CN116434885B - Viscoelastic fracture phase field calculation method considering tension and compression anisotropy of polymer composite material - Google Patents
Viscoelastic fracture phase field calculation method considering tension and compression anisotropy of polymer composite material Download PDFInfo
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Abstract
The invention provides a viscoelastic fracture phase field simulation analysis method considering tension and compression anisotropy of a polymer composite material, which comprises the steps of establishing a viscoelastic phase field fracture simulation model and initializing; obtaining stress-strain response of the polymer composite material containing the damage by adopting a standard finite element method by means of a power-assisted balance equation and a generalized Maxwell model; calculating elastic strain energy density after tensile and compressive decomposition based on a strain tensor spectrum decomposition method, and decomposing elastic strain energy of balanced and unbalanced branches into a stretching part and a compression part; calculating dissipation energy density and phase field driving energy; solving a phase field control equation by means of finite element discrete mode to obtain the degree of freedom of a phase field; substituting the calculated phase field degree of freedom into a solution decay function and repeatedly solving until a stop condition is reached. The method provided by the invention can accurately simulate the tension and compression anisotropy of the material, simulate the static and isostatic dependence of the viscoelastic material, and treat the loading rate effect.
Description
Technical Field
The invention belongs to the field of composite material fracture simulation analysis, and particularly relates to a polymer composite material viscoelasticity fracture phase field simulation analysis considering tension and compression anisotropy.
Background
Polymer composites have been increasingly used in a wide range of engineering fields such as aerospace, civil engineering, biomedical and electrical industries, and the like. The mechanical properties of the polymer composite exhibit more pronounced rate and time dependence, such as loading rate effects, creep and stress relaxation, than metallic materials. In addition, compressive strength is generally higher than tensile strength. These mechanical properties can influence the breaking behaviour of the material to a large extent. For example, experimental evidence suggests that polymer composites fail under long-term low tensile load conditions; instead, they can withstand long-term compressive loads. In order to ensure safe operation and reliability of critical structures utilizing such materials, their breaking behavior must be thoroughly understood to ensure accurate predictions. Therefore, the inclusion of these influencing factors into the fracture process model is of great importance for reducing risk.
In all crack simulation methods, the fracture phase field method (Phase Field Method to Fracture) does not need to track crack geometry, so that the advantages of complex fracture behaviors such as bifurcation fusion and the like are easy to calculate, and the fracture phase field method is rapidly developed in the last twenty years. In the fracture phase field method of the linear elastic material, elastic strain energy drives crack initiation and evolution. In order to avoid impractical cracking, it is generally assumed that (generalized) compressive stress does not cause material damage, breaking down elastic potential energy into tensile and compressive portions, and that only the elastic potential energy of the tensile portion is believed to drive the evolution of material damage and the formation of cracks. There are two general modes of tensile-compressive decomposition, one is body bias decomposition proposed by Amor, namely, according to the body amount and bias amount of strain, positive body amount strain and elastic strain energy caused by all bias amount strain are considered to drive crack generation, namely, only volume expansion and shear deformation can generate cracks, and elastic strain energy caused by negative body amount strain cannot drive crack generation; another is the strain tensor spectral decomposition proposed by Miehe, i.e., describing the elastic strain energy in terms of principal strain (and volume strain), which is believed to drive crack initiation, while negative principal strain and volume strain-induced elastic strain energy does not. The former decomposition treatment method has three disadvantages: (1) the Γ convergence criterion is not satisfied, i.e., it is not clear at present what energy and physical processes are recovered when the critical dimension tends to 0; (2) cracks and stiffness decay still occur under the condition of three-way negative strain, which are inconsistent with the existing test and theoretical analysis results (such as compression failure cannot occur under the condition of dynamic and static pressure); (3) uniaxial tensile compression strength ratio obtained by body bias decomposition modeGreatly limiting its application in pulling and pressing asymmetric materials. While the second method avoids these three disadvantages and is therefore employed in large quantities in practical applications.
