Applied Mathematics and Mechanics was founded in 1980 by CHIEN Wei-zang, a celebrated Chinese scientist in mechanics and mathematics. The current editor in chief is Professor LU Tianjian from Nanjing University of Aeronautics and Astronautics. The Journal was a quarterly in the beginning, a bimonthly the next year, and then a monthly ever since 1985. It carries original research papers on mechanics, mathematical methods in mechanics and interdisciplinary mechanics based on artificial intelligence mathematics. It also strengthens attention to mechanical issues in interdisciplinary fields such as mechanics and information networks, system control, life sciences, ecological sciences, new energy, and new materials, making due contributions to promoting the development of new productive forces.
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2024, Volume 45, Issue 10
publish date：October 01 2024
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2024, 45(10): 1243-1255.
doi: 10.21656/1000-0887.450113
Abstract:
The rapid development of smart wearable devices makes a higher requirement for the power supply components, including endurance, convenience and lightweight and so on. The thermoelectric devices can directly convert the thermal energy released by human metabolism into electricity, which can be further used to continuously power the wearable devices. With the global-local and micro-macro combined analysis method, the thermo-electro-mechanical coupling bending behavior and strength failure of a negative Poisson's ratio thermoelectric device (NPR-TEG) were analyzed. Firstly, the macroscopic bending characteristics and the section with the largest stress were given through the establishment a homogeneous analysis model for the NPR-TEG. Then, the force analysis model for the thermoelectric honeycomb was built. The critical load for the strength failure of a mesoscopic cell wall was also derived with the thermodynamic strength theory. The results show that, the stress level of the thermoelectric honeycomb decreases first and then increases with the re-entrant angle. For the NPR-TEG, the strength failure occurred first in the middle part of the device. For the thermoelectric device with the traditional hexagonal honeycomb, the strength failure occurs at the end of the device rather than the middle part. With the fracture failure occurring in the thermoelectric device, the critical crack length of the middle fracture approximately equals that of the end fracture. The critical crack length could be fitted as an exponential function of the re-entrant angle.
The rapid development of smart wearable devices makes a higher requirement for the power supply components, including endurance, convenience and lightweight and so on. The thermoelectric devices can directly convert the thermal energy released by human metabolism into electricity, which can be further used to continuously power the wearable devices. With the global-local and micro-macro combined analysis method, the thermo-electro-mechanical coupling bending behavior and strength failure of a negative Poisson's ratio thermoelectric device (NPR-TEG) were analyzed. Firstly, the macroscopic bending characteristics and the section with the largest stress were given through the establishment a homogeneous analysis model for the NPR-TEG. Then, the force analysis model for the thermoelectric honeycomb was built. The critical load for the strength failure of a mesoscopic cell wall was also derived with the thermodynamic strength theory. The results show that, the stress level of the thermoelectric honeycomb decreases first and then increases with the re-entrant angle. For the NPR-TEG, the strength failure occurred first in the middle part of the device. For the thermoelectric device with the traditional hexagonal honeycomb, the strength failure occurs at the end of the device rather than the middle part. With the fracture failure occurring in the thermoelectric device, the critical crack length of the middle fracture approximately equals that of the end fracture. The critical crack length could be fitted as an exponential function of the re-entrant angle.
2024, 45(10): 1256-1267.
doi: 10.21656/1000-0887.450244
Abstract:
The singularities of the coupled fields near the apex of a piezoelectric/piezomagnetic composite wedge under antiplane deformation were studied. With the complex variable function theory and the eigenfunction expansion method, the explicit expressions of the eigenfunction equations for singularity orders were derived for 16 combinations of mechanically free or clamped, electrically closed or open and magnetically closed or open ones. Based on the obtained eigenequations, some numerical results were given to show the influences of the wedge angles, boundary conditions and material combination types on the singular behaviors. The results show that, the singularities of the coupled fields in a piezoelectric/piezomagnetic composite wedge are more complicated than those in a piezoelectric counterpart.
