CN112665962A - Prediction method of material refined crack - Google Patents
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Abstract
The invention provides a prediction method of a material refined crack, which comprises the following steps: obtaining relevant mechanical parameters of the material; modeling according to the actual situation of the object to be simulated, and substituting the parameters obtained in the step 1 into the established model; establishing a fracture theory of the phase field model in a complex failure mode so that the phase field model can effectively simulate fracture conditions of multiple failure modes; establishing a refinement process theory so as to accurately simulate a crack path and an expansion process; applying an actual loading process to the model; and solving the cracking process of the model according to the established theory. The invention can implicitly obtain the cracking damage process of a simulated object under various external force conditions, does not need to increase additional judgment criteria, can provide theoretical basis and guidance for whether the material cracks under a certain environment and how to crack after cracking, and solves the problems that the traditional method can only carry out crack fitting but not crack prediction and the simulated crack is too coarse.
Description
Technical Field
The invention belongs to the technical field of material mechanics research, and particularly relates to a prediction method of a refined crack of a material.
Background
In recent years, various fracture numerical simulation methods have been developed, including a finite element expansion method, a boundary element method, a gridless method, etc., but these methods must explicitly track the path of the crack, that is, additional judgment criteria must be used to determine whether the material is cracked. In addition, most crack simulation theories existing at present are still in the stage of crack fitting, that is, before a crack path is obtained, the grid refinement division of a finite unit cannot be performed on a numerical model, and the division process of the grid needs to be improved manually by continuously repeating calculation. In addition, the value of the fracture energy directly influences the calculation result, and the fracture energy is difficult to be accurately measured in a physical experiment, so that the parameter acquisition is influenced, and the efficiency of numerical simulation is reduced.
In summary, the existing methods and techniques for simulating fracture have the following disadvantages: (1) the simulated damage mode is single, and the complicated damage process cannot be simulated; (2) the crack path must be explicitly solved by a judgment criterion; (3) the finite elements in the numerical model cannot be automatically and reasonably refined, so that the calculation precision is limited; (4) it is difficult to accurately measure the fracture energy.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides the prediction method of the material fine cracks, which can accurately simulate the fine cracks of brittle materials such as rocks and metals and quasi-brittle materials, and simultaneously improve the crack paths and the precision of the calculation results.
In order to solve the technical problems, the invention adopts the following technical scheme:
a prediction method of material refined cracks comprises the following steps:
step 1: carrying out a tensile test to obtain relevant mechanical parameters of the material;
step 2: modeling according to the actual situation of the object to be simulated, and substituting the parameters obtained in the step 1 into the established model;
and step 3: establishing a fracture theory of the phase field model in a complex failure mode so that the phase field model can effectively simulate fracture conditions of multiple failure modes;
and 4, step 4: establishing a refinement process theory so as to accurately simulate a crack path and an expansion process;
and 5: applying an actual loading process to the model;
step 6: and solving the cracking process of the model according to the theories established in the steps 3 and 4.
Further, the parameters include modulus of elasticity, poisson's ratio, axial tensile energy to break, tensile strength, and the ratio of critical shear strength to critical tensile strength to break.
Further, the axial tensile failure energy of the material is obtained by the following formula:
Gcfor the fracture energy, dU is the energy dissipation during fracture of the test piece and dA is the fracture area.
Further, the axial tensile failure energy of the material is obtained by the following formula: and for the fracture area dA, monitoring the crack propagation state and the crack area in real time by using an ultrasonic transmitter, and inputting the monitoring result into a computer for data analysis, thereby obtaining the real crack state and area at any loading moment.
