A method for establishing a large deformation viscoelastic constitutive relation based on prony series

Abstract

This invention presents a method for establishing large deformation viscoelastic constitutive relations based on Prony technology. It addresses the problems of existing viscoelastic constitutive models lacking universality, and the complexity and programming difficulties associated with their mathematical relationships. The invention includes: decomposing the stress tensor into the sum of the deviatoric stress tensor and the mean stress; decomposing the strain tensor into the sum of the deviatoric strain tensor and the mean strain; and providing a convolutional expression for the deviatoric stress tensor and the mean stress. The increment of the deviatoric stress tensor is expressed as a function of the deviatoric strain increment, and the increment of the mean stress is expressed as a function of the mean strain increment. The tensor form of the incremental constitutive relation is written in matrix form. This method has been programmed, and the numerical results obtained from the program agree well with the experimental results in specimen-level experiments on solid rocket motor propellant grains.

Description

Technical Field

[0001] This invention relates to the field of viscoelastic mechanical analysis technology of propellant grains, specifically to a method for establishing large deformation viscoelastic constitutive relations based on Prony series. Background Technology

[0002] The viscoelastic mechanical analysis of solid rocket motor propellant grains has always been a key factor in analyzing and predicting the structural integrity and lifespan of propellant grains. To accurately predict the mechanical properties of the propellant grain structure, it is necessary to establish universal and effective constitutive equations. Solid propellants are typical viscoelastic materials, exhibiting nonlinear (geometric and physical) characteristics under certain loads. Existing analytical methods often consider physical nonlinearity while neglecting geometric nonlinearity, leading to significant discrepancies between the analytical results and actual values ​​under large deformation conditions.

[0003] Linear viscoelastic constitutive models define linear viscoelasticity as a condition where stress and strain exhibit a linear relationship under a given stress limit, and this relationship changes with time and loading rate. When the deformation of a solid propellant is small, it can be considered a linear viscoelastic material. Linear viscoelastic constitutive models can be used to analyze the mechanical response of solid propellants under load. Linear viscoelastic constitutive models are currently the most widely used model in engineering, and mathematically, they can be divided into differential and integral forms.

[0004] Differential models use differential equations to describe constitutive relations. Differential viscoelastic constitutive models are simple in form, have clear physical meaning, and are suitable for solving analytical solutions for viscoelastic materials. However, when using these differential constitutive equations for calculations, time derivatives of orders ten or even higher appear, causing considerable computational difficulties. Furthermore, many constants in the model must be calculated, further increasing the difficulty of its application. Therefore, considering the ease of equation calculation and the accuracy of the model, differential linear elastic constitutive models are rarely used in engineering computational analysis.

[0005] Currently, the structural integrity analysis of solid propellant grains increasingly employs integral linear viscoelastic constitutive models because they not only accurately reflect the viscoelasticity of the material but also require fewer material parameters in their constitutive equations, which are relatively easy to obtain experimentally. Integral linear viscoelastic constitutive models based on the Boltzmann superposition principle are commonly used for the viscoelastic analysis of solid propellants.

[0006] In general, the integral linear viscoelastic constitutive model is suitable for finite element numerical calculations of linear viscoelastic materials. It can be used to characterize the mechanical properties of solid propellants under relatively small strain conditions and is widely used in the integrity analysis of composite propellant materials.

[0007] Linear viscoelastic differential and integral constitutive models are applicable to the analysis of problems in different fields. Although they differ in their expression forms, they are essentially the same, and their equations can be derived from another form of equation.

[0008] Further research on solid propellants using nonlinear viscoelastic constitutive models has revealed significant nonlinearity in propellant materials at large strains, and even at small strains. This makes existing linear viscoelastic models inadequate for accurately and effectively characterizing the mechanical properties of solid propellants. Numerous studies on nonlinear viscoelastic constitutive models have been conducted in recent decades. Based on the different causes of nonlinearity, viscoelastic constitutive models can be divided into damage-free and damage-inducing nonlinear constitutive models. Damage-free nonlinear viscoelastic constitutive models assume that the nonlinearity is caused by the material's inherent nonlinearity, while damage-inducing models attribute it to large deformations. When the deformation is less than 15%, the nonlinearity is primarily caused by the material itself.

