A constitutive model establishment and numerical calculation method for SiC fiber reinforced titanium matrix composites

Abstract

The present invention discloses a constitutive model establishment and numerical calculation method for SiC fiber reinforced titanium-based composite materials; the model of the present invention can obtain f / Ti composite material structure mechanical analysis of the homogenized constitutive equation, improve the efficiency of composite material structure mechanical analysis, and can obtain SiC f / Ti composite material microstructure stress / strain field, to ensure the accuracy of composite material mechanical analysis, convenient for SiC f Multi-scale mechanical analysis of SiC / Ti composites. f The matrix plasticity and interface debonding damage of Ti / Ti composites are both included in the tangent modulus matrix, and no additional terms appear in the overall stiffness equation. The form is simple and easy to program.

Description

Technical Field

[0001] The present invention belongs to the technical field of multi-scale mechanical property prediction of composite materials, and particularly relates to a constitutive model establishment and numerical calculation method of SiC fiber reinforced titanium-based composite materials, specifically a multi-scale stress-strain response prediction method of SiC fiber reinforced titanium-based composite materials. Background Art

[0002] SiC fiber reinforced titanium matrix composites (hereinafter referred to as SiC f / Ti) has the advantages of high specific strength, high specific modulus and high thermal stability, and is an ideal material for aircraft engine compressor rotor components. f The titanium (Ti) matrix of the SiC / Ti composite material has the characteristic of plastic deformation, and the matrix and the fiber are not completely bonded. Under the service environment, the matrix is ​​prone to plastic deformation and interface debonding damage, which restricts the SiC f / Ti composite materials. In order to effectively f Mechanical analysis of SiC / Ti composite components is required f / Ti composite material constitutive equations simultaneously take into account matrix plasticity and interface debonding damage, and ensure high efficiency and accuracy of mechanical analysis. At present, the multi-scale calculation method for equivalent performance of complex composite structures disclosed in patent CN106066913B converts composite materials into anisotropic homogeneous materials, gives the equivalent elastic modulus of composite materials, and can improve the efficiency of mechanical analysis of composite materials. However, this method cannot guarantee the accuracy of mechanical analysis of composite materials, because the method does not give the microscopic stress field inside the composite material. In addition, the constitutive equations of the components of the composite material described in the patent are elastic constitutive equations, which cannot be used for SiC f Mechanical analysis of Ti / Ti composites.

[0003] The journal article "Cohesive Zone-Based Damage Evolution in Periodic Materials Via Finite-Volume Homogenization" established a mesomechanical model of composite materials with interface debonding based on the finite volume direct average micromechanical model (FVDAM). Its control equation contains both mesoscopic variables and macroscopic variables, which can clearly show multi-scale characteristics and obtain the homogenized constitutive equation and mesoscopic stress field of the composite material. However, this model is based on the FVDAM model in secant form. The plastic deformation of the matrix and interface debonding are expressed as nonlinear additional terms of the FVDAM control equation in this model, which makes programming difficult. The journal article "A tangent finite-volume direct averaging micromechanics framework for elastoplastic porous materials: theory and validation" uses the FVDAM model in tangent form to introduce the elastoplastic mechanics constitutive equation into the tangent modulus of the composite material. There is no nonlinear additional term in the control equation. However, this model is only used to calculate elastoplastic porous materials rather than SiC. f / Ti composites, without considering the characterization of composite interface debonding damage in the tangent modulus. Summary of the invention

[0004] In order to solve the problems in the prior art, the present invention provides a constitutive model establishment and numerical calculation method of SiC fiber reinforced titanium matrix composite material. The present invention is based on the finite volume direct average micromechanical model (FVDAM) and considers SiC f The elastic-plastic properties of the titanium matrix and the interface debonding between the matrix and the fiber of the SiC / Ti composites were established. f Multi-scale mechanical analysis model of / Ti.

