A modeling method based on coupled equations of damaged hyperelastic materials
By constructing a set of coupled equations based on damaged hyperelastic materials, the problem of inaccurate ACL damage assessment in existing technologies has been solved, enabling accurate prediction and quantification of anterior cruciate ligament damage, and improving the objectivity and reliability of detection.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- XIAN UNIV OF TECH
- Filing Date
- 2026-03-26
- Publication Date
- 2026-07-14
AI Technical Summary
Existing technologies are insufficient to accurately characterize the occurrence and development of ACL damage under physiological loads, resulting in low accuracy in ACL damage assessment.
A coupled equation modeling method based on damaged hyperelastic materials is adopted. By constructing a highly nonlinear solitary wave propagation model and a Neo-Hookean constitutive model, and combining scalar damage variables and uniaxial tension boundary conditions, the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury is established. Then, by using the Helmholtz free energy function and Cauchy stress transformation, the stress-strain relationship is constructed, the effective elastic modulus is obtained, and a coupled equation system is formed.
It improves the prediction accuracy of anterior cruciate ligament injury, simplifies the model structure, avoids interference from complex physiological structures on the detection results, quantifies the mechanical properties of materials, and enhances the objectivity and repeatability of injury characteristic parameters.
Smart Images
Figure CN122392725A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of nondestructive testing, and in particular to a modeling method based on a set of coupled equations for damaged hyperelastic materials. Background Technology
[0002] The knee joint can withstand body weight and the impact of exercise, and enables normal walking, running, and jumping through flexion and extension movements. The complex structure of the knee joint makes it more susceptible to injury and disease. The anterior cruciate ligament (ACL) in the knee joint maintains knee stability by controlling the anterior displacement and rotation of the tibia. Damage to the ACL can lead to severe knee instability, subsequently causing pathological changes such as knee effusion, meniscus tears, and articular cartilage wear. Therefore, developing rapid and accurate diagnostic methods for ACL injuries is crucial for timely intervention and treatment, preventing secondary knee injuries, and improving patient prognosis.
[0003] Existing technologies focus on the mechanical damage evolution of ACL materials, establishing material fatigue damage models based on cyclic loading conditions, characterizing damage parameters using the elastic modulus method, and assessing the degree of ACL damage by quantifying the decay of the elastic modulus. However, the single mechanical model in existing technologies is insufficient to accurately characterize the damage occurrence and development process of ACL under physiological loads, resulting in low accuracy in damage assessment. Summary of the Invention
[0004] Therefore, it is necessary to provide a modeling method based on the coupled equations of damaged hyperelastic materials to address the above-mentioned technical problems.
[0005] The present invention adopts the following technical solution: This invention provides a modeling method based on coupled equations for damaged hyperelastic materials, including: Based on Hertz's contact law governing the generation and propagation of highly nonlinear isolated waves in a one-dimensional particle chain, a formula corresponding to the propagation model of highly nonlinear isolated waves is constructed. The anterior cruciate ligament is simplified as a hyperelastic material. Based on the assumption of incompressible isotropic hyperelasticity, combined with the Neo-Hookean constitutive model, scalar damage variables, and uniaxial tension boundary conditions, the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury is constructed through the Helmholtz free energy function and Cauchy stress transformation. The effective elastic modulus of the damaged hyperelastic material is obtained by taking the first derivative of the strain in the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury. Substituting the effective elastic modulus of the damaged hyperelastic material into the relationship between the force and the material compression yields the force expression. Substituting the force expression into the formula corresponding to the highly nonlinear solitary wave propagation model, we obtain a set of coupled equations based on highly nonlinear solitary waves and damaged hyperelastic materials.
[0006] Alternatively, the formula corresponding to the highly nonlinear solitary wave propagation model is: ; in, The equation of motion for the first particle is... For the first i The differential equation of motion for each particle For the first N The differential equation of motion for each particle Let be the stiffness constant of the particle. For the mass of the particles, This represents the number of particles in the particle chain. This represents taking the maximum value. For the first particle in t The displacement from its equilibrium position at any given moment. For the first i -1 particle in t The displacement from its equilibrium position at any given moment. For the first i Each particle t The displacement from its equilibrium position at any given moment. For the first i+ 1 particle in t The displacement from its equilibrium position at any given moment. For the first N -1 particle in t The displacement from its equilibrium position at any given moment. No. N Each particle t The displacement from its equilibrium position at any given moment. , This represents the Hertzian contact force between the particle and the material.
[0007] Optionally, the stress-strain relationship is: ; in, For stress, For ligament injury variables, In response to the situation; The effective elastic modulus of the damaged hyperelastic material is: ; in, The effective elastic modulus of the damaged hyperelastic material.
[0008] Alternatively, the expression for the force is: ; in, For the action force, Where is the particle radius, The Poisson's ratio of the particles. Let be the elastic modulus of the particle. For hyperelastic materials, Poisson's ratio To reduce the effective elastic modulus of the damaged hyperelastic material, This represents the amount of material compression.
[0009] Optionally, the coupling equations based on highly nonlinear solitary waves and damaged hyperelastic materials are as follows: ; in, The equation of motion for the first particle is... For the first i The differential equation of motion for each particle For the first N The differential equation of motion for each particle Let be the stiffness constant of the particle. For the mass of the particles, This represents the number of particles in the particle chain. This represents taking the maximum value. For the first particle in t The displacement from its equilibrium position at any given moment. For the first i -1 particle in t The displacement from its equilibrium position at any given moment. For the first i Each particle t The displacement from its equilibrium position at any given moment. For the first i+ 1 particle in t The displacement from its equilibrium position at any given moment. For the first N -1 particle in t The displacement from its equilibrium position at any given moment. No. N Each particle t The displacement from its equilibrium position at any given moment. , The effective elastic modulus of the damaged hyperelastic material.
[0010] Optionally, the method further includes: Transformation of the coupled equations based on highly nonlinear isolated waves and damaged hyperelastic materials; Initial conditions were set, and the transformed equations were numerically solved using the fourth-order Runge-Kutta method. The results showed a correspondence between the amplitude attenuation, time delay, and disappearance threshold of the time delay of the secondary and tertiary rebound waves, the degree of anterior cruciate ligament injury, and the effective elastic modulus.
[0011] Optionally, the coupling equations between the transformed highly nonlinear solitary wave and the hyperelastic material are as follows: ; in, The Poisson's ratio of the first particle. For the first i Poisson's ratio of each particle For the first N Poisson's ratio of each particle For the first particle in t The displacement from its equilibrium position at any given moment. For the second particle in t The displacement from its equilibrium position at any given moment. For the first i -1 particle in t The displacement from its equilibrium position at any given moment. For the first i Each particle t The displacement from its equilibrium position at any given moment. For the first i+ 1 particle in t The displacement from its equilibrium position at any given moment. For the first N -1 particle in t The displacement from its equilibrium position at any given moment. No. N Each particle t The displacement from its equilibrium position at any given moment. Where is the particle radius, For the mass of the particles, To reduce the effective elastic modulus of the damaged hyperelastic material, This refers to the amount of material compression. The Poisson's ratio of the particles. Poisson's ratio for hyperelastic materials.
