A method for simulating the intrinsic strength of structural ceramics considering grain microstructure
By establishing a dual-scale model and combining Python and Abaqus software, the microstructure and macroscopic model of polycrystalline ceramic materials are simulated, and the crack propagation resistance is calculated. This solves the problem of inaccurate simulation of the intrinsic strength of polycrystalline ceramic materials and achieves rapid and accurate strength assessment.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- SOUTH CHINA UNIV OF TECH
- Filing Date
- 2024-04-17
- Publication Date
- 2026-07-17
AI Technical Summary
Existing technologies cannot accurately simulate the intrinsic strength of polycrystalline ceramic materials, and the difficulty in controlling the initial defect size in experiments leads to inaccurate strength tests.
A dual-scale model was adopted, combining Python to generate Voronoi polygons and Abaqus software to establish microstructure and macrostructure models. The intrinsic strength of the material was simulated by calculating crack propagation resistance, and the influence of microstructure on fracture toughness was quantitatively analyzed using the dual-scale model.
The influence of microstructure on fracture toughness was quantitatively analyzed by a dual-scale model, which solved the problem of inaccurate calculations in traditional models and provided a fast and accurate evaluation method for the research and development of polycrystalline ceramic materials, overcoming the limitation of difficulty in controlling the size of initial defects in experiments.
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Figure CN118280488B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of microstructure analysis technology for polycrystalline ceramics, and in particular to a method for simulating the intrinsic strength of structural ceramics that takes into account grain microstructure. Background Technology
[0002] As is well known, brittleness is a typical characteristic of ceramic materials, meaning that ceramic materials are very sensitive to defects. Clarifying the influence of microstructure on the strength of ceramic materials is the key to materials research and development.
[0003] In actual production, there are numerous types of defects, making it difficult to analyze all types using a single model. Indentation has been widely used in previous studies to simulate initial defect cracks. Research results show that when the defect size is large, the logarithmic distribution relationship between crack size and flexural strength is linear. As the number of indented cracks decreases, the relationship between strength and indented crack size becomes nonlinear, until a plateau is reached where strength is no longer sensitive to the initial crack. Based on this experiment, it can be found that the strength of a material is closely related to the defect size; therefore, the strength corresponding to a large defect is the intrinsic strength of the material. Due to limitations in material detection amplification, it is difficult to detect all defects in real materials. Therefore, the definition of material strength seems somewhat imprecise, and the intrinsic strength of the material and how the initial crack affects the strength test value are unknown. According to the definition of strength (fracture toughness) in fracture mechanics, the critical state of material fracture is unstable crack propagation. Therefore, the strength corresponding to the critical unstable crack propagation is the intrinsic strength of the material. Summary of the Invention
[0004] The purpose of this invention is to provide a method for simulating the intrinsic strength of structural ceramics that considers grain microstructure. By calculating crack propagation resistance, the relationship between the material's microstructure and intrinsic fracture strength is studied, providing a basis for the intrinsic strength control of high-strength and high-toughness structural ceramics.
[0005] To achieve the above objectives, the present invention provides a method for simulating the intrinsic strength of structural ceramics considering grain microstructure, comprising the following steps:
[0006] S1. Establish a dual-scale model, which includes a microstructure model and a macroscopic model;
[0007] S2. Calculate the damage of the microstructure model and obtain the damage mechanical parameters of the material;
[0008] S3. Transfer the damage mechanics parameters of the microstructure model to the macroscopic model, and calculate the crack propagation resistance curve in the material through the fracture of the macroscopic model.
[0009] S4. Calculate the intrinsic strength of the material using the crack propagation resistance curve.
[0010] Preferably, the establishment of the microstructure model in the dual-scale model of S1 includes the following steps:
[0011] Step 1: Use Python to generate Voronoi polygons. Voronoi polygons are used for grain simulation, and the edges of Voronoi polygons are used for grain boundary simulation.
[0012] Step 2: Mesh the microstructure model using Abaqus software. To achieve arbitrary crack propagation, based on the Abaqus secondary development interface, generate zero-thickness cohesive elements covering the entire model for the meshed model. Before the damage initiation and evolution of the cohesive elements, follow the linear elastic response relationship. Input material mechanical parameters and apply boundary conditions. Perform simulated tensile and shear tests through static analysis to obtain the damage mechanical parameters of the microstructure model.
[0013] Preferably, the damage mechanical parameters in S2 include tensile strength, shear strength, tensile fracture energy, shear fracture energy, and material parameters under tensile and shear stress.