[0005] To achieve the above object, the technical solution adopted by the present invention is:

[0006] A constitutive model establishment and numerical calculation method for SiC fiber reinforced titanium matrix composite material, comprising the following steps:

[0007] Step 1: According to the SiC fiber reinforced titanium matrix composite (SiC f / Ti) volume fraction and interface thickness, and establish SiC f / Ti composite unit cell model, and assign mechanical property parameters of each component of the composite material in the unit cell model;

[0008] Step 2: Divide the unit cell model described in step 1 into n qarbitrary hexahedral subcells, numbered from 1 to n q , any one subcell is called subcell (q); subcell (q) is transformed into a standard cube subcell through isoparametric transformation, and the corresponding relationship between the coordinates of subcell (q) and the coordinates of the standard cube subcell is established;

[0009] Step 3: Set the displacement increment of the subcell (q) The expression, superscript (q) is the subcell number, subscript i is the coordinate component symbol, which can be 1, 2, or 3; displacement increment The macroscopic displacement increment divided into subcells (q) and the microscopic displacement increment of the daughter cell (q) The sum of the two terms,

[0010] Step 4: Introduce the average displacement increment of the daughter cell (q) on the outer surface (p) and average surface force increment Where: the superscript (q,p) represents the outer surface (p) of the subcell (q), and then the incremental form local stiffness equation of the subcell (q) is established;

[0011] Step 5: According to the continuity of displacement and surface force, the local stiffness equation of each subcell is assembled into the global stiffness equation of the composite unit cell;

[0012] Step 6: Solve the global stiffness equation of the composite unit cell to obtain the tensor increment for each strain concentration and

[0013] Step 7: From the subcell volume V (q) , strain concentration tensor increment and and the subcell tangent modulus matrix Get the macroscopic tangent modulus matrix of the composite material Macroscopic equivalent thermal expansion coefficient α H and homogenized constitutive equations;

[0014] Step 8: Set the macroscopic stress increment of the composite unit cell at each load step and the temperature change increment Δ(δT), the macroscopic strain increment is calculated according to the homogenized constitutive equation at a certain load step

[0015] Step 9: Increment the macroscopic strain Substitute into the overall stiffness equation and solve it to obtain the stress increment Δσ of the subcell (q) (q) , strain increment Δε (q) , matrix plastic strain increment Δε pl ( q) and interface damage variable d(q) ;

[0016] Step 10: According to the plastic strain increment Δε of the matrix pl ( q) and interface damage variable d (q) , update the tangent modulus matrix of the subcell (q) Global stiffness equation and macroscopic tangent modulus matrix

[0017] Step 11: Repeat steps 6 to 10 until the tangent modulus matrix convergence;

[0018] Step 12: Output the microscopic field and macroscopic tangent modulus matrix of the composite unit cell under this load step Macro stress and macroscopic strain

[0019] Step 13: The macroscopic tangent modulus matrix obtained from the previous step Used for mechanical analysis of composite structural parts; a series of macroscopic stresses obtained from the loading history Macro strain The stress-strain response of the composite material under a certain loading process is obtained.

[0020] Furthermore, in step 1: SiC f The macroscopic coordinate system of the unit cell model of the / Ti composite material is Ox 1 x 2 x 3 , where: O is the origin, x 1 , x 2 , x 3 It is called the macroscopic coordinate component.

[0021] Furthermore, in step 2: the coordinate system of any hexahedral subcell (q) is O (q) -y (q)1 y (q)2 y (q)3 , where: (q) is the origin, y (q)1 ,y (q)2 ,y (q)3 is called the microscopic coordinate component; the coordinate system of the standard cubic subcell is O s -ξ 1 ξ 2 ξ 3 , where: s is the origin, ξ 1 ,ξ 2 ,ξ 3 is a standard cubic subcell, whose range is -1≤ξ 1 ,ξ2 ,ξ 3 ≤1; macroscopic coordinate component x 1 ,x 2 ,x 3 and the microscopic coordinate component y (q)1 ,y (q)2 ,y (q)3 The relationship is in: From point O to O (q) Vector.

[0022] Furthermore, in step 3: the macro displacement increment The expression is in: is the macroscopic strain increment component, x j is the macroscopic coordinate, the subscripts i and j are the coordinate component symbols, which are 1, 2, and 3; the microscopic displacement increment The expression is:

[0023]

[0024] in: and is the unknown coefficient in the microscopic displacement increment, the subscript i is the coordinate component sign, and takes 1, 2, or 3.