Further, the control equation of the phase field model established in step 3 is:
in the formula: s is an element of [0,1 ]]As a crack phase field variable, /)0In order to obtain a diffuse width of the crack,
for tensile stresses perpendicular to the plane of failure,
n is a direction vector perpendicular to the plane of fracture, m is a direction vector parallel to the plane of fracture, psi0Energy density before degradation,. psi0IIIs psi0Is a shear part ofI(s) and ωII(s) are two degenerate functions, the first derivative of a variable, e.g. ω, being added by a primeI'(s) is omegaI(s) a first derivative with respect to s,
is the Hamiltonian, σ is the degenerated stress tensor, D is the degenerated elastic matrix, ε is the strain tensor,
α(s) is the geometric fracture function, E is the elastic modulus, μ is the shear modulus, GcIAnd GcIIRespectively, I-type crack energy and II-type crack energy, tausAnd σtUltimate tensile stress and ultimate shear stress, respectively, and defining variable x ═ taus/σtAnd is and
ωI(s)、ωIIthe relationship of(s) to the phase field variable s is written as:
wherein: : bI、bIIThe ability to control the degradation function is in [0,1 ]]Internal value taking; in a pure tensile failure state, bI＝1，b II0; in the pure shearing state, bI＝0，b II1 is ═ 1; in general, b can also be usedI＝bIIThe mixed model which is 1 improves the iteration efficiency while ensuring the calculation precision, phi(s) and Q(s) are polynomials of phase field variable s and are used for determining the form of a degradation function, p and aiThe parameters to be determined are determined from the material parameters and the specific softening curve.
Further, the establishment of the refinement process theory in step 4 is to enable the model to obtain the most reasonable finite element distribution through a mesh refinement algorithm, and the specific theory is as follows:
a multi-node quadrilateral unit is established, wherein the number of nodes in the unit ranges from 4 to 8, and the shape function of the unit is constructed as follows:
wherein N isn(xi, η) is a shape function of any point in the unit at the node n, and for nodes No. 5, 6, 7 and 8, when the shape function value does not exist, the corresponding shape function value becomes 0; (xi, η) is the local coordinate of any point in the unit, and the local coordinate of each node is:
further, the refinement process in step 4 further includes establishing a set of refinement criteria, and the specific strategy is as follows:
when the cells are initially divided, all quadrilateral 4-node cells are adopted, when the phase field value of a certain node reaches a certain critical value, a node is added on the boundary of the cell adjacent to the cell, and when the number of the nodes of any cell reaches 8, the cell is divided into 4 four-node small cells;
for the node phase field critical value, five-level division of 0.2/0.4/0.6/0.8/1.0 is adopted, each level is divided once, and meanwhile, the difference between adjacent units cannot exceed two levels.
Further, in the step 6, an iterative method or a step-by-step decoupling algorithm is adopted to solve the cracking process of the model.
Furthermore, the step-by-step decoupling algorithm is that in each loading step, after the phase field control process is solved, a phase field model control equation and a grid refinement algorithm need to be carried out in a circulating mode until the grid refinement degree meets the simulation precision requirement, the crack expansion state and the stress displacement data in the step are obtained, the data are substituted into the next loading step for analysis until the test piece is completely broken, and the data obtained in each loading step are read, so that the crack expansion process and the distribution state can be obtained.
Compared with the prior art, the invention has the beneficial effects that:
1) the invention can implicitly obtain the cracking damage process of a simulated object under various external force conditions, does not need to increase additional judgment criteria, can provide theoretical basis and guidance for whether the material cracks under certain environment and how to crack after cracking, and solves the problems that the traditional method can only carry out crack fitting but not crack prediction and the simulated crack is too coarse;
2) the universal phase field method for simulating different failure modes of the brittle material provided by the invention has the advantages that an energy equation is divided into a tension part, a sheared part and a compression part for simulation, double degradation functions are adopted to respectively represent the contribution degrees of normal stress and tangential stress to crack expansion, a unified fracture criterion is adopted, the relation between the compressive strength and the tensile strength of the material is brought into theory, a mixed model is created, the calculation accuracy is ensured, and the iteration efficiency is greatly improved; the method for accurately calculating the crack direction is established, so that the problem that the crack direction is difficult to accurately predict by the traditional method can be solved;
3) the universal phase field method for simulating different failure modes of the brittle material is established based on the relationship of tensile, compressive and shear resistance of the material, the constraint that the phase field theory can only simulate the failure mode under the tensile condition is broken through by applying the unified failure theory, and the accurate simulation of the failure process and the crack path of the brittle material under the complex boundary condition is realized under other possible failure modes (pure shear, tension shear and compression shear); the parameters obtained by the uniaxial tension test and the uniaxial compression test are utilized to realize the prediction of the crack propagation direction under any loading path;
4) the method has the advantages of simple implementation process, strong practicability and wide application range, can provide theoretical and technical support for crack propagation simulation problems of brittle materials and quasi-brittle materials in different failure modes, and promotes the finite element and phase field models to accurately simulate the brittle solid materials such as rocks, metals, ceramics and the like.