[0014] Preferably, the macroscopic model in the dual-scale model of S1 is a macroscopic finite element model with pre-existing cracks, which is established by the following steps: establishing a three-point bending finite element model with a length and width of 40mm*4mm, arranging global mesh seed points for the model, and generating a mesh of appropriate size.
[0015] Preferably, in order to achieve arbitrary crack propagation, the macroscopic model uses extended finite element method to simulate crack propagation. The damage initiation adopts the maximum normal stress criterion, and the damage evolution adopts the linear degradation criterion, specifically referring to the maximum normal stress criterion and linear degradation criterion of the microscopic model. Its input parameters are derived from the damage mechanical parameters of the microscopic representative volume element. The material mechanical parameters are input to the macroscopic model and a load is applied. The correct mesh type is assigned, and a three-point bending condition is performed through static analysis to obtain the load at which crack propagation begins.
[0016] Preferably, the crack propagation resistance curve in S3 is based on the load at the start of crack propagation in a macroscopic finite element model with a pre-existing crack. The stress intensity factor is calculated according to the three-point bending formula in the ASTM standard. By using the stress intensity factor for different initial crack lengths, the crack propagation resistance curve is obtained. The calculation formula for the crack propagation resistance curve is as follows:
[0017]
[0018] Where K max It is the saturation value of the stress intensity factor, ΔK max It is the increase in fracture toughness due to the stable propagation of the crack, where a0 is the initial crack length and λ is a factor related to the increased resistance to crack propagation.
[0019] Preferably, during the crack propagation process of the macroscopic model, when the stress intensity factor applied to the crack tip is equal to the crack propagation resistance, and simultaneously when the stress intensity factor at the crack tip and the crack propagation resistance increase at the same rate with increasing crack size, this is considered a critical condition for unstable crack propagation. The unstable crack propagation condition is evaluated through the following relationship:
[0020]
[0021] Where a is the crack length, K R For the crack propagation resistance curve, K app The stress intensity factor applied at the crack tip is given by the following formula:
[0022]
[0023] Where σ is the applied stress, Y is the geometric factor, and the geometric factor of the crack is referenced as follows:
[0024]
[0025] Q = 1 + 1.464(2b / a) 1.65 (5)
[0026]
[0027] H=1-(1.22+0.24b / a)b / h+[0.55-1.05(2b / a) 0.75 +0.47(2b / a) 1.5 (b / h) 2 (7)
[0029] Where b is the crack width, h is the sample height, Q is the crack shape factor, M is the surface correction factor, and H is a polynomial used to correct the stress intensity factor.
[0030] Preferably, in step S4, when calculating the intrinsic strength of the material, the combined formulas (1)-(3) yield the following:
[0031]
[0032]
[0033] so:
[0034]
[0035] because much smaller and so:
[0036]
[0037] When the initial crack is less than λ / 2, the strength increases with crack propagation. When the crack propagates to λ / 2, the strength reaches its maximum value. If the crack needs to propagate more stably, the load needs to be reduced to stabilize the crack again. If the load continues to increase, the crack will propagate unstably. If the initial crack is greater than λ / 2, the crack will not propagate stably with increasing load and will directly enter the unstable propagation stage. The maximum strength, i.e., the intrinsic strength, is a function of λ and Y.
[0038]
[0039] Therefore, the present invention employs the above-mentioned method for simulating the intrinsic strength of structural ceramics that considers grain microstructure, and has the following beneficial effects:
[0040] (1) By establishing a dual-scale model, the influence of microstructure on fracture toughness was quantitatively analyzed, which solved the problem of inaccurate fracture toughness calculation caused by the shortcomings of traditional microscale model, such as short pre-crack, limited crack propagation length, and limited grains involved in the calculation. It also solved the problem of the macro model being separated from the microstructure.
[0041] (2) The intrinsic strength of the material can be obtained by calculation, which solves the problem of inaccurate strength test caused by the difficulty in controlling the size of the initial defects in the experiment. It provides a rapid evaluation method for the research, preparation and application of polycrystalline ceramic materials, and plays a role in saving time and effort, being fast, efficient and accurate.
[0042] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description
[0043] Figure 1 This is a microstructure model diagram of an embodiment of the intrinsic strength simulation method for structural ceramics considering grain microstructure according to the present invention;
[0044] Figure 2 This is a macroscopic model diagram of an embodiment of the intrinsic strength simulation method for structural ceramics considering grain microstructure according to the present invention;
[0045] Figure 3 This is an embodiment of the intrinsic strength simulation method for structural ceramics considering grain microstructure of the present invention, which uses microscale finite element models with different grain sizes to calculate crack propagation resistance curves.