[0009] In damage-free nonlinear viscoelastic constitutive models, temperature and strain rate are the two main factors affecting the viscoelasticity of solid propellants. Experiments show that the nonlinear viscoelastic stress-strain curves differ with different strain rates. Based on the strain rate range of deformation, damage-free nonlinear viscoelastic constitutive models can be further divided into low-strain-rate and high-strain-rate models.

[0010] When studying constitutive models of solid propellants at low strain rates, the methods commonly used to establish nonlinear viscoelastic constitutive models, depending on the research methods and approaches, include empirical, theoretical, and semi-empirical methods. Among these, semi-empirical nonlinear viscoelastic constitutive models are the most widely used in engineering practice. Many nonlinear viscoelastic constitutive models have been established based on various methods and approaches, including differential, single-integral, multiple-integral, and power-law representations. Among these, the viscoelastic single-integral constitutive model is widely used in practice. Two commonly used single-integral constitutive models based on semi-empirical methods are the Leaderman nonlinear viscoelastic constitutive model and the Schapery thermodynamic constitutive model. The Leaderman constitutive model is suitable for the mechanical analysis of viscoelastic materials with low strain rates and small deformations. However, the Leaderman constitutive model does not consider thermodynamic factors, and the influence of the deformation thermodynamic field on the nonlinear constitutive theory cannot be ignored. Based on irreversible thermodynamic theory, Schapery derived the relaxation and creep nonlinear viscoelastic constitutive equations with reduced time under isothermal uniaxial stress using the Gibbs free energy form.

[0011] Thermodynamic constitutive models are frequently used for the viscoelastic analysis of hydroxyl-butadiene composite solid propellants under low strain rates. The constitutive equations introduce reduced time, apply stress-strain functions with respect to time, and embody the time-temperature equivalence principle. This makes the material parameters easier to obtain experimentally and facilitates finite element method programming. These models are based on simple and practical theories derived from experimental findings, and can accurately describe the nonlinear mechanical characteristics of solid propellants under small deformation and low strain rates.

[0012] When studying the mechanical performance response of solid propellants under high strain rates, the Zhu-Wang-Tang (ZWT) nonlinear viscoelastic constitutive model (applicable to strain ≤7%, strain rate 10⁻⁴ s⁻¹ - 10³ s⁻¹) and the viscoelastic-hyperelastic constitutive model (applicable to strain ≤30%, strain rate ≤1450 s⁻¹) are often favored by researchers at home and abroad.

[0013] The ZWT nonlinear viscoelastic constitutive model was proposed by Tang Zhiping, Zhu Zhaoxiang, and Wang Lili based on the Green-Rivlin multiple product nonlinear viscoelastic constitutive theory when studying the mechanical properties of polymers under high strain rates. This model can be viewed as consisting of a nonlinear spring-shaped constitutive model and two Maxwell models connected in parallel, corresponding to the first three terms and the last two terms on the right-hand side of the equation, respectively. The first three terms describe the nonlinear elastic response of the material, while the last two terms describe the viscoelastic response of the material at quasi-static low strain rates and dynamic high strain rates, respectively.

[0014] The ZWT constitutive model effectively reflects the influence of strain rate on material nonlinearity and has been widely applied in materials such as polymers, explosives, and double-base propellants. However, this model cannot effectively describe the entire deformation process of the propellant. When the strain exceeds 7% under quasi-static conditions or the deformation approaches material failure, the model fails to meet engineering requirements.

