International Journal of Mechanical Sciences最新文献

英文中文

Generative deep learning for designing irregular metamaterials with programmable nonlinear mechanical responses

IF 7.1 1区工程技术 Q1 ENGINEERING, MECHANICAL

International Journal of Mechanical Sciences

Pub Date : 2025-03-09 DOI: 10.1016/j.ijmecsci.2025.110123

Zhuoyi Wei , Jiaxin Chen , Kai Wei

Introducing irregularities found in natural materials enables metamaterials to achieve excellent properties or special functions, while it still poses challenges in customizing their nonlinear responses due to the huge design space and complex intrinsic structure-property relationships. Here, we originally propose a generative deep learning framework, which synergistically comprises a variational encoder and a property predictor, to construct a continuous and meaningful latent space for designing irregular metamaterials with programmable strain-stress curves. Specifically, irregular metamaterials are created through two types of building blocks with opposite deformation directions, and thus rich stress-strain curve responses are achieved benefitting from their different spatial arrangements. Furthermore, we obtain new irregular metamaterials beyond the dataset by simple manipulations in the latent space. Finally, we deploy an optimization method to flexibly achieve the inverse design of irregular metamaterials to satisfy targeted strain-stress curves. Particularly, we successfully identify multiple irregular metamaterials hitting the same nonlinear properties. Our established framework provides a new approach to meet prescribed yet complex nonlinear mechanical behavior and contributes to deep learning-aided materials design.

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引用次数: 0

Positive and negative feedback of entangled lightning multiphysics on composites

IF 7.1 1区工程技术 Q1 ENGINEERING, MECHANICAL

International Journal of Mechanical Sciences

Pub Date : 2025-03-08 DOI: 10.1016/j.ijmecsci.2025.110022

Yue Wu , Fusheng Wang , Chenyang Lv , Jinru Sun , Xiangteng Ma , Chenguang Huang , Zhiqiang Fan , Shaozhen Wang , Chenglin Wang , Yunpeng Gao , Zemin Duan , Xueling Yao

Lightning strike will have intertwined multi-physical effects on carbon fiber reinforced polymer. According to experiments and high-fidelity simulations in this study, distinctive positive and negative feedback is originally identified between the entangled lightning effects in composites damage driving. The intricate mesoscopic feedback mechanisms are groundbreakingly revealed through the anisotropic equivalent circuit. Positive feedback exists between ablation and mechanical effects, while thermal strain is counteracted by overpressure and ampere force effects. These feedback relationships are formed due to dissimilar action mechanisms and energy transformations, which dynamically change with time and location. They jointly cause complex non-uniform damage to composites.

引用次数: 0

Closed-form solutions for wave propagation in hexagonal diatomic non-local lattices 六边形二原子非局部晶格中波传播的闭式解法

IF 7.1 1区工程技术 Q1 ENGINEERING, MECHANICAL

International Journal of Mechanical Sciences

Pub Date : 2025-03-08 DOI: 10.1016/j.ijmecsci.2025.110095

F. Ongaro , P.H. Beoletto , F. Bosia , M. Miniaci , N.M. Pugno

Periodic mass–spring lattices are commonly used to investigate the propagation of waves in elastic systems, including wave localisation and topological protection in phononic crystals and metamaterials. Recent studies have shown that introducing non-neighbouring (i.e., beyond nearest neighbour) connections in these chains leads to multiple topologically localised modes, while generating roton-like dispersion relations. This paper focuses on the theoretical analysis of elastic wave propagation in hexagonal diatom mass–spring systems in which both neighbouring and non-neighbouring interactions occur through linear elastic springs. Closed-form expression for the dispersion equations are derived, up to an arbitrary order of beyond-the-nearest connections for both in-plane and out-of-plane mass displacements. This allows to explicitly determine the influence of the order of non-neighbouring interactions on the band gaps, the local minima and the slope inversions in the first Brillouin zone for the considered unit cell. All analytical solutions are numerically verified. Finally, examples are provided on how non-neighbouring connections can be exploited to enhance the localisation of topologically-protected edge modes in waveguides constructed using mirror symmetric diatomic lattices constituted by two regions with different unit cell orientations. The study provides further insight on how to design phononic crystals generating roton-like behaviour and to exploit them for topologically protected waveguiding.