[0025] Furthermore, in step 4: the average displacement increment of the daughter cell (q) on the outer surface (p) for:

[0026] Where: i,m,n=1,2,3

[0027] in: is the microscopic displacement increment of the daughter cell (q) on the outer surface (p), ξ m , n is the standard cube coordinate component, the subscripts i, m, n are the coordinate component symbols, and the values ​​are 1, 2, and 3;

[0028] Average surface force increment for:

[0029]

[0030] in: is the increment of the surface force of the daughter cell (q) on the outer surface (p), The matrix is ​​represented as The subscripts i, m, and n are all coordinate component symbols, and are 1, 2, and 3;

[0031] The local stiffness equation in incremental form is:

[0032]

[0033] in: N (q,p) is the normal vector matrix, containing the outer normal vector n of the subcell (q) on the outer surface (p) (q,p) ; is the tangent modulus matrix; is the macroscopic strain increment; Γ (q) is the thermal stress array, including the tangent modulus and thermal expansion coefficient of the subcell (q); Δ(δT) is the temperature change increment; is the tangent local stiffness matrix of the subcell (q) on the outer surface (p); is the average displacement increment of all external surfaces of the subcell (q) Composed of arrays.

[0034] Furthermore, in step 5: the overall stiffness equation of the composite material is:

[0035]

[0036] in: K t is the overall tangent stiffness matrix; It is an array composed of the average displacement increments of the outer surfaces of all sub-cells on the composite unit cell; The macroscopic strain increment The resulting load matrix is: D t Contains the difference in tangent modulus of adjacent subcells; ΓΔ(δT) is the load array generated by the temperature change increment Δ(δT), where: the matrix Γ contains the difference in tangent modulus and thermal expansion coefficient of adjacent subcells.

[0037] Furthermore, in step 6: the strain concentration tensor increment and The strain increment Δε of the composite subcell is established (q) and macroscopic strain increment The relationship between the temperature change increment Δ(δT) is:

[0038]

[0039] Further, in step 7: the composite macroscopic tangent modulus matrix and the macroscopic equivalent thermal expansion coefficient α H The expressions are:

[0040]

[0041] Where: v qis the volume fraction of the daughter cell (q) in the whole unit cell, is the daughter cell tangent modulus matrix;

[0042] The homogenized constitutive equation is

[0043] in: is the macro stress increment.

[0044] Compared with the prior art, the present invention has the following beneficial effects:

[0045] The model of the present invention can obtain both f / Ti composite material structure mechanical analysis of the homogenized constitutive equation, improve the efficiency of composite material structure mechanical analysis, and can obtain SiC f / Ti composite material microstructure stress / strain field, to ensure the accuracy of composite material mechanical analysis, convenient for SiC f Multi-scale mechanical analysis of SiC / Ti composites. f The matrix plasticity and interface debonding damage of Ti / Ti composites are both included in the tangent modulus matrix, and no additional terms appear in the overall stiffness equation. The form is simple and easy to program. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 SiC based on the FVDAM model f Multi-scale mechanical analysis process of / Ti composites;

[0047] Figure 2 SiC f Unit cell model and divided model of / Ti composite material;

[0048] Figure 3 is the bilinear stress-strain relationship of interface debonding damage;

[0049] Figure 4 It is a schematic diagram of an arbitrary hexahedral subcell and coordinate system, and a standard cubic subcell and coordinate system;

[0050] Figure 5 is a schematic diagram of periodic boundary conditions;

[0051] Figure 6 SiC f Cloud diagram of microscopic equivalent stress, plastic strain and damage variable of Ti / Ti composite material under transverse stress state;

[0052] Figure 7 SiC f Stress-strain curves of / Ti composite materials under transverse stress state. DETAILED DESCRIPTION

[0053] The present invention will be further described below in conjunction with the embodiments.

[0054] Example 1

[0055] A constitutive model establishment and numerical calculation method for SiC fiber reinforced titanium matrix composite material, comprising the following steps:

[0056] Step 1: According to the SiC fiber reinforced titanium matrix composite (SiC f / Ti) volume fraction and interface thickness, and establish SiC f / Ti composite unit cell model, and assign mechanical property parameters of each component of the composite material in the unit cell model; the SiC f The macroscopic coordinate system of the unit cell model of the / Ti composite material is Ox 1 x 2 x 3 , where: O is the origin, x 1 , x 2 , x 3 It is called the macroscopic coordinate component.