For the viscoelastic phase field fracture model, the existing work mostly does not carry out tension-compression decomposition on elastic strain energy and cannot consider tension-compression anisotropy. The elastic strain energy is decomposed in a body bias decomposition mode in part of the work, and the reason is that the constitutive model of the viscoelastic material is usually described in a body bias mode, so that the tensile and compressive decomposition energy can be obtained more conveniently in the mode. However, it is conceivable that the viscoelastic fracture phase field method based on body bias decomposition still has three defects in the line elastic fracture phase field method based on body bias decomposition, so that the tension and compression anisotropy of the polymer composite material is difficult to accurately consider, and the fracture behavior simulation analysis of the polymer composite material cannot be accurately performed.
Disclosure of Invention
Aiming at the defects of the prior art, the invention needs to establish a fracture phase field model of the viscoelastic polymer composite material, and based on a generalized Maxwell model, the viscoelasticity of the volume strain and the viscoelasticity of the deflection strain are considered at the same time, the balance item and the unbalance item in the model are decomposed in a strain tensor spectrum decomposition mode, so that the problems that the viscoelasticity three-way negative strain still generates cracks and rigidity attenuation, the situation of high tension and compression dissimilarity degree cannot be considered are solved, and the mechanical behaviors of the polymer composite material under different static pressures are accurately considered.
The invention provides a simulation analysis method of a viscoelastic fracture phase field of a polymer composite material considering tension and compression anisotropy, which comprises the following steps:
s1, establishing a viscoelastic fracture phase field simulation model of a polymer composite material and initializing;
s2, obtaining total strain epsilon of the polymer composite material under the current time step through a force balance equation and a generalized Maxwell viscoelastic model and a standard finite element method ij Volumetric strain vol ε ii Deflection strain e ij Volume elastic strain of unbalanced itemUnbalanced term volume viscous strain->Unbalanced item elastic bias strain->And unbalanced item viscosity bias strain->
S3, based on the total strain epsilon of the polymer composite material obtained in the step S2 under the current time step ij Volumetric strain vol ε ii Deflection strain e ij Volume elastic strain of unbalanced itemUnbalanced term volume viscous strain->Unbalanced item elastic bias strain->And unbalanced item viscosity bias strain->Acquiring phase field driving H (t) of the polymer composite material;
s31, a strain tensor spectrum decomposition method is introduced to obtain elastic strain energy density of the polymer composite material after tensile compression decomposition;
s311, defining the elastic strain energy density psi of the balance item of the polymer composite material e,eq The method comprises the following steps:
wherein ,λeq Is balanced by the Ramez constant and hasK eq To balance the term bulk modulus, μ eq To balance the term shear modulus, ε is the total strain ε ij The main strain, tr (·) obtained is the trace sign;
s312, carrying out tension-compression decomposition on the elastic strain energy density of the balance item of the polymer composite material obtained in the S311 by adopting a strain tensor spectrum decomposition method, and obtaining the elastic strain energy density of the balance item of the polymer composite material after tension-compression decomposition:
wherein, + represents a tensile portion, -represents a compressive portion,<·> ± ＝(·±|·|)/2，tensile and compressive portions that balance the elastic strain energy density of the term;
s313 relaxation time of the mth volume Maxwell elementAnd relaxation time of mth offset Maxwell element +.>Taking the same, the relationship between the mth individual quantity Maxwell element and the mth offset Maxwell element is established:
wherein ,mτ Offsetting the relaxation time of the Maxwell element combinations for the mth volume;
s314, describing the total strain of the polymer composite material under the current time step according to elastic deformation and viscous deformation, and expressing the total strain of the polymer composite material as the sum of elastic strain and viscoelastic strain for a bulk and offset Maxwell assembly;
wherein ,indicating elastic strain and having-> and />Elastic volume strain and elastic bias strain, respectively, of the mth unbalanced item, +.>Indicates viscoelastic strain and has +.> and />Viscous volumetric strain and viscous partial strain, respectively, of the mth unbalanced term;
s315, obtaining the unbalanced term elastic strain energy density of the mth individual mass deflection Maxwell assembly:
wherein ,m λ ne is the m unbalanced item and has m K ne For the mth unbalanced term bulk modulus, m μ ne for the mth unbalanced term shear modulus, m ε e for passing->The resulting principal strain;
s316, carrying out tension-compression decomposition on the unbalanced term elastic strain energy density obtained in the S315 by adopting a strain tensor spectrum decomposition method, and obtaining the unbalanced term elastic strain energy density of the mth body mass deviation Maxwell combination after tension-compression decomposition:
wherein ,tensile and compressive portions that are the mth unbalanced term elastic strain energy density;
