, if not eliminated, by the application of micro-thermal expansion materials, and bi-material beams are widely utilized in the design of micro-thermal expansion structures, thereby exhibiting great application potentials. The elasticity solution of bi-material curved beam under free thermal expansion has been proposed by scholars. Based on this solution, the simplified form is proposed in this paper, and extended to the case where the rotation angles at both ends of the circular arc are constrained under thermal loads. Besides, the geometric parameters and the nonlinear problems of the thermal expansion of bi-material curved beam are analyzed. In addition, a novel type of negative thermal expansion material has been designed by applying the bi-material curved beam to the tetra chiral and anti-tetra chiral materials. The proposed material has greater negative thermal expansion effect than the traditional tetra and anti-tetra chiral materials that are with straight beams.

Peridynamic analysis of materials interface fracture with thermal effect

A thermomechanical peridynamic framework is established for fracture analysis of dissimilar material interface coupled with residual thermal effect. In this frame, an extended peridynamic mixed-mode failure model considering thermal effect is proposed for interfacial fracture prediction, and the peridynamics-based methods for interface crack energy release rate and mode mixity computations are given. Meanwhile, a new peridynamic contact model is proposed for frictional contact modeling of crack surfaces. Then, three examples of the single edge-notched bimaterial (SENB), asymmetric bimaterial cantilever beams (ABCB) and four-point shearing (FPS) tests, are analyzed for the model verification and application. The elastic and interfacial fracture behaviors of these tests are predicted by the proposed peridynamic models, and compared to the analytical and the FEM solutions. The results show that the proposed peridynamic models can successfully predict the failure characteristics of materials interface, and capture the role of residual thermal effect on the interfacial fracture.

Biomimetic bi-material designs for additive manufacturing

Superior material properties have been recently exhibited under the concept of biomimetic designs, where the material architectures are inspired by nature. In this study, a computational framework is developed to present novel architectured bi-material structures with tunable stiffness, strength, and toughness to be used for additive manufacturing (AM). The structure of natural nacre is mimicked to design robust multilayered structures constructed from hexagonal brittle and hard building blocks bonded with soft materials and supports. A set of computational models consisting of fully bonded zones, while allowing for interlayer interactions are created to accurately mimic the interplay between the hard and soft organic phases. As required for such complex designs, the numerical constraints are properly set to run quasi-static non-linear explicit analysis, which allow for a 3× faster analysis with higher efficiency and 2× lower computational cost, when compared to static analysis. The models are used to assess the stiffness, strength and toughness of bi-material beams when subjected to a flexural three-point bending load. The influence of structural features like the soft-to-hard volume ratio (i.e. the distance between each building block, its aspect ratio, and overlap length), material features (e.g. the stiffness ratio of the hard-to-soft phases), the plastic strain failure of soft phase, and AM features (e.g. different types of within-layer/sandwiched supports) are systematically investigated. The results revealed that the toughness of the architectured beams was enhanced by up to 25% when compared to a monolithic structure. This improvement is due to the frictional tile sliding in the brittle phase and the extensive shear plastic deformation of the soft interfaces. This work provides compatible designs to facilitate the AM of nacre-based bi-martial structures with balanced/tailored mechanical performance and to understand the influence of the architectural parameters.

Nonlinear vibrations of a bi-material beam under thermal and mechanical loadings

Nonlinear vibration of a two layers (bi-material) beam under thermal and mechanical loadings is studied in the paper. The model of the beam is based on the extended Timoshenko beam theory taking into account geometric nonlinearities. The derived nonlinear partial differential equations are reduced to ordinary differential equations for the first three vibration modes. The reduced model is compared with results based on modal experimental analysis and the finite element method and a very good agreement was found. The complete bifurcation analysis is carried out for the reduced 3 DOF system for the clamped–clamped boundary conditions. The obtained resonance curves demonstrated the hardening phenomenon with instability zones in a case of large vibrations. The elevated temperature leaded to buckling, stronger modes interaction, an increase of instability zones occurring on the resonance curves and to chaotic oscillations coexisting with the periodic response.