[0057] Step 2: Divide the unit cell model described in step 1 into n q arbitrary hexahedral subcells, numbered from 1 to n q , any one subcell is called subcell (q); subcell (q) is transformed into a standard cube subcell through isoparametric transformation, and the corresponding relationship between the coordinates of subcell (q) and the coordinates of the standard cube subcell is established;

[0058] In step 2: the coordinate system of any hexahedral subcell (q) is O (q) -y (q)1 y (q)2 y (q)3 , where: (q) is the origin, y (q)1 ,y (q)2 ,y (q)3 is called the microscopic coordinate component; the coordinate system of the standard cubic subcell is O s -ξ 1 ξ 2 ξ 3 , where: s is the origin, ξ 1 ,ξ 2 ,ξ 3 is a standard cubic subcell, whose range is -1≤ξ 1 ,ξ 2 ,ξ 3 ≤1; macroscopic coordinate component x 1 ,x 2 ,x 3 and the microscopic coordinate component y(q)1 ,y (q)2 ,y (q)3 The relationship is in: From point O to O (q) Vector.

[0059] Step 3: Set the displacement increment of the subcell (q) The expression, superscript (q) is the subcell number, subscript i is the coordinate component symbol, which can be 1, 2, or 3; displacement increment The macroscopic displacement increment divided into subcells (q) and the microscopic displacement increment of the daughter cell (q) The sum of the two terms,

[0060] In step 3: macro displacement increment The expression is in: is the macroscopic strain increment component, x j is the macroscopic coordinate, the subscripts i and j are the coordinate component symbols, which are 1, 2, and 3; the microscopic displacement increment The expression is:

[0061]

[0062] in: and is the unknown coefficient in the microscopic displacement increment, the subscript i is the coordinate component sign, and takes 1, 2, or 3.

[0063] Step 4: Introduce the average displacement increment of the daughter cell (q) on the outer surface (p) and average surface force increment Where: the superscript (q,p) represents the outer surface (p) of the subcell (q), and then the incremental form local stiffness equation of the subcell (q) is established;

[0064] In step 4: the average displacement increment of the daughter cell (q) on the outer surface (p) for:

[0065]

[0066] in: is the microscopic displacement increment of the daughter cell (q) on the outer surface (p), ξ m , n is the standard cube coordinate component, the subscripts i, m, n are the coordinate component symbols, and the values ​​are 1, 2, and 3;

[0067] The average surface force increment of the daughter cell (q) on the outer surface (p) for:

[0068]

[0069] in: is the increment of the surface force of the daughter cell (q) on the outer surface (p), The matrix is ​​represented as The subscripts i, m, and n are all coordinate component symbols, and are 1, 2, and 3;

[0070] The local stiffness equation in incremental form is:

[0071]

[0072] in: N (q,p) is the normal vector matrix, containing the outer normal vector n of the subcell (q) on the outer surface (p) (q,p) ; is the tangent modulus matrix; is the macroscopic strain increment; Γ (q) is the thermal stress array, including the tangent modulus and thermal expansion coefficient of the subcell (q); Δ(δT) is the temperature change increment; is the tangent local stiffness matrix of the subcell (q) on the outer surface (p); is the average displacement increment of all external surfaces of the subcell (q) Composed of arrays.

[0073] Step 5: According to the continuity of displacement and surface force, the local stiffness equation of each subcell is assembled into the global stiffness equation of the composite unit cell;

[0074] In step 5: the overall stiffness equation of the composite material is:

[0075]

[0076] in: K t is the overall tangent stiffness matrix; It is an array composed of the average displacement increments of the outer surfaces of all sub-cells on the composite unit cell; The macroscopic strain increment The resulting load matrix is: D t Contains the difference in tangent modulus of adjacent subcells; ΓΔ(δT) is the load array generated by the temperature change increment Δ(δT), where: the matrix Γ contains the difference in tangent modulus and thermal expansion coefficient of adjacent subcells.