s317, combining the decomposed polymer composite material balance term elastic strain energy density obtained in the step S312And the unbalanced term elastic strain energy density of the decomposed mth individual mass deviation Maxwell combination calculated in the step S316 +.>The total elastic strain energy density of the polymer composite material after tensile and compressive decomposition is obtained:
wherein n is the unbalanced term number of the Maxwell model,tensile and compressive portions that are the total elastic strain energy density;
s32, obtaining the dissipation energy density psi in the viscoelastic fracture phase field model of the polymer composite material v ；
S33, acquiring phase field driving energy H (t) of polymer composite material damage evolution and crack formation according to the total elastic strain energy density of the polymer composite material after tension and compression decomposition obtained in the step S31 and the dissipated energy density in the polymer composite material viscoelastic fracture phase field model obtained in the step S32:
wherein ,βv For the dissipation energy contribution factor, t is the current solving step time, and τ is the history time;
s4, acquiring a polymer composite material phase field control equation and a phase field degree of freedom d by means of a finite element discrete mode on the premise of phase field driving H (t) of polymer composite material damage evolution and crack formation obtained in the step S3; the phase field control equation is:
s5, solving an attenuation function h (d) = (1-d) through the obtained phase field degree of freedom d 2 Then, the process goes to step S2, and the process is performed in a loop until the set stop condition is reached.
Further, the step S1 specifically includes the following steps:
s11, building a viscoelastic phase field fracture finite element simulation model of the polymer composite material according to finite element simulation requirements, dividing grids and setting boundary conditions;
s12, setting material parameters including generalized Maxwell viscoelastic constitutive model parameters and critical energy release rate G c Crack propagation width l c Crack formation dissipation energy contribution factor beta v ∈[0,1]；
S13, initializing the degree of freedom d of the viscoelastic fracture phase field of the polymer composite material to 0.
Further, the generalized Maxwell viscoelastic constitutive model parameters described in step S12 include an unbalanced term number n, and a balanced term bulk modulus K eq Shear modulus μ of balance term eq Bulk modulus of unbalanced term m K ne Shear modulus of unbalanced term m μ ne And relaxation time m τ。
Preferably, in step S2, the constitutive model based on the generalized Maxwell model is:
wherein , and />The balance term volume stress, balance term bias stress, mth unbalanced term volume stress and mth unbalanced term bias stress, respectively, which are equal to the total stress sigma ij The relation is that and />The volume viscosity system and the offset viscosity coefficient are respectively; the strain relation isg (d) is an attenuation function, which can be taken as g (d) = (1-d) 2 。
Preferably, the force balance equation in step S2 is:
wherein ,is the divergence operator and σ is the stress tensor.
Preferably, the step S32 specifically includes the following steps:
s321, acquiring each unbalanced viscous dissipation energy density in a viscoelastic fracture phase field model of the polymer composite material;
wherein , and />Respectively representing the mass dissipation energy density and the offset dissipation energy density of the mth unbalanced item,/> and />The volume and offset viscosity coefficients of the mth unbalanced term, respectively.
S321, converting each unbalanced viscous dissipation energy density in a viscoelastic fracture phase field model of the polymer composite material into a non-rate form;
s322 relaxation time of the volume introducing the mth unbalanced termAnd the relaxation time of the offset of the mth unbalanced term +.>Converting each unbalanced viscous dissipation energy density in the polymer composite viscoelastic fracture phase field model to,
wherein ,relaxation time of the volume of the mth unbalanced item and have +.> Relaxation time being the bias of the mth unbalanced term and having +.>
S323, obtaining the viscous dissipation energy psi of the polymer composite material fracture phase field model according to the unbalanced item quantity and the offset dissipation energy density in the polymer composite material viscoelastic fracture phase field model obtained in the step S322 v ；
wherein ,ψv Indicating the total dissipated energy density.
Compared with the prior art, the invention has the following beneficial effects:
(1) The invention is based on a generalized Maxwell model, and simultaneously considers the volume strain viscoelasticity and the deflection strain viscoelasticity, and decomposes the balance item and the unbalance item elasticity in the model by adopting a strain tensor spectrum decomposition mode, thereby solving the problems that the viscoelasticity three-way negative strain still generates cracks and stiffness attenuation and the situation that the tension-compression degree is higher cannot be considered;
(2) The method provided by the invention not only can accurately simulate the tension and compression anisotropy of the polymer composite material, but also can accurately simulate the mechanical behaviors of the polymer composite material under different static pressures;
(3) The invention can accurately simulate the influence of viscoelasticity on the mechanical response and fracture behavior of the polymer composite material, including loading rate effect, creep deformation, relaxation and other behaviors.
