A numerical simulation method for shape memory polymers based on hyper-viscoelastic constitutive model

Abstract

The invention discloses a shape memory polymer numerical simulation method based on hyperviscoelastic constitutive theory. The hyperviscoelastic constitutive model and the nonlinear finite element analysis method are respectively used as starting points. First, a shape memory polymer constitutive model based on hyperviscoelastic theory is designed and implemented in the finite element analysis software ABAQUS. Then, the thermo-solid coupling behavior of the shape memory polymer is simulated by finite element. The shape memory behavior of common structures such as beams, rods, plates and shells can be numerically simulated more accurately to a certain extent, and the commonly used mechanical response quantities such as stress, strain and displacement inside the structure can be obtained. The invention is simple and convenient, does not require experiments, saves costs, and can be used to control the shape memory function of the material in actual engineering.

Description

Technical Field

[0001] The invention belongs to the technical field of numerical simulation, and in particular relates to a numerical simulation method for thermo-solid coupling behavior of a shape memory polymer. Background Art

[0002] Shape memory polymer (SMP) is an emerging intelligent polymer material. It has attracted great attention from scientists due to its unique advantages such as large deformation, good biocompatibility, and human absorption. If you want to describe the shape memory behavior of the specific structure of SMP under complex stress states, it is difficult to directly apply the thermodynamic constitutive model of SMP. The most effective way is to use the finite element method to perform numerical simulation calculations and analysis to obtain the required results. However, since SMP is a new type of intelligent material, there is no corresponding material model in the existing finite element analysis software, so it is impossible to directly apply the software to perform numerical simulation and analysis on the mechanical behavior of the complex structure of SMP under complex stress states. Therefore, it is more urgent to develop a complete set of finite element simulation method systems.

[0003] Patent publication number CN103552696A proposes a frame-type spatially deployable structure based on shape memory polymers, which includes multiple shape memory polymer retractable units. The shape memory polymer sheet is heated by a heating element, thereby driving the telescopic sleeve to extend and deform, thereby realizing the deployment of a cubic spatially deployable frame structure. However, its structure is relatively simple, and no specific value of the driving force is given, so it is impossible to accurately and quantitatively control the deployment of the structure.

[0004] Patent publication number CN103303497A proposes a shape memory polymer intelligent torsion release mechanism with a spatial unfolding structure, which divides a hollow round rod into two parts, one of which is a shape memory polymer round rod segment that can be twisted, and the other is a shape memory polymer round rod segment that can be stretched. The structure is simple, but it only considers stretching and torsion deformation, and does not consider the deformation of the structure under complex stress conditions, which has certain limitations. Summary of the invention

[0005] In order to overcome the shortcomings of the prior art, the present invention provides a shape memory polymer numerical simulation method based on hyperviscoelastic constitutive theory, taking the hyperviscoelastic constitutive model and the nonlinear finite element analysis method as the starting point, firstly, a shape memory polymer constitutive model based on hyperviscoelastic theory is designed, and implemented in the finite element analysis software ABAQUS; then, the finite element simulation of the thermo-solid coupling behavior of the shape memory polymer is performed, and the shape memory behavior of common structures such as beams, rods, plates and shells can be numerically simulated more accurately to a certain extent, and the commonly used mechanical response quantities such as stress, strain and displacement inside the structure can be obtained. The present invention is simple and convenient, does not require experiments, saves costs, and can be used to control the shape memory function of the material in actual engineering.

[0006] The technical solution adopted by the present invention to solve the technical problem includes the following steps:

[0007] Step 1: Construct a hyper-viscoelastic constitutive model;

[0008] Step 1-1: Make the following assumptions about the SMP material properties: ① SMP is isotropic, homogeneous, and nonlinear during deformation; ② SMP is incompressible;

[0009] Step 1-2: Express the stress response of the material as the product of the relaxation part g(t) with time as the independent variable and the stress part σ0(ε) with strain as the independent variable, and construct a hyper-viscoelastic constitutive model:

[0010] σ(ε,t)=σ0(ε)g(t) (1)

