Hyperelastic Material Research Articles

Experimental investigation of the movement patterns and deformation of the pigs when passing through a pipe elbow and reducer

Experimentally investigate the movement dynamics of the pigs having various geometric shapes through the pipeline elbows and adapters manufactured from various hyperelastic materials, and assess the risks of their getting stuck in such elements. Experimentally determine the required pressure in the behind-pig space for the experimental pig prototypes to pass through the pipeline elbows and adapters.Pig prototypes of various geometric shapes (cup-type, cylindrical two-disc type, multi-disc type, dumbbell disc type and three-ball dumbbell type) were designed in order to carry out the experimental investigation. Based on pigs' 3D models, the 3D models of casting mould have been designed and printed on a 3D printer. Pig prototypes were manufactured by filling the casting moulds with silicone compound with hardness of 30 units by Shore A hardness scale and polyurethane with hardness of 80 units by Shore A hardness scale.An experimental glass pipeline was designed and mounted to monitor the dynamics of the solid-cast pig prototype movement through the pipeline elbows. Video recordings of the process allowed us to identify and describe the patterns of pig prototype deformations in the glass pipeline elbow. Pressure was measured in the behind-pig space during the movement of pig prototypes through straight sections, the elbow and the adapter of the experimental pipeline made of metal. Measurements were taken for dry and wet inner walls of glass and metal pipelines.Cup-type pigs made of silicone compound showed best results in passing through the elbows at the lowest pressure in the behind-pig space (0.33 kgf/cm2). However, suppose the inner wall of the pipeline is dry. In that case, the pig tightness is lost in the pipeline elbow due to significant deformation of the pig, which causes the behind-pig space pressure to increase to 0.71 kgf/cm2 and augments the risk of the pig getting stuck. The dumbbell disc-type and three-ball dumbbell-type pigs made of silicone compound also show good results in passing through the elbows with low pressure in the behind-pig space (up to 0.5 kgf/cm2). Polyurethanepigs are highly rigid; therefore, for them to pass through the pipeline elbows, the pressure in the behind-pig space should be 2-4 times higher than for pigs made of silicone compound.Only the cup-type pig made of silicone compound can pass through the reducing pipe adapters with pressure in the behind-pig space being at least 8 kgf/cm2.The investigation was performed in experimental mode. Further investigation will entail mathematical and numerical modelling of the pig prototypes movement through the pipeline elbows and adapters.The results obtained during the investigation will help to develop a more thorough approach to planning the process of using the pigs to clean the pipelines with elbows and reduce adapters. They allow us to choose the geometric shape and material of the pigs, taking into account the pipeline operating parameters (inlet pressure and flow rate). It is especially appropriate during the first planned pipeline cleaning with pigs. It allows us to minimize the risk of pigs getting stuck in the pipeline.The subject of investigation is patterns of the pig`s friction coefficient, material hardness and geometric shape, impacting its ability to pass through the elbows and reducing adapters, and the value of the required pressure in the behind-pig space.

Ablation catheter–induced mechanical deformation in myocardium: computer modeling and ex vivo experiments

Cardiac catheter ablation requires an adequate contact between myocardium and catheter tip. Our aim was to quantify the relationship between the contact force (CF) and the resulting mechanical deformation induced by the catheter tip using an ex vivo model and computational modeling. The catheter tip was inserted perpendicularly into porcine heart samples. CF values ranged from 10 to 80 g. The computer model was built to simulate the same experimental conditions, and it considered a 3-parameter Mooney-Rivlin model based on hyper-elastic material. We found a strong correlation between the CF and insertion depth (ID) (R2 = 0.96, P < 0.001), from 0.7 ± 0.3 mm at 10 g to 6.9 ± 0.1 mm at 80 g. Since the surface deformation was asymmetrical, two transversal diameters (minor and major) were identified. Both diameters were strongly correlated with CF (R2 ≥ 0.95), from 4.0 ± 0.4 mm at 20 g to 10.3 ± 0.0 mm at 80 g (minor), and from 6.4 ± 0.7 mm at 20 g to 16.7 ± 0.1 mm at 80 g (major). An optimal fit between computer and experimental results was achieved, with a prediction error of 0.74 and 0.86 mm for insertion depth and mean surface diameter, respectively.Graphical