[0012] Optionally, the initial conditions are: and .
[0013] This invention provides a modeling apparatus based on a set of coupled equations for damaged hyperelastic materials, comprising: The first building module is used to construct the formula corresponding to the highly nonlinear isolated wave propagation model based on Hertz's contact law, which governs the generation and propagation of highly nonlinear isolated waves in a one-dimensional particle chain. The second building module is used to simplify the anterior cruciate ligament as a hyperelastic material. Based on the assumption of incompressible isotropic hyperelasticity, combined with the Neo-Hookean constitutive model, scalar damage variables, and uniaxial tension boundary conditions, the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury is constructed through the Helmholtz free energy function and Cauchy stress transformation. The first determining module is used to calculate the first derivative of the strain in the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury, and to obtain the effective elastic modulus of the damaged hyperelastic material. The second determining module is used to substitute the effective elastic modulus of the damaged hyperelastic material into the relationship between the force and the material compression to obtain the force expression, and then substitute the force expression into the formula corresponding to the highly nonlinear solitary wave propagation model to obtain a set of coupled equations based on highly nonlinear solitary waves and damaged hyperelastic materials.
[0014] The present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the above-described method for predicting anterior cruciate ligament injury status based on nonlinear solitary waves.
[0015] The present invention provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, it implements the above-mentioned method for predicting the anterior cruciate ligament injury status based on nonlinear solitary waves.
[0016] The above-mentioned at least one technical solution adopted in this invention can achieve the following beneficial effects: This invention simplifies the anterior cruciate ligament (ACL) as a hyperelastic material. When the stress in the ACL exceeds a preset stress threshold, damage to the ACL is determined. The stress-strain relationship of the hyperelastic material after ACL damage is obtained, and the first derivative of the strain in the stress-strain relationship is calculated to obtain the effective elastic modulus of the hyperelastic material. The effective elastic modulus is substituted into the relationship between the force and the material compression to obtain the force. The force is then substituted into the formula corresponding to the highly nonlinear solitary wave propagation model to obtain a set of coupled equations based on highly nonlinear solitary waves and damaged hyperelastic materials. This invention first simplifies the anterior cruciate ligament (ACL) as a hyperelastic material and uses a stress threshold to determine injury occurrence, aligning with the fundamental principles of biomaterial mechanics. The simplified model retains the core mechanical properties of the ligament material while avoiding interference from complex physiological structures in the detection results. Second, by differentiating the stress-strain relationship after injury, the effective elastic modulus is obtained, quantifying the material's mechanical properties mathematically. This eliminates the influence of subjective experience on parameter acquisition in traditional detection methods, making the extraction of injury characteristic parameters objective and repeatable. Finally, the construction of a highly nonlinear solitary wave coupled with the hyperelastic material injury equations establishes a precise mapping relationship between the granular chain wave signal and the ligament injury state. This method improves the prediction accuracy of ACL injuries. Attached Figure Description
[0017] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings:
[0018] Figure 1 A schematic diagram of a modeling method based on coupled equations of damaged hyperelastic materials provided by the present invention; Figure 2 The time-velocity curve provided for this invention; Figure 3 The displacement curves of each particle provided for this invention are shown in the following figures: (a) is the time velocity diagram of the thirteenth particle, (b) is the time displacement diagram of each particle, (c) is the time velocity diagram of each particle, and (d) is the force diagram of the thirteenth particle. Figure 4 The following are displacement curves of each particle when the elastic modulus is 150, provided by the present invention; wherein, (a) is the velocity diagram of the thirteenth particle, (b) is the displacement diagram of each particle, (c) is the velocity diagram of each particle, and (d) is the force diagram of the thirteenth particle. Figure 5 The following are displacement curves of each particle when the elastic modulus is 250, provided by the present invention; wherein, (a) is the velocity diagram of the thirteenth particle, (b) is the displacement diagram of each particle, (c) is the velocity diagram of each particle, and (d) is the force diagram of the thirteenth particle. Figure 6 The velocity diagram of the thirteenth particle provided by the present invention, wherein (a) is a velocity diagram of v s The velocity diagram of the thirteenth particle when v = 0.2, (b) shows the velocity of v. s Velocity diagram of the thirteenth particle at 0.4; Figure 7 The velocity curves and internal force diagrams provided for this invention are as follows: (a) is the velocity diagram of the thirteenth particle when the effective elastic modulus is 4.875; (b) is the displacement diagram of each particle when the effective elastic modulus is 4.875; (c) is the velocity diagram of each particle when the effective elastic modulus is 4.875; and (d) is the force diagram of the thirteenth particle when the effective elastic modulus is 4.875. Figure 8 The velocity curves and internal force diagrams provided for this invention include, in which (a) is the velocity diagram of the thirteenth particle when the effective elastic modulus is 4, (b) is the displacement diagram of each particle when the effective elastic modulus is 4, (c) is the velocity diagram of each particle when the effective elastic modulus is 4, and (d) is the force diagram of the thirteenth particle when the effective elastic modulus is 4. Figure 9 The velocity diagram of the 13th particle provided by this invention. Figure 9 Figure (a) shows the elastic modulus. Es The velocity diagram of the thirteenth particle at 150 GPa is shown. In Figure (9), (b) represents the elastic modulus. Es The velocity diagram of the thirteenth particle at 250 GPa; Figure 10 The force diagram of the thirteenth particle under various elastic moduli provided by this invention; Figure 11 The elastic modulus provided by the present invention Es The curve is shown when =11.55, where (a) is the elastic modulus. Es The velocity diagram of the thirteenth particle at 11.55, and (b) the elastic modulus. Es Particle displacement diagram when =11.55, (c) diagram shows the elastic modulus. Es Particle velocity diagram at 11.55; Figure 12 The elastic modulus provided by the present invention Es The curves are shown when the elastic modulus is 11.5, where (a) is the elastic modulus. The velocity diagram of the thirteenth particle at 39.24, (b) shows the elastic modulus. Particle displacement diagram at 39.24, (c) shows the elastic modulus. Particle velocities at 39.24; Figure 13 The elastic modulus provided by the present invention The curves at =6.5, where (a) shows the elastic modulus. The velocity diagram of the thirteenth particle at =6.5, (b) shows the elastic modulus. Particle displacement diagram at =6.5, (c) diagram shows the elastic modulus. Particle velocity diagram at 6.5; Figure 14 The elastic modulus provided by the present invention The curve is shown when =22.425, where (a) is the elastic modulus. The velocity diagram of the thirteenth particle at =22.425, (b) shows the elastic modulus. Particle displacement diagram at =22.425, (c) diagram shows the elastic modulus. Particle velocity diagram at 22.425; Figure 15 The elastic modulus provided by the present invention The curve when =1.625, where (a) is the elastic modulus. The velocity diagram of the thirteenth particle when the elastic modulus is 1.625, and Figure (b) shows the elastic modulus. Particle displacement diagram when =1.625, (c) diagram shows the elastic modulus. Particle velocities at a value of 1.625; Figure 16 The elastic modulus provided by the present invention The curve when = 5.606, where (a) is the elastic modulus. The velocity diagram of the thirteenth particle at 5.606, and Figure (b) shows the elastic modulus. Particle displacement diagram when = 5.606, (c) diagram shows the elastic modulus. Particle velocities at 5.606; Figure 17 A schematic diagram of a modeling device based on a set of coupled equations for damaged hyperelastic materials provided by the present invention. Detailed Implementation
[0019] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention.
