International Journal of Non-Linear Mechanics最新文献_第3页

Energy-constrained Lagrangian neural networks for data-driven modeling of robotic systems: Method and prosthetic applications 数据驱动机器人系统建模的能量约束拉格朗日神经网络：方法和假肢应用

IF 3.2 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-10-10 DOI: 10.1016/j.ijnonlinmec.2025.105283

Qidi Wu , Wen Zhang , Xingbiao Xie , Shu Zhang , Xiaoxu Zhang , Jian Xu

Dynamic models are crucial for model-based control of multi-degree-of-freedom systems, particularly in robotic applications. While traditional methodologies, including Newtonian and Lagrangian mechanics, have been widely employed, they exhibit limitations in accurately capturing unstructured factors such as joint friction and transmission flexibility. Inspired by the physical interpretability inherent in Lagrangian mechanics and the universal approximation capabilities of neural networks, this paper introduces a novel Energy-constrained Lagrangian Neural Network (EnLNN) modeling framework. The proposed EnLNN decomposes the acceleration field into conservative and nonconservative components, with the former represented by a Lagrangian Neural Network (LNN) and the latter by a feedforward neural network (FNN). A distinctive feature of the EnLNN is its incorporation of energy constraints, which allows the conservative component to preserve energy to the greatest extent, thereby mitigating the misallocation of force fields between conservative and nonconservative components. This approach yields a more precise Lagrangian representation than conventional LNNs. The efficacy of the EnLNN modeling approach was evaluated through numerical simulation of a double pendulum system and experimental validation on a lower limb prosthesis. The results substantiate that the EnLNN framework effectively distinguishes conservative and nonconservative components from empirical data while maintaining high modeling accuracy and demonstrating robust extrapolation capabilities.

动态模型对于多自由度系统的基于模型的控制至关重要，特别是在机器人应用中。虽然包括牛顿力学和拉格朗日力学在内的传统方法已被广泛采用，但它们在准确捕捉关节摩擦和传动灵活性等非结构化因素方面存在局限性。受拉格朗日力学固有的物理可解释性和神经网络的通用逼近能力的启发，提出了一种新的能量约束拉格朗日神经网络（EnLNN）建模框架。提出的EnLNN将加速度场分解为保守和非保守分量，其中保守分量由拉格朗日神经网络（LNN）表示，非保守分量由前馈神经网络（FNN）表示。EnLNN的一个显著特点是它结合了能量约束，这使得保守分量能够最大程度地保留能量，从而减轻了保守和非保守分量之间力场的错配。这种方法产生了比传统LNNs更精确的拉格朗日表示。通过双摆系统的数值模拟和下肢假肢的实验验证，对EnLNN建模方法的有效性进行了评估。结果表明，EnLNN框架有效地从经验数据中区分了保守和非保守成分，同时保持了较高的建模精度，并展示了强大的外推能力。

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引用次数: 0

Axial–torsional dynamic model of a drill-string considering thermal-assisted drilling 考虑热辅助钻井的钻柱轴扭动力学模型

IF 3.2 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-10-10 DOI: 10.1016/j.ijnonlinmec.2025.105279

Yuri B.S. Moralles , Daniel A. Castello , Luiz F. Bez , Thiago G. Ritto

Thermal-assisted drilling, a recent advancement in drilling technology, aims to enhance process efficiency but its effects on drill-string dynamics remain largely unexplored. This paper presents an axial–torsional model with nonlinear bit-rock interaction coupling to investigate these effects. The first contribution is the proposition of a regularization method for a widely used nonlinear bit-rock interaction model. To validate the method, model parameters are calibrated against experimental data, demonstrating strong alignment with observed results. The second contribution is the integration of thermal effects into the bit-rock interaction, using thermal load inputs informed by previous experimental and numerical studies. Results show that under the most critical thermal load, the rate of penetration (ROP) increases by approximately 43%, highlighting the potential of thermal-assisted drilling to enhance drilling efficiency. Additionally, axial and torsional forces at the bit, along with the rotational speed of the bit, exhibit minimal variations under thermal conditions. These results indicate that thermal-assisted drilling can substantially improve ROP without compromising dynamic stability, offering new strategies for efficient and effective drilling operations.