Drawings
FIG. 1 is a flowchart of a method for predicting a refined crack of a material according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of dimensions and boundary conditions of a uniaxially pressed sample in accordance with an embodiment of the present invention;
FIG. 3 is a graph of actual crack distribution measured in an indoor uniaxial compression test according to an embodiment of the invention;
FIG. 4 is a monitoring roadmap for ultrasound infrared techniques according to embodiments of the present invention;
FIG. 5 is a schematic diagram illustrating the result of the crack propagation path simulation of a uniaxial compression test specimen according to an embodiment of the invention;
FIG. 6 is a schematic diagram of a material fine crack predicted by an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be described clearly and completely with reference to the following embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict.
The present invention is further illustrated by the following examples, which are not to be construed as limiting the invention.
Referring to fig. 1, the present invention provides a method for predicting a refined crack of a material, comprising the following steps:
step 1: carrying out a tensile test to obtain relevant mechanical parameters of the material; in this example, taking a simulation of the compressive failure process of two uniaxial compressive concrete samples as an example, the dimensions of the sample are as shown in fig. 2, and three initial cracks are formed in the central area of the sample, where θ is 45 °, S is 12.7mm, and C is 0mm, which is actually a test process in which the compressive shear and tensile shear failure modes coexist, and the test result is as shown in fig. 3.
The parameters of the material required by the method are obtained by performing an indoor uniaxial tensile test, and the following parameters are obtained by measuring the material: (1) elastic modulus E is 5.96GPa, (2) Poisson's ratio v is 0.15, and (3) uniaxial tensile breaking energy Gc15N/m, (4) tensile strength ft3.2MPa, and (5) a ratio χ of a critical shear fracture strength to a critical tensile fracture strength of 5.0.
Wherein the tensile energy to break of the material can be obtained by the following formula:
Gcfor the fracture energy, dU is the energy dissipation during fracture of the test piece and dA is the fracture area.
The fracture area dA is difficult to measure a real numerical value due to the fact that a fracture surface is not flat, therefore, as shown in FIG. 4, a technical ultrasonic infrared method is used for monitoring the crack expansion state and the crack area in real time, specifically, an ultrasonic emitter monitors and obtains the crack expansion condition of a material in real time, and a monitoring result is input into a computer for data analysis, so that the real crack state and area at any loading moment are obtained.
And 2, carrying out equal-proportion modeling in numerical simulation software according to the actual situation of the object to be simulated, wherein commercial software or self-programming software such as ANSYS, ABAQUS and the like can be adopted, and the parameters obtained in the step 1 are substituted into the model.
