应用数学和力学

Cover And Contents

2024, 45(11)

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Preface

2024, 45(11): ⅰ-ⅱ.

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A Hamiltonian System Solution Method for Planar Problems of 2D Quasicrystals

LI Tong, QU Jianlong, WANG Wei, WANG Chenlong, XU Xinsheng

2024, 45(11): 1359-1371. doi: 10.21656/1000-0887.450204

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Aimed at the planar problem of 2D quasicrystals, the problem was transformed into one of symplectic eigenvalues and symplectic eigensolutions through introduction of the Hamiltonian system. In the Hamiltonian system, the solution to this problem was expressed by a series of symplectic eigensolutions. With the symplectic conjugate orthogonality relationship between symplectic eigensolutions, the solving problem satisfying boundary conditions can be reduced to a problem of solving algebraic equations, thus to form an analytical solution method. The proposed method can be directly extended to solve the problems of mixed boundary conditions and segmented boundary conditions.

Explicit Topology Optimization of Multi-Material Active Structures Based on Piezoelectric Actuation

YAN Xiaoye, LAI Qi, MENG Yao, ZHANG Weisheng

2024, 45(11): 1372-1380. doi: 10.21656/1000-0887.450197

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The lightweight structure design is a crucial consideration in the industrial field. Unlike passive structures that rely solely on material stiffness to resist external loads, active structures achieve lightweighting by active change of the internal force-driven deformation. An explicit topology optimization method was introduced for piezoelectric multi-material active structures with the moving morphable components (MMC) method. The proposed method minimizes the total mass of the active structure by simultaneously optimizing the structure topology and the distribution of piezoelectric actuators while satisfying the displacement constraints. To optimize the polarization characteristics for adaptive piezoelectric actuation under complex loading conditions, 3 independent sets of MMCs were utilized. These components describe the distributions of elastic and piezoelectric materials as well as the corresponding polarization characteristics, resulting in a composite active structure with explicit geometric descriptions. Numerical examples demonstrate that, compared to passive structures, multi-material active structures based on piezoelectric actuation can realize structural lightweighting more efficiently.

Interaction Between Flexoelectric Fields Associated With Microholes in Solids

HAO Yihan, TIAN Xinpeng, DENG Qian

2024, 45(11): 1381-1391. doi: 10.21656/1000-0887.450208

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The flexoelectric fields' interactions between microholes of common defects in solid materials are studied. With the collocation mixed finite element method, the distributions of the stress, the strain gradient, and the flexoelectric field around the hole of the single hole and the double holes, respectively, are compared. The numerical simulation results indicate that, the flexoelectric fields' interaction around the double holes emerges with the gradual decrease of the distance between the double holes. In addition, the effects of the distance between holes and the size of holes on the flexoelectric fields' interaction between microholes are explored. The results show that, reducing the distance between double holes and shrinking the size of holes will induce to an enhanced interaction of the flexoelectric field between double holes.

Study on the Electret Film Crumpling Deformation Theory and Flexoelectric-Like Responses

LIU Yaohua, LI Jun, FENG Xiaoman, MA Jianhua, WANG Binglei

2024, 45(11): 1392-1404. doi: 10.21656/1000-0887.450195

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The flexoelectric energy harvesters face such challenges as the monotonous energy harvesting mode, the low electromechanical coupling coefficient, the only prominent effect on microscales, and the limited energy conversion efficiency on macroscales. The electret, as a dielectric material with embedded charges, exhibits significant flexoelectric-like responses induced by non-uniform deformation. The crumpled film in complex bidirectional contraction, provides a novel efficient energy harvesting approach due to high strain gradients on macroscales. Herein the strong macroscopic electromechanical coupling properties of electrets were combined with the advantageous high strain gradients of crumpling, to establish a deformation theory for crumpled flexoelectret films. Based on this model, the flexoelectric-like responses and energy harvesting characteristics of crumpled flexoelectret films were analyzed for different charge densities, supporting cup radii, film thicknesses, and scales. The results indicate that, for a 1 mm thick flexoelectret film, the effective flexoelectric-like intensity is 2 orders higher than that of the intrinsic flexoelectric effect of the pure PDMS film, with the charge density q=-0.2 mC · m^-2.

Bending Wave Analysis of Porous Flexoelectric Metamaterial Plates

ZHAO Haoyang, HE Zhuangzhuang, ZHANG Chunli

2024, 45(11): 1405-1415. doi: 10.21656/1000-0887.450282

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Porous dielectric metamaterials exhibit spatially non-uniform strain distribution due to internal pores with strain gradients particularly pronounced at the pore edges, leading to significant flexoelectric coupling effects. As a result, porous dielectric metamaterials represent a class of smart materials characterized by flexoelectric-type electromechanical coupling, showing great potential for various applications. A mixed finite element method (M-FEM) was employed to investigate the propagation characteristics of bending waves in porous flexoelectric metamaterial plates, aimed at the effects of pore sizes, pore numbers, and gradient distribution parameters of pore diameters within unit cells on the elastic wave bandgap structure. The results reveal that, the flexoelectric coupling effect enhances the effective stiffness of the structure, and thus increases the bending wave bandgap frequency; as the pore size increases, the bending wave bandgap frequency will decrease and the bandgap width will narrow; as the number of pores increases, the bandgap frequency will gradually decrease, and the bandgap will exhibit opening and closing phenomena; for porous dielectric metamaterial plates with gradient distributions of pore sizes, the larger the gradient index is, the wider the bending wave bandgap will be.