Drawings
FIG. 1 is a flow chart of a simulation analysis of a viscoelastic fracture phase field of a polymer composite material with tension and compression anisotropy considered in the invention;
FIG. 2 is a schematic diagram of a generalized Maxwell model of the present invention;
FIG. 3 is a graph comparing stress-strain curves of PBX9502 polymer composites obtained by the experiments and simulations of the present invention;
FIG. 4 is a schematic illustration of crack initiation, propagation and fracture processes during dumbbell stretching of the PBX9502 polymer composite material of the invention;
FIG. 5 is a graph showing compressive stress strain at various static pressures for EDC37 polymer composites obtained by the present invention;
FIG. 6 is a graph showing compressive stress strain at different static pressures for EDC37 polymer composites simulated in accordance with the present invention;
FIG. 7 is a graph showing the compressive strength of EDC37 polymer composites of the present invention at various hydrostatic pressures;
FIG. 8 is a schematic illustration of a geometric sample of a polymer composite in an embodiment of the loading rate influencing according to the present invention;
FIG. 9 is a schematic diagram of fracture morphology of a polymer composite material simulated in a loading rate-affecting example of the present invention;
FIG. 10 is a graph of load-elongation curves for polymer composites at different loading rates in the loading rate influencing examples of the present invention.
Detailed Description
Exemplary embodiments, features and aspects of the present invention will be described in detail below with reference to the attached drawings. In the drawings, like reference numbers indicate identical or functionally similar elements. Although various aspects of the embodiments are illustrated in the accompanying drawings, the drawings are not necessarily drawn to scale unless specifically indicated.
The invention provides a viscoelastic fracture phase field simulation analysis of a polymer composite material considering tension and compression anisotropy, which is shown in fig. 1, and specifically comprises the following steps:
s1, establishing a viscoelastic fracture phase field simulation model of a polymer composite material and initializing;
s11, building a viscoelastic phase field fracture finite element simulation model of the polymer composite material according to finite element simulation requirements, dividing grids and setting boundary conditions;
s12, setting material parameters including generalized Maxwell viscoelastic constitutive model parameters, critical energy release rate Gc and crack propagation width l c Crack formation dissipation energy contribution factor beta v ∈[0,1]Generalized Maxwell viscoelastic constitutive model parameters include non-flatNumber of balance terms n, bulk modulus of balance term K eq Shear modulus μ of balance term eq Bulk modulus of unbalanced term m K ne Shear modulus of unbalanced term m μ ne And relaxation time m τ。
S13, initializing the degree of freedom d of the viscoelastic fracture phase field of the polymer composite material to 0.
S2, obtaining total strain epsilon of the polymer composite material under the current time step through a force balance equation and a generalized Maxwell viscoelastic model and a standard finite element method ij Volumetric strain vol ε ii Deflection strain e ij Volume elastic strain of unbalanced itemUnbalanced term volume viscous strain->Unbalanced item elastic bias strain->Viscous bias strain of unbalanced item
The constitutive model based on the generalized Maxwell model is as follows:
wherein , and />The balance term volume stress, balance term bias stress, mth unbalanced term volume stress and mth unbalanced term bias stress, respectively, which are equal to the total stress sigma ij The relation is that and />The volume viscosity system and the offset viscosity coefficient are respectively; the strain relation is->g (d) is the decay function, taken as g (d) = (1-d) 2 。
A generalized maxwell model taking into account both volume deformation viscoelasticity and shear deformation viscoelasticity is shown in fig. 2.
The force balance equation is:
wherein ,is a divergence operator, and σ is a stress tensor.