The singularities of the coupled fields near the apex of a piezoelectric/piezomagnetic composite wedge under antiplane deformation were studied. With the complex variable function theory and the eigenfunction expansion method, the explicit expressions of the eigenfunction equations for singularity orders were derived for 16 combinations of mechanically free or clamped, electrically closed or open and magnetically closed or open ones. Based on the obtained eigenequations, some numerical results were given to show the influences of the wedge angles, boundary conditions and material combination types on the singular behaviors. The results show that, the singularities of the coupled fields in a piezoelectric/piezomagnetic composite wedge are more complicated than those in a piezoelectric counterpart.
2024, 45(10): 1268-1278.
doi: 10.21656/1000-0887.450190
Abstract:
Based on the couple stress theory, the axisymmetric contact problem between a rigid insulating spherical punch and a transversely isotropic piezoelectric half-space was studied. With the Hankel integral transform and the integral least squares approach, the analytical solutions of the contact pressure were obtained. The effects of the characteristic material length on the contact pressure distribution, the contact radius and the indentation depth were discussed. The results indicate that, the contact pressure obtained based on the couple-stress theory is significantly greater than the classical results.
Based on the couple stress theory, the axisymmetric contact problem between a rigid insulating spherical punch and a transversely isotropic piezoelectric half-space was studied. With the Hankel integral transform and the integral least squares approach, the analytical solutions of the contact pressure were obtained. The effects of the characteristic material length on the contact pressure distribution, the contact radius and the indentation depth were discussed. The results indicate that, the contact pressure obtained based on the couple-stress theory is significantly greater than the classical results.
2024, 45(10): 1279-1287.
doi: 10.21656/1000-0887.450088
Abstract:
The performances of piezoelectric PN junction photovoltaic cells are closely related to the internal potential barrier configurations and the distributions of carriers, and can be tuned through carrier transport characteristic changes by the piezopotentials under the piezo-effect. However, the classical PN junction model fails to describe the coupling effect between multiple physical fields and carriers in the potential barrier zone due to the depletion layer assumptions and others, and in turn gives severely distorted results. Herein a mechanics-electricity-photonics-carrier global multi-field coupling model was developed to investigate the tuning mechanism for mechanical loadings on the output characteristics of ZnO photovoltaic cells. The numerical results indicate that, the short-circuit current, the open-circuit voltage, and the maximum output power of the photovoltaic cell increase with the compressive stress under a fixed light intensity, while tensile stresses are not conducive to improving the performances of photovoltaic cells. In addition, a better tuning effect occurs with a loading region wider than the illuminated region, or with both these two external fields acting in the same side of the n/p-zone.
The performances of piezoelectric PN junction photovoltaic cells are closely related to the internal potential barrier configurations and the distributions of carriers, and can be tuned through carrier transport characteristic changes by the piezopotentials under the piezo-effect. However, the classical PN junction model fails to describe the coupling effect between multiple physical fields and carriers in the potential barrier zone due to the depletion layer assumptions and others, and in turn gives severely distorted results. Herein a mechanics-electricity-photonics-carrier global multi-field coupling model was developed to investigate the tuning mechanism for mechanical loadings on the output characteristics of ZnO photovoltaic cells. The numerical results indicate that, the short-circuit current, the open-circuit voltage, and the maximum output power of the photovoltaic cell increase with the compressive stress under a fixed light intensity, while tensile stresses are not conducive to improving the performances of photovoltaic cells. In addition, a better tuning effect occurs with a loading region wider than the illuminated region, or with both these two external fields acting in the same side of the n/p-zone.