An equivalent modeling method for coupling stiffness of metal braided sleeve based on hyper-elastic material function

Abstract Metal hoses are important connecting components in pipelines, used for compensation in length and vibration reduction in system. Due to the complex structure of the metal braided sleeve on the outer layer of the metal hose and the nonlinear stiffness caused by coupling with the metal corrugated pipe during the combination process, modeling and calculation are difficult. In this paper, an equivalent modeling method for the coupling stiffness of metal braided sleeve based on Hyper-elastic material function is proposed. The parameters of Hyper-elastic material function are obtained through experiments. Experiments and simulations show that this method has a high accuracy, which propose an efficient way to solve the problems in theoretical analysis and simulation modeling of metal hose.

The Effect of Hyperelasticity and Nonlinearity on the Dynamic Behaviors of Hyperelastic Functionally Graded Beams on Nonlinear Elastic Foundation

Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

Inverse Hessian by stochastic projection and application to system identification in nonlinear mechanics of solids

Parameter identification in nonlinear mechanics of solids, a class of non-convex optimization problems of significant interest, often has to balance fast convergence with enhanced exploration. Local optimization schemes based on Newton or quasi-Newton updates cannot tread this fine balance. A similar limitation also applies to many derivative-free global evolutionary searches that typically end up with slower convergence. Notwithstanding a relatively rapid convergence around a local optimum, schemes using classical gradients require that the objective function be smooth. Thus, applied to real-world scenarios, these algorithms face challenges such as non-smoothness including discontinuities in the objective function and costly evaluations of gradients or Hessians, whenever possible. Based on a novel stochastic projection to estimate inverse Hessians, this work aims at generalizing a fast Newton-type local search whilst retaining capabilities of noise-assisted exploration in non-convex optimization. In principle, such a meeting of the best of both worlds in optimization should provide for a powerful paradigm shift, offering a versatile and robust tool for solving complex identification and variational problems arising in nonlinear mechanics of solids. The approach is well-suited even for cases where the objective function is implicitly defined and not amenable to explicit evaluation. In a first implementation of our proposal, we assess its performance against a few benchmark non-convex problems. Finally, we solve a parameter estimation problem for an isotropic and incompressible hyperelastic material.

3D-Printed Ultracompact Multicore Fiber-Tip Probes for Simultaneous Measurement of Nanoforce and Temperature.

Optical fiber force sensing has attracted considerable interest in biological, materials science, micromanipulation, and medical applications owing to its compact and cost-efficient configuration. However, the glass fiber has an intrinsic high Young's modulus, resulting in force sensors being generally less sensitive. While hyperelastic polymer materials can be utilized to enhance the force sensitivity, the thermodynamic properties of the polymer may weaken the sensing accuracy and reliability. Herein, we demonstrate ultracompact three-dimensional (3D)-printed multicore fiber (MCF) tip probes for simultaneous measurement of nanoforce and temperature with high sensitivity. The sensor is highly sensitive to force-induced deformation due to the special geometric features of the polymer microcantilever, and the high-temperature sensitivity can be implemented through the poly(dimethylsiloxane) (PDMS) microcavity on the same fiber facet. Moreover, the sensitivities of the fiber interferometers are remarkably enhanced by introducing the optical analogue of the Vernier effect. Such a device exhibits a force sensitivity of 56.35 nm/μN, which is more than 103 times that of all-silica fiber force sensors. The PDMS microcavity provides a temperature sensitivity of 1.447 nm/°C, measuring the local temperature of the probe and compensating for temperature crosstalk of the force detection. The proposed compact MCF-tip sensor can simultaneously measure nanoforce and temperature with high sensitivity, facilitating multiparameter sensing in a restricted space environment and showing the potential in miniaturized all-fiber multiparameter sensors.

