CN115114816A - Numerical simulation method for crack propagation of multi-interface non-uniform material under strong transient thermal load - Google Patents
Numerical simulation method for crack propagation of multi-interface non-uniform material under strong transient thermal load Download PDFInfo
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Abstract
The invention belongs to the technical field of transient thermal fracture of non-uniform materials, and discloses a numerical simulation method for crack propagation of a multi-interface non-uniform material under strong transient thermal load, which comprises the following steps: a numerical model of the inhomogeneous material containing the cracks is established in Matlab, thermodynamic parameters of the material are set, a transient temperature field is obtained based on a linear acceleration principle, and transient stress intensity factors of the tips of the cracks in the multi-interface inhomogeneous material are extracted through transient interaction integration. And obtaining crack propagation key parameters in the heterogeneous material based on a fracture theory. The numerical simulation method for the crack propagation of the multi-interface non-uniform material under the transient heat load has the characteristics of high efficiency, low cost, accurate calculation and good engineering application prospect. The method can consider the heat transfer delay effect in the transient heat transfer process, conveniently and quickly simulate the crack propagation of the multi-interface non-uniform material under the transient heat load, and provide a design basis for the engineering design and evaluation of the non-uniform material.
Description
Technical Field
The invention belongs to the technical field of transient thermal fracture of non-uniform materials, and particularly relates to a numerical simulation method for crack propagation of a multi-interface non-uniform material under transient thermal load.
Background
Compared with a uniform material with single component, the non-uniform material has unique performance and becomes an indispensable key material for the development of the national aerospace industry. Fracture failure is a typical failure mode in the use of heterogeneous materials, and the research on crack propagation problems in recent years is a hot spot of the research on fracture behavior of heterogeneous materials. On one hand, due to extreme service environments (such as thermal shock, thermal shock and thermal ablation) of the heterogeneous materials, the Fourier heat conduction theory implies the assumption of infinite heat propagation speed and is not suitable for a strong transient heat conduction process, so that a new physical model is required to be re-interpreted and defined. On the other hand, the heterogeneous material has a complex structure, and multiple interfaces are easy to appear in the material, so that the crack propagation process becomes more complex. The development of thermal shock fracture experimental studies is time and capital intensive. In the aspect of crack propagation calculation simulation, Wang B.L.Han J.C.Du S.Y.thermal shock resistance analysis method of ceramic coating/metallic substrate systems [ J ] Engineering crack Mechanics,2010,77(6):939-950) obtains I-type crack propagation process key parameters based on Fracture theory, but cannot solve the problem of mixed crack propagation in materials, so that a numerical simulation method is urgently needed to analyze the problem of mixed crack propagation of multi-interface non-uniform materials.
Disclosure of Invention
In view of the above, a first objective of the present invention is to provide a numerical simulation method for crack propagation of a multi-interface non-uniform material under a strong transient thermal load, which can effectively solve the problem of evaluating key parameters of a hybrid crack propagation process of the multi-interface non-uniform material.
In order to achieve the first purpose, the invention provides the following technical scheme:
establishing a multi-interface non-uniform material model in Matlab, setting material parameters, and prefabricating a mixed type crack, wherein the method is based on a finite element expansion method, grids do not need to be divided again for the discontinuous body evolution problem, and the crack is independent of the grids, and specifically comprises the following steps: based on an extended finite element method, discontinuous bodies (crack surfaces and particles) are provided with jump functions, the units of the method are divided into three types, namely a conventional unit, a crack surface penetrating unit and a crack tip unit, wherein the singularity of the crack tip unit is represented by an angle function extracted from a crack tip field analytical solution; solving a transient temperature field, a transient thermal stress field and a transient stress intensity factor in the inhomogeneous material plate; refining the crack tip based on an extended finite element method; crack propagation is based on the criteria of fracture.