2024, 45(10): 1288-1299.
doi: 10.21656/1000-0887.450070
Abstract:
Piezoelectric semiconductors (PSs) possess the characteristics of coexistence of piezoelectric and semiconductor properties and have broad application prospects in new multifunctional electronic/optoelectronic devices. It is very important to theoretically analyze multi-field coupling mechanical responses of PS structures under external loads. However, the governing equations describing the multi-field coupling mechanical behaviors of PS structures contain physically nonlinear current equations. On the other hand, many semiconductor devices typically operate under large deformation, which raises a geometrically nonlinear problem. The presence of physical and geometric nonlinearity poses challenges to the solution of the problem. Herein, for PS beam structures, a method based on physics informed neural networks (PINNs) was established to efficiently solve their nonlinear multi-field coupling responses. Through successive elimination of carrier-related terms and piezoelectricity-related terms from the constructed PINNs, the proposed method can be reduced to the cases of piezoelectric and pure elastic structures, respectively. With the proposed PINNs, the multi-field coupling responses of a PS beam under static uniform pressure were predicted. Numerical results show that, the proposed method can effectively solve the nonlinear multi-field coupling problems of the PS, piezoelectric and pure elastic structures. Relatively, it exhibits higher accuracy in solving piezoelectric and pure elastic structures.
Piezoelectric semiconductors (PSs) possess the characteristics of coexistence of piezoelectric and semiconductor properties and have broad application prospects in new multifunctional electronic/optoelectronic devices. It is very important to theoretically analyze multi-field coupling mechanical responses of PS structures under external loads. However, the governing equations describing the multi-field coupling mechanical behaviors of PS structures contain physically nonlinear current equations. On the other hand, many semiconductor devices typically operate under large deformation, which raises a geometrically nonlinear problem. The presence of physical and geometric nonlinearity poses challenges to the solution of the problem. Herein, for PS beam structures, a method based on physics informed neural networks (PINNs) was established to efficiently solve their nonlinear multi-field coupling responses. Through successive elimination of carrier-related terms and piezoelectricity-related terms from the constructed PINNs, the proposed method can be reduced to the cases of piezoelectric and pure elastic structures, respectively. With the proposed PINNs, the multi-field coupling responses of a PS beam under static uniform pressure were predicted. Numerical results show that, the proposed method can effectively solve the nonlinear multi-field coupling problems of the PS, piezoelectric and pure elastic structures. Relatively, it exhibits higher accuracy in solving piezoelectric and pure elastic structures.
2024, 45(10): 1300-1312.
doi: 10.21656/1000-0887.450200
Abstract:
The surface elastic and flexoelectric effects significantly influence the mechanical behaviors of nanoscale materials and structures. The static buckling behaviors of piezoelectric semiconductor (PS) beams were studied through the establishment of an Euler-Bernoulli beam theoretical model in view of the Steigmann-Ogden surface elasticity and flexoelectricity. The governing equations and associated boundary conditions were derived under the Hamiltonian variational principle. In combination with the conservation equations for electrostatics and linear drift-diffusion equations, the analytical solutions of the effective elastic constants and critical buckling loads were obtained under both short and open circuit conditions. Numerical calculations were carried out to explore the effective elastic behaviors of the nanoscale PS beam under the effects of flexoelectricity, surface elasticity and shielding of charge carriers. This work provides a valuable guidance for designing high-performance electronic devices with piezoelectric semiconductor beams.
The surface elastic and flexoelectric effects significantly influence the mechanical behaviors of nanoscale materials and structures. The static buckling behaviors of piezoelectric semiconductor (PS) beams were studied through the establishment of an Euler-Bernoulli beam theoretical model in view of the Steigmann-Ogden surface elasticity and flexoelectricity. The governing equations and associated boundary conditions were derived under the Hamiltonian variational principle. In combination with the conservation equations for electrostatics and linear drift-diffusion equations, the analytical solutions of the effective elastic constants and critical buckling loads were obtained under both short and open circuit conditions. Numerical calculations were carried out to explore the effective elastic behaviors of the nanoscale PS beam under the effects of flexoelectricity, surface elasticity and shielding of charge carriers. This work provides a valuable guidance for designing high-performance electronic devices with piezoelectric semiconductor beams.