[0077] Step 6: Solve the global stiffness equation of the composite unit cell to obtain the tensor increment for each strain concentration and

[0078] In step 6: Strain concentration tensor increment and The strain increment Δε of the composite subcell is established (q) and macroscopic strain increment The relationship between the temperature change increment Δ(δT) is:

[0079]

[0080] Step 7: From the subcell volume V (q) , strain concentration tensor increment and and the subcell tangent modulus matrix Get the macroscopic tangent modulus matrix of the composite material Macroscopic equivalent thermal expansion coefficient α H and homogenized constitutive equations;

[0081] In step 7: composite macroscopic tangent modulus matrix and the macroscopic equivalent thermal expansion coefficient α H The expressions are:

[0082]

[0083] Where: v q is the volume fraction of the daughter cell (q) in the whole unit cell, is the daughter cell tangent modulus matrix;

[0084] The homogenized constitutive equation is

[0085] in: is the macro stress increment.

[0086] Step 8: Set the macroscopic stress increment of the composite unit cell at each load step and the temperature change increment Δ(δT), the macroscopic strain increment is calculated according to the homogenized constitutive equation at a certain load step

[0087] Step 9: Increment the macroscopic strain Substitute into the overall stiffness equation and solve it to obtain the stress increment Δσ of the subcell (q) (q) , strain increment Δε (q) , matrix plastic strain increment Δε pl(q) and interface damage variable d (q) ;

[0088] Step 10: According to the plastic strain increment Δε of the matrix pl(q) and interface damage variable d(q) , update the tangent modulus matrix of the subcell (q) Global stiffness equation and macroscopic tangent modulus matrix

[0089] Step 11: Repeat steps 6 to 10 until the tangent modulus matrix convergence;

[0090] Step 12: Output the microscopic field and macroscopic tangent modulus matrix of the composite unit cell under this load step Macro stress and macroscopic strain

[0091] Step 13: The macroscopic tangent modulus matrix obtained from the previous step Used for mechanical analysis of composite structural parts; a series of macroscopic stresses obtained from the loading history Macro strain The stress-strain response of the composite material under a certain loading process is obtained.

[0092] Example 2

[0093] A constitutive model establishment and numerical calculation method for SiC fiber reinforced titanium matrix composite material, comprising the following steps:

[0094] Step 1: According to the volume fraction of the unit cell and the interface thickness, establish Figure 2 SiC shown f The unit cell model of / Ti composite material. The macro coordinate system is defined as Ox 1 x 2 x 3 , where: O is the origin, x 1 , x 2 , x 3 It is called the macroscopic coordinate component; it also defines the local curvilinear coordinate system O of the interface. c -atn, where: O c is the origin, a, t, n are the three directions of the local curvilinear coordinate system, directions a, t, n are also called the axial direction, tangential direction and normal direction along the composite material interface. The fiber volume fraction is 35%, and the interface thickness w is 2.5% of the fiber radius r, that is, w = 0.025r. The mechanical properties parameters of each component of the composite material are given. Among them: The thermoelastic properties of each component are shown in the following table:

[0095]

[0096] For Ti-6Al-4V matrix, its plastic properties are reflected in the model as follows: Plastic mechanics model with power hardening:

[0097]

[0098] Where: eqv is the equivalent stress, σ Y is the yield limit, is the equivalent plastic strain, h and n are the power hardening parameters, and the specific values ​​of the three parameters are shown in the following table:

[0099]

[0100] For the interface, the debonding damage is reflected in the model as follows: Figure 3 The bilinear interface debonding damage evolution law is shown in Figure 2, where: and are the stress component and strain component of the subcell (q) located at the interface in the local curvilinear coordinate system, is the daughter cell strain threshold when the interface begins to be damaged, is the subcell strain threshold when the interface is completely damaged. When the subscript i′j′ takes aa, tt, nn, it is the component along the axial, tangential and normal directions respectively; when the subscript i′j′ takes tn, an, at, it is the shear component on the axial-tangential plane, axial-normal plane and tangential-normal plane respectively. These parameters can be obtained by experimental methods or molecular dynamics simulation methods. The specific values ​​of these parameters in this embodiment are shown in the table below;