A crack propagation numerical simulation method under strong transient thermal load comprises the following steps:
the method comprises the following steps: establishing a numerical model of the inhomogeneous material containing the cracks, wherein the numerical model comprises grid division, node and unit definition, thermal expansion coefficient, thermal conductivity, Young modulus, density and specific heat material parameters are defined as any analytic function form of a space coordinate, and the length and the angle of the cracks are set; applying strong transient thermal load to the numerical model of the inhomogeneous material, solving the transient temperature field of the numerical model of the inhomogeneous material based on the linear acceleration principle, wherein the numerical iteration formula is
Wherein T is the temperature within the non-uniform material, [ K ] 1 ]Is a heat conduction matrix of the heat conduction equation, [ K ] 2 ]Is a convective matrix of heat transfer equations, [ K 3 ]Is a matrix of heat capacities of the heat conduction equation, [ M]Is a thermal delay matrix of the thermal conduction equation,
is a matrix of equivalent stiffness values that is,
is the equivalent thermal load matrix, Δ t is the iteration step, n is the iteration step, δ 1 ～δ 6 Is a constant coefficient parameter;
step two: the temperature in the inhomogeneous material obtained in the first step needs to be superimposed into the mechanical strain of the inhomogeneous material in the form of thermal strain, and the total strain of the inhomogeneous material is finally formed, wherein the thermal constitutive equation of the temperature and the strain is as follows:
ε heat ＝αΔT[1 1 1 0 0 0] T
in the formula, epsilon heat Is the thermal strain induced by temperature, α is the coefficient of thermal expansion, Δ T is the temperature difference within the non-uniform material;
further calculating an actual field in the inhomogeneous material, wherein the actual field comprises an actual stress field, an actual displacement field and an actual strain field, and the actual strain is formed by superposing thermal strain and mechanical strain; introducing an auxiliary field (Yu HJ, Wang J, Shimada T, Wu HP, Wu LZ, Kuna M, Kitamura T.An I-integral method for crack-tip interaction factor variation from auxiliary to main switching in a rotary displacement-crystals, J.Mech.Phys.solids 2016; 94: 207-:
where δ is the kronecker symbol, σ is the stress in the crack tip integration zone, u is the displacement in the crack tip integration zone, and ε is the crack tip integration zoneInternal strain, subscripts i, j, k, l are the dummy mark value ranges of the tensor are 1, 2, 3 and 4, omega is the crack tip integral area, q is the weight function value range is 0-1, T is the temperature in the non-uniform material, and T is the temperature in the non-uniform material 0 Initial temperature within the heterogeneous material, α is the coefficient of thermal expansion of the heterogeneous material, θ 1 The included angle between an interface curve coordinate system and a rectangular coordinate system is shown, an upper mark aux represents auxiliary field quantity, an upper mark act represents real field quantity, and upper marks 1, 2 and 3 represent different materials on two sides of an interface;
based on the equivalent relation of interaction integration, an energy principle and the transient stress intensity factor, further extracting the transient thermal stress intensity factor of the crack tip:
wherein G is the energy release rate, E is the Young's modulus, K I And K II Respectively is an I-type thermal stress intensity factor and an II-type thermal stress intensity factor; the transient thermal stress intensity factor of the non-uniform material is directly extracted through transient interaction integration, and an integral peripheral region omega of a crack tip in the method comprises multiple materials, so that the method is suitable for the multi-interface material;
step three: providing the instantaneous crack propagation length and the crack cracking angle of the mixed type crack by utilizing the transient stress intensity factor extracted in the second step and combining the maximum cyclic stress criterion and the fracture criterion, providing key parameters for crack propagation simulation under strong transient thermal load, and substituting the key parameters into the non-uniform material numerical model in the first step to realize the crack propagation simulation under the transient thermal load;
the crack propagation key parameters are determined by the maximum cyclic stress criterion and the fracture criterion, all instantaneous fracture parameters participate in crack judgment under strong transient thermal load in the third step, and simultaneously, the crack angle of the crack is calculated based on the maximum cyclic stress criterion:
in the formula, K Ie (a) Is the equivalent transient thermal stress intensity factor, θ c Is the crack opening angle and a is the crack length.
The invention has the beneficial effects that: the numerical simulation method for the multi-interface non-uniform material crack propagation under the transient thermal load has the characteristics of high efficiency, low cost and accurate calculation, and has good engineering application prospect. The method can consider the heat transfer delay effect in the transient heat transfer process, conveniently and quickly simulate the crack propagation of the multi-interface non-uniform material under the transient heat load, and provide a design basis for the engineering design and evaluation of the non-uniform material.
Drawings
FIG. 1 is a flow chart illustrating a numerical simulation method for crack propagation in a non-uniform material under transient thermal load in accordance with an embodiment of the present invention.
FIG. 2 is a schematic view of various combinations of interfaces within a non-uniform material.
FIG. 3 is a schematic view of transient crack cracking length extraction.
FIG. 4 is a schematic diagram of a model of a dual interface heterogeneous material.
FIG. 5 is a graph illustrating the variation of the transient stress intensity factor in a dual interface non-uniform material sheet with crack length under transient thermal load: (a) the initial stage of transient heat load; (b) late transient thermal load.
Detailed Description
The following further describes a specific embodiment of the present invention with reference to the drawings and technical solutions.