{"title":"Closed-form solutions for wave propagation in hexagonal diatomic non-local lattices","authors":"F. Ongaro , P.H. Beoletto , F. Bosia , M. Miniaci , N.M. Pugno","doi":"10.1016/j.ijmecsci.2025.110095","DOIUrl":"10.1016/j.ijmecsci.2025.110095","url":null,"abstract":"<div><div>Periodic mass–spring lattices are commonly used to investigate the propagation of waves in elastic systems, including wave localisation and topological protection in phononic crystals and metamaterials. Recent studies have shown that introducing non-neighbouring (i.e., beyond nearest neighbour) connections in these chains leads to multiple topologically localised modes, while generating roton-like dispersion relations. This paper focuses on the theoretical analysis of elastic wave propagation in hexagonal diatom mass–spring systems in which both neighbouring and non-neighbouring interactions occur through linear elastic springs. Closed-form expression for the dispersion equations are derived, up to an arbitrary order of beyond-the-nearest connections for both in-plane and out-of-plane mass displacements. This allows to explicitly determine the influence of the order of non-neighbouring interactions on the band gaps, the local minima and the slope inversions in the first Brillouin zone for the considered unit cell. All analytical solutions are numerically verified. Finally, examples are provided on how non-neighbouring connections can be exploited to enhance the localisation of topologically-protected edge modes in waveguides constructed using mirror symmetric diatomic lattices constituted by two regions with different unit cell orientations. The study provides further insight on how to design phononic crystals generating roton-like behaviour and to exploit them for topologically protected waveguiding.</div></div>","PeriodicalId":56287,"journal":{"name":"International Journal of Mechanical Sciences","volume":"291 ","pages":"Article 110095"},"PeriodicalIF":7.1,"publicationDate":"2025-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143629077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Continuous high-throughput characterization of mechanical properties via deep learning

IF 7.1 1区工程技术 Q1 ENGINEERING, MECHANICAL

International Journal of Mechanical Sciences

Pub Date : 2025-03-08 DOI: 10.1016/j.ijmecsci.2025.110137

Gengxuan Zhu , Xueyan Hu , Ronghao Bao , Weiqiu Chen

High-throughput experiments (HTE) aim to acquire extensive chemical or physical properties in a single experiment, thereby enhancing testing efficiency. To simplify the extraction of diverse properties from one specimen, samples have moved from “discrete” arrays to “continuous” gradient ones. Despite this, complex responses of “continuous” gradient samples have impeded the development of continuous HTE. Full-field data, which can be obtained with Digital Image Correlation (DIC), is necessary for mechanical property characterizations. Traditional inversion methods for calculating property distributions from this data are slow and error-prone. Deep learning (DL) offers a faster and more accurate alternative for characterizing properties. Therefore, based on convolutional neural networks (CNNs), this article establishes a mapping model to obtain the modulus distribution directly from the full-field displacement. In view of the cost of time, simulation data are used to replace DIC data. However, fine mesh must be used to obtain the precise responses of gradient samples which unfortunately making the DL model face the challenge of time-consuming dataset generation and high-dimensional data mapping. To alleviate the difficulties, the isoparametric graded finite element (IGFE) formulation is introduced in this article, which offers an efficient way to generate datasets with low-dimension but high-fidelity. Results show that our framework not only has high prediction accuracy (with the L₁-error of 1.38%) but also enables fast characterization (within 12 ms), providing methodological support for high-throughput characterization based on gradient samples.

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引用次数: 0

Optimized synchronization efficiency in micromechanical arch beams 优化微机械拱形梁的同步效率

IF 7.1 1区工程技术 Q1 ENGINEERING, MECHANICAL

International Journal of Mechanical Sciences

Pub Date : 2025-03-07 DOI: 10.1016/j.ijmecsci.2025.110098

Zunhao Xiao , Zhan Shi , Qiangfeng Lv , Xuefeng Wang , Xueyong Wei , Ronghua Huan

Synchronization phenomena in MEMS devices are extensively studied due to their critical applications and intricate dynamics. Nevertheless, research on synchronization time – key to sensor performance – remains sparse. Current optimization efforts are predominantly focused on device fabrication and signal transmission, while dynamics approaches are limited to perfected straight beams, which can deviate in practical applications. In this study, we explore the dynamics of synchronization in a clamped–clamped micromechanical arch beam, modulated by electrothermal currents. Initially, we employed electrothermal currents to achieve an optimal synchronization time. Our theoretical analysis demonstrated that reducing equivalent nonlinearity leads to a shorter synchronization time. This effect was experimentally verified by manipulating the static DC voltage in electrostatic excitation to control the nonlinearity. By combining electrothermal current regulation and nonlinearity control, we substantially reduced synchronization time by 84%, from 1.170 s to 0.182 s. These results introduce a novel strategy for enhancing the detection efficiency of synchronization sensors, with broad implications for sensor technology.