[0101]

[0102] Step 2: Divide the unit cell model into a number of arbitrary hexahedral subcells: f The unit cell of the / Ti composite material is divided into 340 arbitrary hexahedral subcells such as Figure 2 As shown, a standard cubic subcell is established at the same time. The coordinate system of any hexahedral subcell (q) is O (q) -y (q)1 y (q)2 y (q)3 , where: (q) is the origin, y (q)1 ,y (q)2 ,y (q)3 is called the microscopic coordinate component; the coordinate system of the standard cubic subcell is O s -ξ 1 ξ 2 ξ 3 , where: s is the origin, ξ 1 ,ξ 2 ,ξ 3 is a standard cubic subcell, whose range is -1≤ξ 1 ,ξ 2 ,ξ 3 ≤1. Establish the coordinate y of any hexahedral subcell (q) through isoparametric transformation(q)1 ,y (q)2 ,y (q)3 With the standard cube subcell coordinate ξ 1 ,ξ 2 ,ξ 3 relationship, such as Figure 4 As shown, we get the physical quantity U versus y (q)1 ,y (q)2 ,y (q)3 The derivative of 1 ,ξ 2 ,ξ 3 The relationship between the derivatives of is:

[0103]

[0104] in: is the Jacobian matrix, which is used in FVDAM theory. The average value of the volume in the daughter cell replaces itself;

[0105] Step 3: According to the FVDAM model, set the displacement increment of the subcell (q) Expression, displacement increment The macroscopic displacement increment divided into subcells (q) and the microscopic displacement increment of the daughter cell (q) The sum of the two items is:

[0106]

[0107] Where: Macro displacement increment Regardless of the subcell number, the expression is:

[0108]

[0109] in: is the macroscopic strain increment component, x j is the macro coordinate, the subscripts i and j are the coordinate component symbols, both of which are 1, 2, and 3;

[0110] Micro displacement increment It is caused by the inhomogeneity of the microstructure of the composite material and is related to the subcell number. The expression is:

[0111]

[0112] in: and is the unknown coefficient in the microscopic displacement increment, the subscript i is the coordinate component sign, and takes 1, 2, or 3.

[0113] Step 4: Introduce the average displacement increment of the outer surface of the subcell:

[0114]

[0115] in: is the microscopic displacement increment of the daughter cell (q) on the outer surface (p), ξ m , n is the standard cube coordinate component, the subscripts i, m, n are the coordinate component symbols, and the values ​​are 1, 2, and 3;

[0116] Average surface force increment

[0117]

[0118] in: is the displacement increment of the daughter cell (q) on the outer surface (p), is the surface force increment of the daughter cell (q) on the outer surface (p), and the subscripts i, m, and n are all coordinate component symbols, which are 1, 2, and 3. Using the geometric equations, equilibrium equations, and incremental form of Hooke's law of elastic-plastic theory, the unknown coefficients and The average displacement increment of the daughter cell (q) on the outer surface (p) is It is expressed as:

[0119]

[0120] Where: ΔW (q) The coefficient to be determined and The array composed of is the average displacement increment of the daughter cell (q) on the outer surface (p) The array composed of Z For To ΔW (q) The transfer matrix of .

[0121] Then the local stiffness equation of the subcell (q) in incremental form is given as:

[0122]

[0123] in: N (q,p) is the normal vector matrix, containing the outer normal vector n of the subcell (q) on the outer surface (p) (q,p) ; is the tangent modulus matrix; is the macroscopic strain increment, and are the macroscopic normal strain increment components along directions 1, 2 and 3, respectively. and are the macroscopic shear strain increment components on the 2-3 plane, 1-3 plane and 1-2 plane respectively; Γ (q) is the thermal stress array, including the tangent modulus and thermal expansion coefficient of the subcell (q); Δ(δT) is the temperature change increment; is the tangent local stiffness matrix of the subcell (q) on the outer surface (p);

[0124] Step 5: Assemble the global stiffness equation of the composite unit cell from the continuity and periodic boundary conditions of displacement and traction. The continuity equation is:

[0125]

[0126] Among them: daughter cells (q) and (q') are adjacent and have a common outer surface (p). is the average displacement component of the daughter cell (q) on the outer surface (p), is the average displacement component of the daughter cell (q') on the outer surface (p); is the average surface force component of the daughter cell (q) on the outer surface (p), is the average surface force component of the daughter cell (q') on the outer surface (p).