And step 3: and establishing a fracture theory of the complex failure mode of the phase field model so that the phase field model can effectively simulate fracture conditions of various failure modes. In order to obtain a crack propagation path implicitly, a recently developed phase field fracture model is adopted, but a traditional phase field model can only simulate a tensile failure process, wherein a phase field model capable of simulating a tensile failure mode and a shear failure mode is adopted, a criterion capable of simulating tensile failure and shear failure in a Unifield fracture criterion is introduced, and a result in an A under phase-field approach for the mechanics of damage and quasi-britle failure is applied to eliminate instability of a calculation result caused by a diffusion width of the phase field model, and a specific theoretical derivation is as follows:
the energy density function is decomposed into three parts, different failure modes can be considered, and different degradation functions are adopted for each part for degradation:
wherein: psi is a degraded energy density function, psi0In order to be the energy density before the degradation,
is psi0Is tensioned portion of psi0IIIs psi0The part to be cut of (a) is,
is psi0The other part of (a) or (b),
and psi0IAnd psi0IIIs defined as
Wherein:
for tensile stresses perpendicular to the plane of failure,
the equivalent shear stress parallel to the failure surface is shown, n is a direction vector vertical to the failure surface, and m is a direction vector parallel to the failure surface; omegaI(s) and ωII(s) are each independently of the other
And psi0IIS is e [0,1 ∈]The crack phase field variable is used for expressing the failure degree of the material, E is the elastic modulus, and mu is the shear modulus.
Wherein, in order to improve the calculation efficiency, a mixed mode, omega, is providedI(s)、ωIIThe relationship of(s) to the phase field variable s is written as:
wherein: bI、bIIThe ability to control the degradation function is in [0,1 ]]Internal value taking; in a pure tensile failure state, bI＝1，b II0; in the pure shearing state, bI＝0，b II1 is ═ 1; in general, b can also be usedI＝bIIThe mixed model which is 1 improves the iteration efficiency while ensuring the calculation precision, phi(s) and Q(s) are polynomials of phase field variable s and are used for determining the form of a degradation function, p and aiThe parameters to be determined can be determined from the material parameters and the specific softening curve.
In the constitutive relation, in order to improve the convergence and convergence efficiency of the calculation, a modified hybrid model is adopted to calculate a degenerated stress tensor and a degenerated elastic matrix, and the calculation method comprises the following steps:
wherein: sigma is degenerated stress tensor, D is degenerated elastic matrix, epsilon is strain tensor, and the other symbols have the same meanings as above.
Wherein, in order to simulate the damage form of the compression shear, the following crack evolution equation is established
Wherein: gcIAnd GcIIRespectively is I-type crack brokenEnergy of rupture and type II crack energy of rupture, l0In order to obtain a diffuse width of the crack,
α(s) is the geometric fracture function, α'(s) is the first derivative of α(s) with respect to s, ωI'(s) is omegaI(s) a first derivative with respect to s,
is a hamiltonian.
If the crack propagation direction of the tensile failure or the compressive failure is to be calculated, the crack propagation direction can be solved by a new function defined as follows:
wherein: θ can be obtained by the following equation:
in the formula: parameter(s)
And
first and second principal normal stresses, respectively, theta being
Direction and first principal normal stress
Angle between directions, and n ═ cos (θ + θ)0),sin(θ+θ0)]，θ0Is that
With the coordinate axis x1The angle between the axes, the direction m of the cracks being perpendicular to n, n being
In the direction of, τsAnd σtUltimate tensile stress and ultimate shear stress, respectively, and defining variable x ═ taus/σt。
Wherein, in order to solve the crack evolution equation smoothly and avoid the crack healing, the following two historical field variables respectively corresponding to the tensile energy and the sheared energy are defined
And
wherein: m is the m-th loading step,
for the history field variable of the mth loading step, the initial loading step has
Wherein:
wherein:
wherein:
energy at break, k, for uniaxial tensile test1Is defined as follows:
for bI＝1，bII0 and bI＝0，b II1, the control equation of tension failure and shear failure can be obtained respectively, the two cases are widely used, and b is adoptedI＝bII1.0, the application range of the phase field model can be greatly expanded, and a phase field model control equation capable of calculating the damage under the simultaneous action of shearing and stretching is obtained:
and 4, step 4: and establishing a refinement process theory so as to accurately simulate the crack path and the propagation process.
