Bi-material Plate Research Articles

Fracture behavior of thermal mismatch in functionally graded materials using phase-field modeling

A phase-field modeling is presented to study the complex thermal mismatch fracture behavior in functionally graded materials. The fracture resistance in functionally graded materials is homogenized by the rule of mixtures of the energy release rate. For validation, the bi-material plates and rectangular plates coated with functionally graded materials under thermal load are examined and the obtained results are compared with existing theoretical and experimental data. In the bi-material plates, the model captures commonly experimentally observed crack patterns such as the perpendicular crack, benching, and the formation of two pairs of parallel cracks. Additionally, a detailed comparison of the correlation between crack depth and film thickness is conducted among theoretical, experimental, and phase-field model data. Notably, the results of the phase-field model exhibit good agreement with experiments compared to the theoretical model. Subsequently, the model is applied to examine the thermal mismatch fracture behavior in plates of different shapes. The analysis emphasizes three key aspects: (i) examining the damage field to understand crack initiation and propagation, (ii) analyzing the major principal stress field in the coatings to identify fracture-critical regions, and (iii) tracking the evolution of elastic/fracture/total energies to investigate the fracture/deformation behavior. This analysis enhances a fundamental understanding of thermal fracture mechanisms in functionally graded materials and provides a foundation for optimizing materials and structures in various engineering applications.

Application of a hierarchical quadrature element method to interface crack analysis in bi-material systems

This paper presents an extension of the hierarchical quadrature element method (HQEM) with p-convergence to evaluating fracture mechanical parameters of bi-material interface cracks. The numerical implementation of HQEM in combination with the virtual crack closure method (VCCM) is comprehensively illustrated, and the formulas for calculating the strain energy release rate (SERR) are rederived. The relationship between components of SERR and the complex stress intensity factors (SIFs) is established. In addition, the effectiveness of an auxiliary parameter related to the dual energy release rates in measuring the accuracy of the present method is examined.The feasibility of applying HQEM to the interface crack problems is demonstrated by investigating the benchmark problem of an infinite bi-material plate with a central interface crack. Two loading cases are considered, including pure tensile loading and mixed tensile-shear loading. The influence of the crack tip element size Δa on the convergence characteristics and computational accuracy of fracture parameters is quantitatively investigated to provide guidance for choosing the most reasonable value of Δa. Numerical results show that even with very coarse meshes, together with a virtual crack extension consisting of only one element, HQEM yields highly accurate results for the total SERR and the complex SIFs, making it suitable for complex, large-scale interface crack problems in dissimilar media.

A modified phase-field model for cohesive interface failure in quasi-brittle solids

We formulate a modified phase-field model for cohesive interface failure in quasi-brittle solids. Our model has two novel features: (i) a traction–separation-damage law for damage process; (ii) an energetic degradation function controlled by critical gap ratio. This modification offers an attractive approach to simulate the cohesive interface failure process. We also provide a robust numerical solution strategy to treat the spatio-temporal evolution of cohesive interface failure. Our model is validated by two benchmark problems including the constant strain patch test and mode-I delamination test. The numerical simulations are compared with some published data. We proceed to apply this model to study the complex failure mechanism of a peeling test on bi-material plates and crack impinging on interfaces in different scenarios, where the effects of critical gap ratios and interface inclination angles are discussed.

Near-zero thermal mismatch flexure metastructure with high-resonant frequency

Heterogeneous mechanical metastructures, especially fabricated by additive manufacturing techniques, have attracted attention in lightweight multifunctional material engineering. Near-zero thermal expansion materials play an important role in precision equipment. However, undesired thermal mismatch flexure distortion occurs when the precision equipment is mounted on the main load-bearing structures of spacecraft and the environmental temperature changes. The present study presents a near-zero thermal mismatch flexure metastructure, including the design strategy inspired by plants with axisymmetric distributing anisotropic microstructures, the thermal deformation constitutive relation derived by the flexibility method, and the fabrication of this metastructure by selective laser melting. Thermal deformation experiments demonstrate that the coefficient of thermal flexure reduces by 92.4% compared with traditional bi-material plates connected by screws. Mechanical vibration experiments demonstrate that the fundamental resonant frequency of this metastructure can be as high as 956 Hz, which is 5.8 times larger than the common requirements of functional components for spacecraft to avoid mechanical resonance with the main structure during launch. The excellent thermal mismatch shape stability and high-resonant frequency properties make it a potential candidate for thermal geometric stable structures working in variable thermal and mechanical environments.