[0011] For SMP, the stress part σ0(ε) with strain as the independent variable adopts the Neo-Hookean hyperelastic model to characterize its mechanical behavior, and the strain energy function expression is:

[0012]

[0013] Among them, U corresponds to the strain energy per unit volume, C 10 and D1 are both temperature-dependent material parameters, is the first deviatoric strain invariant, J el Indicates the elastic volume ratio of the material;

[0014] The relaxation part with time as the independent variable is expressed in the form of Prony series:

[0015]

[0016]

[0017] where g ∞ and g jBoth represent dimensionless constants; τ j represents the relaxation time of each Prony series term, and N represents the number of Prony series terms;

[0018] Step 1-3: Introduce the WLF equation to express the relationship between the translation function and temperature within 80°C above the critical transition temperature of the polymer:

[0019]

[0020] Where C1 and C2 are the basic constants of the WLF equation; T ref is the reference temperature of the material; α T (T) represents the horizontal shift factor, which is a function of temperature, and T represents the current temperature;

[0021] Step 2: Numerical simulation of thermo-solid coupling behavior of shape memory polymer;

[0022] Step 2-1: According to the thermo-mechanical cycle of SMP, the finite element simulation process is divided into the following four steps: setting a predefined temperature field T1 and setting a fixed constraint on the left end of the SMP specimen;

[0023] Step 2-2: During the high temperature deformation stage, an external load is applied to the SMP specimen to give it a pre-strain of ε m , keeping the temperature T1 constant, setting an axial elongation displacement Δl that increases linearly with time at the free end;

[0024] Step 2-3: Stress freezing stage, loading to pre-strain ε m After that, the SMP specimen is cooled down to make it glassy; the above conditions of the SMP specimen are kept unchanged, and the temperature field decreases to T2 over time;

[0025] Step 2-4: In the low-temperature unloading stage, the SMP specimen is unloaded and its shape is fixed; the temperature T2 is kept constant and the tensile displacement of the free end is removed;

[0026] Step 2-5: During the temperature rise recovery phase, the temperature of the SMP specimen is raised to T1 under the condition of zero stress, and the deformation of the SMP specimen gradually recovers and returns to its initial shape.

[0027] Preferably, C1 and C2 are 6.9 and 87.9 respectively, T ref The value is 80℃.

[0028] Preferably, T1 = 60°C, ε m =0.1, Δl=10mm, T=20℃.

[0029] Preferably, the specific steps of implementing the hyper-viscoelastic constitutive model in the finite element analysis software ABAQUS include:

[0030] S1: In ABAQUS, the viscoelastic behavior is described by the built-in generalized Maxwell model; the hyperelastic behavior is described by the built-in Neo-Hookean model; when setting the material properties, the viscoelastic parameters of the material are converted into discrete Prony series parameters g i ,τ i , g i ,τ i represent the shear modulus and relaxation time of SMP, respectively;

[0031] S2: Set up the Neo-Hookean model to describe the hyperelastic behavior of polymers and input the model material parameters C 10 , D1;

[0032] S3: Set up a generalized Maxwell model to describe the viscoelasticity of the polymer, inputting the relaxation time and shear modulus

[0033] S4: Set the coefficients in the WLF equation that describes the time-temperature equivalence principle of polymers, where T ref =80℃, C1=6.9, C2=87.9;

[0034] S5: Set the thermal expansion coefficient to describe the thermal effect of the polymer; complete the implementation of the hyper-viscoelastic constitutive model.

[0035] Preferably, the specific steps of performing finite element simulation on the thermosetting coupling behavior of the shape memory polymer are:

[0036] b1: Establish a three-dimensional model of the SMP structure;

[0037] b2: Define the material properties of SMP and create cross sections to assign to the structure to be calculated;

[0038] b3: Divide the structure into unit grids, using different types of units for different structures;

[0039] b4: define the four analysis steps in the shape memory process, namely high temperature deformation, stress freezing, low temperature unloading and heating recovery;

[0040] b5: Set corresponding boundary conditions for each analysis step, including constraints and loads;

[0041] b6: Set the ambient temperature field around the structure;

[0042] b7: Create a Job file and submit it for calculation.