S3, based on the total strain epsilon of the polymer composite material obtained in the step S2 under the current time step ij Volumetric strain vol ε ii Deflection strain e ij Volume elastic strain of unbalanced itemUnbalanced term volume viscous strain->Unbalanced item elastic bias strain->And unbalanced item viscosity bias strain->Acquiring phase field driving H (t) of the polymer composite material;
s31, a strain tensor spectrum decomposition method is introduced to obtain elastic strain energy density of the polymer composite material after tensile compression decomposition;
s311, defining the elastic strain energy density psi of the balance item of the polymer composite material e,eq The method comprises the following steps:
wherein ,λeq Is balanced by the Ramez constant and hasK eq To balance the term bulk modulus, μ eq To balance the term shear modulus, ε is the total strain ε ij The main strain, tr (·) obtained is the trace sign;
s312, carrying out tension-compression decomposition on the elastic strain energy density of the balance item of the polymer composite material obtained in the S311 by adopting a strain tensor spectrum decomposition method, and obtaining the elastic strain energy density of the balance item of the polymer composite material after tension-compression decomposition;
wherein, + represents a tensile portion, -represents a compressive portion,<·> ± ＝(·±|·|)/2，tensile and compressive portions that balance the elastic strain energy density of the term;
s313 relaxation time of the mth volume Maxwell elementAnd relaxation time of mth offset Maxwell element +.>Taking the m-th volume Maxwell element as the same, and establishing a connection between the m-th volume Maxwell element and the m-th offset Maxwell element;
wherein ,m τ is the relaxation time of the mth bulk bias Maxwell element combination;
s314, describing the total strain of the polymer composite material under the current time step according to elastic deformation and viscous deformation, and expressing the total strain of the polymer composite material as the sum of elastic strain and viscoelastic strain for a bulk and offset Maxwell assembly;
wherein ,indicating elastic strain and having-> and />Elastic volume strain and elastic bias strain, respectively, of the mth unbalanced item, +.>Indicates viscoelastic strain and has +.> and />Viscous volumetric strain and viscous partial strain, respectively, of the mth unbalanced term;
s315, obtaining the unbalanced term elastic strain energy density of the mth individual mass deflection Maxwell assembly;
wherein ,m λ ne is the m unbalanced item and has m K ne For the mth unbalanced term bulk modulus, m μ ne for the mth unbalanced term shear modulus, m ε e for passing->The resulting principal strain;
s316, carrying out tension-compression decomposition on the unbalanced term elastic strain energy density obtained in the S315 by adopting a strain tensor spectrum decomposition method, and obtaining each unbalanced term elastic strain energy density of the mth body mass deviation Maxwell combination after tension-compression decomposition;
wherein ,tensile and compressive portions of elastic strain energy density for the mth imbalance term.
S317, combining the decomposed polymer composite material balance term elastic strain energy density obtained in the step S312And the unbalanced term elastic strain energy density of the decomposed mth individual mass deviation Maxwell combination calculated in the step S316 +.>The total elastic strain energy density of the polymer composite material after the tensile and compressive decomposition is obtained;
wherein n is the unbalanced term number of the Maxwell model,is the tensile and compressive portion of the total elastic strain energy density.
S32, obtaining the dissipation energy density psi in the viscoelastic fracture phase field model of the polymer composite material v ；
S321, converting each unbalanced viscous dissipation energy density in a viscoelastic fracture phase field model of the polymer composite material into a non-rate form;
wherein , and />Respectively representing the mass dissipation energy density and the offset dissipation energy density of the mth unbalanced item,/> and />The volume and offset viscosity coefficients of the mth unbalanced term, respectively.
S321, converting each unbalanced viscous dissipation energy density in a viscoelastic fracture phase field model of the polymer composite material into a non-rate form;
s322 relaxation time of the volume introducing the mth unbalanced termAnd the relaxation time of the offset of the mth unbalanced term +.>Bonding a polymer composite materialThe unbalanced viscous dissipation energy densities in the elastic fracture phase field model are converted into,
wherein ,relaxation time of the volume of the mth unbalanced item and have +.> Relaxation time being the bias of the mth unbalanced term and having +.>
S323, obtaining the viscous dissipation energy psi of the polymer composite material fracture phase field model according to the unbalanced item quantity and the offset dissipation energy density in the polymer composite material viscoelastic fracture phase field model obtained in the step S322 v ；
S33, acquiring phase field driving energy H (t) of polymer composite material damage evolution and crack formation according to the total elastic strain energy density of the polymer composite material after tension and compression decomposition obtained in the step S31 and the dissipated energy density in the polymer composite material viscoelastic fracture phase field model obtained in the step S32;
wherein ,βv To dissipate the energy contribution factor, t is the current solution step time and τ is the history time.