Numerical simulation of interfacial and subinterfacial crack propagation by using extended peridynamics

The interaction between crack propagation and interfaces is one of the critical problems in bimaterial systems. In this paper, interfacial and subinterfacial fracture behaviors are simulated to investigate this interaction. The modeling approach for bimaterial structures is first introduced, where the extended bond-based peridynamic model is adopted. Then a convergence study is carried out with a bimaterial plate containing two asymmetric inclusions under tensile load. The comparison with the finite element solution validates the proposed modeling approach. Subsequently, several interfacial and subinterfacial fracture cases are considered to pinpoint the influence of interfaces on crack propagation. The numerical results show that the proposed method can satisfactorily capture the interplay between interfaces and crack propagation. Furthermore, in the subinterfacial fracture case, an equilibrium state of mode-I crack growth is successfully obtained, implying that a specific loading condition can counteract the effect of the interface.

Fracture analysis of interfacial crack in piezoelectric bimaterial by XIGA approach using Bézier extraction of NURBS

This work presents analyses of interfacial crack in piezoelectrics bimaterial plate under electromechanical loading environment by extended isogeometric analysis (XIGA) approach using Bézier extraction of NURBS. Asymptotic crack–tip branch functions and discontinuous Heaviside function are implemented for local enrichment in the approximation of isogeometric analysis for capturing singularity at cracktip and discontinuity across surfaces of a crack. Bézier extraction of NURBS–based basis functions are implemented in discretization of domain for approximating both unknown field solution and geometries. The Bézier extraction operation decomposes the NURBS basis functions into a set of linear combination of Bernstein polynomials and a set of C 0 –continuity Bézier elements. Domain–form interaction integral is implemented to obtain generalized fracture parameters. To assess the robustness and accuracy in results obtained from XIGA, numerical results are first validated from analytical ones and then additionally, few numerical problems are solved for calculating the fracture parameters in terms of normalized intensity factors with respect to shear loads, electrical loads, crack orientation angle and angle of polarization.

Analytical fracture parameters of two unequal collinear interface cracks in an orthotropic bimaterial

This paper presents an analytical solution for the fracture parameters of two unequal collinear interface cracks (UCIC) in an infinite orthotropic or isotropic bimaterial subject to in-plane loading. The analytical expressions for the stress intensity factors (SIF), Jk-integrals and crack opening displacements of two UCIC in an infinite orthotropic bimaterial plate are derived using the complex variable method and evaluated in conjunction with the Jacobian elliptic functions and elliptic integrals. The presented theory is verified using the boundary element method. The results show that the fracture parameters are greatly influenced by mismatch of the engineering constants and proximity of the cracks. The inner tip of the longer crack always experiences the maximum SIFs and Jk values. These fracture parameters will increase exponentially when the distance between the cracks inner tips decreases. This study provides a baseline for the verification of future numerical and experimental studies of two UCIC which is not currently available in the SIFs handbooks.

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.

Analysis of horizontally polarized shear waves on a highly inhomogeneous loaded bi-material plate

<abstract><p>The current manuscript critically examines the propagation of horizontally polarized shear waves on the dispersion of a highly inhomogeneous thin bonded bi-material plate when a load due to the Winkler's elastic foundation is prescribed. An analytical procedure of solution is deployed for the study; in addition to the exploitation of effective boundary conditions approach for the asymptotic examination. The overall inference of the current study is the realization of the fact that the vibrational displacements in both layers are enhanced by an increase in the inhomogeneity parameter; at the same time lessened with an increment in the foundation parameter. Moreover, a perfect approximation of the dispersion relation has been realized, with its validity extending to almost the entire low-frequency range. Lastly, the influence of the material inhomogeneity has been noted to affect fundamental mode, as against the presence of the foundation parameter which affects the first harmonic curve. More so, an increase in the two parameters narrows the chances of low-frequency propagation.</p></abstract>

Stochastic Fracture Analysis Using Scaled Boundary Finite Element Methods Accelerated by Proper Orthogonal Decomposition and Radial Basis Functions

This paper presents a stochastic analysis method for linear elastic fracture mechanics using the Monte Carlo simulations (MCs) and the scaled boundary finite element method (SBFEM) based on proper orthogonal decomposition (POD) and radial basis functions (RBF). The semianalytical solutions obtained by the SBFEM enable us to capture the stress intensity factors (SIFs) easily and accurately. The adoption of POD and RBF significantly reduces the model order and increases computation efficiency, while maintaining the versatility and accuracy of MCs. Numerical examples of cracks in homogeneous and bimaterial plates are provided to demonstrate the effectiveness and reliability of the proposed method, where the crack inclination angles are set as uncertain variables. It is also found that the larger the scale of the problem, the more advantageous the proposed method is.