[0043] The beneficial effects of the present invention are as follows:

[0044] 1. The present invention proposes a constitutive relationship for shape memory polymers based on the Neo-Hookean hyperelastic model and the generalized Maxwell viscoelastic model, and successfully implements the constitutive relationship in the finite element analysis software ABAQUS. The constitutive model implementation method does not require the writing of complex and cumbersome user material subroutines in Fortran language, but only requires the input of corresponding material parameters. The method is simple and easy to operate, which greatly shortens the development process.

[0045] 2. The present invention proposes a complete set of simulation processes for numerically simulating the thermosolid coupling behavior of shape memory polymer structures. The distribution of the internal stress field, strain field and displacement field of the SMP structure during the shape memory process can be obtained through the nonlinear finite element calculation function of the software ABAQUS. This method does not require experiments, is simple and convenient, reduces capital expenditure, and is conducive to engineering practice. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 Flowchart for setting up the hyper-viscoelastic constitutive model of the present invention in ABAQUS.

[0047] Figure 2 Flow chart of the SMP thermo-solid coupling finite element analysis of the present invention. DETAILED DESCRIPTION

[0048] The present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0049] The purpose of the present invention is first to use a hyper-viscoelastic constitutive model to define the complex constitutive relationship of thermotropic shape memory polymers, avoiding the need to write complex and cumbersome user material subroutines to achieve association with finite element analysis software. Secondly, the shape memory effect of the shape memory polymer structure under the action of thermal-mechanical coupling is simulated, and no experimental operation is required. Some commonly used thermodynamic response quantities such as stress, strain, displacement, etc. at each moment in the entire shape memory process can be obtained through finite element numerical simulation. Finally, a comparative analysis is performed with relevant experimental data to determine that the numerical simulation results can be used as a reference for the correct application of shape memory polymer engineering structures.

[0050] 1) Hyper-viscoelastic constitutive model

[0051] In order to describe the hyperelastic and viscoelastic behavior of shape memory polymers, the following assumptions need to be made about the properties of SMP materials: ① SMP is isotropic, homogeneous, and nonlinear during deformation. ② SMP is incompressible. Therefore, the stress response of the material can be expressed as the product of the relaxation part g(t) with time as the independent variable and the stress part σ0(ε) with strain as the independent variable:

[0052] σ(ε,t)=σ0(ε)g(t) (1)

[0053] For SMP, the stress part σ0(ε) with strain as the independent variable can be characterized by the Neo-Hookean hyperelastic model, and its strain energy function expression is:

[0054]

[0055] Among them, U corresponds to the strain energy per unit volume, C 10 and D1 are both temperature-dependent material parameters, is the first deviatoric strain invariant.

[0056] The time-dependent relaxation can be expressed in the form of a Prony series:

[0057]

[0058]

[0059] where g ∞ and g j are dimensionless quantities. Then σ0(ε) can describe the transient response of SMP when it is subjected to external loads. When the external load acts for a certain period of time, SMP will experience stress relaxation. When SMP is completely relaxed, its steady-state stress response is σ0(ε)g ∞ .

[0060] In addition, the WLF equation is introduced, which is obtained through various experiments. It can express the relationship between the translation function and temperature within 80°C above the critical transition temperature of the polymer:

[0061]

[0062] Where C1 and C2 are the basic constants of the WLF equation, which are usually taken as 6.9 and 87.9 respectively. ref It is the reference temperature of the material, which is 80℃ here.

[0063] In ABAQUS, the viscoelastic behavior can be described by the built-in generalized Maxwell model, and the hyperelastic behavior can be described by the built-in Neo-Hookean model. However, when setting the material properties, the viscoelastic parameters of the material must be converted into discrete Prony series parameters g. i ,τ i Among them, g i ,τ i represent the shear modulus and relaxation time of SMP, respectively.