2024, 45(10): 1313-1319.
doi: 10.21656/1000-0887.450192
Abstract:
Pressure has significant influences on the crystal structures and functional properties of perovskite ferroelectric materials, but relatively minor impact on the phase transition temperature, and can serve as an effective means to enhance the dielectric and ferroelectric properties of these materials. Molecular dynamics simulations were conducted based on the first principles to explore the evolution of ferroelectricity in barium titanate (BTO) single crystals subjected to hydrostatic pressures ranging from the atmospheric pressure to 150 GPa. The findings demonstrate that, a non-monotonic trend of the ferroelectricity of BTO occurs with the increase of the pressure. The ferroelectric first weakens, then intensifies, and finally disappears, with a peak at 42 GPa. This behavior can be attributed to the pressure-induced reduction in atomic spacings. This reduction disrupts the delicate balance between long-range Coulomb forces and short-range electron repulsions. The findings elucidate the ferroelectric behavior of BTO single crystals under ultra-high hydrostatic pressure, providing a theoretical foundation for their future applications to devices and offering valuable theoretical guidance for experimental investigations of BTO ferroelectricity under ultra-high pressures.
Pressure has significant influences on the crystal structures and functional properties of perovskite ferroelectric materials, but relatively minor impact on the phase transition temperature, and can serve as an effective means to enhance the dielectric and ferroelectric properties of these materials. Molecular dynamics simulations were conducted based on the first principles to explore the evolution of ferroelectricity in barium titanate (BTO) single crystals subjected to hydrostatic pressures ranging from the atmospheric pressure to 150 GPa. The findings demonstrate that, a non-monotonic trend of the ferroelectricity of BTO occurs with the increase of the pressure. The ferroelectric first weakens, then intensifies, and finally disappears, with a peak at 42 GPa. This behavior can be attributed to the pressure-induced reduction in atomic spacings. This reduction disrupts the delicate balance between long-range Coulomb forces and short-range electron repulsions. The findings elucidate the ferroelectric behavior of BTO single crystals under ultra-high hydrostatic pressure, providing a theoretical foundation for their future applications to devices and offering valuable theoretical guidance for experimental investigations of BTO ferroelectricity under ultra-high pressures.
2024, 45(10): 1320-1331.
doi: 10.21656/1000-0887.450224
Abstract:
The ferroelectric tunneling junction, with a metal-ferroelectric ultra-thin film-metal structure, has different tunneling resistance states through polarization manipulation, leading to potential applications in next-generation information storage devices with low-power consumption, fast reading/writing speed, high storage density, and non-volatility. However, the ferroelectric thin films still experience high-temperature rises with reduced stability due to high driving fields, and reducing the driving electric field is crucial for designing ferroelectric tunneling devices. The ferroelectric thin films with coexisting domains have lowered barriers and decreased driving electric fields for domain switching, which are achieved through substrate manipulation. Herein the substrate effects on the driving field, the tunneling resistance switching ratio and the tunneling properties, were studied based on the WKB approximation combined with the Landau phenomenological theory. The results show that, the ferroelectric tunnel junction with coexisting domains exhibits 3 resistive states corresponding to out-of-plane and in-plane polarizations. The effective driving electric field can be reduced to 25 MV/m, which is 76% lower than that with 2 resistive single domains. The proposed theoretical framework provides a fundamental understanding of the formation of multi-state and reduction of the driving field for low-energy, multi-resistance ferroelectric storage devices.
The ferroelectric tunneling junction, with a metal-ferroelectric ultra-thin film-metal structure, has different tunneling resistance states through polarization manipulation, leading to potential applications in next-generation information storage devices with low-power consumption, fast reading/writing speed, high storage density, and non-volatility. However, the ferroelectric thin films still experience high-temperature rises with reduced stability due to high driving fields, and reducing the driving electric field is crucial for designing ferroelectric tunneling devices. The ferroelectric thin films with coexisting domains have lowered barriers and decreased driving electric fields for domain switching, which are achieved through substrate manipulation. Herein the substrate effects on the driving field, the tunneling resistance switching ratio and the tunneling properties, were studied based on the WKB approximation combined with the Landau phenomenological theory. The results show that, the ferroelectric tunnel junction with coexisting domains exhibits 3 resistive states corresponding to out-of-plane and in-plane polarizations. The effective driving electric field can be reduced to 25 MV/m, which is 76% lower than that with 2 resistive single domains. The proposed theoretical framework provides a fundamental understanding of the formation of multi-state and reduction of the driving field for low-energy, multi-resistance ferroelectric storage devices.