S4, acquiring a polymer composite material phase field control equation and a phase field degree of freedom d by means of a finite element discrete mode on the premise of phase field driving H (t) of polymer composite material damage evolution and crack formation obtained in the step S3; the phase field control equation is:
s5, solving an attenuation function g (d) = (1-d) through the obtained phase field degree of freedom d 2 Then, the process jumps to step S2, and the solution is circularly carried out until the set stop condition is reached.
The effects of the present invention are shown below in three examples.
And (3) displaying the pulling and pressing opposite effect: polymer composite with object named PBX9502 with strain rate of 10 -4 The material parameters are shown in Table 1. The simulation calculation adopts a central symmetry model, beta v ＝0，l 0 =0.2 mm, cell size is taken as l 0 /h e ＝2。
TABLE 1
| K eq | μ eq | 1 μ ne | 1 τ | G c |
| 1425MPa | 380MPa | 1130.5MPa | 25.12s | 13J/m2 |
As shown in fig. 3, the calculated tensile compressive stress strain curve is well matched with the test data. The tensile strength and the compressive strength obtained by calculation are respectively 4.65MPa and 12.28MPa, and the compression tensile strength ratio is 2.64; the tensile strength and the compressive strength obtained by the experiment are respectively 4.53MPa and 12.52MPa, and the compressive tensile strength is 2.76; the tensile and compressive strength errors were 1% and-2.2%, respectively. In the embodiment, the tensile curve and the compression curve of the material can be accurately described by only adopting a first-order generalized Maxwell model. Fig. 4 shows crack initiation and propagation to fracture during dumbbell stretching. By means of simulation analysis of uniaxial compression-tension fracture behaviors, the tension-compression opposite characteristic of the PBX material can be accurately described by adopting a phase field method of strain tensor spectrum decomposition in combination with a proper constitutive model.
Mechanical property display under static pressure: the object was a polymer composite named EDC37, the test results of which are shown in fig. 5, whose mechanical properties under different static pressures are not uniform. The greater the static pressure, the higher the compressive strength, the greater the maximum compressive stress corresponds to the compressive strain, the more stress is maintained beyond the maximum pressure, and the less likely it is to yield. The calculation model is carried out by adopting a three-dimensional unit, the x-, y-and z-planes are simply supported and constrained, the x+ and y+ planes are free, static and equal pressure is firstly applied to the x+, y+ and z+ planes in the loading process, on the basis, the z+ plane is further loaded with equal strain rate through displacement, and EDC37 material parameters are shown in a table 2.
TABLE 2
| K eq | μ eq | 1 μ ne | 1 τ | G c |
| 444.6MPa | 263.6MPa | 167.7MPa | 25.12s | 5J/m2 |
The stress-strain curves obtained by simulation and test under different static pressures are shown in figure 6, and the rules of the stress-strain curves are consistent. As can be further seen from fig. 7, the compressive strength increases linearly with increasing static isostatic pressure; the slopes of the compression strength obtained by experiments and simulation along with the change of static pressure and equal pressure are 2.11MPa/MPa and 2.19MPa/MPa respectively, and the error is 2.8%; by the simulation method, the compressive stress strain curves and compressive strength under different static equal pressures can be accurately described.
Demonstration of load Rate Effect: the object of this example is a two-sided crack polymer composite slab stretch breaking process. The geometry is shown in fig. 8, and the initial cracks with a length of a=7mm are preformed on the left and right sides of the middle, wherein the width is 30mm, the length is 75mm, and the thickness is 2 mm. The bottom is fixed, and the top is subjected to displacement loading and stretching to fracture by adopting different loading rates (the loading rates are 25,50,75,100,200 mm/min). The simulation model adopts a two-dimensional plane stress model, constraint conditions are the same as the experimental process, the bottom is completely fixed, the top is used for constraining displacement in the horizontal direction, and meanwhile, the fixed displacement loading rate is installed for stretching loading. The crack propagation width is taken as l 0 Cell size at the central fracture region (5 mm high) was taken as h =0.5 mm e =0.1667 mm, other region unit size is taken as h e =0.6667 mm, material parameters are shown in table 3.