[0015] The application of the ZWT constitutive model to study the stress-strain curves of double-base propellants under quasi-static and dynamic loading conditions revealed that the ZWT model can only describe the mechanical properties of materials with strains within 2% at low strain rates and within 5% at high strain rates, but it cannot be used to describe the mechanical response of propellants over a large strain range. For NEPE propellants, which are soft materials with large deformation and significant rate-dependent, temperature-dependent, and history-dependent characteristics, the ZWT nonlinear viscoelastic constitutive model and the basic hyperelastic constitutive model cannot accurately describe their mechanical properties at high strain rates. Therefore, considering the large deformation and rate-dependent characteristics of NEPE propellants, a viscoelastic-hyperelastic constitutive model suitable for describing the mechanical properties of NEPE propellants needs to be established.

[0016] Damaged nonlinear viscoelastic constitutive models address the nonlinear properties of materials under large deformations, which continuum mechanics attributes to material damage. However, existing models fail to account for this damage and thus cannot accurately describe the mechanical properties of solid propellants. Therefore, some researchers have proposed damaged nonlinear viscoelastic constitutive models, focusing on constructing a function relating to damage. Schapery, based on the elastic-viscoelastic correspondence principle, considers material damage and introduces a damage function relating to macroscopic intrinsic parameters to characterize the nonlinearity caused by damage, presenting a three-dimensional damaged nonlinear viscoelastic constitutive model.

[0017] This study investigates the constitutive model of HTPB composite solid propellant under damage and aging. Based on the Schappery nonlinear viscoelastic constitutive model with damage, and considering the aging properties of the material, a nonlinear viscoelastic constitutive model of HTPB composite propellant with damage and aging was established through acoustic emission damage detection experiments. The calculation results are in good agreement with the results calculated using the Schappery nonlinear constitutive equation with damage and the experimental results. This model effectively characterizes the nonlinear characteristics of HTPB composite solid propellant, and the parameters in the equation are easily obtained experimentally, facilitating engineering applications.

[0018] For a general ZWT nonlinear viscoelastic constitutive model, the material dynamic constitutive behavior considering damage evolution at high strain rates incorporates the internal damage evolution of the material, introducing damage to establish a damage-based ZWT model applicable to larger material deformations. In studying the high strain rate mechanical properties of dual-base propellants and their damage-incorporated ZWT constitutive models, comparisons between predicted stress values ​​obtained from the ZWT constitutive model and the damage-based ZWT constitutive model and experimental values ​​revealed that the non-damage-based ZWT constitutive model can only predict the viscoelastic segment of the propellant with a strain range of 0–0.03, while the damage-based ZWT model can predict the viscoelastic segment and the plastic flow segment after yielding within the strain range of 0–0.14, but the damage parameters are difficult to determine.

[0019] In engineering practice, semi-empirical single-integral methods are frequently used to study nonlinear constitutive models of viscoelastic materials. As can be seen from the methods and theories above, single-integral constitutive relations are characterized by simple models, efficient calculation methods, and strong applicability. They can be used as numerical methods to describe the nonlinear behavior of various viscoelastic materials, addressing both quasi-static and dynamic viscoelastic problems.

[0020] Nonlinear behavior can occur even with small strains during the creep process of viscoelastic materials. Leaderman modified the Boltzmann superposition principle to establish Leaderman theory. Pipkin and Rogers further extended it to the three-dimensional case. Schapery derived a nonlinear viscoelastic constitutive equation with a reduced time factor based on irreversible thermodynamics. This equation considers thermodynamic factors and is therefore also called the thermodynamic constitutive equation. Schapery theory embodies time-temperature equivalence and allows for both horizontal and vertical shifts of the time coordinate, and has been widely applied in recent years. The reduced time used in Schapery's constitutive equation is an important method for introducing generalized time in solid mechanics, but Leaderman constitutive relations have greater advantages in terms of simplicity and predictability.

[0021] Existing viscoelastic constitutive models are all designed for specific types of materials and are not applicable to other types of materials, thus lacking universality. Furthermore, the mathematical relationships in some constitutive models are quite complex, making them difficult to program. Summary of the Invention

[0022] To address the aforementioned problems, this invention provides a method for establishing large deformation viscoelastic constitutive relations based on Prony series, thereby solving the problems of the lack of universality of existing viscoelastic constitutive models, as well as the complexity of mathematical relations and the difficulty in programming existing constitutive models.