热辅助钻井是钻井技术的最新进展，旨在提高工艺效率，但其对钻柱动力学的影响在很大程度上仍未被探索。本文提出了一个具有非线性钻头-岩石相互作用耦合的轴扭模型来研究这些影响。第一个贡献是提出了一种广泛使用的非线性钻头-岩石相互作用模型的正则化方法。为了验证该方法，根据实验数据校准了模型参数，证明与观测结果有很强的一致性。第二个贡献是将热效应整合到钻头-岩石相互作用中，使用了以前的实验和数值研究提供的热负荷输入。结果表明，在最临界热负荷下，钻速（ROP）提高了约43%，凸显了热辅助钻井提高钻井效率的潜力。此外，在热条件下，钻头的轴向力和扭转力以及钻头的转速变化最小。这些结果表明，热辅助钻井可以在不影响动态稳定性的情况下大幅提高机械钻速，为高效钻井作业提供了新的策略。

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引用次数: 0

Physics-informed harmonic balance identification of high-dimensional structures with nonlinear stiffness and damping 具有非线性刚度和阻尼的高维结构的物理谐波平衡辨识

IF 3.2 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-10-10 DOI: 10.1016/j.ijnonlinmec.2025.105282

Rui Jin, Qinghua Liu, Yinhang Ma, Xiangli Shen, Yusheng Wang, Hui Qian, Dong Jiang

Nonlinear mechanical structures are widely used in aerospace applications, such as morphing wings and vibration attenuation systems, where accurate prediction of dynamic characteristics and effective structural health monitoring critically depend on the precise identification of nonlinear parameters. However, parameter identification for high-dimensional structures with nonlinear stiffness and damping faces practical challenges including strong parameter coupling, computational inefficiency, and noise sensitivity, which limit the applicability of existing methods. Traditional techniques such as optimization-based algorithms or purely data-driven approaches often struggle to balance accuracy, efficiency, and interpretability. To address these issues, this study proposes a novel hybrid approach that integrates the Restoring Force Surface (RFS) method with the Harmonic Balance Method (HBM), referred to as RFSHB. By combining the data-driven advantages of the RFS and the frequency-domain algebraic framework of HBM, the method achieves adaptive determination of nonlinear orders and enables efficient estimation of high-dimensional nonlinear stiffness and damping forces. Numerical simulations on a bistable nonlinear energy sink show that RFSHB achieves high accuracy, with errors below 3 % for nonlinear stiffness coefficients and under 10 % for nonlinear damping coefficients at a noise level of 40 dB. Applied to a six-degree-of-freedom nonlinear system with local hardening stiffness, the method reduces computation time by over 70 % compared to optimization algorithms while limiting parameter errors to within 2 %. Even under strong noise interference (20 dB), RFSHB restricts parameter errors to below 30 %, significantly outperforming traditional RFS and optimization-based methods. By incorporating cross-validation and spectral regularization, RFSHB balances physical interpretability with high computational efficiency, offering a robust foundation for inverse design in complex nonlinear systems.

非线性机械结构广泛应用于航空航天领域，如变形机翼和减振系统，其动态特性的准确预测和有效的结构健康监测关键依赖于非线性参数的精确识别。然而，具有非线性刚度和阻尼的高维结构的参数识别面临着参数耦合强、计算效率低和噪声敏感等现实挑战，限制了现有方法的适用性。传统技术，如基于优化的算法或纯数据驱动的方法，往往难以平衡准确性、效率和可解释性。为了解决这些问题，本研究提出了一种新的混合方法，将恢复力面（RFS）方法与谐波平衡方法（HBM）相结合，称为RFSHB。该方法结合了RFS的数据驱动优势和HBM的频域代数框架，实现了非线性阶数的自适应确定，实现了高维非线性刚度和阻尼力的高效估计。在双稳态非线性能量池上的数值模拟表明，在噪声水平为40 dB时，RFSHB的非线性刚度系数误差小于3%，非线性阻尼系数误差小于10%，具有较高的精度。将该方法应用于具有局部硬化刚度的六自由度非线性系统，与优化算法相比，计算时间缩短了70%以上，参数误差控制在2%以内。即使在强噪声干扰（20 dB）下，RFSHB也能将参数误差控制在30%以下，显著优于传统的RFS和基于优化的方法。通过结合交叉验证和谱正则化，RFSHB在物理可解释性和高计算效率之间取得了平衡，为复杂非线性系统的反设计提供了坚实的基础。