[0020] Devices such as desktop computers, servers, and laptops are capable of executing the present invention. For ease of explanation, the following description will focus on servers as the executing entity.
[0021] The knee joint is the largest synovial joint in the human body, composed of the distal femur, proximal tibia, and patella, forming a closed space joint with a joint capsule and ligaments. The knee joint contains cartilage and synovial fluid, which cushion the impact of movement and reduce injury. The anterior cruciate ligament (ACL) is one of the most important and vulnerable ligaments in the knee joint, playing a crucial role in stabilizing the joint and transmitting loads. Injuries to the ACL increase the risk of long-term degenerative changes in the knee joint, severely impacting patients' work and quality of life. Diagnosing an ACL injury is also a complex procedure.
[0022] One-dimensional particle chains are typical nonlinear dynamic systems. Experimental results show that the generation and propagation of highly nonlinear solitary waves in one-dimensional particle chains follow Hertz's contact law, exhibiting significant robustness and support properties that allow solitary waves to propagate long distances while maintaining almost unchanged velocity and shape. The fundamental properties of solitary waves propagating from particle chains are highly adjustable. For example, the displacement between any two contacting particles, the particle velocity and acceleration, as well as the amplitude, propagation speed, and total energy of the solitary wave, can all be adjusted by changing the particle diameter, density, elastic modulus, and particle arrangement. Furthermore, the waveform and velocity of the solitary wave can be finely adjusted by regulating the pre-compression force. Job et al., by embedding piezoelectric materials between particles, measured the propagation process of solitary waves in one-dimensional particle chains and obtained the changes in inter-particle internal forces. They found that the reflection of solitary waves strongly depends on the mechanical properties, geometry, and boundary conditions of the contacting objects, providing theoretical guidance for the application of solitary waves in the field of nondestructive testing. In addition, there has been considerable research on granular chains in the biomedical field, such as measuring intraocular pressure, detecting the elastic modulus of the femoral head and the trabeculae between the femur, and identifying defects in porous solids near the proximal femoral head.
[0023] Within the field of nondestructive testing (NDT), solitary waves demonstrate immense application potential due to their superior information transmission efficiency. Accurate NDT and evaluation can be achieved by analyzing the reflection characteristics of solitary waves on the surface of the structure under test. Currently, NDT based on highly nonlinear solitary waves has been applied to some materials, but research related to biomaterials is limited. This invention uses theoretical calculations and numerical simulations to study the coupling effect between solitary waves in a particle chain and a damaged anterior cruciate ligament (ACL). By selecting ligaments with different degrees of damage and varying the material parameters of the incident particles, this invention investigates the influence of different parameters of the damaged material on the propagation delay and amplitude of the reflected solitary waves, achieving NDT for the ACL through comparative analysis.
[0024] The technical solutions provided by the various embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
[0025] Figure 1This is a schematic diagram of a modeling method for a coupled equation system based on damaged hyperelastic materials in this invention, which specifically includes the following steps: S101: Based on Hertz's contact law governing the generation and propagation of highly nonlinear isolated waves in a one-dimensional particle chain, construct the formula corresponding to the propagation model of highly nonlinear isolated waves.
[0026] The construction process of a highly nonlinear solitary wave propagation model specifically includes: constructing the motion differential equation of the first particle; constructing the second... The differential equations of motion for each particle; where... , N The number of particles in the particle chain; construct the first N The differential equation of motion for the first particle; based on the differential equation of motion for the second particle... The differential equation of motion for the nth particle, the nth N By applying the differential equations of motion of individual particles and Newton's second law, a highly nonlinear solitary wave propagation model is obtained.
[0027] Specifically, the differential equation of motion for the first particle is constructed; the differential equation of motion for the second particle is constructed. The differential equations of motion for each particle; where... , N The number of particles in the particle chain; construct the first N The differential equation of motion for the first particle; based on the differential equation of motion for the second particle... The differential equation of motion for the nth particle, the nth N By applying the differential equations of motion of individual particles and Newton's second law, a highly nonlinear solitary wave propagation model is obtained.
[0028] Assume a one-dimensional particle chain consists of Composed of individual particles, the granular crystal is modeled using a discrete element method. Each particle is arranged in close contact without pre-compression, and the particle chain satisfies Hertz's contact law. The direction of motion is constrained along... A discrete element model of the particle chain is obtained by applying fixed constraints to the other directions of motion, with uniaxial motion. Figure 2 A schematic diagram of the propagation motion equation of a one-dimensional particle chain provided by the present invention is shown below. Figure 2 As shown, under the impact of the first particle, due to the collision, kinetic energy is excited between each particle along the path. x A highly nonlinear solitary wave propagating in a specific direction. An inter-particle model is established based on the discrete element method, and a set of differential equations of motion for the particle chain is constructed.
[0029] According to Hertzian contact theory, two elastic spheres in contact, when subjected to a small load, can... Under the influence of the force, the contact area changes from point contact to surface contact. and compression The relationship between them is given by formula (1): (1); in, F The force between the two particles. For compression amount, Let be the radius of the first elastic sphere. Let the radius of the second elastic sphere be . Let be the elastic modulus of the first elastic sphere. The elastic modulus of the second elastic sphere. Let Poisson's ratio be the ratio of the first elastic sphere. Both are Poisson's ratios for elastic spheres, which are essentially particles.
[0030] Assuming there is no pre-compression between the particles, the differential equation of motion for the first particle can be expressed as formula (2): (2); in, The equation of motion for the first particle is... Let be the stiffness constant of the particle. This represents taking the maximum value. For the first particle in t The displacement from its equilibrium position at any given moment. For the second particle in t The displacement from its equilibrium position at any given moment. This represents taking the maximum value.
[0031] Then the first i The differential equation of motion for a particle can be expressed as formula (3): (3); in, For the first i The differential equation of motion for each particle Let be the stiffness constant of the particle. For the first i -1 particle in t The displacement from its equilibrium position at any given moment. For the first i Each particle t The displacement from its equilibrium position at any given moment. .
[0032] The last particle is simultaneously subjected to the forces of the previous particle and the medium it is in contact with, so the differential equation of motion for the last particle is expressed as formula (4): (4); in, For the first NThe differential equation of motion for each particle For particle mass, This represents the number of particles in the particle chain. This represents the Hertzian contact force between the particle and the material.