[0023] To address the aforementioned technical problems, one objective of this invention is to provide a method for establishing an incremental viscoelastic constitutive relation, the specific steps of which are as follows:

[0024] S1: The Kirchhoff stress tensor S ij Decomposed into the deviatoric stress tensor s ij and average stress The sum of these will give rise to the Green's strain tensor E. ij Decomposed into partial strain tensor e ij and mean strain sum;

[0025] S2: Deviatoric stress tensor s ij (t) and the deviatoric strain tensor e ij (t), mean stress s(t) and mean strain The convolutional constitutive relations between them are as follows:

[0026]

[0027] In the formula, G(t) and K(t) are the shear relaxation modulus and the volume relaxation modulus, respectively;

[0028] S3: Kirchhoff stress tensor S ijIncrement ΔS ij (t m ), that is, t m+1 Kirchhoff stress tensor S at time t ij (t m+1 ) and t m Kirchhoff stress tensor S at time t ij (t m ) difference;

[0029] S4: Using the relationships between deviatoric stress and deviatoric strain, and between mean stress and mean strain in S2, the increment Δs of the deviatoric stress is obtained. ij (t m The relationship between ) and deviatoric strain, and the increment of mean stress Δs(t) m The relationship between the strain and the mean strain can be approximated by a simple difference method, yielding the following relationship:

[0030]

[0031] In the formula

[0032] ΔG(t m )=G(t m+1 )-G(t m ),

[0033]

[0034] Δe ij (t m ) = e ij (t m+1 )-e ij (t m ),

[0035] ΔE kk (t m ) = E kk (t m+1 )-E kk (t m );

[0036] S5: Using the Prony series expansions of E(t), G(t), and K(t)

[0037]

[0038] And it is assumed that the time step is constant (i.e., Δt). m =Δt, m = 0, 1, 2, ...), and the relationship in S4 (the increment of deviatoric stress Δs) ij (t m The relationship between ) and deviatoric strain, and the increment of mean stress Δs(t) mThe relationship between the mean strain and the mean strain can be rewritten in the following incremental form:

[0039]

[0040] In the formula

[0041]

[0042] S6: For isotropic materials, the incremental constitutive relation in S5 can be written in matrix form as follows:

[0043]

[0044] In the formula

[0045]

[0046]

[0047] Furthermore, the S2 shear relaxation modulus and volumetric relaxation modulus are calculated using the tensile relaxation modulus E(t) and Poisson's ratio μ.

[0048]

[0049] Furthermore, S3 will increment ΔS ij (t m ) can be expressed as the increment of deviatoric stress Δs ij (t m ) and the increment of mean stress Δs(t) m The sum of ) and the increment of deviatoric stress Δs ij (t m ) equals t m+1 The deviatoric stress at time t and t m The difference in deviatoric stress at time t, the increment of mean stress Δs(t) m ) equals t m+1 The average stress at time t m The difference in average stress at time points.

[0050] The above-mentioned one or more technical solutions of the present invention have at least one or more of the following technical effects: The present invention provides a method for establishing an incremental viscoelastic constitutive relation using Prony series. This constitutive relation is applicable to both linear and nonlinear constitutive relations, and is highly versatile, simple and reliable. Attached Figure Description

[0051] Figure 1 : Schematic diagram of sample shape and size;

[0052] Figure 2 First-order and second-order Prony series fitting of tensile modulus,

[0053] Where: the red curve represents the first-order Prony series, and the blue curve represents the second-order Prony series;

[0054] Figure 3 Comparison of experimental and calculated values ​​of deformation.