[0046] Figure 4 This invention provides an embodiment of a method for simulating the intrinsic strength of structural ceramics that considers grain microstructure. The intrinsic strength is calculated using microscale finite element models with different grain sizes. Detailed Implementation
[0047] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.
[0048] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.
[0049] Example 1
[0050] As shown in the figure, this invention provides a method for simulating the intrinsic strength of structured ceramics considering grain microstructure. The method employs a dual-scale model, which includes a microstructure model and a macroscopic model. The specific steps include:
[0051] S1. Establish microstructure model: Generate representative volume elements of polycrystalline ceramics with average grain sizes of 0.1μm, 0.5μm, 1μm, 5μm, and 10μm. Figure 1 A representative volumetric element model with microstructural features is presented, and the specific process is as follows:
[0052] Voronoi polygons, generated using Python, are used to simulate the microstructure of polycrystalline ceramics. The Voronoi polygons simulate the ceramic grains, while their edges simulate the grain boundaries. The regions of the Voronoi polygons are defined as follows:
[0053] V i ={x∈R│l_(x r -p i ) <l_(x-p j ),i≠j}
[0054] Among them, V i Let x represent the i-th Voronoi polygon region, R be the set of all points in the Voronoi polygon, and x be the set of all points in the Voronoi polygon. r P is any point belonging to R. i and P j Let l_i and j_j represent unique seed points in the i-th and j-th Voronoi polyhedra, respectively. The number of randomly distributed seed points corresponds to the number of polygons in space S. i () represents point x and seed point p. i The distance between them, l_(xp) j () represents point x and seed point p. j The distance between them. Minimum repulsion distance l min As shown below:
[0055] l_(p i -p j)>l_min
[0056] l_(p i -p j ) is the seed point p i and seed point p j The distance between seed points, l_min is the minimum distance between seed points.
[0057] S2. Using Abaqus software, global mesh seed points are set for the microstructure model to generate a mesh of appropriate size. To achieve arbitrary crack propagation, based on the Abaqus secondary development interface, zero-thickness cohesive elements are inserted into the meshed model. Before the damage initiation and evolution of the cohesive elements, the elements obey linear elastic relations, and the constitutive equations are as follows:
[0058]
[0059] t n For normal stress, t s For tangential stress, σ n For normal strain, σ s For two tangential strains, K nn Let K be the normal stiffness in direction 1. ns K represents the shear stiffness in 12 directions. sn K represents the shear stiffness in the 21st direction. ss The normal stiffness is in two directions.
[0060] Damage initiation is based on the maximum normal stress criterion, which takes the following form:
[0061]
[0062] Where: t n0 For normal intensity, t s0 This represents the shear strength.
[0063] After the damage initiation condition of the cohesive element is satisfied, the stiffness decreases linearly, and the damage variable d m It will increase from 0 to 1:
[0064]
[0065] This refers to the stress during the damage evolution stage.
[0066] S3. Input the material mechanical parameters and apply the load, assign the correct mesh type, and perform simulated tensile and shear tests through static analysis; obtain the damage mechanical parameters of the microscopic representative volume element, including tensile strength, shear strength, tensile fracture energy, and shear fracture energy.
[0067] S4. Establish a macroscopic model with pre-existing cracks: Taking three-point bending as an example, such as... Figure 2 As shown, a three-point bending finite element model with dimensions of 40mm x 4mm was established. Global mesh seed points were set for the model to generate a mesh of appropriate size. To achieve arbitrary crack propagation, the extended finite element method was used to simulate crack propagation. The displacement field in the solution domain can be expressed as:
[0068]
[0069] Where N represents the node set of the normal element, N s N is the set of nodes of element t that is completely penetrated by the crack. t It is a set of nodes of elements with crack tips. N i (x), N j (x) and N k (x) is the shape function; u i These are the degrees of freedom for nodal displacement. j and b k It is a vector with rich degrees of freedom. F(x) is the asymptotic function, and H(x) is the jump function, which can be defined as follows:
[0070] H(x) = sign(f(x)) - sign(f j (x))
[0071] Where f(x) is the level set function, f j (x) is another level set function to eliminate the nodal enhancement of the mixed element near the crack. The level set function can be defined as:
[0072] f(x) = sign[n·(xx)] * )]·min||xx * ||
[0073] Where x is the coordinate of any point, x * It is the coordinate on the crack surface closest to x, where n is x. * The normal vector at the crack is used. Damage initiation adopts the maximum normal stress criterion, and damage evolution adopts the linear degradation criterion, specifically referring to the maximum normal stress criterion and linear degradation criterion of the microscopic model. Its input parameters are derived from the damage mechanics parameters of the microscopic representative volume element. The material mechanics parameters are input into the macroscopic model and a load is applied, the correct mesh type is assigned, and static analysis is performed to simulate tensile and shear tests under three-point bending conditions to obtain the load at which the material begins to fail.