Finite element graft stress for anteromedial portal, transtibial, and hybrid transtibial femoral drillings under anterior translation and medial rotation: an exploratory study

Stress concentration on the Anterior Cruciate Ligament Reconstruction (ACLr) for femoral drillings is crucial to understanding failures. Therefore, we described the graft stress for transtibial (TT), the anteromedial portal (AM), and hybrid transtibial (HTT) techniques during the anterior tibial translation and medial knee rotation in a finite element model. A healthy participant with a non-medical record of Anterior Cruciate Ligament rupture with regular sports practice underwent finite element analysis. We modeled TT, HTT, AM drillings, and the ACLr as hyperelastic isotropic material. The maximum Von Mises principal stresses and distributions were obtained from anterior tibial translation and medial rotation. During the anterior tibia translation, the HTT, TT, and AM drilling were 31.5 MPa, 34.6 Mpa, and 35.0 MPa, respectively. During the medial knee rotation, the AM, TT, and HTT drilling were 17.3 MPa, 20.3 Mpa, and 21.6 MPa, respectively. The stress was concentrated at the lateral aspect of ACLr,near the femoral tunnel for all techniques independent of the knee movement. Meanwhile, the AM tunnel concentrates the stress at the medial aspect of the ACLr body under medial rotation. The HTT better constrains the anterior tibia translation than AM and TT drillings, while AM does for medial knee rotation.

Mechanical behavior of full-thickness burn human skin is rate-independent

Skin tissue is recognized to exhibit rate-dependent mechanical behavior under various loading conditions. Here, we report that the full-thickness burn human skin exhibits rate-independent behavior under uniaxial tensile loading conditions. Mechanical properties, namely, ultimate tensile stress, ultimate tensile strain, and toughness, and parameters of Veronda–Westmann hyperelastic material law were assessed via uniaxial tensile tests. Univariate hypothesis testing yielded no significant difference (p > 0.01) in the distributions of these properties for skin samples loaded at three different rates of 0.3 mm/s, 2 mm/s, and 8 mm/s. Multivariate multiclass classification, employing a logistic regression model, failed to effectively discriminate samples loaded at the aforementioned rates, with a classification accuracy of only 40%. The median values for ultimate tensile stress, ultimate tensile strain, and toughness are computed as 1.73 MPa, 1.69, and 1.38 MPa, respectively. The findings of this study hold considerable significance for the refinement of burn care training protocols and treatment planning, shedding new light on the unique, rate-independent behavior of burn skin.

A new method for determining the ogden parameters of soft materials using indentation experiments

A full understanding of the material properties of skin tissue is crucial for exploring its tribo-mechanical behaviour. It has been widely accepted that the mechanical behaviour of skin tissue for both small and large deformations can be accurately described using a hyperelastic model, such as the one developed by Ogden. However, obtaining these Ogden parameters for in-vivo skin by in-vivo experiments no matter the indentation or suction tests is a significant challenge. The mathematical model used to describe the material behaviour during the test should consider not only the material nonlinearity but also the geometrical confinement of the tissue, the large deformations induced, and the fact that the specimens are relatively thin.A range of contact models is available to describe the contact behaviour during the indentation test. However, none of them can be used for hyperelastic materials with small thickness under large deformations. Simultaneously explaining material nonlinearity and geometric nonlinearity, either through theoretical equations or numerical calculations, poses a significant challenge.In this research, we propose a pragmatic method to obtain Ogden parameters for in-vivo skin tissue by combining experimental indentation results and numerical simulations. The indentation tests were used to obtain the force-indentation depth curves, while the numerical simulations were used to obtain the strain fields. The method assumes the material behaviour of specimens can be linearized in each small deformation increment, and the contact model developed by Hayes can be applied to accommodate each increment. Then, the linear elastic behaviour in each increment can be described by the elastic modulus E which were obtained using Hayes model, and the principal stresses in each increment were subsequently obtained using Hooke's law. By combining all stress fields, overall stress-strain curves can be constructed, from which the hyperelastic Ogden parameters can be obtained. A second numerical simulation of the hyperelastic indentation was then performed using the obtained Ogden parameters, allowing a comparison of the experimental and simulated relationships between force and indentation.

Simulation of Uterus Active Contraction and Fetus Delivery in ls-dyna.