[0055] Where: ○ represents experimental data points, and * represents calculated values. Detailed Implementation Plan

[0056] This invention provides a method for establishing incremental viscoelastic constitutive relations. The basic content includes: decomposing the stress tensor into the sum of the deviatoric stress tensor and the mean stress; decomposing the strain tensor into the sum of the deviatoric strain tensor and the mean strain; and providing a convolutional expression for the deviatoric stress tensor and the mean stress. The increment of the deviatoric stress tensor is expressed as a function of the deviatoric strain increment, and the increment of the mean stress is expressed as a function of the mean strain increment. The tensor form of the incremental constitutive relation is written in matrix form. This method has been programmed, and the numerical results obtained by the program agree well with the experimental results in specimen-level experiments on solid rocket motor propellant grains.

[0057] The technical solutions of the present invention will be clearly and completely described below with reference to the embodiments and accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments obtained. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention are within the scope of protection of the present invention.

[0058] This invention provides a method for establishing a large deformation viscoelastic constitutive relation based on the Prony series, the specific steps of which are as follows:

[0059] Step 1. Convert the Kirchhoff stress tensor S ij Decomposed into the deviatoric stress tensor s ij and average stress The sum of the Green's strain tensor E. ij Decomposed into partial strain tensor e ij and mean strain sum.

[0060] Step 2. Write out the deviatoric stress tensor s. ij (t) and the deviatoric strain tensor e ij (t), mean stress s(t) and mean strain E kk Convolutional constitutive relations between (t) and (t).

[0061]

[0062] In the formula, G(t) and K(t) are the shear relaxation modulus and volumetric relaxation modulus, respectively. They can be calculated using the tensile relaxation modulus E(t) and Poisson's ratio μ.

[0063]

[0064]

[0065] Step 3. Write down the Kirchhoff stress tensor S. ij Increment ΔS ij (t m ), that is, t m+1 Kirchhoff stress tensor S at time t ij (t m+1 ) and t m Kirchhoff stress tensor S at time t ij (t m The difference. Further, the increment ΔS ij (t m ) can be expressed as the increment of deviatoric stress Δs ij (t m ) and the increment of mean stress Δs(t) m The sum of ) and the increment of deviatoric stress Δs ij (t m ) equals t m+1 The deviatoric stress at time t and t m The difference in deviatoric stress at time t, the increment of mean stress Δs(t) m ) equals t m+1 The average stress at time t m The difference in average stress at time points.

[0066] Step 4. Using the relationships between deviatoric stress and deviatoric strain, and between mean stress and mean strain from Step 2, write down the increment of deviatoric stress Δs. ij (t m The relationship between ) and deviatoric strain, and the increment of mean stress Δs(t) m The relationship between the mean strain and the mean strain can be obtained by making a simple difference approximation.

[0067]

[0068] In the formula

[0069] ΔG(t m )=G(t m+1 )-G(t m ),

[0070] ΔK(t m )=K(t m+1 )-K(t m ).

[0071] Δt m =tm+1 -t m ,

[0072] Δe ij (t m ) = e ij (t m+1 )-e ij (t m ),

[0073] ΔE kk (t m ) = E kk (t m+1 )-E kk (t m ).

[0074] Step 5. Use the Prony series expansions of E(t), G(t), and K(t)

[0075]

[0076] And it is assumed that the time step is constant (i.e., Δt). m =Δt, m = 0, 1, 2, ...), and the relationship in step 4 (the increment of deviatoric stress Δs) ij (t m The relationship between ) and deviatoric strain, and the increment of mean stress Δs(t) m The relationship between the mean strain and the mean strain can be rewritten in the following incremental form:

[0077]

[0078] In the formula

[0079]

[0080] Step 6. For isotropic materials, the incremental constitutive relation from Step 5 can be written in matrix form as follows:

[0081]

[0082] In the formula

[0083]

[0084] ΔS(t m )=

[0085] [ΔS 11 (t m ) ΔS 22 (t m ) ΔS 33 (t m ) ΔS 23(t m ) ΔS 31 (t m ) ΔS 12 (t m )] T ,

[0086] ΔE(t m )=

[0087] [ΔE 11 (t m ) ΔE 22 (t m ) ΔE 33 (t m ) 2ΔE 23 (t m ) 2ΔE 31 (t m ) 2ΔE 12 (t m )] T ,

[0088]

[0089] Numerical simulations were performed using the viscoelastic constitutive relation obtained by this invention. The calculation results were compared with the experimental data of the specimens, and the calculation results and experimental results were in good agreement.