The refinement process theory enables the numerical model to obtain the most reasonable finite element distribution through a grid refinement algorithm, and the specific theory is as follows:
a multi-node quadrilateral unit (the number of nodes in the unit is varied from 4 to 8) is established, and the shape function of the unit is constructed as follows:
wherein N isn(xi, η) is a shape function of any point in the unit at the node n, and for nodes No. 5, 6, 7 and 8, when the shape function value does not exist, the corresponding shape function value becomes 0; (xi, eta) is a local coordinate of an arbitrary point in the cell, and the local coordinate of each node is
When the cells are initially divided, all quadrilateral 4-node cells are adopted, when the phase field value of a certain node reaches a certain critical value, a node is added on the boundary of the cell adjacent to the cell, and when the number of nodes of any cell reaches 8, the cell is divided into 4 four-node small cells.
For the node phase field critical value, five-level division of 0.2/0.4/0.6/0.8/1.0 is adopted, each level is divided once, and meanwhile, the difference between adjacent units cannot exceed two levels.
And 5: applying an actual load process to the numerical model, and in order to simulate a static load process, adopting sufficiently small load steps, wherein each displacement load step is set to be 5.0 multiplied by 10 equal to delta u-5And mm, gradually applying displacement, establishing a crack identification system, respectively processing the boundary of the model and the initial crack, and dividing the boundary of the crack into two parts to prevent the crack from healing and influencing the simulation result.
Step 6: and solving the cracking process of the model according to the theories established in the step 3 and the step 4. And solving the cracking process of the simulated object by adopting an iterative method or a step-by-step decoupling algorithm, wherein in the iterative method, the phase field control process is continuously and repeatedly calculated until the result of the last iteration meets the error requirement, so that the method has higher solving precision and slower calculating efficiency. The distributed decoupling algorithm does not need to repeatedly calculate a phase field control process, the accuracy of the method is not as good as that of an iterative method, the calculation accuracy can still be guaranteed under the condition that the load step is small enough, the calculation efficiency is very high, and the step-by-step decoupling algorithm is adopted for solving. In each load step, after a phase field control process is solved, a crack expansion area needs to be refined, in order to ensure the calculation precision after refinement, a phase field model control equation and a grid refinement algorithm need to be circularly carried out until the grid refinement degree meets the simulation precision requirement, the crack expansion state and the stress displacement data in the step are obtained, the data are substituted into the next load step for analysis until a test piece is completely broken, the data obtained in each load step are substituted into business software such as ABAQUS and the like for reading processing, the expansion process and the distribution state of cracks can be obtained, and the graph 5 and the graph 6 are a crack evolution process schematic diagram in a numerical simulation process and self-adaptive grid distribution in a crack evolution process.
In conclusion, the method can provide theoretical and technical support for crack propagation simulation problems of the brittle material and the quasi-brittle material in different failure modes, and promotes the accurate simulation of the finite element and the phase field model on the brittle solid materials such as rock, metal, ceramic and the like.
Claims (9)
1. A prediction method for refined cracks of a material is characterized by comprising the following steps:
step 1: carrying out a tensile test to obtain relevant mechanical parameters of the material;
step 2: modeling according to the actual situation of the object to be simulated, and substituting the parameters obtained in the step 1 into the established model;
and step 3: establishing a fracture theory of the phase field model in a complex failure mode so that the phase field model can effectively simulate fracture conditions of multiple failure modes;
and 4, step 4: establishing a refinement process theory so as to accurately simulate a crack path and an expansion process;
and 5: applying an actual loading process to the model;
step 6: and solving the cracking process of the model according to the theories established in the steps 3 and 4.
2. The method of predicting material microcracks according to claim 1, wherein the parameters include modulus of elasticity, Poisson's ratio, axial tensile failure energy, tensile strength, and ratio of critical shear failure strength to critical tensile failure strength.
3. The method for predicting the refined crack of the material as claimed in claim 2, wherein the axial tensile fracture energy of the material is obtained by the following formula:
Gcfor the fracture energy, dU is the energy dissipation during fracture of the test piece and dA is the fracture area.
4. The method for predicting the refined crack of the material as claimed in claim 3, wherein the axial tensile fracture energy of the material is obtained by the following formula: and for the fracture area dA, monitoring the crack propagation state and the crack area in real time by using an ultrasonic transmitter, and inputting the monitoring result into a computer for data analysis, thereby obtaining the real crack state and area at any loading moment.