[0064] The present invention is based on the hyper-viscoelastic constitutive model of SMP and is implemented in the finite element analysis software ABAQUS. The specific steps include: S1: setting a Neo-Hookean model that describes the hyperelastic behavior of polymers, wherein the model material parameters that need to be input are C 10 =1.1×10 8 , D1=1.7×10 -9 ; S2: Set the generalized Maxwell model to describe the viscoelasticity of the polymer. The corresponding relaxation time and shear modulus need to be input. The relevant data are shown in Table 1; S3: Set the coefficients in the WLF equation that describes the time-temperature equivalence principle of the polymer, where T ref =80℃, C1=6.9, C2=87.9; S4: Set the thermal expansion coefficient to describe the thermal effect of the polymer. The relevant data are shown in Table 2. The flow chart is as follows Figure 1 shown.

[0065] Table 1 Viscoelastic material parameters of SMP

[0066]

[0067] Table 2 Thermal expansion coefficient of SMP

[0068]

[0069] 2) Numerical simulation of thermo-solid coupling behavior of shape memory polymers

[0070] In addition to proposing a method for implementing the hyper-viscoelastic constitutive model in ABAQUS, the present invention also proposes a numerical simulation method for completing the shape memory process of various commonly used SMP structures under complex stress states based on the constitutive model and nonlinear finite element analysis.

[0071] Taking the SMP rod structure as an example, its shape memory process is as follows:

[0072] According to the thermo-mechanical cycle of SMP, the finite element simulation process can be divided into the following four steps. Since SMP is a viscoelastic material, the SMP specimen is in a rubber state at high temperature in the initial state and is prone to deformation. The predefined temperature field T = 60 ° C is set, and a fixed constraint is set on the left end of the specimen.

[0073] (1) In the high temperature deformation stage, an external load is applied to the SMP specimen to make its pre-strain ε m = 0.1. Keep the temperature T = 60°C, and set the axial elongation displacement Δl = 10 mm at the free end, which increases linearly with time.

[0074] (2) Stress freezing stage, loading to prestrain ε m=0.1, the sample is cooled to make it glassy. Keeping the above boundary conditions of the SMP sample unchanged, the temperature field slowly decreases to T = 20°C over time.

[0075] (3) Low-temperature unloading stage: The SMP specimen is slowly unloaded and its shape is fixed. The temperature T = 20°C is kept constant and the tensile displacement of the free end is slowly removed.

[0076] (4) Temperature recovery stage: when the stress is zero, the temperature of the specimen is raised to T = 60 °C, and its deformation gradually recovers and returns to its initial shape.

[0077] The shape memory process of other SMP structures is similar to the above, except that the deformation method is slightly different. However, the complex deformation of any structure can be composed of the five basic deformations in mechanics: tension, compression, bending, torsion and shear, and thus can be simulated accordingly in finite element software.

[0078] The specific steps include: b1: establish a three-dimensional model of the SMP structure; b2: define the material properties of the SMP, and create a cross section to assign to the structure to be calculated; b3: divide the structure into unit grids, and use different types of units for different structures; b4: define the four analysis steps in the shape memory process, namely high temperature deformation, stress freezing, low temperature unloading and temperature recovery; b5: set corresponding boundary conditions for each analysis step, including constraints and loads; b6: set the ambient temperature field around the structure; b7: create a Job file and submit it for calculation. The flow chart is as follows Figure 2 shown.

[0079] This method can be used to calculate the common mechanical response quantities such as stress, strain and displacement in the four stages of the SMP structure's shape memory process, and the numerical simulation results of this method have been verified many times and are in good agreement with the experimental results. This method does not require testing, and the deformation inside the structure can be clearly understood through the finite element analysis software ABAQUS, which has wide applicability.

[0080] In summary, the present invention provides a method for simulating various SMP structures to complete the shape memory process by means of finite element numerical simulation, and provides a complete set of calculation processes for simulating the thermo-solid coupling behavior of shape memory polymers in the finite element analysis software ABAQUS. Specific embodiment:

[0082] Shape memory polymer is a kind of stimulus-responsive polymer material, which has the characteristics of active controllable large deformation (20%-500%), degradability, various driving modes, good biocompatibility, etc. It is expected to replace some traditional metal materials as mechanical driving functions in the future. However, most of the current research on engineering structures made of shape memory polymer materials still relies on multiple tests, and the test costs generally consume a lot of manpower, material and financial resources, resulting in a large amount of material waste, and may not necessarily obtain ideal test results.