{"title":"Optimized synchronization efficiency in micromechanical arch beams","authors":"Zunhao Xiao , Zhan Shi , Qiangfeng Lv , Xuefeng Wang , Xueyong Wei , Ronghua Huan","doi":"10.1016/j.ijmecsci.2025.110098","DOIUrl":"10.1016/j.ijmecsci.2025.110098","url":null,"abstract":"<div><div>Synchronization phenomena in MEMS devices are extensively studied due to their critical applications and intricate dynamics. Nevertheless, research on synchronization time – key to sensor performance – remains sparse. Current optimization efforts are predominantly focused on device fabrication and signal transmission, while dynamics approaches are limited to perfected straight beams, which can deviate in practical applications. In this study, we explore the dynamics of synchronization in a clamped–clamped micromechanical arch beam, modulated by electrothermal currents. Initially, we employed electrothermal currents to achieve an optimal synchronization time. Our theoretical analysis demonstrated that reducing equivalent nonlinearity leads to a shorter synchronization time. This effect was experimentally verified by manipulating the static DC voltage in electrostatic excitation to control the nonlinearity. By combining electrothermal current regulation and nonlinearity control, we substantially reduced synchronization time by 84%, from 1.170 s to 0.182 s. These results introduce a novel strategy for enhancing the detection efficiency of synchronization sensors, with broad implications for sensor technology.</div></div>","PeriodicalId":56287,"journal":{"name":"International Journal of Mechanical Sciences","volume":"291 ","pages":"Article 110098"},"PeriodicalIF":7.1,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143620318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Topology optimization with multi-phase length-scale control

IF 7.1 1区工程技术 Q1 ENGINEERING, MECHANICAL

International Journal of Mechanical Sciences

Pub Date : 2025-03-07 DOI: 10.1016/j.ijmecsci.2025.110086

A. Asadpoure , M.M. Rahman , S.A. Nejat , L. Javidannia , L. Valdevit , J.K. Guest , M. Tootkaboni

We present a multi-material density-based topology optimization framework that offers full length-scale control on each of the material phases involved. We represent different material phases by different sets of independent design variables, while avoiding a prohibitive number of constraints, and devise a consistent penalization tailored to multimaterial design. The independent design variables are passed through multi-phase Heaviside projections and the modified material model with penalization to define element densities and material properties. Overfilling is avoided via constraints on element densities which are handled through “sum of powers” aggregation and smoothing to curtail the need for local constraints and the associated computational burden. The proposed framework enables the imposition of individual length scales while avoiding, to a large extent, the issues related to phase mixing at boundaries. It is also amenable to gradient-based optimizers and thus capable of solving large-scale multi-material topology optimization problems. Multiple topology optimization problems, including compliance minimization and design of compliant mechanisms are provided to demonstrate the effectiveness of the proposed framework to cleanly enforce specified length-scales on individual material phases.

{"title":"Topology optimization with multi-phase length-scale control","authors":"A. Asadpoure , M.M. Rahman , S.A. Nejat , L. Javidannia , L. Valdevit , J.K. Guest , M. Tootkaboni","doi":"10.1016/j.ijmecsci.2025.110086","DOIUrl":"10.1016/j.ijmecsci.2025.110086","url":null,"abstract":"<div><div>We present a multi-material density-based topology optimization framework that offers full length-scale control on each of the material phases involved. We represent different material phases by different sets of independent design variables, while avoiding a prohibitive number of constraints, and devise a consistent penalization tailored to multimaterial design. The independent design variables are passed through multi-phase Heaviside projections and the modified material model with penalization to define element densities and material properties. Overfilling is avoided via constraints on element densities which are handled through “sum of powers” aggregation and smoothing to curtail the need for local constraints and the associated computational burden. The proposed framework enables the imposition of individual length scales while avoiding, to a large extent, the issues related to phase mixing at boundaries. It is also amenable to gradient-based optimizers and thus capable of solving large-scale multi-material topology optimization problems. Multiple topology optimization problems, including compliance minimization and design of compliant mechanisms are provided to demonstrate the effectiveness of the proposed framework to cleanly enforce specified length-scales on individual material phases.</div></div>","PeriodicalId":56287,"journal":{"name":"International Journal of Mechanical Sciences","volume":"291 ","pages":"Article 110086"},"PeriodicalIF":7.1,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Transferability of machine-learning interatomic potential to α-Fe nanocrystalline deformation