5. The method for predicting material refined cracks according to claim 1, wherein the control equation of the phase field model established in step 3 is:
in the formula: s is an element of [0,1 ]]As a crack phase field variable, /)0In order to obtain a diffuse width of the crack,
for tensile stresses perpendicular to the plane of failure,
n is a direction vector perpendicular to the plane of fracture, m is a direction vector parallel to the plane of fracture, psi0Energy density before degradation,. psi0IIIs psi0Is a shear part ofI(s) and ωII(s) are two respective degradation functions, the first derivative of which is represented by the addition of a prime to the variable, e.g. ωI'(s) is omegaI(s) a first derivative with respect to s,
is the Hamiltonian, σ is the degenerated stress tensor, D is the degenerated elastic matrix, ε is the strain tensor,
α(s) is the geometric fracture function, E is the elastic modulus, μ is the shear modulus, GcIAnd GcIIRespectively, I-type crack energy and II-type crack energy, tausAnd σtUltimate tensile stress and ultimate shear stress, respectively, and defining variable x ═ taus/σtAnd is and
wherein: bI、bIIThe ability to control the degradation function is in [0,1 ]]Internal value taking; in a pure tensile failure state, bI＝1，bII0; in the pure shearing state, bI＝0，bII1 is ═ 1; in general, b can also be usedI＝bIIThe mixed model which is 1 improves the iteration efficiency while ensuring the calculation precision, phi(s) and Q(s) are polynomials of phase field variable s and are used for determining the form of a degradation function, p and aiThe parameters to be determined are determined from the material parameters and the specific softening curve.
6. The method for predicting material refined cracks according to claim 1, wherein the theory of the refinement process established in step 4 is that the model can obtain the most reasonable finite element distribution through a mesh refinement algorithm, and the specific theory is as follows:
a multi-node quadrilateral unit is established, wherein the number of nodes in the unit ranges from 4 to 8, and the shape function of the unit is constructed as follows:
wherein N isn(xi, η) is a shape function of any point in the unit at the node n, and for nodes No. 5, 6, 7 and 8, when the shape function value does not exist, the corresponding shape function value becomes 0; (xi, η) is the local coordinate of any point in the unit, and the local coordinate of each node is:
7. the method for predicting material refinement cracks according to claim 6, wherein the refinement process in step 4 further comprises establishing a set of refinement criteria, and the specific strategy is as follows:
when the cells are initially divided, all quadrilateral 4-node cells are adopted, when the phase field value of a certain node reaches a certain critical value, a node is added on the boundary of the cell adjacent to the cell, and when the number of the nodes of any cell reaches 8, the cell is divided into 4 four-node small cells;
for the node phase field critical value, five-level division of 0.2/0.4/0.6/0.8/1.0 is adopted, each level is divided once, and meanwhile, the difference between adjacent units cannot exceed two levels.
8. The method for predicting the refined cracks of the material as claimed in claim 1, wherein the step 6 is implemented by solving the cracking process of the model by using an iterative method or a step-by-step decoupling algorithm.
9. The method for predicting the refined cracks of the material according to claim 8, wherein the step-by-step decoupling algorithm is used for solving, in each loading step, after the phase field control process is solved, the phase field model control equation and the grid refinement algorithm are required to be carried out in a circulating mode until the grid refinement degree meets the requirement of simulation accuracy, the crack propagation state and the stress displacement data in the step are obtained, the data are substituted into the next loading step for analysis until the test piece is completely broken, and the data obtained in each loading step are read and processed, so that the crack propagation process and the distribution state can be obtained.
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| CN115510712A (en) * | 2022-09-29 | 2022-12-23 | 武汉大学 | Accurate calibration method for fracture energy for eliminating loading rate effect |
| CN115544834A (en) * | 2022-09-30 | 2022-12-30 | 东南大学 | Concrete material damage evolution simulation method based on phase field model |
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