[0083] The present invention proposes a numerical simulation method for the thermo-solid coupling behavior of shape memory polymers based on a hyper-viscoelastic constitutive model, using the Neo-Hookean model and the generalized Maxwell model in the form of Prony series to characterize the hyperelastic and viscoelastic behaviors of SMPs respectively, and is implemented in the finite element analysis software ABAQUS. In addition, a complete set of numerical simulation methods for the shape memory process of SMPs is proposed, and the powerful nonlinear problem processing capability of ABAQUS is used to obtain the mechanical response quantities of some commonly used SMP structures through simulation calculations, including the stress, strain and displacement in the four stages of the shape memory process, which has important engineering practical significance.

[0084] 1. Implementation of hyper-viscoelastic constitutive model in ABAQUS

[0085] In step S1, open the Edit Material interface and select the Hyperelastic model. Then select Neo Hooke in the Strain energy potential and set the Moduli time scale to Instantaneous. Finally, enter C in the Data field. 10 and the value of D1.

[0086] In step S2, select the Viscoelastic model in the Edit Material interface, use the Prony series in the time domain to characterize the generalized Maxwell model, and enter the data in Table 1 in the Data field.

[0087] In step S3, add a Trs item in the Viscoelastic module to describe the time-temperature equivalence principle of the SMP material, select WLF in the Suboption Editor, and enter the values ​​of Theta 0, C1, and C2 in Data.

[0088] In step S4, select the Expansion thermal expansion model in the Edit Material interface, use the temperature-dependent thermal expansion coefficient, and enter the data in Table 2 in Data.

[0089] 2. Numerical simulation of SMP thermo-solid coupling based on nonlinear finite element analysis

[0090] In step b1, the SMP structural model to be calculated is established through the modeling function in ABAQUS software, and the model can also be imported from outside.

[0091] In step b2, after setting up the constitutive model of the SMP, create a material cross section and assign the cross section to the structural model.

[0092] In step b3, for general three-dimensional solid structures, select the unit type as C3D8RH, that is, the eight-node rectangular reduced integration unit, set the grid density to a moderate level, and automatically divide the grid by default.

[0093] In step b4, in addition to the software's default initial analysis step, four Visco viscosity analysis steps need to be created to describe the four stages of the shape memory process. Enter real time in Time period in seconds, turn on the Nlgeomon geometric nonlinearity switch, and set the analysis step load increment to Ramp linearly over step to increase over time.

[0094] In step b5, set the degree of freedom constraints of the boundary. The load is generally applied by setting a certain boundary displacement. The unloading function can be achieved by selecting Deactivated in the analysis step of the unloading phase.

[0095] In step b6, select Temperature in Create Predefined Field to create a temperature field, and assume that the temperature is evenly distributed around the structure. Ignore the heat conduction effect, and also set the corresponding temperature size in different analysis steps.

[0096] In step b7, create a Job work path and submit the calculation.