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引用次数: 0

A non-linear thermoelastic constitutive model for isotropic materials based on Gibbs free energy 基于吉布斯自由能的各向同性材料非线性热弹性本构模型

IF 3.2 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-10-04 DOI: 10.1016/j.ijnonlinmec.2025.105274

Edgár Bertóti

This paper presents a Gibbs free energy-based constitutive framework for large thermoelastic deformations of isotropic materials. The independent constitutive variables are the Kirchhoff stress tensor and the temperature, while the dependent variables are the spatial logarithmic Hencky strain tensor and the entropy. The approach relies on the multiplicative decomposition of the deformation gradient, which naturally leads to an additive decomposition of the Hencky strain into uncoupled deviatoric, volumetric, and thermal parts. The Gibbs potential is likewise additively decomposed into elastic and thermal parts. The elastic part of the Gibbs free energy per unit intermediate volume can be taken as any of the complementary energy potentials developed for isothermal, non-linear elastic deformations, with the additional assumption that the material coefficients are temperature-dependent. The thermal part of the Gibbs free energy depends on the spherical component of the Kirchhoff stress tensor and the temperature. General forms of the constitutive equations for the Hencky strain and entropy are derived for compressible materials. In the case of incompressible materials, the inherently implicit constitutive model yields explicit relations between the Hencky strain and Cauchy stress components. Special forms of the thermoelastic constitutive equations are derived and investigated for two cases: (i) a modified Hencky-type model suitable for moderately large strains, and (ii) a power-law form of the elastic Gibbs free energy expressed in terms of stress invariants. The predictive capabilities of these models, particularly with respect to the thermoelastic inversion effect and structural heating in rubber-like materials, are evaluated through parameter fitting to experimental data. Comparisons are also made with predictions from a thermoelastic extension of Ogden’s constitutive model.

提出了一种适用于各向同性材料大热弹性变形的吉布斯自由能本构框架。独立本构变量为基尔霍夫应力张量和温度，因变量为空间对数亨奇应变张量和熵。该方法依赖于变形梯度的乘法分解，这自然导致henky应变的加性分解为不耦合的偏差、体积和热部分。吉布斯势也同样相加分解为弹性和热两个部分。单位中间体积的吉布斯自由能的弹性部分可以作为等温非线性弹性变形的任何互补能势，并附加假设材料系数与温度相关。吉布斯自由能的热部分取决于基尔霍夫应力张量的球分量和温度。导出了可压缩材料的henky应变和熵本构方程的一般形式。在不可压缩材料的情况下，固有的隐式本构模型产生了亨基应变和柯西应力分量之间的显式关系。本文推导并研究了两种情况下热弹性本构方程的特殊形式：(i)适用于中等大应变的修正henky型模型，以及（ii）弹性Gibbs自由能的幂律形式，即应力不变量。通过对实验数据的参数拟合来评估这些模型的预测能力，特别是关于橡胶类材料的热弹性反转效应和结构加热。还与奥格登本构模型的热弹性扩展预测进行了比较。

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引用次数: 0

A multiple scales approach to analyze the nonlinear dynamics of the high-temperature superconducting magnetic levitation train 高温超导磁悬浮列车非线性动力学分析的多尺度方法

IF 3.2 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-10-03 DOI: 10.1016/j.ijnonlinmec.2025.105268

Giovanni Migliaccio , Francesco D’Annibale , Haitao Li , Zigang Deng , Francesco dell’Isola , Gino D’Ovidio

This paper investigates the nonlinear dynamics of the High-Temperature Superconducting (HTS) pinning magnetic levitation (MAGLEV) transit system under development at the University of L’Aquila. Due to its inherently weak damping characteristics, the MAGLEV system is particularly susceptible to external disturbances, such as mechanical or magnetic irregularities along the guideway. To analytically characterize its complex nonlinear dynamics, a simplified nonlinear single-degree-of-freedom model is developed, and the Multiple Scales Method (MSM) is employed as a solution technique. This approach enables the evaluation of how key design parameters influence the system’s dynamic response. The analysis highlights the emergence of both primary and secondary resonances, which arise depending on system parameters and the nonlinear nature of the levitation force, potentially impacting not only performance but also stability. Finally, the analytical findings are validated against benchmark solutions obtained through direct numerical integration of the system’s nonlinear equation of motion.