TABLE 3 Table 3
| K eq | μ eq | 1 μ ne | 1 τ | G c |
| 24.91MPa | 1.549MPa | 1.028MPa | 2.84s | 3.8N/mm |
The fracture morphology obtained by simulation is shown in fig. 9. As can be seen from fig. 10, with increasing loading rate (from 25mm/min to 200 mm/min), initial stiffness increases gradually, peak load increases gradually, and elongation at break decreases gradually, which is consistent with the mechanical properties of the viscoelastic material, demonstrating that the method can effectively react to the viscoelastic stress-strain response and fracture characteristics of the material.
The above examples are only illustrative of the preferred embodiments of the present invention and are not intended to limit the scope of the present invention, and various modifications and improvements made by those skilled in the art to the technical solution of the present invention should fall within the scope of protection defined by the claims of the present invention without departing from the spirit of the present invention.
Claims (6)
1. A polymer composite viscoelastic fracture phase field simulation analysis method considering tension and compression anisotropy comprises the following steps:
s1, establishing a viscoelastic fracture phase field simulation model of a polymer composite material and initializing;
s2, obtaining total strain epsilon of the polymer composite material under the current time step through a force balance equation and a generalized Maxwell viscoelastic model and a standard finite element method ij Volumetric strain vol ε ii Deflection strain e ij Volume elastic strain of unbalanced itemUnbalanced term volume viscous strain->Unbalanced item elastic bias strain->Viscous bias strain of unbalanced item
S3, based on the total strain epsilon of the polymer composite material obtained in the step S2 under the current time step ij Volumetric strain vol ε ii Deflection strain e ij Volume elastic strain of unbalanced itemUnbalanced term volume viscous strain->Unbalanced term elastic deflection strainAnd unbalanced item viscosity bias strain->Acquisition ofPhase field driving H (t) of the polymer composite;
s31, a strain tensor spectrum decomposition method is introduced to obtain elastic strain energy density of the polymer composite material after tensile compression decomposition;
s311, defining the elastic strain energy density psi of the balance item of the polymer composite material e,eq The method comprises the following steps:
wherein ,λeq Is balanced by the Ramez constant and hasK eq To balance the term bulk modulus, μ eq To balance the term shear modulus, ε is the total strain ε ij The main strain, tr (·) obtained is the trace sign;
s312, carrying out tension-compression decomposition on the elastic strain energy density of the balance item of the polymer composite material obtained in the S311 by adopting a strain tensor spectrum decomposition method, and obtaining the elastic strain energy density of the balance item of the polymer composite material after tension-compression decomposition:
wherein, + represents a tensile portion, -represents a compressive portion,<·> ± ＝(·±|·|)/2，tensile and compressive portions that balance the elastic strain energy density of the term;
s313 relaxation time of the mth volume Maxwell elementAnd relaxation time of mth offset Maxwell elementTaking the same to establishAssociation of mth volume Maxwell element and mth offset Maxwell element:
wherein ,m τ is the relaxation time of the mth bulk bias Maxwell element combination;
s314, describing the total strain of the polymer composite material at the current time step according to elastic deformation and viscous deformation, and for a bulk and offset Maxwell combination, expressing the total strain as the sum of elastic strain and viscoelastic strain:
wherein ,indicating elastic strain and having-> and />Elastic volume strain and elastic bias strain, respectively, of the mth unbalanced item, +.>Indicates viscoelastic strain and has +.> Andviscous volumetric strain and viscous partial strain, respectively, of the mth unbalanced term;
s315, obtaining the unbalanced term elastic strain energy density of the mth individual mass deflection Maxwell assembly:
wherein ,m λ ne is the m unbalanced item and has m K ne For the mth unbalanced term bulk modulus, m μ ne for the mth unbalanced term shear modulus, m ε e for passing->The resulting principal strain;
s316, carrying out tension-compression decomposition on the unbalanced term elastic strain energy density obtained in the S315 by adopting a strain tensor spectrum decomposition method, and obtaining the unbalanced term elastic strain energy density of the mth body mass deviation Maxwell combination after tension-compression decomposition:
wherein ,tensile and compressive portions that are the mth unbalanced term elastic strain energy density;
s317, combining the decomposed polymer composite material balance term elastic strain energy density obtained in the step S312And the unbalanced term elastic strain energy density of the decomposed mth individual mass deviation Maxwell combination calculated in the step S316 +.>The total elastic strain energy density of the polymer composite material after tensile and compressive decomposition is obtained:
wherein n is the unbalanced term number of the Maxwell model,tensile and compressive portions that are the total elastic strain energy density;
s32, obtaining the dissipation energy density psi in the viscoelastic fracture phase field model of the polymer composite material v ；
S33, acquiring phase field driving energy H (t) of polymer composite material damage evolution and crack formation according to the total elastic strain energy density of the polymer composite material after tension and compression decomposition obtained in the step S31 and the dissipated energy density in the polymer composite material viscoelastic fracture phase field model obtained in the step S32:
wherein ,βv For the dissipation energy contribution factor, t is the current solving step time, and τ is the history time;
s4, acquiring a polymer composite material phase field control equation and a phase field degree of freedom d by means of a finite element discrete mode on the premise of phase field driving H (t) of polymer composite material damage evolution and crack formation obtained in the step S3; the phase field control equation is:
s5, solving an attenuation function g (d) = (1-d) through the obtained phase field degree of freedom d 2 Then, the process goes to step S2, and the process is performed in a loop until the set stop condition is reached.