Vaginal childbirth is the final phase of pregnancy when one or more fetuses pass through the birth canal from the uterus, and it is a biomechanical process. The uterine active contraction, causing the pushing force on the fetus, plays a vital role in regulating the fetus delivery process. In this project, the active contraction behaviors of muscle tissue were first modeled and investigated. After that, a finite element method (FEM) model to simulate the uterine cyclic active contraction and delivery of a fetus was developed in ls-dyna. The active contraction was driven through contractile fibers modeled as one-dimensional truss elements, with the Hill material model governing their response. Fibers were assembled in the longitudinal, circumferential, and normal (transverse) directions to correspond to tissue microstructure, and they were divided into seven regions to represent the strong anisotropy of the fiber distribution and activity within the uterus. The passive portion of the uterine tissue was modeled with a Neo Hookean hyperelastic material model. Three active contraction cycles were modeled. The cyclic uterine active contraction behaviors were analyzed. Finally, the fetus delivery through the uterus was simulated. The model of the uterine active contraction presented in this paper modeled the contractile fibers in three-dimensions, considered the anisotropy of the fiber distribution, provided the uterine cyclic active contraction and propagation of the contraction waves, performed a large deformation, and caused the pushing effect on the fetus. This model will be combined with a model of pelvic structures so that a complete system simulating the second stage of the delivery process of a fetus can be established.

Nonlinear dynamics of cantilevered hyperelastic pipes conveying fluid: Comparative study of linearelasticity and hyperelasticity

In light of the softness and lightweight characteristics, hyperelastic pipes are prone to large deformation. Intending to provide accurate predictions of their large deformation vibrations, the present study aims to establish a rotation-angle-based geometrically exact model for cantilevered pipes conveying fluid composed of hyperelastic materials. Additionally, given the presence of nonlinear geometric relations, it is imperative to assess the role of hyperelasticity in the nonlinear dynamics of fluid-conveying pipes in comparison to linearelastic pipes. The modeling process integrates the Mooney-Rivlin hyperelastic model with a geometrically exact description of the pipe. And the final governing equation is developed through energy analysis and the extended Hamilton's principle. Subsequently, Galerkin's method and the finite difference method are employed to address the governing equation. Following the verification of the proposed model, parametric studies are conducted in the form of planar oscillation diagrams, time histories, phase trajectories, and bifurcation diagrams. The findings of the present study highlight three critical insights. Firstly, the hyperelastic model introduces numerous complicated nonlinear terms as function of the rotation angle with high-order nonlinearity. In the present study, at least fifth-order nonlinear terms should be preserved to adequately predict large deformation vibrations. Secondly, the Mooney-Rivlin hyperelastic model imparts a hardening effect in comparison to the linearelastic model, significantly reducing vibration amplitudes. Lastly, the third-order approximate model is demonstrated to be inadequate in predicting large deformation vibrations at high flow velocity, as it contains the approximation on not only the geometric description but the hyperelasticity-related terms.

Stress–displacement stabilized finite element analysis of thin structures using Solid-Shell elements, Part II: Finite strain hyperelasticity

This work is the second of a two-part research project focused on modeling solid-shell elements using a stabilized two-field finite element formulation. The first part introduces a stabilization technique based on the Variational Multiscale framework, which is proven to effectively address numerical locking in infinitesimal strain problems. The primary objective of the study was to characterize the inherent numerical locking effects of solid-shell elements in order to comprehensively understand their triggers and how stabilized mixed formulations can overcome them. In this current phase of the work, the concept is extended to finite strain solid dynamics involving hyperelastic materials. The aim of introducing this method is to obtain a robust stabilized mixed formulation that enhances the accuracy of the stress field. This improved formulation holds great potential for accurately approximating shell structures undergoing finite deformations. To this end, three techniques based in the Variational Multiscale stabilization framework are presented. These stabilized formulations allow circumventing the compatibility restriction of interpolating spaces of the unknowns inherent to mixed formulations, thus allowing any combination of them. The accuracy of the stress field is successfully enhanced while maintaining the accuracy of the displacement field. These improvements are also inherited to the solid-shell elements, providing locking-free approximation of thin structures.