Numerical investigation of bimaterial interfacial crack with interaction of voids and inclusions using XFEM

This article is presented to investigate the influence of several discontinuity like interfacial-crack, void and inclusion in the bimaterial plate, to analyze the normalized mixed-mode stress intensity factor (NMMSIFs) using XFEM under the uniaxial tensile load with considering the several numerical problems. It is investigated the normalized stress intensity factors (NSIFs) for an isotropic bimaterial plate and NMMSIFs for an isotropic-orthotropic bimaterial plate through varying the single-multi voids/inclusions position while its above, aligned and away from the interfacial edge crack and center crack of the bimaterial plate. It is also analyzed the void/inclusion several shape influence in the bimaterial plate. It is observed higher and fluctuating SIFs obtained with more number of voids/inclusions, influence of elliptical and diamond shape of void/inclusion is higher.

Analytical stress intensity factors and Jk‐integrals of periodic and collinear interface cracks between dissimilar orthotropic materials

AbstractThis paper introduces an analytical expression for the stress intensity factors at the crack tips of a row of periodic and collinear cracks between infinite dissimilar orthotropic materials. The analytical Jk‐integrals of these traction‐free cracks have been presented for different material combinations, ‘crack length’/‘interval width’ ratios (λ) and various in‐plane loading conditions. Some of the analytical Jk‐integrals have been compared with the numerical values computed using the boundary element crack shape sensitivities which are in good agreement. For any in‐plane loading condition of a bimaterial plate with periodic and collinear interface cracks, the normalised J1‐integral is almost independent of λ, but there is a slight variation in the normalised J2‐integral with respect to λ. This is provided that the normalising factor is the strain energy release rate of a homogeneous plate made from one of the materials with the same ratio of λ and subject to the same loading condition.

Peridynamic modeling of Lamb wave propagation in bimaterial plates

In this study the nonlocal bond-based peridynamic (PD) theory was applied to simulate Lamb wave transmission in 2D bimaterial plates. Plates of unlike materials were jointed end-to-end to make a bimaterial plate. The surface of the plate was then excited tangentially by single-frequency and multi-frequency force signals. The influence of changing material properties on travelling of the symmetric and antisymmetric Lamb waves were studied in detail. The phase and group velocities of travelling wave in bimaterial plates were calculated and it was found that dissimilar material properties of two joint plates can significantly alter the amplitude and arrival time of the wave. Moreover, Fast Fourier Transform (FFT) of the time history of symmetric Lamb wave velocity was calculated and verified. Accuracy and consistency of PD wave model was confirmed entirely using Spectral Finite Element Method (SFEM). The comparison between the results of two applied methods shows a good agreement.

A Circular Economy based Decision Support System for the Assembly/Disassembly of Multi-Material Components

Recent advancements in the aptness of multi-material structures in automated production lines have allowed for the assimilation of circular economy strategies. This creates the need of an intermediate platform, able to bridge the gap between the end-user and the complexity of these manufacturing processes. This study presents a modular Design Support System that receives a specified design point for a component and provides the user with Assembly/Disassembly related variables and scenarios. These scenarios are combined with the prospects of reusability for the multi-material component’s individual parts. A bi-material plate has been adopted for the presentation of the proposed platform’s functions and results.

A polling direction influence on fracture parameters of a limited permeable interface crack in a piezoelectric bi-material

An arbitrary polling direction of the upper material is considered for a plane-strain problem of a crack between two different piezoelectric materials subject to a tensile mechanical stress and an electrical displacement that are applied at infinity. Limited permeable electrical conditions at the crack faces are considered.The problem is reduced to the problem of linear relationship by use of complex function theory. Formulas for stresses at the interface as well as the intensity factors at the crack tips are presented in the form of simple analytical formulas.A finite-sized piezoelectric bi-material plate with interface crack, which length is substantially less than the plate dimension, was analysed by the finite elements method. The influence of the polling direction on the fracture parameters is studied. The results are presented for different polarisation angles and for different combinations of the materials. A good agreement between analytical and numerical results is shown.

Stress Intensity Factors of Slanted Cracks in Bi-Material Plates

In this study, the stress intensity factors (SIF) of slanted cracks in bi-material plates subjected to mode I loading is numerically solved. Based on the literature survey, tremendous amount of research works are available studying the normal cracks in both similar and dissimilar plates. However, lack of SIF behavior for slanted cracks especially when it is embedded in bi-material plates. The slanted cracks are then modelled numerically using ANSYS finite element program. Two plates of different in mechanical properties are firmly bonded obliquely and then slanted edge cracks are introduced at the lower inclined edge. Isoparametric singular element is used to model the crack tip and the SIF is determined which is based on the domain integral method. Three mechanical mismatched and four slanted angles are used to model the cracks. According to the present results, the effects of mechanical mismatch on the SIF for normal cracks are not significant. However, it is played an important role when slanted angles are introduced. It is suggested that higher SIF can be obtained when the cracks are inclined comparing with the normal cracks. Consequently, accelerating the crack growth at the interface between two distinct materials.