IF 7.1 1区工程技术 Q1 ENGINEERING, MECHANICAL

International Journal of Mechanical Sciences

Pub Date : 2025-03-05 DOI: 10.1016/j.ijmecsci.2025.110132

Kazuma Ito , Tatsuya Yokoi , Katsutoshi Hyodo , Hideki Mori

To improve the mechanical properties of polycrystalline metallic materials, understanding the elementary processes involved in their deformation at the atomic level is crucial. In this study, firstly, we evaluate the transferability of the recently proposed α-Fe machine-learning interatomic potential (MLIP), constructed from mechanically generated training data based on crystal space groups, to the tensile deformation process of nanopolycrystals. The transferability was evaluated by comparing the physical properties and lattice defect formation energies, which are important in the deformation behavior of nanopolycrystals, with those obtained from density functional theory (DFT) and by comprehensively calculating extrapolation grades based on active learning methods for the local atomic environment in the nanopolycrystal during tensile deformation. These evaluations demonstrate the superior transferability of the MLIP to the tensile deformation of the nanopolycrystals. Furthermore, large-scale molecular dynamics calculations were performed using the MLIP and the most commonly used embedded atom method (EAM) potential to investigate the effect of grain size on the deformation behavior of α-Fe polycrystals and the effect of interatomic potentials on them. The uniaxial tensile deformation behavior of the nanopolycrystals obtained from EAM was qualitatively consistent with that obtained from MLIP. This result supports the results of many studies conducted using EAM and is an important conclusion considering the high computational cost of the MLIP. Furthermore, the construction method for the MLIP used in this study is applicable to other metals. Therefore, this study considerably contributes to the understanding and material design of various metallic materials through the construction of highly accurate MLIPs.

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引用次数: 0

A coupled thermal-mechanical-oxidative model for predicting oxidation and stress affected by cracks

IF 7.1 1区工程技术 Q1 ENGINEERING, MECHANICAL

International Journal of Mechanical Sciences

Pub Date : 2025-03-05 DOI: 10.1016/j.ijmecsci.2025.110131

Hongcui Wang , Tiechao Bai , Weijie Li , Xiaoyu Wang , Ying Li

High-temperature thermal structural materials under laser irradiation are often exposed to simultaneous thermal stresses and mechanical loads, and the interaction between these factors may lead to crack propagation, oxide delamination, and even material failure. By establishing a novel coupled thermal-mechanical-oxidative (CTMO) model, this study systematically investigates the effects of crack properties on the oxidation growth and stress evolution of C/SiC composites in a high-temperature-stress-oxidative environment. Unlike the existing studies, this study incorporates several crack characterization parameters, such as crack width, spacing, depth, and inclination angle, into a unified multi-physics field coupling framework. The complex effects of these parameters on oxide formation and stress distribution are analyzed in detail. Through numerical simulations, this paper reveals the interaction mechanism between mechanical loading, oxidation behavior and crack evolution, especially the material degradation behavior under extreme conditions. The results show that the crack width and depth significantly affect the oxide diffusion and stress concentration, while the crack spacing and inclination angle further influence the material failure mode by changing the stress field interactions and oxidant diffusion paths. The CTMO model proposed not only provides theoretical support for the optimization of the performance of high-temperature thermal structural materials in complex environments, but also provides a scientific basis for the material selection and design optimization of laser protection systems. The results reveal the coupling effect between oxide growth and crack extension, which provides a new perspective for understanding the degradation mechanism of composite materials under high-temperature stress oxidation environment.

{"title":"A coupled thermal-mechanical-oxidative model for predicting oxidation and stress affected by cracks","authors":"Hongcui Wang , Tiechao Bai , Weijie Li , Xiaoyu Wang , Ying Li","doi":"10.1016/j.ijmecsci.2025.110131","DOIUrl":"10.1016/j.ijmecsci.2025.110131","url":null,"abstract":"<div><div>High-temperature thermal structural materials under laser irradiation are often exposed to simultaneous thermal stresses and mechanical loads, and the interaction between these factors may lead to crack propagation, oxide delamination, and even material failure. By establishing a novel coupled thermal-mechanical-oxidative (CTMO) model, this study systematically investigates the effects of crack properties on the oxidation growth and stress evolution of C/SiC composites in a high-temperature-stress-oxidative environment. Unlike the existing studies, this study incorporates several crack characterization parameters, such as crack width, spacing, depth, and inclination angle, into a unified multi-physics field coupling framework. The complex effects of these parameters on oxide formation and stress distribution are analyzed in detail. Through numerical simulations, this paper reveals the interaction mechanism between mechanical loading, oxidation behavior and crack evolution, especially the material degradation behavior under extreme conditions. The results show that the crack width and depth significantly affect the oxide diffusion and stress concentration, while the crack spacing and inclination angle further influence the material failure mode by changing the stress field interactions and oxidant diffusion paths. The CTMO model proposed not only provides theoretical support for the optimization of the performance of high-temperature thermal structural materials in complex environments, but also provides a scientific basis for the material selection and design optimization of laser protection systems. The results reveal the coupling effect between oxide growth and crack extension, which provides a new perspective for understanding the degradation mechanism of composite materials under high-temperature stress oxidation environment.</div></div>","PeriodicalId":56287,"journal":{"name":"International Journal of Mechanical Sciences","volume":"291 ","pages":"Article 110131"},"PeriodicalIF":7.1,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143641816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Deformation prediction of circular cell honeycomb under fixture-workpiece systems