A phase-field model for thermo-mechanical fracture

In this article, we propose a thermodynamically consistent phase-field model for thermo-mechanical fracture and provide an open-source implementation of the proposed model using a recently developed finite element toolbox, Gridap in Julia. Here, we have derived the balance equations for the thermo-mechanical fracture by invoking the virtual power principle and determined the constitutive relations for the thermodynamic fluxes based on the satisfaction of the thermodynamic laws. Our proposed formulation provides an equation of temperature evolution that can easily accommodate dissipative effects such as viscous damping. We provide very compact and user-friendly open-source codes for implementing the proposed model using Gridap in Julia that requires very low memory usage and gives a high degree of flexibility to the users in defining weak forms of the governing partial differential equations (PDEs). We have validated the proposed model and its implementation against such standard results available in the literature as crack propagation in the cruciform shape material, single edge notched plate, bi-material beam, and a quenching test.

Softening-Spring Phenomenon in Large Amplitude Vibration of Two-Layer Bi-Material Beams

The purpose of this study is to analyze the nonlinear vibration behavior of two-layer bi-material beams by using the Euler beam theory and Timoshenko beam theory. It is assumed that material properties satisfy a power law distribution along the thickness direction. The differential quadrature (DQ) method in conjunction with an iterative algorithm is employed to solve the nonlinear governing equations for two-layer bi-material beams. Numerical results are presented to show the influence of Young’s modulus ratio and thickness ratio on nonlinear frequency ratio–amplitude curves of two-layer bi-material beams. It is observed that the nonlinear frequency ratio–amplitude curves are unsymmetrical with the hinged end because of bending-extension coupling effect in two-layer beams. It is also observed that the hinged–hinged two-layer beams may occur the softening-spring nonlinearity when the amplitude of oscillation is small. This softening-spring nonlinearity is dependent on the thickness ratio, Young’s modulus ratio, vibration amplitude, layer number and boundary condition. The greater the difference of material properties between the two layers is, the more obvious the softening-spring nonlinearity becomes.

Peridynamic modeling of elastic bimaterial interface fracture

A general peridynamics-based framework for elastic bimaterial interface fracture analysis is established. In this frame, the formation of peridynamic interface bond force is presented for nonlocal interface modeling, and the energy release rate and mode mixity of interface cracks are computed with peridynamic model. A modified critical energy density (MCED) criterion considering both interfacial and materials fracture toughness is proposed for peridynamic bond failure analysis, and the bond failure competition strategy is considered for crack path selection of an interface crack along or kinking out of the interface. Then, four tests of a bimaterial plate under uniform tensile stress, asymmetric bimaterial cantilever beams (ABCB), bimaterial single edge notched (BSEN) and four-point shearing (FPS) tests are investigated for the model verification and application. The deformation and fracture behaviors of these specimens are predicted with the proposed peridynamic models, and compared to those from FEM, theoretical and experimental solutions. The results show that the proposed peridynamics-based method can successfully capture the characteristics of bimaterial interface fracture, including the delamination fracture, kinking nucleation and crack path selection of interface cracks.

Multi-material adhesive joints with thick bond-lines: Crack onset and crack deflection

This study investigates the fracture onset and crack deflection in multi-material adhesive joints with thick bond-lines (≈10 mm) under global mode I loading. The role of adherend-adhesive modulus-mismatch and pre-crack length are scrutinized. The parameters controlling the crack path directional stability are also discussed. Single-material (i.e. steel-steel and GFRP-GFRP) and bi-material (i.e. steel-GFRP) double-cantilever beam joints bonded with a structural epoxy adhesive are tested. The joints are modelled analytically, considering a beam on elastic-plastic foundation, to include characteristic length scales of the problem (e.g. adhesive thickness, plastic zone) and numerically using Finite Element Model. An empirical relation, in terms of geometrical and material properties of the joints, that defines the transition between non-cohesive and cohesive fracture onset is found. Above a specific pre-crack length the stress singularity at pre-crack tip rules over the stress singularity near bi-material corners, resulting in mid-adhesive thickness cohesive fracture onset. However, the cracking direction rapidly deflects out from the adhesive layer centre-line. Positive T-stress along the crack tip is found to be one of the factors for the unstable crack path.