Operator learning for homogenizing hyperelastic materials, without PDE data

In this work, we address operator learning for stochastic homogenization in nonlinear elasticity. A Fourier neural operator is employed to learn the map between the input field describing the material at fine scale and the deformation map. We propose a variationally-consistent loss function that does not involve solution field data. The methodology is tested on materials described either by piecewise constant fields at microscale, or by random fields at mesoscale. High prediction accuracy is obtained for both the solution field and the homogenized response. We show, in particular, that the accuracy achieved with the proposed strategy is comparable to that obtained with the conventional data-driven training method.

Mechanical and morphological characterization of the emphysematous lung tissue

Irreversible alveolar airspace enlargement is the main characteristic of pulmonary emphysema, which has been extensively studied using animal models. While the alterations in lung mechanics associated with these morphological changes have been documented in the literature, the study of the mechanical behavior of parenchymal tissue from emphysematous lungs has been poorly investigated. In this work, we characterize the mechanical and morphological properties of lung tissue in elastase-induced emphysema rat models under varying severity conditions. We analyze the non-linear tissue behavior using suitable hyperelastic constitutive models that enable to compare different non-linear responses in terms of hyperelastic material parameters. We further analyze the effect of the elastase dose on alveolar morphology and tissue material parameters and study their connection with respiratory-system mechanical parameters. Our results show that while the lung mechanical function is not significantly influenced by the elastase treatment, the tissue mechanical behavior and alveolar morphology are markedly affected by it. We further show a strong association between alveolar enlargement and tissue softening, not evidenced by respiratory-system compliance. Our findings highlight the importance of understanding tissue mechanics in emphysematous lungs, as changes in tissue properties could detect the early stages of emphysema remodeling. Statement of significanceGas exchange is vital for life and strongly relies on the mechanical function of the lungs. Pulmonary emphysema is a prevalent respiratory disease where alveolar walls are damaged, causing alveolar enlargement that induces harmful changes in the mechanical response of the lungs. In this work, we study how the mechanical properties of lung tissue change during emphysema. Our results from animal models show that tissue properties are more sensitive to alveolar enlargement due to emphysema than other mechanical properties that describe the function of the whole respiratory system.

Fracture analysis of hyperelastic membrane using bond-associated non-ordinary state-based peridynamics

In this paper, a non-ordinary state-based peridynamics model (NOSB PD) has been proposed to simulate the large deformation and failure analysis of hyperelastic membranes. To account for the curving horizon of the membrane, a non-local membrane theory has been developed by approximating it as a flat surface and applying the plane stress assumption locally. Thus, the membrane structure is simulated using a single layer of material points, which simplifies implementation, improves efficiency, and avoids volume locking. The Gent model has been employed to simulate the hyperelastic material constitutively, as it has an elegant mathematical framework and can achieve the entire range of stretches. Furthermore, Bond-associate NOSB PD is utilized to overcome zero-energy modes without affecting the material properties. A modification has been addressed to overcome the boundary effect by adding a complementary force density to the boundary nodes. The accuracy of large deformation and incompressibility of the developed models are verified by the corresponding finite element analyses. Finally, the proposed method is demonstrated by presenting some benchmark examples, and the results illustrate the accuracy and efficiency of the proposed method for the large deformation, fracture, and off-plane tearing analyses of hyperelastic membranes.

Effect of particle aspect ratio in targeted drug delivery in abdominal aortic aneurysm

Aneurysm is a permanent irreversible bulge in the artery that can occur with higher prevalence among elderly individuals. Although invasive surgical procedures can prevent their development, they come with considerable side effects. Recently, treatments based on targeted drug delivery have gained a lot of attention to suppress aneurysm growth. Numerical simulations have been shown to be of great role in the prediction of blood hemodynamics and vascular wall behaviour in the case of an aneurysm. Moreover, the utilization of high-fidelity approaches such as the Lagrangian frame of reference can address the motion characteristics of microbubble (MB) contrast agents in particulate flows. This study aims to investigate the effect of particle aspect ratio on the adhesion of oblate spheroid particles to the vascular wall. Accordingly, a two-way fluid–structure interaction (FSI) method consisting of a hyperelastic material model for the vessel along with a non-Newtonian, compressible model for blood was employed to simulate an abdominal aortic aneurysm (AAA). Moreover, the ligand–receptor binding concept has been utilized to address the quantification of MBs adhesion. Five sets of aspect ratios ranging from 1 to 9 have been investigated and results indicated that with the increase of the aspect ratio the rate of adhesion decreases. Two drastic changes in the particle number occurred due to the diastolic peak and negative velocity profile, respectively. However, it was concluded that the hydrodynamic of the MBs in terms of velocity and wall distance is rather insensible to the particle shape.