[0033] According to formulas (1)-(4) and Newton's second law, the set of differential equations of motion for the particle chain is given by formula (5): (5); in, The equation of motion for the first particle is... For the first i The differential equation of motion for each particle For the first N The differential equation of motion for each particle Let be the stiffness constant of the particle. For the mass of the particles, This represents the number of particles in the particle chain. The Hertzian contact force between the particles and the material. For the first particle in t The displacement from its equilibrium position at any given moment. For the first i -1 particle in t The displacement from its equilibrium position at any given moment. For the first i Each particle t The displacement from its equilibrium position at any given moment. For the first i+ 1 particle in t The displacement from its equilibrium position at any given moment. For the first N -1 particle in t The displacement from its equilibrium position at any given moment. No. N Each particle t The displacement from its equilibrium position at any given moment. , N This represents the number of particles in the particle chain. This represents taking the maximum value.
[0034] S102: The anterior cruciate ligament is simplified as a hyperelastic material. Based on the assumption of incompressible isotropic hyperelasticity, combined with the Neo-Hookean constitutive model, scalar damage variables, and uniaxial tension boundary conditions, the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury is constructed through the Helmholtz free energy function and Cauchy stress transformation.
[0035] The anterior cruciate ligament is simplified as an isotropic hyperelastic material, assuming a stress threshold beyond which damage occurs. Damage is modeled using a scalar variable, therefore the damage range of the material behavior is also isotropic.
[0036] Currently, in finite element simulations of the knee joint, many researchers simplify the properties of ligaments to those of linear elastic materials, rarely considering the hyperelastic or viscoelastic material properties of ligament tissue. Hyperelastic materials possess good elasticity and resilience, and their properties differ from those of linear elastic materials; they exhibit nonlinear changes. Setting the material properties of ligaments as hyperelastic materials more closely approximates the actual mechanical properties of knee joint ligaments. In such materials, strain energy density is a function of strain, and stress can be obtained by differentiating the strain energy density with respect to strain. Commonly used constitutive models for hyperelastic materials using strain energy density functions include the Neo-Hookean model, polynomial model, Mooney-Rivlin model, and Yeoh model. The Mooney-Rivlin and Neo-Hookean forms of energy density function constitutive models are frequently used by researchers for hyperelastic materials. When studying the biomechanical properties of the human knee joint, researchers mostly adopt the Neo-Hookean constitutive model.
[0037] The Neo–Hookean model is defined by formula (6): (6); in, As the first strain invariant, The parameters of the Neo–Hookean model are related to the shear modulus. Related, that is .
[0038] Principal stresses were established based on the Neo–Hookean model. ,strain With parameters The relationship between them. The energy density function of hyperelastic materials. It is about strain invariants The function can be defined by formula (7):
[0039] (7); in, As the second strain invariant, As the third strain invariant, For the energy density function of the hyperelastic material.
[0040] The first strain invariant, the second strain invariant, the third strain invariant, and the elongation ratio in the principal tensile direction are given by formula (8): (8); in, For the first i Elongation ratio in the principal tensile direction, For the first i Strain in the principal tensile direction.
[0041] As can be seen from the stress-strain form defined by Piola-Kirchhoff and Cauchy-Green, the strain energy density function... For the main draw ratio Taking the partial derivative, we obtain the stress-strain relationship of the hyperelastic material, also known as the Piola-Kirchhoff stress and Cauchy-Green strain, which is in the form of formula (9): (9); in, For Piola-Kirchhoff stress, For Cauchy-Green strain.
[0042] From formulas (7) to (9), the principal stresses under uniaxial tension of hyperelastic materials can be obtained. The expression is formula (10): (10); in, Principal stresses in uniaxial tension of hyperelastic materials.
[0043] Based on the characteristics of the uniaxial tensile test, its tensile ratio is... , and The relationship is given by formula (11): (11); in, It is the tensile ratio in the tensile direction of the test, representing the numerical relationship between uniaxial elongation and uniaxial strain. , It is the elongation ratio in the single-axis direction. It is uniaxial engineering strain, when the material is not deformed. , .
[0044] From formulas (8) and (11), the first strain invariant, the second strain invariant, and the third strain invariant in the uniaxial tensile test can be obtained as formula (12): (12); From formulas (10), (11), and (12), the principal stresses can be obtained. For formula (13): (13); Will Substituting into formula (13), we can obtain the principal stresses. ,strain With parameters The relationship between them is given by formula (14): (14); This invention takes parameters The value of 1.95 is used because it is a parameter calculated based on the anterior cruciate ligament of healthy adult males and is generally representative.
[0045] Will Substituting 1.95 into formula (14), we obtain the stress-strain relationship as follows: .
[0046] strain The elastic modulus can be obtained by taking the first derivative. The elastic modulus of the hyperelastic material is a function of strain. initial elastic modulus The value is 11.7 MPa.
[0047] The derivation of the stress-strain relationship of hyperelastic materials after anterior cruciate ligament injury includes: 1. Assume that the general form of the Helmholtz free-energy function exists as formula (15): (15); in, Let be the deformation gradient tensor. Since the elastic range is isotropic, equation (15) can be simplified to equation (16): (16); in, For the right stretch tensor ( C is the right Cauchy-Green tensor. For Green-Lagrange strain tensor ( ).
[0048] Assuming that There are restrictions on formulas (17) and (18): (17); (18); Based on the assumptions, the concept of scalar damage is introduced as formula (19): (19); in, It is a scalar damage variable. It is an effective free energy function. From formulas (17) and (18), we can obtain formula (20):
[0049] (20); The Clausius-Planck inequality based on the assumption of a purely mechanical process can be written as formula (21): (twenty one); (twenty two); in, For dissipation density, For internal energy density, S For the second Piola-Kirchhoff stress tensor ( , P (This is the first Piola-Kirchhoff stress tensor).
[0050] Therefore, formula (23) can be obtained as follows: (twenty three); From formula (23), we can obtain formula (24) as follows: (twenty four); Therefore, formula (25) is satisfied: (25); in, It is an effective second Piola-Kirchhoff stress tensor.
[0051] 2. Assume that the energy function in formula (19) takes the form of the Ogden model, i.e., formula (26): (26); in, , , It is the principal stretch (the principal value of U). , ( P =1,2…… N ) is the material parameter of the Ogden model. In the Neo-Hookean model... N =1, =2. That is, formula (27):
[0052] (27); 3. Because it is an incompressible hyperelastic material, the constitutive model of formula (27) can be rewritten in the form of the main tensile direction, that is... , which is formula (28): (28); in, yes The main value, For hydrostatic pressure (Lagrange multiplier, used to apply incompressible conditions, assuming the material is incompressible and its volume does not change). .
[0053] Based on formula (28), the principal Cauchy stress is converted into formula (29): (29); in These are the principal values of the Cauchy principal stress tensor. , .
[0054] 4. Considering only the overall model of uniaxial tension, the formula (30) is satisfied: (30); Formula (30) can be rewritten in matrix form as formula (31): (31); The corresponding right-hand tensile tensor required for stress calculation is obtained from formulas (28), (29), and U. .