[0090] Table 1. Deviation between experimental and calculated values ​​of deformation.

[0091]

[0092]

[0093] Obviously, those skilled in the art can make various modifications and variations to the embodiments of the present invention without departing from the spirit and scope of the embodiments of the present invention. Thus, if these modifications and variations to the embodiments of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention also intends to include these modifications and variations.

Claims

1. A method for establishing large deformation viscoelastic constitutive relations based on Prony series, characterized in that, The specific steps are as follows: S1: The Kirchhoff stress tensor S ij Decomposed into the deviatoric stress tensor s ij and average stress The sum of these will give rise to the Green's strain tensor E. ij Decomposed into partial strain tensor e ij and mean strain sum; S2: Deviatoric stress tensor s ij (t) and the deviatoric strain tensor e ij (t), mean stress s(t) and mean strain The convolutional constitutive relations between them are as follows: In the formula, G(t) and K(t) are the shear relaxation modulus and the volume relaxation modulus, respectively; S3: the Kirschner stress tensor S ij the increment AS ij (t m ), also the Kirschner stress tensor S m+1 at time t ij (t m+1 ) and the Kirschner stress tensor S m at time t ij (t m ) S4: Using the relationships between deviatoric stress and deviatoric strain, and between mean stress and mean strain in S2, the increment Δs of the deviatoric stress is obtained. ij (t m The relationship between ) and deviatoric strain, and the increment of mean stress Δs(t) m The relationship between the strain and the mean strain can be approximated by a simple difference method, yielding the following relationship: In the formula ΔG(t m )=G(t m+1 )-G(t m ), ΔK(t m )=K(t m+1 )-K(t m ), Δt m =t m+1 -t m , Δe ij (t m )=e ij (t m+1 )-e ij (t m ), ΔE kk (t m )=E kk (t m+1 )-E kk (t m ); S5: Using the Prony series expansions of E(t), G(t), and K(t) And it is assumed that the time step is constant (i.e., Δt). m =Δt, m = 0, 1, 2, ...), and the relationship in S4 (the increment of deviatoric stress Δs) ij (t m The relationship between ) and deviatoric strain, and the increment of mean stress Δs(t) m The relationship between the mean strain and the mean strain can be rewritten in the following incremental form: In the formula S6: For isotropic materials, the incremental constitutive relation in S5 can be written in matrix form as follows: In the formula ΔS(t m )= [ΔS 11 (t m )ΔS 22 (t m )ΔS 33 (t m )ΔS 23 (t m )ΔS 31 (t m )ΔS 12 (t m )] T , ΔE(t m )= [ΔE 11 (t m )ΔE 22 (t m )ΔE 33 (t m )2ΔE 23 (t m )2ΔE 31 (t m )2ΔE 12 (t m )] T , 2. The method for establishing large deformation viscoelastic constitutive relations based on Prony series according to claim 1, characterized in that: The S2 shear relaxation modulus and volumetric relaxation modulus are calculated using the tensile relaxation modulus E(t) and Poisson's ratio μ.

3. The method for establishing large deformation viscoelastic constitutive relations based on Prony series according to claim 1, characterized in that: The S3 will increment ΔS ij (t m ) can be expressed as the increment of deviatoric stress Δs ij (t m ) and the increment of mean stress Δs(t) m The sum of ) and the increment of deviatoric stress Δs ij (t m ) equals t m+1 The deviatoric stress at time t and t m The difference in deviatoric stress at time t, the increment of mean stress Δs(t) m ) equals t m+1 The average stress at time t m The difference in average stress at time points.

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