IF 7.1 1区工程技术 Q1 ENGINEERING, MECHANICAL

International Journal of Mechanical Sciences

Pub Date : 2025-03-05 DOI: 10.1016/j.ijmecsci.2025.110120

Yan Wang, Zhigang Dong, Junchao Tian, Yan Bao, Renke Kang, Yan Qin

Carbon fiber reinforced plastic (CFRP) circular cell honeycombs are increasingly used in lightweight structures, but their weak radial stiffness makes them highly susceptible to deformation during clamping, reducing machining accuracy. Accurately predicting this deformation is essential for improving machining precision and ensuring structural integrity. In this study, a numerical model based on planar beam theory is developed to investigate the deformation mechanism of CFRP circular cell honeycombs. Additionally, a finite element analysis (FEA) model is established to incorporate various clamping factors, providing a more comprehensive prediction framework. Both approaches consider actual workpiece characteristics and clamping conditions. The predicted deformations are quantitatively compared with measured surface profiles, showing that the proposed method achieves a prediction error within 10 %. This validates the accuracy of the approach and confirms its applicability to practical machining conditions. The findings of this study offer valuable guidance for achieving high-precision machining of CFRP circular cell honeycombs, contributing to enhanced machining accuracy and reduced workpiece deformation.

{"title":"Deformation prediction of circular cell honeycomb under fixture-workpiece systems","authors":"Yan Wang, Zhigang Dong, Junchao Tian, Yan Bao, Renke Kang, Yan Qin","doi":"10.1016/j.ijmecsci.2025.110120","DOIUrl":"10.1016/j.ijmecsci.2025.110120","url":null,"abstract":"<div><div>Carbon fiber reinforced plastic (CFRP) circular cell honeycombs are increasingly used in lightweight structures, but their weak radial stiffness makes them highly susceptible to deformation during clamping, reducing machining accuracy. Accurately predicting this deformation is essential for improving machining precision and ensuring structural integrity. In this study, a numerical model based on planar beam theory is developed to investigate the deformation mechanism of CFRP circular cell honeycombs. Additionally, a finite element analysis (FEA) model is established to incorporate various clamping factors, providing a more comprehensive prediction framework. Both approaches consider actual workpiece characteristics and clamping conditions. The predicted deformations are quantitatively compared with measured surface profiles, showing that the proposed method achieves a prediction error within 10 %. This validates the accuracy of the approach and confirms its applicability to practical machining conditions. The findings of this study offer valuable guidance for achieving high-precision machining of CFRP circular cell honeycombs, contributing to enhanced machining accuracy and reduced workpiece deformation.</div></div>","PeriodicalId":56287,"journal":{"name":"International Journal of Mechanical Sciences","volume":"290 ","pages":"Article 110120"},"PeriodicalIF":7.1,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A phase-field fracture model for piezoelectrics in hydrogen-rich environment

IF 7.1 1区工程技术 Q1 ENGINEERING, MECHANICAL

International Journal of Mechanical Sciences

Pub Date : 2025-03-05 DOI: 10.1016/j.ijmecsci.2025.110092

Yu Tan , Fan Peng , Peidong Li , Chang Liu , Jianjun Zhao , Xiangyu Li

Piezoelectric materials are often serviced in various extreme environments and exhibit complex fracture behaviors. Past studies usually focus on the electro-mechanical coupling behavior of piezoelectric materials, ignoring the influence of environmental factors. In this paper, a phase-field model for brittle fracture in piezoelectrics under hydrogen-rich environment is developed, and the coupling effects among the elastic, electric and chemical fields have been considered. A phenomenological model is developed to characterize the deterioration of fracture toughness in hydrogen-rich environment. To solve this problem numerically, a robust staggered scheme is proposed via a hybrid manner. Numerical simulations are performed to discuss the influences of hydrogen concentration and external electric field on the fracture behaviors of piezoelectrics. It is found that the existence of hydrogen atoms will reduce fracture loads and promote the cracking of piezoelectric specimens significantly. This study will provide theoretical support for the reliability assessment of piezoelectric devices in hydrogen-rich environment.

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