[0055] The relevant stress state is obtained based on the neo-Hookean model, from which… Formulas (32) and (33) are given: (32); (33); Apply boundary conditions: Towards free boundary conditions Hydrostatic pressure can be obtained Therefore, we can obtain formula (34): (34); According to formula (29), and assuming that only the principal direction is considered, the Cauchy stress of the model is given by formula (35): (35); in, The right-hand tensor is a second-order tensor. The tensor is converted into uniaxial tensile strain by... Formula (36) can be obtained: (36); Depend on Therefore, the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury is given by formula (37): (37).
[0056] Among them, scalar damage variables .
[0057] S103: Take the first derivative of the strain in the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury to obtain the effective elastic modulus of the damaged hyperelastic material.
[0058] The effective elastic modulus is obtained by taking the first derivative of the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury, i.e., formula (37). For formula (38): (38).
[0059] S104: Substitute the effective elastic modulus of the damaged hyperelastic material into the relationship between the force and the material compression to obtain the force expression. Substitute the force expression into the formula corresponding to the highly nonlinear solitary wave propagation model to obtain the coupling equation set based on highly nonlinear solitary waves and damaged hyperelastic materials.
[0060] A coupling model of the particle chain and the hyperelastic material is established. Based on Hertz's modified contact theory, the force is obtained as formula (39): (39); in, For the action force, Where is the particle radius, The Poisson's ratio of the particles. Let be the elastic modulus of the particle. For hyperelastic materials, Poisson's ratio The effective elastic modulus of a hyperelastic material, This represents the amount of material compression.
[0061] Depend on The compressibility can be obtained as a relationship with strain, that is, the relationship between the applied force and the material compressibility is formula (40): (40); Substituting equation (40) into the equation corresponding to the highly nonlinear solitary wave propagation model, i.e., equation (5), we obtain the coupling equation set based on highly nonlinear solitary waves and damaged hyperelastic materials, which is equation (41): (41); in, This represents taking the maximum value. .
[0062] In an exemplary embodiment, the method further includes: transforming the coupled equations based on highly nonlinear isolated waves and damaged hyperelastic materials; setting initial conditions and numerically solving the transformed equations using the fourth-order Runge-Kutta method to obtain the corresponding relationship between the amplitude attenuation, time delay and time delay disappearance threshold of the secondary and tertiary rebound waves, the degree of anterior cruciate ligament damage and the effective elastic modulus.
[0063] The coupling equations between highly nonlinear solitary waves and damaged hyperelastic materials were transformed. Initial conditions were set, and the transformed equations were numerically solved using the fourth-order Runge-Kutta method. The correlation between the amplitude attenuation, time delay, and time delay vanishing threshold of secondary and tertiary rebound waves, the degree of anterior cruciate ligament (ACL) damage, and the effective elastic modulus was obtained. Based on the correlation between the amplitude attenuation, time delay, and time delay vanishing threshold of multiple secondary and tertiary rebound waves, the degree of ACL damage, and the effective elastic modulus, a database was constructed.
[0064] Transform the system of equations in formula (41) into formula (42): (42); The initial conditions are: and , The constant is used. The fourth-order Runge-Kutta method is employed to numerically solve the system of equations.
[0065] In an exemplary embodiment, the fourth-order Runge-Kutta method was used to solve the differential equation (42), and the effects of different particle material properties and different damage levels on the solitary wave were studied. The parameters of the particle chain were set as follows: assuming the particle chain consists of 25 stainless steel spheres, and the radius of each particle in the particle chain is... mm, density is kg / m3, Poisson's ratio is The elastic modulus is Gpa. The first particle impacts the particle chain at a speed of 447.2 μm / ms, and the speed of the 13th particle is used to represent the propagation of the particle solitary wave.
[0066] The effect of the elastic modulus of the anterior cruciate ligament on solitary waves.
[0067] 1. The effect of changes in solitary wave parameters.
[0068] Pick =4.875, with other parameters remaining unchanged, the interaction between the one-dimensional particle chain and the material, the displacement curves of each particle, the velocity curve of the 13th particle, and the velocity and displacement curves of each particle are shown in the figure. Figure 2As shown. Figure 2 The time-velocity curve provided for this invention, such as Figure 2 As shown, before 0.5 ms, the velocity curve experiences a sudden positive change, indicating that the isolated wave has propagated to the 13th particle at this point, defined as the incident wave IW. Subsequently, the isolated wave undergoes three negative abrupt changes of different values, indicating that due to the constraint of the ligament, the isolated wave propagates in the opposite direction along the particle chain to the 13th particle. These three rebounds are defined as the initial rebound wave PSW, the secondary rebound wave SSW, and the tertiary rebound wave TSW. The time difference between PSW and IW is defined as T1; the time difference between SSW and IW is defined as T2; and the time difference between TSW and IW is defined as T3. The amplitude of the rebound isolated wave is A. ref ,definition f =( A ref / A in ) 2 To measure the energy ratio of rebound waves. The energy ratio of PSW is... f 1 The energy ratio of SSW is f 2 The energy ratio of TSW is f 3 .
[0069] The specific analysis is as follows: When =4.875, the material parameters of the particle chain remain unchanged, such as Figure 2 As shown, at 0.2114 ms, the incident wave reaches the 13th particle with a velocity amplitude of 304 μm / ms. When the incident wave reaches the terminal particle, due to the constraint of the ligament, the isolated wave rebounds once. At 0.6556 ms, the initial rebound wave reaches the 13th particle with a velocity amplitude of 303 μm / ms. The velocity perturbation curve of the 13th particle has three peaks with different negative values. When the structure is subjected to lateral impact by the particle chain, if the elastic modulus of the structure is small, its compression will be large. Due to the existence of compression, the two terminal particles will still rebound after separating from the particle chain, which leads to the terminal particles colliding with the particle chain again, thus generating secondary and tertiary rebound waves.
[0070] The influence of the elastic modulus of the particle chain on the rebound solitary wave is analyzed as follows: Figure 3 The displacement curves of each particle provided by the present invention are shown in Figure (3). Figure (a) in Figure (3) is the time velocity diagram of the thirteenth particle, Figure (b) in Figure (3) is the time displacement diagram of each particle, Figure (c) in Figure (3) is the time velocity diagram of each particle, and Figure (d) in Figure (3) is the force diagram of the thirteenth particle. When the elastic modulus of the particle is... Es At 150 GPa, such as Figure 3As shown in Figure (a), when the elastic modulus of the particle chain changes, the incident wave remains completely unchanged, indicating that the incident wave is independent of the elastic modulus of the particles. However, as the elastic modulus gradually decreases, the primary rebound wave remains almost constant with a very slight change in amplitude, while the amplitudes of the secondary and tertiary rebound waves gradually decrease. The velocity amplitude of the secondary rebound wave decreases from 46.0468 μm / ms to 33.47247 μm / ms and then to 10.8937 μm / ms, and the tertiary rebound wave decreases from 21.6719 μm / ms to 16.2531 μm / ms and then disappears completely. The displacement curve shows that as the elastic modulus increases, the two end particles initially separate and then collide before rebounding as a whole with the entire particle chain. This explains why the secondary and tertiary rebound waves gradually disappear. Therefore, when a one-dimensional particle chain probes a hyperelastic semi-infinite space, the elastic modulus of the particles should be selected... Es ≥200GPa is more accurate.