Example (b):
referring to fig. 1, a numerical simulation method for crack propagation of a multi-interface non-uniform material includes the following steps:
step1, establishing a numerical model of the two-dimensional multi-interface non-uniform material plate in Matlab, setting material properties including grid division, defining nodes and units, and defining any analytic function form of thermal expansion coefficient, thermal conductivity, Young modulus, density and specific heat material thermodynamic parameters as space coordinates, wherein the distribution of the non-uniform material thermodynamic parameters adopts an exponential function form, and a program is written to realize the distribution form of the non-uniform material thermodynamic parameters, FIG. 2 is a schematic diagram of various interfaces in the non-uniform material, when the thermodynamic parameters and derivatives thereof at the material interfaces are continuous, the ideal interfaces are defined, when the thermodynamic parameters and derivatives thereof are continuous, the weak interfaces are defined, and when the thermodynamic parameters and derivatives thereof are discontinuous, the strong interfaces are defined, so that the double-interface non-uniform material can have 9 combination forms. Setting the length and the angle of the crack, prefabricating the mixed type crack based on the finite element expansion method, not needing to divide the grid again when the discontinuous body evolves, the crack is independent of the grid, adopting a mode of increasing the freedom degree of the node, increasing four freedom degrees of the node at the tip of the crack, and increasing one freedom degree of the node penetrated by the crack. In fig. 3, open triangles indicate jump nodes on the crack surface, and open circles indicate crack tip reinforcing nodes.
Solving the transient temperature field in the non-uniform material plate, writing a Gaussian integral function and a heat conduction boundary condition program code, and increasing the solving difficulty of the temperature field due to the introduction of a heat delay term in a heat conduction equation. According to the invention, the second-order heat conduction differential equation is dispersed through a linear acceleration principle, and the temperature field in the non-uniform material under transient heat load is directly solved without converting the second-order heat conduction differential equation into a first-order differential equation set. Solving the transient temperature field of the numerical model of the inhomogeneous material based on the linear acceleration principle, wherein the numerical iteration formula is
Wherein T is the temperature within the non-uniform material, [ K ] 1 ]Is a heat conduction matrix of the heat conduction equation, [ K ] 2 ]Is a convective matrix of heat transfer equations, [ K 3 ]Is a matrix of heat capacities of the heat conduction equation, [ M]Is a thermal delay matrix of the thermal conduction equation,
is an equivalent stiffness matrix，
Is the equivalent thermal load matrix, Δ t is the iteration step, n is the iteration step, δ 1 ～δ 6 Is a constant coefficient parameter;
step2, adding the temperature obtained in Step1 in the inhomogeneous material into the mechanical strain of the inhomogeneous material in the form of thermal strain, and finally forming the total strain of the inhomogeneous material, wherein the thermal constitutive equation of the temperature and the strain is as follows:
ε heat ＝αΔT[1 1 1 0 0 0] T
in the formula, epsilon heat Is the thermal strain induced by temperature, α is the coefficient of thermal expansion, Δ T is the temperature difference within the non-uniform material;
further calculating an actual field in the inhomogeneous material, wherein the actual field comprises an actual stress field, an actual displacement field and an actual strain field, and the actual strain is formed by superposing thermal strain and mechanical strain; introducing an auxiliary field (Yu HJ, Wang J, Shimada T, Wu HP, Wu LZ, Kuna M, Kitamura T.An I-integral method for crack-tip interaction factor variation from auxiliary to main switching in a rotary displacement-crystals, J.Mech.Phys.solids 2016; 94: 207-:
wherein δ is a kronecker symbol, σ is stress in the crack tip integration region, u is displacement in the crack tip integration region, ε is strain in the crack tip integration region, subscripts i, j, k, l are the dummy values of the above tensor as 1, 2, 3, 4, Ω is the crack tip integration region, q is the weight function as 0-1, T is temperature in the non-uniform material, and 0 initial temperature within heterogeneous material, alpha isCoefficient of thermal expansion, theta, of homogeneous material 1 The included angle between an interface curve coordinate system and a rectangular coordinate system is shown, an upper mark aux represents auxiliary field quantity, an upper mark act represents real field quantity, and upper marks 1, 2 and 3 represent different materials on two sides of an interface;
based on the equivalent relation of interaction integration, an energy principle and the transient stress intensity factor, further extracting the transient thermal stress intensity factor of the crack tip:
wherein G is the energy release rate, E is the Young's modulus, K I And K II Respectively is an I-type thermal stress intensity factor and an II-type thermal stress intensity factor; the transient thermal stress intensity factor of the non-uniform material is directly extracted through transient interaction integration, and an integral peripheral region omega of a crack tip in the method comprises multiple materials, so that the method is suitable for the multi-interface material;
step3, taking model figure 4 as an example to carry out numerical calculation of crack tip fracture parameters in the dual-interface heterogeneous material, and figure 5 is a calculation result of extracting a crack tip stress intensity factor by utilizing an interaction integration method, wherein crack propagation criteria are carried out based on fracture criteria in the method, and the fracture criteria are in the following forms
K(x,t)≤K IC (x)
K IC (x) The fracture toughness of the inhomogeneous material is obtained, once the instantaneous stress intensity factor in the material is higher than the fracture toughness of the material, the crack will be expanded, and as shown in fig. 5a, the projections of the intersection point of the fracture toughness and the stress intensity factor on the horizontal axis when t is 0.002 and t is 0.004 are corresponding crack expansion lengths respectively, so that the instantaneous crack expansion length can be given, and the initial crack expansion of the transient thermal load can be divided into two stages of stable expansion and unstable expansion as can be seen from fig. 5 a.