2. The method for simulating the viscoelastic fracture phase field analysis of a polymer composite material taking account of stretch-compression anisotropy according to claim 1, wherein the step S1 specifically comprises the steps of:
s11, building a viscoelastic phase field fracture finite element simulation model of the polymer composite material according to finite element simulation requirements, dividing grids and setting boundary conditions;
s12, setting material parameters including generalized Maxwell viscoelastic constitutive model parameters and critical energy release rate G c Crack propagation width l c Crack formation dissipation energy contribution factor beta v ∈[0,1]；
S13, initializing the degree of freedom d of the viscoelastic fracture phase field of the polymer composite material to 0.
3. The method for simulating analysis of a viscoelastic fracture phase field of a polymer composite material with consideration of tension and compression anisotropy according to claim 1, wherein the generalized Maxwell viscoelastic constitutive model parameters in step S12 include an unbalanced term number n, a balanced term bulk modulus K eq Shear modulus μ of balance term eq Bulk modulus of unbalanced term m K ne Shear modulus of unbalanced term m μ ne And relaxation time m τ。
4. The method for simulating and analyzing the viscoelastic fracture phase field of the polymer composite material with consideration of the tension and compression anisotropy according to claim 1, wherein the constitutive model based on the generalized Maxwell model in the step S2 is as follows:
wherein , and />The balance term volume stress, balance term bias stress, mth unbalanced term volume stress and mth unbalanced term bias stress, respectively, which are equal to the total stress sigma ij The relationship is-> and />The volume viscosity system and the offset viscosity coefficient are respectively; the strain relation is epsilon ij ＝ vol ε ij +e ij ,/>g (d) is the decay function, taken as g (d) = (1-d) 2 。
5. The method for simulating analysis of viscoelastic fracture phase field of polymer composite material according to claim 1, wherein the force balance equation in step S2 is:
wherein ,is a divergence operator, and σ is a stress tensor.
6. The method for simulating analysis of viscoelastic fracture phase field of polymer composite material according to claim 1, wherein step S32 comprises the steps of:
s321, acquiring each unbalanced viscous dissipation energy density in a viscoelastic fracture phase field model of the polymer composite material;
wherein , and />Respectively representing the mass dissipation energy density and the offset dissipation energy density of the mth unbalanced item,/> and />The volume and offset viscosity coefficients of the mth unbalanced term respectively;
s321, converting each unbalanced viscous dissipation energy density in a viscoelastic fracture phase field model of the polymer composite material into a non-rate form;
s322 relaxation time of the volume introducing the mth unbalanced termAnd the relaxation time of the offset of the mth unbalanced term +.>Converting each unbalanced viscous dissipation energy density in the polymer composite viscoelastic fracture phase field model to:
wherein ,relaxation time of the volume of the mth unbalanced item and have +.> Relaxation time being the bias of the mth unbalanced term and having +.>
S323, obtaining the viscous dissipation energy psi of the polymer composite material fracture phase field model according to the unbalanced item quantity and the offset dissipation energy density in the polymer composite material viscoelastic fracture phase field model obtained in the step S322 v ；
wherein ,ψv Indicating the total dissipated energy density.
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