本文研究了拉奎拉大学正在开发的高温超导（HTS）钉钉式磁悬浮（MAGLEV）运输系统的非线性动力学。由于其固有的弱阻尼特性，磁悬浮系统特别容易受到外部干扰，例如沿导轨的机械或磁性不规则。为了解析其复杂的非线性动力学特性，建立了一个简化的非线性单自由度模型，并采用多尺度法（MSM）作为求解技术。这种方法能够评估关键设计参数如何影响系统的动态响应。分析强调了主共振和次共振的出现，这取决于系统参数和悬浮力的非线性性质，不仅可能影响性能，还可能影响稳定性。最后，将分析结果与系统非线性运动方程直接数值积分得到的基准解进行了验证。

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引用次数: 0

Unilateral vibration transmission in mechanical systems with bilinear coupling 双线性耦合机械系统的单边振动传递

IF 3.2 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-10-01 DOI: 10.1016/j.ijnonlinmec.2025.105272

Ali Kogani, Behrooz Yousefzadeh

Unilateral vibration transmission refers to a scenario in which the transmitted response of a system is in purely tensile or compressive oscillations, despite the system being driven by a symmetric harmonic input. This intriguing transmission phenomenon can arise in systems that exhibit different effective elasticity in compression and tension; i.e., bilinear elasticity. We present a computational investigation of unilateral transmission in the steady-state response of harmonically driven mechanical systems with bilinear coupling. Starting with two bilinearly coupled oscillators, we find that breaking the mirror symmetry of the system, in either elastic or inertial properties, facilitates unilateral transmission by allowing it to occur near a primary resonance. This asymmetry also enables nonreciprocal transmission to occur. We then investigate the nonreciprocal dynamics of the system, including linear stability analysis, with a focus on unilateral transmission. We also extend our discussion to a bilinear periodic structure, for which we investigate the influence of the number of units and energy dissipation on unilateral transmission. We report on the existence of stable nonreciprocal unilateral transmission near primary and internal resonances of the system, as well as other nonreciprocal features such as period-doubled and quasiperiodic response characteristics.

单侧振动传递是指尽管系统由对称谐波输入驱动，但系统的传递响应是纯拉伸或压缩振动的情况。这种有趣的传输现象可能出现在压缩和拉伸表现出不同有效弹性的系统中；即双线性弹性。本文对双线性耦合谐波驱动机械系统稳态响应中的单边传动进行了计算研究。从两个双线性耦合振荡器开始，我们发现打破系统的镜像对称性，在弹性或惯性特性中，通过允许它发生在主共振附近，促进单边传输。这种不对称也使非互传发生。然后，我们研究了系统的非互易动力学，包括线性稳定性分析，重点是单边传输。我们还将讨论扩展到双线性周期结构，研究了单元数和能量耗散对单侧传输的影响。我们报道了系统在主共振和内部共振附近存在稳定的非互易单侧传输，以及其他非互易特征，如倍周期和准周期响应特征。

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引用次数: 0

Crashworthiness of composite sandwich structures with tailored Poisson's ratios under bird strike 定制泊松比复合材料夹层结构在鸟击下的耐撞性

IF 3.2 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-10-01 DOI: 10.1016/j.ijnonlinmec.2025.105273

Shaochen Ning, Yadong Zhou

Bird strike has emerged as one of the most critical soft-body impact scenarios encountered by modern aircraft during take-off and landing. Among crashworthiness designs, composite sandwich structures have received considerable attention due to their superior energy absorption capacity and design flexibility. This study explores the combined influence of Poisson's ratios and unit cell alignment directions. Finite Element Analysis (FEA) was used to evaluate the non-linear structure responses of five cell designs with tailored positive (PPR), negative (NPR), and zero (ZPR-1, ZPR-2 and ZPR-3) Poisson's ratios under three orthogonal directions, in terms of energy absorptions, deformation modes, and damage degrees. Among all configurations, ZPR-3 demonstrates the most balanced and robust performance, achieving kinetic energy reductions of 7.43 %, 6.91 %, and 11.63 % greater than the conventional PPR structure in three orthogonal directions, respectively. Moreover, ZPR-3 maintains the lowest core equivalent plastic strain (PEEQ), with peak values as low as 0.26, and consistently exhibits wide but slight plastic zones. Its ability to balance elastic recovery, plastic deformation, and resistance to repeated bird strikes highlights its strong potential for crashworthiness sandwich components in aerospace applications.