Claims

1. A numerical simulation method for shape memory polymer based on hyper-viscoelastic constitutive law, characterized in that: The steps include: Step 1: Construct a hyper-viscoelastic constitutive model; Step 1-1: Make the following assumptions about the SMP material properties: ① SMP is isotropic, homogeneous, and nonlinear during deformation; ② SMP is incompressible; Step 1-2: Express the stress response of the material as the product of the relaxation part g(t) with time as the independent variable and the stress part σ0(ε) with strain as the independent variable, and construct a hyper-viscoelastic constitutive model: σ(ε,t)=σ0(ε)g(t) (1) For SMP, the stress part σ0(ε) with strain as the independent variable adopts the Neo-Hookean hyperelastic model to characterize its mechanical behavior, and the strain energy function expression is: Among them, U corresponds to the strain energy per unit volume, C 10 and D1 are both temperature-dependent material parameters, is the first deviatoric strain invariant, J el Indicates the elastic volume ratio of the material; The relaxation part with time as the independent variable is expressed in the form of Prony series: where g ∞ and g j Both represent dimensionless constants; τ j represents the relaxation time of each Prony series term, and N represents the number of Prony series terms; Step 1-3: Introduce the WLF equation to express the relationship between the translation function and temperature within 80°C above the critical transition temperature of the polymer: Where C1 and C2 are the basic constants of the WLF equation; T ref is the reference temperature of the material; α T (T) represents the horizontal shift factor, which is a function of temperature, and T represents the current temperature; Step 2: Numerical simulation of thermo-solid coupling behavior of shape memory polymer; Step 2-1: According to the thermo-mechanical cycle of SMP, the finite element simulation process is divided into the following four steps: setting a predefined temperature field T1 and setting a fixed constraint on the left end of the SMP specimen; Step 2-2: During the high temperature deformation stage, an external load is applied to the SMP specimen to give it a pre-strain of ε m , keeping the temperature T1 constant, setting an axial elongation displacement △l that increases linearly with time at the free end; Step 2-3: Stress freezing stage, loading to pre-strain ε m After that, the SMP specimen is cooled down to make it glassy; the above conditions of the SMP specimen are kept unchanged, and the temperature field decreases to T2 over time; Step 2-4: In the low-temperature unloading stage, the SMP specimen is unloaded and its shape is fixed; the temperature T2 is kept constant and the tensile displacement of the free end is removed; Step 2-5: During the temperature rise recovery phase, the temperature of the SMP specimen is raised to T1 under the condition of zero stress, and the deformation of the SMP specimen gradually recovers and returns to its initial shape.

2. The method for numerical simulation of shape memory polymer based on super viscoelastic constitutive law according to claim 1, characterized in that: The C1 and C2 are 6.9 and 87.9 respectively, T ref The value is 80℃.

3. The method for numerical simulation of shape memory polymer based on super viscoelastic constitutive law according to claim 1, characterized in that: T1 = 60 ° C, ε m =0.1, △l=10mm, T=20℃.

4. The method for numerical simulation of shape memory polymer based on super viscoelastic constitutive law according to claim 1, characterized in that: The specific steps of implementing the hyper-viscoelastic constitutive model in the finite element analysis software ABAQUS include: S1: In ABAQUS, the viscoelastic behavior is described by the built-in generalized Maxwell model; the hyperelastic behavior is described by the built-in Neo-Hookean model; when setting the material properties, the viscoelastic parameters of the material are converted into discrete Prony series parameters g i ,τ i , g i ,τ i represent the shear modulus and relaxation time of SMP, respectively; S2: Set up the Neo-Hookean model to describe the hyperelastic behavior of polymers and input the model material parameters C 10 , D1; S3: Set up a generalized Maxwell model to describe the viscoelasticity of the polymer, inputting the relaxation time and shear modulus S4: Set the coefficients in the WLF equation that describes the time-temperature equivalence principle of polymers, where T ref =80℃, C1=6.9, C2=87.9; S5: Set the thermal expansion coefficient to describe the thermal effect of the polymer; complete the implementation of the hyper-viscoelastic constitutive model.

5. The method for numerical simulation of shape memory polymer based on super viscoelastic constitutive law according to claim 1, characterized in that: The specific steps of finite element simulation of the thermo-solid coupling behavior of the shape memory polymer are as follows: b1: Establish a three-dimensional model of the SMP structure; b2: Define the material properties of SMP and create cross sections to assign to the structure to be calculated; b3: Divide the structure into unit grids, using different types of units for different structures; b4: define the four analysis steps in the shape memory process, namely high temperature deformation, stress freezing, low temperature unloading and heating recovery; b5: Set corresponding boundary conditions for each analysis step, including constraints and loads; b6: Set the ambient temperature field around the structure; b7: Create a Job file and submit it for calculation.

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