2024, 45(10): 1332-1344.
doi: 10.21656/1000-0887.450180
Abstract:
Based on the Gurtin-Murdoch surface elasticity theory and the magnetoelectroelasticity (MEE) theory, the fracture behaviors of MEE materials containing nanoscale lip-shaped orifice with 2 asymmetric cracks under anti-plane mechanical loads and in-plane electromagnetic loads were investigated with the analytic function conformal mapping technique. Analytical solutions for the generalized MEE stress fields around defects (the lip-shaped orifice and cracks), as well as the crack tip MEE intensity factors and energy release rates, were given. Under special conditions, the obtained results would degenerate into existing results or offer new insights. Numerical examples reveal that, the defect surface effects on the MEE intensity factors are dependent on the radii of nano-sized circular holes, the size of the lip-shaped orifice, the size of secondary cracks originating from the lip-shaped orifice, and the applied MEE loads. Under the surface effect, the dimensionless energy release rate varies with the lip width, the infinity mechanical load, the infinity electrical load and the infinity magnetic load.
Based on the Gurtin-Murdoch surface elasticity theory and the magnetoelectroelasticity (MEE) theory, the fracture behaviors of MEE materials containing nanoscale lip-shaped orifice with 2 asymmetric cracks under anti-plane mechanical loads and in-plane electromagnetic loads were investigated with the analytic function conformal mapping technique. Analytical solutions for the generalized MEE stress fields around defects (the lip-shaped orifice and cracks), as well as the crack tip MEE intensity factors and energy release rates, were given. Under special conditions, the obtained results would degenerate into existing results or offer new insights. Numerical examples reveal that, the defect surface effects on the MEE intensity factors are dependent on the radii of nano-sized circular holes, the size of the lip-shaped orifice, the size of secondary cracks originating from the lip-shaped orifice, and the applied MEE loads. Under the surface effect, the dimensionless energy release rate varies with the lip width, the infinity mechanical load, the infinity electrical load and the infinity magnetic load.
2024, 45(10): 1345-1358.
doi: 10.21656/1000-0887.450006
Abstract:
A new PD-FEM (peridynamics-finite element method) hybrid model method was proposed for solving thermal-mechanical coupling problems. Its solution region was divided into peridynamics and finite element subregions. The hybrid bonds introduced to mixed these two subregions were composed of finite element nodes and peridynamics material points. The hybrid model was used to simulate damage behavior of alumina ceramic plates under thermal shock loads. Calculation results showed that the cracks initiation and propagation obtained by the hybrid model were in good agreement with experiment results, which validated the accuracy and availability of the hybrid model. The PD-FEM hybrid model inherits the advantage of peridynamics in dealing with discontinuous problems. Because the finite element method is introduced, the model significantly improves the efficiency of studying thermal-mechanical coupling problems using peridynamics method.
A new PD-FEM (peridynamics-finite element method) hybrid model method was proposed for solving thermal-mechanical coupling problems. Its solution region was divided into peridynamics and finite element subregions. The hybrid bonds introduced to mixed these two subregions were composed of finite element nodes and peridynamics material points. The hybrid model was used to simulate damage behavior of alumina ceramic plates under thermal shock loads. Calculation results showed that the cracks initiation and propagation obtained by the hybrid model were in good agreement with experiment results, which validated the accuracy and availability of the hybrid model. The PD-FEM hybrid model inherits the advantage of peridynamics in dealing with discontinuous problems. Because the finite element method is introduced, the model significantly improves the efficiency of studying thermal-mechanical coupling problems using peridynamics method.
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Founded in 1980 ( monthly )
Supervisor：Chongqing Jiaotong University
Founder：CHIEN Weizang
Honorary Chief Editor：ZHONG Wanxie
Editor-in-Chief：LU Tianjian
ISSN 1000-0887
CN 50-1060/O3