Step4, not only the crack propagation direction can be obtained by utilizing the maximum circumferential stress criterion, but also the mixed type thermal stress intensity factor can be equivalent, and the I type stress intensity factor K can appear at the mixed type crack tip I And IIType stress intensity factor K II The type I stress intensity factor K is determined by the maximum cyclic stress criterion I And type II stress intensity factor K II Carrying out the equivalence of the raw materials,
in the formula, K Ie (a) Is the equivalent transient thermal stress intensity factor, θ c Is the crack opening angle and a is the crack length.
K IC (x) Is the fracture toughness of non-uniform material, and the equivalent fracture criterion form is as follows
K Ie (x,t)≤K IC (x)
Once the instantaneous equivalent stress intensity factor K in the inhomogeneous material Ie (a) To achieve fracture toughness K IC (x) The crack will propagate and further the instantaneous crack propagation length is determined by Step3, and the hybrid crack angle is determined by the maximum hoop stress criterion.
The numerical simulation method for crack propagation of the heterogeneous material is suitable for the mixed crack multi-interface heterogeneous material, the setting of cracks, thermal boundary conditions and heterogeneous material parameters is realized by adopting subprograms, the transient thermal stress intensity factor is extracted by a transient interaction integration method, the efficiency is improved, meanwhile, for a mixed crack propagation finite element model and the function form (Young modulus, thermal expansion coefficient, thermal conductivity, density, specific heat, plate size, fracture toughness, crack length and deflection direction) of the thermodynamic parameters of the heterogeneous material, only one-time analysis model needs to be established, and the rest work is completed by the subprograms.
The embodiments are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are mutually referred to. The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
In conclusion, in the simulation process, a C language interface in Matlab is adopted to write a subprogram, so that the numerical simulation of crack expansion in the non-uniform material containing mixed cracks is realized, the extraction of crack expansion parameters of the multi-interface non-uniform material is realized by utilizing an expansion finite element method, an interaction integral and a fracture criterion, the method has the advantages of convenience and easiness in operation, high calculation efficiency and high universality, the extraction of key parameters of thermal shock cracks of the multi-interface non-uniform material under transient thermal load can be realized, and the thermal delay effect in the heat transfer process is considered; when the parameters are researched, only one modeling is needed, and the rest of work can be completed in a subprogram, so that the modeling efficiency is improved. Therefore, the method has the characteristics of high modeling efficiency, low cost, simplicity, convenience, feasibility and accurate calculation, and has a good engineering application prospect.
The above description is only for the preferred embodiments of the present invention, but the protection scope of the present invention is not limited thereto, and any person skilled in the art can substitute or change the technical solution and the inventive concept of the present invention within the scope of the present invention.