鸟撞已经成为现代飞机在起飞和降落过程中遇到的最关键的软体撞击场景之一。在抗碰撞设计中，复合材料夹层结构因其优越的吸能能力和设计灵活性而备受关注。本研究探讨了泊松比和单元胞排列方向的综合影响。采用有限元分析（FEA）对正（PPR）、负（NPR）和零（ZPR-1、ZPR-2和ZPR-3）泊松比设计的5种单元格在3个正交方向上的非线性结构响应进行了评价，包括能量吸收、变形模式和损伤程度。在所有构型中，ZPR-3表现出最平衡和稳健的性能，在三个正交方向上分别比传统的PPR结构减少了7.43%、6.91%和11.63%的动能。ZPR-3保持最低的芯等效塑性应变（PEEQ），峰值低至0.26，且始终呈现较宽但较弱的塑性区。其平衡弹性恢复、塑性变形和抵抗反复鸟击的能力突出了其在航空航天应用中作为耐撞夹层组件的强大潜力。

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引用次数: 0

Tension–compression asymmetry in viscoelastic mechanical behavior and micro–mesoscale coupling mechanisms of composite propellant 复合推进剂粘弹性力学行为的拉压不对称性及微-介尺度耦合机理

IF 3.2 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-09-30 DOI: 10.1016/j.ijnonlinmec.2025.105271

Xu Zhang , Jiangtao Wang , Xiangyang Liu , Yuan Zhang , Ningfei Wang , Xiao Hou

Solid propellants are particle-filled polymeric materials exhibiting nonlinear viscoelastic properties. In this study, viscoelastic compression tests are conducted on solid propellants. The compression nominal stress–strain curves exhibit a J shaped. Relaxation time and viscous stress increase with increasing deformation. The Tension–compression asymmetry in viscoelastic behavior is analyzed. Compared with tension, compression induces higher stress, longer relaxation time, and a larger viscous part. The mechanisms of nonlinear relaxation and Tension–compression asymmetry are analyzed through free volume theory and mesoscale simulations. At the microscopic scale, the limited free volume hinders the rearrangement of molecular networks and chain segments under large deformations. Less free volume and more coiled chain segments under compression lead to higher stress and longer relaxation time. At the mesoscopic scale, the patterns of interface debonding and damage evolution differ under tension and compression. The strength disparity between interfaces and particles leads to distinct tension and compression modulus. The micro–mesoscale coupling mechanisms result in Tension–compression asymmetry in the macroscopic viscoelastic mechanical behavior. The methodology and findings provide insights for multiscale investigations of other particle-filled composites.

固体推进剂是颗粒填充的高分子材料，具有非线性粘弹性特性。本研究对固体推进剂进行了粘弹性压缩试验。压缩标称应力-应变曲线呈J型。松弛时间和粘性应力随变形的增大而增大。分析了粘弹性中的拉压不对称性。与拉伸相比，压缩产生更大的应力，更长的松弛时间和更大的粘性部分。通过自由体积理论和中尺度模拟分析了非线性松弛和拉压不对称的机理。在微观尺度上，有限的自由体积阻碍了大变形下分子网络和链段的重排。压缩下的自由体积越小，螺旋链段越多，应力越大，松弛时间越长。在细观尺度上，拉伸和压缩作用下的界面剥离和损伤演化模式不同。界面和颗粒之间的强度差异导致了不同的拉伸和压缩模量。微观-中观耦合机制导致宏观粘弹性力学行为的拉压不对称。该方法和发现为其他颗粒填充复合材料的多尺度研究提供了见解。

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引用次数: 0

Nonlinear dynamic study of rotary running of casing string in deep horizontal wells 深水平井套管柱旋转下入的非线性动力学研究

IF 3.2 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-09-29 DOI: 10.1016/j.ijnonlinmec.2025.105270