A Guided-Wave-Based Method for Detecting Defects in the Adhesive Layer of the Steel-Epoxy-Steel Sandwich Structure

WANG Xishun, GUO Hao, ZHU Qingfeng, DU Yuehao, MIAO Hongchen

2024, 45(11): 1416-1427. doi: 10.21656/1000-0887.450193

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Steel epoxy sleeves are widely used to repair oil and gas pipelines. The integrity of the epoxy layer between the sleeve and the pipeline directly determines the quality of the repair. Due to the unique sandwich structure formed by the sleeve, the epoxy layer, and the pipeline, traditional nondestructive testing methods have difficulty effectively identifying defects in the epoxy layer. Therefore, there is an urgent need to develop new nondestructive testing methods. A guided-wave-based method was developed for detecting defects in the adhesive layer of the steel-epoxy-steel sandwich structure. Firstly, the dispersion curves of guided waves in the sandwich structure were calculated with the semi-analytical finite element method. The LS₁ wave was selected to detect defects in the adhesive layer based on the dispersion characteristics, waveform structures, and attenuation properties. Subsequently, a piezoelectric transducer capable of exciting LS₁ waves was designed. The effectiveness of the transducer was verified through numerical simulations and experiments. Then, numerical simulations and experiments were conducted to study the interaction between LS₁ waves and cavity defects in the adhesive layer. The results show that, with a defect length within 4 times of the wavelength, the amplitude of the LS₁ wave's reflected wave would change approximately linearly with the defect length. Based on this, a signal processing method is proposed, which can effectively identify defect reflection signals when the defect size is not less than twice the wavelength.

Phase Field Simulation of Dielectric Breakdown in Ferroelectric Composites

WANG Zihe, MI Zhang, WANG Jie

2024, 45(11): 1428-1439. doi: 10.21656/1000-0887.450266

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The ferroelectric composite material with ferroelectric polymer as the matrix and ferroelectric ceramic as the filler overcomes the inverted relationship between high polarization strength and high breakdown strength of single-phase ferroelectric materials, exhibits excellent multi-field coupling properties such as piezoelectricity and energy storage. Recently, it draws increasing attention. However, the stress and electric field concentration at the interface of ferroelectric composite materials can cause the electromechanical coupling failure of the materials, and the dielectric breakdown is one of the main failure modes of ferroelectric composite materials. Therefore, understanding the effects of ceramic fillers on the dielectric breakdown performances of ferroelectric composite materials is crucial for their application in high-performance energy conversion and storage devices. Aimed at the multi-field coupling failure of ferroelectric composite materials, a phase field model involving polarization, strain, and breakdown order parameters was established to study the dielectric breakdown behavior of ferroelectric composite materials under electrical loads. The phase field simulation results indicate that, as the particle size of the ceramic filler increases, the electrical breakdown path will avoid the ceramic particles, and the maximum electric field inside the material will gradually increase, resulting in a lower breakdown strength of the composite material. In addition, a nonlinear relationship exists between the dielectric breakdown strength and the particle size of the filler. The work provides a certain theoretical basis for the design of dielectric breakdown strength of ferroelectric composite materials.

A Phase-Field Model for Interfacial Fracture in 2D Quasicrystal Bimaterials

GUAN Gaofei, LI Tong, NIE Xueyang, ZHANG Yingrui, XU Xinsheng, SUN Jiabin, ZHOU Zhenhuan

2024, 45(11): 1440-1454. doi: 10.21656/1000-0887.450203

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A phase-field model for the interfacial fracture of 2D decagonal quasicrystal (QC) bimaterials was proposed to predict the crack propagation path. Firstly, the discrete interface was transformed into a smeared interface through introduction of an interface phased field, and therefore the interface phase field governing equations and corresponding boundary conditions were obtained. The continuous distribution of the interface phased field was obtained with the finite element method, and in turn the singularity of material properties at the interface was eliminated. Subsequently, the governing equations for 2D QC bimaterials were obtained based on the Francfort-Marigo variational principle, and solved with the staggered solution scheme. In numerical examples, the present results were compared with existing references and excellent agreements were observed. In addition, the effects of the phason field on the crack propagation path were investigated, with the evolution of multiple cracks explored.

2024 Vol. 45, No. 11