Claims (1)
1. A crack propagation numerical simulation method under strong transient thermal load is characterized by comprising the following steps:
the method comprises the following steps: establishing a numerical model of the inhomogeneous material containing the cracks, wherein the numerical model comprises grid division, node and unit definition, thermal expansion coefficient, thermal conductivity, Young modulus, density and specific heat material parameters are defined as any analytic function form of a space coordinate, and the length and the angle of the cracks are set; applying strong transient thermal load to the numerical model of the inhomogeneous material, solving the transient temperature field of the numerical model of the inhomogeneous material based on the linear acceleration principle, wherein the numerical iteration formula is
Wherein T is the temperature within the non-uniform material, [ K ] 1 ]Is a heat conduction matrix of the heat conduction equation, [ K ] 2 ]Is a convective matrix of heat transfer equations, [ K 3 ]Is a matrix of heat capacities of the heat conduction equation, [ M]Is a thermal delay matrix of the thermal conduction equation,
is a matrix of equivalent stiffness values that is,
is the equivalent heat load matrix, Δ t is the iteration step, n is the iteration step, δ 1 ～δ 6 Is a constant coefficient parameter;
step two: the temperature in the inhomogeneous material obtained in the first step needs to be superimposed into the mechanical strain of the inhomogeneous material in the form of thermal strain, and the total strain of the inhomogeneous material is finally formed, wherein the thermal constitutive equation of the temperature and the strain is as follows:
ε heat ＝αΔT[1 1 1 0 0 0] T
in the formula, epsilon heat Is the thermal strain induced by temperature, α is the coefficient of thermal expansion, Δ T is the temperature difference within the non-uniform material;
further calculating an actual field in the inhomogeneous material, wherein the actual field comprises an actual stress field, an actual displacement field and an actual strain field, and the actual strain is formed by superposing thermal strain and mechanical strain; introducing an auxiliary field into an integral area of the crack tip, wherein the auxiliary field comprises an auxiliary stress field, an auxiliary displacement field and an auxiliary strain field, and introducing the auxiliary field and an actual field in the integral area of the crack tip into an interaction integral, so as to further obtain an interaction form containing a temperature term and an interface term as follows:
wherein δ is a kronecker symbol, σ is stress in the crack tip integration region, u is displacement in the crack tip integration region, ε is strain in the crack tip integration region, subscripts i, j, k, l are the dummy values of the above tensor as 1, 2, 3, 4, Ω is the crack tip integration region, q is the weight function as 0-1, T is temperature in the non-uniform material, and 0 initial temperature within the heterogeneous material, α is the coefficient of thermal expansion of the heterogeneous material, θ 1 The included angle between an interface curve coordinate system and a rectangular coordinate system is shown, an upper mark aux represents auxiliary field quantity, an upper mark act represents real field quantity, and upper marks 1, 2 and 3 represent different materials on two sides of an interface;
based on the interaction integral, the energy principle and the equivalence relation of the transient stress intensity factor, further extracting the transient thermal stress intensity factor of the crack tip:
wherein G is the energy release rate, E is the Young's modulus, K I And K II Respectively an I-type thermal stress intensity factor and an II-type thermal stress intensity factor; the transient thermal stress intensity factor of the non-uniform material is directly extracted through transient interaction integration, and an integral peripheral region omega of a crack tip in the method comprises multiple materials, so that the method is suitable for the multi-interface material;
step three: providing the instantaneous crack propagation length and the crack cracking angle of the mixed type crack by utilizing the transient stress intensity factor extracted in the second step and combining the maximum cyclic stress criterion and the fracture criterion, providing key parameters for crack propagation simulation under strong transient thermal load, and substituting the key parameters into the non-uniform material numerical model in the first step to realize the crack propagation simulation under the transient thermal load;
the crack propagation key parameters are determined by the maximum cyclic stress criterion and the fracture criterion, all instantaneous fracture parameters participate in crack judgment under strong transient thermal load in the third step, and simultaneously, the crack angle of the crack is calculated based on the maximum cyclic stress criterion:
in the formula, K Ie (a) Is the equivalent transient thermal stress intensity factor, θ c Is the crack opening angle and a is the crack length.
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| CN117150858A (en) * | 2023-09-06 | 2023-12-01 | 天津大学 | Crack-containing continuous fiber reinforced resin matrix composite three-dimensional finite element modeling method |
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| US20030024323A1 (en) * | 2001-06-25 | 2003-02-06 | Jy-An Wang | Fracture toughness determination using spiral-grooved cylindrical specimen and pure torsional loading |
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Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN117150858A (en) * | 2023-09-06 | 2023-12-01 | 天津大学 | Crack-containing continuous fiber reinforced resin matrix composite three-dimensional finite element modeling method |
| CN117150858B (en) * | 2023-09-06 | 2024-03-26 | 天津大学 | Crack-containing continuous fiber reinforced resin matrix composite three-dimensional finite element modeling method |
| CN117150822A (en) * | 2023-10-30 | 2023-12-01 | 中南大学 | Method and system for calculating thermal coupling stress intensity factor of interface crack |
| CN117150822B (en) * | 2023-10-30 | 2024-02-06 | 中南大学 | Method and system for calculating thermal coupling stress intensity factor of interface crack |
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Application publication date: 20220927 |