Solitary waves in slightly dispersive quasi-incompressible hyperelastic materials

Based on the classical theory of simple materials of differential type and the results on the analytical form of constitutive models consistent with the laws of thermodynamics, we introduce a very general response function for the Cauchy stress tensor of a dispersive hyperelastic solid. Next, by focusing on the propagation of localised waves in slightly dispersive quasi incompressible solids, we prove the existence of a rich variety of solitary wave solutions as well as kink wave solutions. Our analysis and results can be easily specialised to shape memory alloys.

Impact of Tissue Damage and Hemodynamics on Restenosis Following Percutaneous Transluminal Angioplasty: A Patient-Specific Multiscale Model.

Multiscale agent-based modeling frameworks have recently emerged as promising mechanobiological models to capture the interplay between biomechanical forces, cellular behavior, and molecular pathways underlying restenosis following percutaneous transluminal angioplasty (PTA). However, their applications are mainly limited to idealized scenarios. Herein, a multiscale agent-based modeling framework for investigating restenosis following PTA in a patient-specific superficial femoral artery (SFA) is proposed. The framework replicates the 2-month arterial wall remodeling in response to the PTA-induced injury and altered hemodynamics, by combining three modules: (i) the PTA module, consisting in a finite element structural mechanics simulation of PTA, featuring anisotropic hyperelastic material models coupled with a damage formulation for fibrous soft tissue and the element deletion strategy, providing the arterial wall damage and post-intervention configuration, (ii) the hemodynamics module, quantifying the post-intervention hemodynamics through computational fluid dynamics simulations, and (iii) the tissue remodeling module, based on an agent-based model of cellular dynamics. Two scenarios were explored, considering balloon expansion diameters of 5.2 and 6.2mm. The framework captured PTA-induced arterial tissue lacerations and the post-PTA arterial wall remodeling. This remodeling process involved rapid cellular migration to the PTA-damaged regions, exacerbated cell proliferation and extracellular matrix production, resulting in lumen area reduction up to 1-month follow-up. After this initial reduction, the growth stabilized, due to the resolution of the inflammatory state and changes in hemodynamics. The similarity of the obtained results to clinical observations in treated SFAs suggests the potential of the framework for capturing patient-specific mechanobiological events occurring after PTA intervention.

Experimental Studies on the Dynamics of the Movement of Cleaning Pigs Through Tee Pipe Fittings

Abstract Experimental setups using steel and transparent glass pipes, along with equally spaced steel stamped and transparent plastic tees, were designed for testing. The experimental results provided insights into the influence of flow directions, pig length, material properties, and air consumption on the dynamics and strength of the pigs within the tees. Furthermore, the study identified the causes of temporary halting, jamming, and the mechanisms of damage to cylindrical pigs made of hyperelastic materials within pipeline tees.

Analysis of internal pressure and vibration of a flexible expansion ratio elongation nozzle shape

The nozzle made of flexible materials can expand and elongate its profile at different flight altitudes, greatly improving the engine’s endurance and ballistic accuracy. However, this also introduces the problem of structural vibration in the nozzle, which adversely affects engine performance. This paper introduces a high-precision hyperelastic material constitutive model for the flexible elongation nozzle and uses numerical simulation to conduct a detailed analysis of the internal pressure and structural vibration under aerodynamic pressure loads. As the flight altitude increases, the pressure magnitude and distribution on the inner wall of the same elongated flexible expansion section do not change. The pressure distribution on the inner wall of the flexible elongation expansion section exhibits certain nonlinear characteristics, and its magnitude decreases gradually along the axial position. The vibration of the elongated expansion nozzle profile becomes more intense with the increase in flight altitude. At high characteristic flight altitudes, the elongation length of the longer expansion section approaches an undamped vibration state.