Pan Fang , Kai Chen , Chuan Yang , Yongchun Wu , Caijun Yu , Qing Shi

Field experience in running casing indicates that excessive friction is a primary factor impeding the efficient deployment of casing in horizontal wells. To investigate the dynamic characteristics of casing during rotational deployment, a nonlinear dynamic model of the casing string in deep horizontal wells is developed. The model is formulated based on the finite element method and discretized using twelve-degree-of-freedom Euler–Bernoulli beam elements. To more accurately simulate the contact interaction between the casing and the wellbore, an improved contact model is introduced to characterize the constraint effects imposed by the wellbore during the running process, while accounting for both axial and radial contact behaviors under dynamic conditions. The governing equations are solved using a combination of the Newmark-β numerical integration scheme and the Newton–Raphson iterative method. A parametric study is conducted to analyze the effects of casing rotation speed and running speed on the dynamic response and frictional resistance. The results show that increased rotation and running speeds intensify the contact between the casing and the wellbore. In particular, when the rotation speed and running speed reach 20 r/min and 2.5 m/min, respectively, both the frictional resistance and torque increase significantly along the entire wellbore. Model validation against field data indicates that the simulated location of maximum frictional resistance deviates by only 2.3 % from the actual obstruction depth, confirming the reliability of the proposed model.

套管下入的现场经验表明，过大的摩擦力是阻碍水平井有效下入套管的主要因素。为了研究套管在旋转下入过程中的动态特性，建立了深水平井套管柱的非线性动态模型。该模型基于有限元法建立，采用十二自由度欧拉-伯努利梁单元进行离散。为了更准确地模拟套管与井筒之间的接触相互作用，引入了一种改进的接触模型，以表征井筒在运行过程中施加的约束效应，同时考虑了动态条件下的轴向和径向接触行为。控制方程采用Newmark-β数值积分格式和Newton-Raphson迭代法相结合的方法求解。通过参数化研究，分析了套管转速和下入速度对动力响应和摩阻的影响。结果表明，旋转和下入速度的增加加强了套管与井筒之间的接触。特别是当旋转速度和下入速度分别达到20 r/min和2.5 m/min时，整个井筒的摩阻和扭矩都显著增加。根据现场数据对模型进行验证表明，模拟的最大摩擦阻力位置与实际障碍物深度的偏差仅为2.3%，证实了所提出模型的可靠性。

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引用次数: 0

Exact solutions and wave interactions for one-dimensional two-phase thin film model of a perfectly soluble antisurfactant 完全可溶抗表面活性剂一维两相薄膜模型的精确解和波相互作用

IF 3.2 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-09-22 DOI: 10.1016/j.ijnonlinmec.2025.105262

Hari Om Jangid , Subhankar Sil , T. Raja Sekhar

In this article, we obtain some exact solutions to a new hyperbolic system of quasilinear partial differential equations which describes the two-phase thin film model of a perfectly soluble antisurfactant by using symmetry analysis. Lie’s method provides a group of transformations for which the set of solutions remains invariant and through the help of push-forward actions, optimal classes are constructed. The aid of optimal classes facilitates exact solutions of the system. Additionally, we compute some traveling wave solutions of the governing system with the help of special transformations. For each phase, the evolution of the film thickness and concentration gradient is characterized by geometric representation of the solutions. The weak discontinuity behavior across a solution curve is analyzed as time progresses. In addition, the propagation of characteristic shock and the corresponding collision between the characteristic shock and the weak discontinuity are discussed. The reflected and transmitted wave amplitudes, along with the jump in shock acceleration influenced by the incident wave after interaction, are computed.

本文用对称分析的方法，得到了描述完全可溶抗表面活性剂两相薄膜模型的一类新的拟线性偏微分双曲方程组的精确解。李氏方法提供了一组变换，其解集保持不变，并通过前推动作的帮助，构造了最优类。最优类的帮助有助于系统的精确解。此外，我们还利用特殊变换计算了控制系统的一些行波解。对于每个相，膜厚度和浓度梯度的演变是由溶液的几何表示来表征的。随着时间的推移，分析了解曲线上的弱不连续行为。此外，还讨论了特征激波的传播以及相应的特征激波与弱不连续面的碰撞。计算了反射波和透射波振幅，以及入射波作用后冲击加速度的跃变。

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引用次数: 0