[0074] Taking polycrystalline alumina as an example, the elastic modulus is 370 GPa, Poisson's ratio is 0.22, grain fracture work is 2.3 N / m, and grain boundary fracture work is 1 N / m. The solid element is CPS4R, and the cohesive element is COH2D4.
[0075] S5. Calculation of Stress Intensity Factor: The macroscopic model obtains the load at the initial crack propagation point. Based on the ASTM-C-1421 standard, the stress intensity factor of the response crack size is calculated.
[0076]
[0077] Where a is the crack length, F is the load at which the crack begins to propagate, S is the span width, B is the sample width, h is the sample height, and Y is the geometric factor, which can be calculated using the following formula:
[0078]
[0079] Q = 1 + 1.464(2b / a) 1.65
[0080]
[0081] H=1-(1.22+0.24b / a)b / h+[0.55-1.05(2b / a) 0.75 +0.47(2b / a) 1.5 (b / h) 2
[0082] Where b is the crack width, h is the sample height, Q is the crack shape factor, M is the surface correction factor, and H is the polynomial used to correct the stress intensity factor. The final result is:
[0083]
[0084] S6. Calculate the stress intensity factor for initial cracks of different lengths and fit it according to the crack propagation resistance curve formula:
[0085]
[0086] Where K max It is the saturation value of the stress intensity factor, ΔK max It is the increase in fracture toughness due to the stable propagation of the crack, where a0 is the initial crack length and λ is a factor related to the increased resistance to crack propagation.
[0087] like Figure 3 As shown, crack propagation resistance curves for polycrystalline alumina ceramics with different grain sizes were obtained. Figure 3 As can be seen, the stress intensity factor gradually increases until it saturates as the crack length increases.
[0088] During crack propagation, the critical condition for unstable crack propagation is considered to be when the stress intensity factor applied to the crack tip is equal to the crack propagation resistance, and simultaneously when the stress intensity factor at the crack tip and the crack propagation resistance increase at the same rate with increasing crack size. Therefore, the unstable crack propagation condition can be evaluated through the following relationship:
[0089]
[0090] The stress intensity factor applied at the crack tip is given by the following formula:
[0091]
[0092] Where σ is the applied stress and Y is the geometric factor.
[0093] S7. Combining the above formulas, we can obtain:
[0094]
[0095]
[0096] so:
[0097]
[0098] because much smaller and so:
[0099]
[0100] This indicates that when the initial crack size is less than λ / 2, the strength increases with crack propagation. When the crack propagates to λ / 2, the strength reaches its maximum value. If the crack needs further stable propagation, the load needs to be reduced to stabilize it again. If the load continues to increase, the crack will propagate unstably. If the initial crack size is greater than λ / 2, the crack will not propagate stably with increasing load and will directly enter the unstable propagation stage. Therefore, the maximum strength (i.e., intrinsic strength) is a function of λ and Y:
[0101]
[0102] The intrinsic strength of polycrystalline alumina ceramics with different grain sizes is calculated based on the above formula.
[0103] S8. Intrinsic Strength Assessment: Figure 4 The intrinsic intensity calculated using polycrystalline alumina models with different grain sizes is presented. The results show that when the grain size is less than 0.5 μm, the intrinsic intensity decreases significantly with decreasing grain size; conversely, when the grain size is greater than 0.5 μm, the intrinsic intensity increases significantly with decreasing grain size.
[0104] Therefore, this invention employs the aforementioned method for simulating the intrinsic strength of structured ceramics that considers grain microstructure. By using a dual-scale model to quantitatively analyze the influence of microstructure on intrinsic strength, it overcomes the limitation that ceramic material strength testing is easily affected by surface polishing. It also overcomes the problem of the macroscopic model being detached from the microstructure, providing a solution for the relationship between microstructure and macroscopic performance. This provides a rapid evaluation method for the research, preparation, and application of high-strength polycrystalline ceramic materials, achieving a time-saving, labor-saving, fast, efficient, and accurate result.
[0105] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.