[0071] Figure 4 The following are displacement curves of each particle when the elastic modulus is 150, provided by the present invention; wherein, (a) is the velocity diagram of the thirteenth particle, (b) is the displacement diagram of each particle, (c) is the velocity diagram of each particle, and (d) is the force diagram of the thirteenth particle.
[0072] Figure 5 The following are displacement curves of each particle when the elastic modulus is 250, provided by the present invention; wherein, (a) is the velocity diagram of the thirteenth particle, (b) is the displacement diagram of each particle, (c) is the velocity diagram of each particle, and (d) is the force diagram of the thirteenth particle.
[0073] Figure 6 The velocity diagram of the thirteenth particle provided by the present invention, wherein (a) is a velocity diagram of v s The velocity diagram of the thirteenth particle when v = 0.2, (b) shows the velocity of v. s Velocity diagram of the thirteenth particle when =0.4.
[0074] When the elastic modulus of the particles remains constant, and only the Poisson's ratio of the particles is changed, the effect on the rebound wave is analyzed as follows: Based on the conclusion of this invention, the elastic modulus is selected as... Es =200GPa, Poisson's ratio v s =0.2, v s =0.28 and v s =0.4, the material properties of the hyperelastic semi-infinite space body remain unchanged. For example... Figure 3 Figure (a) and Figure 6 As shown in Figure (a), when v s When =0.2, andv s Compared to 0.28, the primary rebound wave remained almost unchanged, with only a slight decrease in amplitude, from 303.05 μm / ms to 301.059 μm / ms. The secondary and tertiary rebound waves also showed slight changes, with their amplitudes decreasing slightly. Figure 3 Figure (a) and Figure 6 As shown in Figure (b), when v s =0.4 and v s Compared to 0.28, the incident wave remained unchanged, but the amplitudes of the secondary and tertiary rebound waves increased slightly with the increase of Poisson's ratio. The velocity amplitude of the secondary rebound wave increased from 33.4726 μm / ms to 39.8156 μm / ms, and that of the tertiary rebound wave increased from 16.2531 μm / ms to 19.105 μm / ms.
[0075] In summary, the amplitude of each rebound wave increases with increasing Poisson's ratio. When the Poisson's ratio decreases to a certain value, the third-order rebound wave gradually weakens and eventually disappears. Therefore, when using particle chains for detection, the Poisson's ratio... v s ≥0.2 accurate.
[0076] 2. The effect of changes in the effective elastic modulus of hyperelastic materials on solitary waves.
[0077] Figure 7 The velocity curves and internal force diagrams provided for this invention are as follows: (a) is the velocity diagram of the thirteenth particle when the effective elastic modulus is 4.875; (b) is the displacement diagram of each particle when the effective elastic modulus is 4.875; (c) is the velocity diagram of each particle when the effective elastic modulus is 4.875; and (d) is the force diagram of the thirteenth particle when the effective elastic modulus is 4.875. Figure 8 The velocity curves and internal force diagrams provided for this invention include: (a) the velocity diagram of the thirteenth particle when the effective elastic modulus is 4; (b) the displacement diagram of each particle when the effective elastic modulus is 4; (c) the velocity diagram of each particle when the effective elastic modulus is 4; and (d) the force diagram of the thirteenth particle when the effective elastic modulus is 4. One-dimensional particle chain and Poisson's ratio are v s =0.4, with an effective elastic modulus of 4.875 MPa and 4 MPa, the interaction of hyperelastic semi-infinite bodies is shown. The perturbation velocity of the thirteenth particle characterizes the propagation of the solitary wave in the chain. The perturbation velocity curve of the thirteenth particle and the displacement curves of each particle are as follows. Figure 7 and Figure 8 As shown. By Figure 7 It can be seen that when the effective elastic modulus At a pressure of 4.875 MPa, the incident solitary wave reaches the thirteenth particle at 0.2114 ms. At this time, the perturbation velocity of the particle is... A in The velocity is 303.992 μm / ms. At 0.6556 ms, the primary rebound wave reaches the thirteenth particle; at this point, the particle's perturbation velocity... A ref The elastic modulus was 303.05 μm / ms, followed by distinct secondary and tertiary rebound waves. As the elastic modulus decreased, the secondary and tertiary rebound waves gradually weakened, such as... Figure 8 As shown, when At 4 MPa, the third-order rebound wave completely disappears, demonstrating the significant impact of changes in the material's effective elastic modulus on solitary waves. Furthermore, the displacement curve shows that the two terminal particles initially separate from the leading particles before colliding, but later rebound as a whole with the leading particles, resulting in a more orderly displacement curve. Theoretically, hyperelastic materials can store and dissipate energy during large deformations. With a lower elastic modulus, the material deforms more easily, absorbing more strain energy at lower stress levels. The two terminal particles will then be insufficient to separate from the particle chain and produce overall rebound, ultimately leading to the disappearance of the secondary rebound wave and only a primary rebound.
[0078] Figure 9 The velocity diagram of the 13th particle provided by this invention. Figure 9 Figure (a) shows the elastic modulus. Es The velocity diagram of the thirteenth particle at 150 GPa is shown. In Figure (9), (b) represents the elastic modulus. Es The velocity diagram of the thirteenth particle at 250 GPa.
[0079] Take the effective elastic modulus =4, other parameters remain unchanged, change the elastic modulus of the particles, from Figure 7 , Figure 8 and Figure 9 It can be seen that when the elastic modulus Es At pressures of 150 GPa, 200 GPa, and 250 GPa, the incident wave and the primary rebound wave show almost no change. As the elastic modulus increases, the amplitude of the secondary rebound wave increases sequentially, but no tertiary rebound wave appears. In summary, regardless of how the effective elastic modulus of the material changes, as the elastic modulus of the particles gradually decreases, the secondary rebound wave also gradually decreases until it disappears.
[0080] Figure 10 This is the force diagram of the thirteenth particle provided by the present invention. Figure 10 The interaction between a one-dimensional particle chain and a hyperelastic semi-infinite space body, such as Figure 10 As shown, with the elastic modulus EsThe internal force variation curve is shown in the figure, which illustrates the change in internal force from 150 GPa to 250 GPa. As the elastic modulus increases, the internal force remains unchanged upon incident. The internal force amplitude is relatively stable during the first rebound, while it gradually increases during the second and third rebounds. T1 shows a gradual decreasing trend, but the change is not significant, while T2 and T3 gradually decrease. In summary, the most reliable range for the interaction between one-dimensional particle chains and a hyperelastic semi-infinite space body is greater than or equal to 200 GPa, consistent with the analysis in this invention.
[0081] The impact of the degree of anterior cruciate ligament damage on solitary waves.