Analysis of the partially closed interface crack in orthotropic materials

The determination of fracture parameters such as mode I and II stress intensity factors, KI and KII, is of considerable practical importance in the analysis and failure prediction of various types of structures. For example, interfacial cracks may develop and propagate in bonded joints or lead to delaminations in composite materials. The finite element method is a powerful technique for analyzing complex structures; however, the singular behavior near crack tips is poorly approximated by conventional finite elements and convergence theorems cease to be applicable at those locations. In a previous paper, a combined analytical and numerical model based on frictionless crack face contact was used to obtain fracture parameters at interfacial cracks in composite laminates in cylindrical bending and subject to transverse loads. In this article, we further consider the method and extend the previous results by considering a variety of shear and normal loads on a cracked bimaterial plate with isotropic or orthotropic materials. Furthermore, additional validation of the method is provided by comparing the results with closed form solutions in the literature for the special case of isotropic materials in a combined tensile and shear field, and good agreement is found. Overall, we conclude that the method provides a practical way to analyze a variety of engineering structures with interfacial cracks.

Measurements of Thermally-Induced Curvatures and Warpages of Printed Circuit Board during a Solder Reflow Process Using Strain Gauges

Measurements of the curvatures and warpages of a printed circuit board (PCB) during a thermal solder reflow process using strain gauges are proposed in this study. In the experiments, a shadow moiré is used for measuring the out-of-plane deformations (or warpage) of a bi-material plate and a PCB with dual in-line memory module (DIMM) sockets during solder reflow heating, while the finite element method (FEM) is used to analyze the thermally-induced deformation of the PCB specimen for ensuring the validity of the measurement. Conventional strain gauges are employed to measure the strains (albeit as in-plane strain data) in both specimens during the solder reflow process. The results indicate that the strain gauge-measured strain data from the top and bottom surfaces of both specimens during the solder reflow can be converted into curvature data with specific equations, and even into global out-of-plane deformations or warpages with a proposed simple beam model. Such results are also consistent with those from the shadow moiré and FEM. Therefore, it has been proved that the strain gauge measurement associated with the simple beam model can provide a method for the real-time monitoring of PCB deformations or warpages with different temperatures during the solder reflow process.

Analysis of semi-elliptical surface cracks in the interface of bimaterial plates under tension and bending

A semi-elliptical surface crack widely exists in the multilayered electronic devices, fibre reinforced laminated composites and solder joints. To ensure the safety of these structural components, three-dimensional interface fracture mechanics analysis is required. In this study, three-dimensional finite element analyses have been conducted to calculate the fracture mechanics parameters, including the stress intensity factors (SIFs), the strain energy release rate G and two phase angles ψ and φ for semi-elliptical interface surface cracks in finite thickness plates. A wide range of crack aspect ratios and relative depths are considered. The SIF computations are presented along the front of a three-dimensional bimaterial interface cracks with a/t values of 0.2, 0.4, 0.6 or 0.8 and a/c values of 0.2, 0.4, 0.6 or 1.0. In particular, far-field loads i.e., remote tension and bending, and various combinations of bimaterials are investigated. These fracture mechanics parameters’ solutions are suitable for fracture and fatigue life prediction for semi-elliptical bi-material interface surface cracks in engineering components.

Stress intensity factor for small edge crack in interfacial singularity region

Fracture of the bonded dissimilar materials generally initiates near the interface, or just from the interface edge due to the edge stress singularity. In this study, the stress intensity factor of an edge crack close to the interface between the dissimilar materials is considered. The small edge crack is strongly dominated by the singular stress field near the interface edge. The analysis of stress intensity factor of small edge crack near the interface in bi-material and butt joint plates is carried out by changing the length and the location of the crack and the region dominated by the interface edge is examined. It is found that the dimensionless stress intensity factor of small crack, normalized by the singular stress at the crack tip point in the bonded plate without the crack, is equal to 1.12, independent of the material combination and adhesive thickness, when the relative crack length with respect to the crack location is less than 0.01. Therefore, the stress intensity factor of the small edge crack can be simply evaluated not only in the bi-material plate but also in the butt joint plate.