Ensuring Accurate Results in Fracture Mechanics Four-Point Bending Interface Characterization

Interface adhesion tests are conducted to obtain a quantitative value of the adhesion strength or interface fracture toughness to forecast initiation and propagation of interface delamination failures in semiconductor packages. One of the common methods of fracture mechanics interface characterization is the four-point bending test. Problems with the experimental setup would result in having inaccurate results. In this study, different issues with the four-point bending test using mold/copper bi-material beam with notch were addressed. It was shown that incorrect anvil alignment and centering could give inaccurate fracture toughness results. The distance between anvil supports and the depth of the notch is also very important. From the study, it can be concluded that issues with experimental setup and the test sample must be addressed to ensure accurate results from four-point bending interface characterization.

Programmable Spiral and Helical Deformation Behaviors of Hydrogel-Based Bi-Material Beam Structures

Soft materials possess magnificent properties which could be harnessed for different potential applications. Compared to other soft materials, hydrogels have some unique advantages which can be used in the shape deformation or shape transformation of structures. This paper aims to investigate the deformation mechanisms of hydrogel-based bi-material beam structures and study the non-uniform geometric effects on the shape transformation including programmable scroll and helical deformations. With a sloped thickness design, the structures could be transformed from an initial quasi-2D beam configuration into some other 2D self-scroll and 3D self-helical configurations. From the hydrogel material model, a modified deformation formula for bi-material beam structures based on the framework of the classical beam theory has been developed to predict the shape morphing behaviors. The relationship between the curvature and the mismatch strain is derived in its explicit form and the theoretical results are verified through several numerical simulations. Furthermore, experiments are carried out to demonstrate the design principles for reconfigurable bi-material beam structures and the experiments show that the structures tend to deform similarly to that predicted by the analytical models. The presented work could provide guidance for future applications of responsive hydrogel-based bi-material beam structures such as in soft actuators and soft robots.

Residually Stressed Bimaterial Beam Specimen for Measuring Environmentally Assisted Crack Growth

Subcritical crack growth can occur in a brittle material when the stress intensity factor is smaller than the fracture toughness if an oxidizing agent (such as water) is present at the crack tip. We present a novel bi-material beam specimen which can measure environmentally assisted crack growth rates. The specimen is “self-loaded” by residual stress and requires no external loading. Two materials with different coefficient of thermal expansion are diffusion bonded at high temperature. After cooling to room temperature a subcritical crack is driven by thermal residual stresses. A finite element model is used to design the specimen geometry in terms of material properties in order to achieve the desired crack tip driving force. The specimen is designed so that the crack driving force decreases as the crack extends, thus enabling the measurement of the crack velocity versus driving force relationship with a single test. The method is demonstrated by measuring slow crack growth data in soda lime silicate glass and validated by comparison to previously published data. The self-loaded nature of the specimen makes it ideal for measuring the very low crack velocities needed to predict brittle failure at long lifetimes.

Free, forced, and random vibrations of a beam composed of highly contrasting materials

This study deals with free, forced undamped vibration and the random, damped vibration of an inhomogeneous beam composed of two materials with significantly different elastic moduli and material densities. Exact solution is obtained for natural frequencies and the mode shapes using Krylov-Duncan functions, along with the comparison with the results furnished by the Galerkin's method. Random vibrations of the bi-material beams are studied for the first time in the literature for the case of proportional damping. Novel element of this study consists in the unintuitive finding that the response of the bi-material beam may considerably exceed the responses of its homogeneous counterparts, composed of a single material. This finding appears of design significance, suggesting that the natural frequencies of the bi-material beam must be made so as to lie above the cutoff frequency of broad-band cutted white noise.