[0082] Figure 11 The elastic modulus provided by the present invention The curve is shown when =11.55, where (a) is the elastic modulus. The velocity diagram of the thirteenth particle at 11.55, and (b) the elastic modulus. Particle displacement diagram when =11.55, (c) diagram shows the elastic modulus. Particle velocities at 11.55.
[0083] Figure 12 The elastic modulus provided by the present invention The curve is shown when the elastic modulus is 39.24, where (a) is the elastic modulus. The velocity diagram of the thirteenth particle at 39.24, (b) shows the elastic modulus. Particle displacement diagram at 39.24, (c) shows the elastic modulus. Particle velocities at a value of 39.24. Material damage is taken as... With other parameters remaining constant, the one-dimensional particle chain is compared with the elastic modulus. =11.55 and The damage material interaction at a value of 39.24, the displacement curves of each particle, the velocity curve of the 13th particle, and the velocity curves of each particle are shown below. Figure 11 and Figure 12 As shown.
[0084] Figure 13 The elastic modulus provided by the present invention The curves at =6.5, where (a) shows the elastic modulus. The velocity diagram of the thirteenth particle at =6.5, (b) shows the elastic modulus. Particle displacement diagram at =6.5, (c) diagram shows the elastic modulus. Velocity diagrams of each particle at 6.5.
[0085] Figure 14 The elastic modulus provided by the present invention The curve is shown when =22.425, where (a) is the elastic modulus. The velocity diagram of the thirteenth particle at =22.425, (b) shows the elastic modulus. Particle displacement diagram at =22.425, (c) diagram shows the elastic modulus. Particle velocities at a value of 22.425. Material damage is taken as... With other parameters remaining constant, the one-dimensional particle chain is compared with the elastic modulus. =6.5 and The material interaction at a damage value of 22.425 is shown in the displacement curves of each particle, the velocity curve of the 13th particle, and the velocity curves of each particle. Figure 13 and Figure 14 As shown.
[0086] Figure 15 The elastic modulus provided by the present invention The curve when =1.625, where (a) is the elastic modulus. Es The velocity diagram of the thirteenth particle at ==1.625, and (b) the elastic modulus. Particle displacement diagram when =1.625, (c) diagram shows the elastic modulus. Particle velocities at 1.625.
[0087] Figure 16 The elastic modulus provided by the present invention The curve when = 5.606, where (a) is the elastic modulus. The velocity diagram of the thirteenth particle at 5.606, and Figure (b) shows the elastic modulus. Particle displacement diagram when = 5.606, (c) diagram shows the elastic modulus. Particle velocities at a value of 5.606. Material damage is taken as... With other parameters remaining constant, the one-dimensional particle chain is compared with the elastic modulus. The interaction between the damaged materials at values of 6.5 and 22.425, the displacement curves of each particle, the velocity curve of the 13th particle, and the velocity curves of each particle are shown below. Figure 15 As shown and Figure 16 As shown.
[0088] This invention investigates the dynamic behavior of the interaction between a one-dimensional particle chain and a semi-infinite space body of a hyperelastic material through numerical simulation. The focus is on analyzing the influence of particle material parameters and hyperelastic material damage on its wave response, leading to the following conclusions: (1) Elastic modulus of the particle chain Es Changes in the elastic modulus of the particle chains have a significant impact on wave propagation behavior. While keeping the parameters of the hyperelastic material constant, the elastic modulus of the particle chains is... EsAs the pressure is gradually reduced from 250 GPa to 150 GPa, the amplitudes of the incident wave and the primary rebound wave remain essentially unchanged, while the amplitudes of the secondary rebound wave and the tertiary rebound wave decrease significantly. When the elastic modulus of the particle chain... Es At 150 GPa, the third-order rebound wave completely disappears. This phenomenon indicates that lower material stiffness inhibits the formation of higher-order rebound waves. From the perspective of internal force response, as... Figure 10 During the incident phase, the internal force response is unaffected by changes in the elastic modulus, and the internal force amplitude is relatively stable during the first rebound phase. However, the internal force amplitudes during the second and third rebound phases decrease significantly as EW decreases. Simultaneously, the rebound time parameters also change: T1 increases slightly but not significantly, while T2 and T3 are significantly prolonged.
[0089] With fixed particle elastic modulus Es Given a Pa of 200 GPa, change its Poisson's ratio. v s The values (ranging from 0.2 to 0.4) show that the incident wave amplitude remains constant, and the overall shapes of the primary, secondary, and tertiary rebound waves are basically the same, but their velocity amplitudes vary. v s The increase and slight improvement indicate that the compressibility of particulate materials has a slight modulating effect on the rebound dynamics.
[0090] (2) Damage to hyperelastic materials has a systematic impact on their wave response. With fixed particle chain parameters ( mm, density is kg / m3, Poisson's ratio is The elastic modulus is Under the condition of Gpa, gradually reduce the elastic modulus of the damaged area. Es At that time, it can be observed that the secondary and tertiary rebound waves gradually weaken, and finally the tertiary rebound wave... Es It disappears completely when it falls below a certain threshold.
[0091] Increased damage does not prevent the generation of secondary rebound waves, but it significantly delays their occurrence and reduces their amplitude. For example... Figure 11 Figure (a) in the middle and Figure 13 As shown in Figure (a), increased damage leads to a longer propagation time and a decreased velocity amplitude of the rebound solitary wave to the 13th particle, reflecting that increased material damage will delay wave propagation and cause energy dissipation. Similarly, from Figure 14 Figure (a) in the middle and Figure 16 A comparison with Figure (a) shows that when the effective elastic modulus of the damaged hyperelastic material is reduced... At higher levels, the increase in damage still leads to similar delay and amplitude reduction, consistent with the aforementioned conclusions, indicating that this response law has a certain robustness to material stiffness.
[0092] In summary, the rebound response characteristics based on highly nonlinear solitary waves can be used to effectively identify damage in hyperelastic materials. By analyzing the occurrence time, amplitude attenuation, and disappearance threshold of higher-order rebound components of the rebound wave, quantitative assessment of material damage can be achieved. This method shows great application potential in the field of biomedical engineering, particularly in the non-destructive testing of soft tissues such as knee ligaments, and is expected to provide a highly efficient, non-invasive damage assessment tool for clinical diagnosis.
[0093] When applying the anterior cruciate ligament injury status prediction method based on nonlinear solitary waves provided by this invention, it is not necessary to rely on... Figure 1 The steps shown are executed in sequence. The specific execution order of each step can be determined as needed, and this invention does not impose any restrictions on it.
[0094] The above describes one or more embodiments of the present invention for predicting the anterior cruciate ligament (ACL) injury status based on nonlinear solitary waves. Based on the same idea, the present invention also provides a corresponding ACL injury status prediction device based on nonlinear solitary waves, such as... Figure 12 As shown.
[0095] Figure 17 A schematic diagram of a modeling device based on a set of coupled equations for damaged hyperelastic materials provided by the present invention includes: The first building module is used to construct the formula corresponding to the propagation model of highly nonlinear isolated waves based on Hertz's contact law, which governs the generation and propagation of highly nonlinear isolated waves in a one-dimensional particle chain.