Claims
1. A method for simulating the intrinsic strength of structural ceramics considering grain microstructure, characterized in that: Includes the following steps: S1. Establish a dual-scale model, which includes a microstructure model and a macroscopic model; S2. Calculate the damage of the microstructure model and obtain the damage mechanical parameters of the material; S3. Transfer the damage mechanics parameters of the microstructure model to the macroscopic model, and calculate the crack propagation resistance curve in the material through the fracture of the macroscopic model. The crack propagation resistance curve in S3 is based on the load at the start of crack propagation in a macroscopic finite element model with a pre-existing crack. The stress intensity factor is calculated according to the three-point bending formula in the ASTM standard. By using the stress intensity factor for different initial crack lengths, the crack propagation resistance curve is obtained. The calculation formula for the crack propagation resistance curve is as follows: （1） Where K max It is the saturation value of the stress intensity factor, ΔK max It is the increase in fracture toughness due to the stable propagation of the crack, where a0 is the initial crack length and λ is a factor related to the increased resistance to crack propagation. In the crack propagation process of the macroscopic model, when the stress intensity factor applied to the crack tip is equal to the crack propagation resistance, and simultaneously when the stress intensity factor at the crack tip and the crack propagation resistance increase at the same rate with increasing crack size, this is considered the critical condition for unstable crack propagation. The unstable crack propagation condition is evaluated through the following relationship: （2） Where a is the crack length, K R The crack propagation resistance curve is shown. The stress intensity factor applied at the crack tip is given by the following formula: （3） Where σ is the applied stress, Y is the geometric factor, and the geometric factor of the crack is referenced as follows: （4） （5） （6） （7） Where b is the crack width, h is the sample height, Q is the crack shape factor, M is the surface correction factor, and H is the polynomial used to correct the stress intensity factor. S4. Calculate the intrinsic strength of the material using the crack propagation resistance curve; When calculating the intrinsic strength of the material in S4, the combined formulas (1)-(3) yield the following: （8） （9） so: （10） because much smaller and ,so: （11） When the initial crack is less than λ / 2, the strength increases with crack propagation. When the crack propagates to λ / 2, the strength reaches its maximum value. If the crack needs to propagate more stably, the load needs to be reduced to stabilize the crack again. If the load continues to increase, the crack will propagate unstably. If the initial crack is greater than λ / 2, the crack will not propagate stably with increasing load and will directly enter the unstable propagation stage. The maximum strength, i.e., the intrinsic strength, is a function of λ and Y. （12）。 2. The method for simulating the intrinsic strength of structural ceramics considering grain microstructure according to claim 1, characterized in that: The establishment of the microstructure model in the dual-scale model of S1 includes the following steps: Step 1: Use Python to generate Voronoi polygons. Voronoi polygons are used for grain simulation, and the edges of Voronoi polygons are used for grain boundary simulation. Step 2: Mesh the microstructure model using Abaqus software. To achieve arbitrary crack propagation, based on the Abaqus secondary development interface, generate zero-thickness cohesive elements covering the entire model for the meshed model. Before the damage initiation and evolution of the cohesive elements, follow the linear elastic response relationship. Input material mechanical parameters and apply boundary conditions. Perform simulated tensile and shear tests through static analysis to obtain the damage mechanical parameters of the microstructure model.
3. The method for simulating the intrinsic strength of structural ceramics considering grain microstructure according to claim 1, characterized in that: The damage mechanics parameters in S2 include tensile strength, shear strength, tensile fracture energy, shear fracture energy, and material parameters under tensile and shear stress.
4. The method for simulating the intrinsic strength of structural ceramics considering grain microstructure according to claim 1, characterized in that: The macroscopic model in the dual-scale model of S1 is a macroscopic finite element model with pre-existing cracks. Its establishment includes the following steps: establishing a three-point bending finite element model with a length and width of 40mm*4mm, arranging global mesh seed points for the model, and generating a mesh of appropriate size.
5. The method for simulating the intrinsic strength of structural ceramics considering grain microstructure according to claim 1, characterized in that: To achieve arbitrary crack propagation, the macroscopic model uses extended finite element method to simulate crack propagation. Damage initiation adopts the maximum normal stress criterion, and damage evolution adopts the linear degradation criterion. For details, refer to the maximum normal stress criterion and linear degradation criterion of the microscopic model. Its input parameters are derived from the damage mechanics parameters of the microscopic representative volume element. The material mechanics parameters are input into the macroscopic model and a load is applied. The correct mesh type is assigned, and a three-point bending condition is performed through static analysis to obtain the load when the crack begins to propagate.