[0096] The second building module is used to simplify the anterior cruciate ligament (ACL) as a hyperelastic material. Based on the assumption of incompressible isotropic hyperelasticity, combined with the Neo-Hookean constitutive model, scalar damage variables, and uniaxial tension boundary conditions, the stress-strain relationship of the hyperelastic material after ACL injury is constructed through the Helmholtz free energy function and Cauchy stress transformation.
[0097] The first determining module is used to calculate the first derivative of the strain in the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury, and obtain the effective elastic modulus of the damaged hyperelastic material.
[0098] The second determining module is used to substitute the effective elastic modulus of the damaged hyperelastic material into the relationship between the force and the material compression to obtain the force expression, and then substitute the force expression into the formula corresponding to the highly nonlinear solitary wave propagation model to obtain a set of coupled equations based on highly nonlinear solitary waves and damaged hyperelastic materials.
[0099] Specific limitations on the modeling apparatus based on the coupled equations of damaged hyperelastic materials can be found in the limitations of the anterior cruciate ligament injury state prediction method based on nonlinear solitary waves mentioned above, and will not be repeated here. Each module in the aforementioned modeling apparatus based on the coupled equations of damaged hyperelastic materials can be implemented entirely or partially through software, hardware, or a combination thereof. These modules can be embedded in or independent of the processor in a computer device, or stored in the memory of a computer device as software, so that the processor can call and execute the corresponding operations of each module.
[0100] The present invention also provides a computer-readable storage medium storing a computer program that can be used to execute the above-described... Figure 1 A modeling method based on coupled equations for damaged hyperelastic materials is provided.
[0101] At the hardware level, this computer device includes a processor, internal bus, network interface, memory, and non-volatile memory, and may also include other hardware required for business operations. The processor reads the corresponding computer program from the non-volatile memory into memory and then executes it to achieve the above. Figure 1 A method for predicting anterior cruciate ligament injury status based on nonlinear solitary waves is provided.
[0102] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the methods described above. Any references to memory, storage, databases, or other media used in the embodiments provided by this invention can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, or optical storage, etc. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM), etc.
[0103] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this invention.
Claims
1. A modeling method based on coupled equations of damaged hyperelastic materials, characterized in that, include: Based on Hertz's contact law governing the generation and propagation of highly nonlinear isolated waves in a one-dimensional particle chain, a formula corresponding to the propagation model of highly nonlinear isolated waves is constructed. The anterior cruciate ligament is simplified as a hyperelastic material. Based on the assumption of incompressible isotropic hyperelasticity, combined with the Neo-Hookean constitutive model, scalar damage variables, and uniaxial tension boundary conditions, the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury is constructed through the Helmholtz free energy function and Cauchy stress transformation. The effective elastic modulus of the damaged hyperelastic material is obtained by taking the first derivative of the strain in the stress-strain relationship of the hyperelastic material after anterior cruciate ligament injury. Substituting the effective elastic modulus of the damaged hyperelastic material into the relationship between the force and the material compression yields the force expression. Substituting this force expression into the formula corresponding to the highly nonlinear solitary wave propagation model, we obtain a set of coupled equations based on highly nonlinear solitary waves and damaged hyperelastic materials.
2. The method as described in claim 1, characterized in that, The formula corresponding to the highly nonlinear solitary wave propagation model is: ； in, The equation of motion for the first particle is... For the first i The differential equation of motion for each particle For the first N The differential equation of motion for each particle Let be the stiffness constant of the particle. For the mass of the particles, This represents the number of particles in the particle chain. This represents taking the maximum value. For the first particle in t The displacement from its equilibrium position at any given moment. For the first i -1 particle in t The displacement from its equilibrium position at any given moment. For the first i Each particle t The displacement from its equilibrium position at any given moment. For the first i+ 1 particle in t The displacement from its equilibrium position at any given moment. For the first N -1 particle in t The displacement from its equilibrium position at any given moment. No. N Each particle t The displacement from its equilibrium position at any given moment. , This represents the Hertzian contact force between the particle and the material.
3. The method as described in claim 2, characterized in that, The stress-strain relationship is as follows: ； in, For stress, For ligament injury variables, In response to the situation; The effective elastic modulus of the damaged hyperelastic material is: ； in, The effective elastic modulus of the damaged hyperelastic material.
4. The method as described in claim 3, characterized in that, The expression for the force is: ； in, For the action force, Where is the particle radius, The Poisson's ratio of the particles. Let be the elastic modulus of the particle. For hyperelastic materials, Poisson's ratio To reduce the effective elastic modulus of the damaged hyperelastic material, This represents the amount of material compression.
5. The method as described in claim 4, characterized in that, The coupling equations based on highly nonlinear isolated waves and damaged hyperelastic materials are as follows: ； in, The equation of motion for the first particle is... For the first i The differential equation of motion for each particle For the first N The differential equation of motion for each particle Let be the stiffness constant of the particle. For the mass of the particles, This represents the number of particles in the particle chain. This represents taking the maximum value. For the first particle in t The displacement from its equilibrium position at any given moment. For the first i -1 particle in t The displacement from its equilibrium position at any given moment. For the first i Each particle t The displacement from its equilibrium position at any given moment. For the first i+ 1 particle in t The displacement from its equilibrium position at any given moment. For the first N -1 particle in t The displacement from its equilibrium position at any given moment. No. N Each particle t The displacement from its equilibrium position at any given moment. , The effective elastic modulus of the damaged hyperelastic material.
6. The method as described in claim 1, characterized in that, The method further includes: The coupling equations based on highly nonlinear isolated waves and damaged hyperelastic materials are transformed. Initial conditions were set, and the transformed equations were numerically solved using the fourth-order Runge-Kutta method. The results showed that there is a correspondence between the amplitude attenuation, time delay, and disappearance threshold of the time delay of the secondary and tertiary rebound waves, the degree of anterior cruciate ligament injury, and the effective elastic modulus.
7. The method as described in claim 6, characterized in that, The coupling equations between the transformed highly nonlinear isolated wave and the hyperelastic material are as follows: ; in, The Poisson's ratio of the first particle. For the first i Poisson's ratio of each particle For the first N Poisson's ratio of each particle For the first particle in t The displacement from its equilibrium position at any given moment. For the second particle in t The displacement from its equilibrium position at any given moment. For the first i -1 particle in t The displacement from its equilibrium position at any given moment. For the first i Each particle t The displacement from its equilibrium position at any given moment. For the first i+ 1 particle in t The displacement from its equilibrium position at any given moment. For the first N -1 particle in t The displacement from its equilibrium position at any given moment. No. N Each particle t The displacement from its equilibrium position at any given moment. Where is the particle radius, For the mass of the particles, To reduce the effective elastic modulus of the damaged hyperelastic material, This refers to the amount of material compression. The Poisson's ratio of the particles. Poisson's ratio for hyperelastic materials.
8. The method as described in claim 7, characterized in that, The initial conditions are: and .