[0127] The periodic boundary conditions are:

[0128]

[0129] Among them: the outer surfaces (p) and (p") of the daughter cells (q) and (q") are located at the boundary of the composite unit cell, and the outer surfaces (p) and (p") are opposite, such as Figure 5 shown. is the average displacement component of the daughter cell (q") on the outer surface (p"); is the average surface force component of the daughter cell (q") on the outer surface (p").

[0130] The overall stiffness equation of the composite unit cell is:

[0131]

[0132] in: K t is the overall tangent stiffness matrix; It is an array composed of the average displacement increments of the outer surfaces of all sub-cells on the composite unit cell; The macroscopic strain increment The resulting load matrix is D tContains the difference in tangent modulus of adjacent subcells; ΓΔ(δT) is the load matrix generated by the temperature change increment Δ(δT), where the matrix Γ contains the difference in tangent modulus and thermal expansion coefficient of adjacent subcells;

[0133] Step 6: Solve the global stiffness equation of the composite unit cell and obtain the strain concentration tensor increment of the subcell (q) and The specific implementation method is:

[0134] Solving equations and get and in, Contains 6 columns, the elements in the 1st to 3rd columns represent the macroscopic strain increment components or The average displacement increment of all subcells on the outer surface of the composite unit cell when the unit cell is 1, and the elements in the 4th to 6th columns represent the macroscopic strain increment components. or The average displacement increment of the outer surface of all subcells on the composite unit cell when the unit cell is 1; It represents the average displacement increment of the outer surface of all daughter cells on the composite unit cell when Δ(δT) is unit 1, that is, when a unit temperature change occurs. Each column element and Distribute to each subcell and substitute into the relationship between the unknown coefficient and the average displacement increment described in step 4 Find the unknown coefficients and substitute them into the displacement increment expression of the subcell, and obtain the strain concentration tensor increment according to the geometric equation and

[0135] Step 7: From the composite unit cell volume V tot , daughter cell volume V (q) , strain concentration tensor increment and Daughter cell tangent modulus matrix The macroscopic tangent modulus matrix of the composite material is obtained:

[0136]

[0137] And the macro equivalent thermal expansion coefficient:

[0138]

[0139] Where: v q =V (q) / V tot is the volume fraction of the daughter cell (q) in the whole unit cell, is the daughter cell tangent modulus matrix;

[0140] Then the homogenized constitutive equation is obtained:

[0141]

[0142] in: is the macro stress increment, and are the macro normal stress increment components along directions 1, 2 and 3, respectively. and are the incremental components of macroscopic shear stress on the 2-3 plane, 1-3 plane and 1-2 plane respectively;

[0143] Step 8: Set the macroscopic stress increment of the composite unit cell under a certain load step and the temperature change increment Δ(δT), and substitute it into the homogenized constitutive equation to obtain the macroscopic strain increment under this load step In this embodiment, SiC f The temperature increment Δ(δT) = -923.3°C is applied to the unit cell of the SiC / Ti composite material to obtain the residual thermal stress field of the composite material. f Transverse stress applied to unit cell of / Ti composite The stress increment applied in each loading step was 2 MPa;

[0144] Step 9: Increment the macroscopic strain Substituting the temperature change increment Δ(δT) into the overall stiffness equation, we get the array consisting of the average displacement increments of the outer surfaces of all subcells on the composite unit cell. Distribute to each daughter cell and find the unknown coefficient of the daughter cell (q) and Then the stress increment of the daughter cell (q) is obtained Strain increment Matrix plastic strain increment and interface damage variables in: and Along Figure 3 The normal stress increment components in directions 1, 2, and 3 are shown. and are the incremental components of shear stress on the 2-3 plane, 1-3 plane and 1-2 plane respectively; and are the positive strain increment components along directions 1, 2 and 3 respectively, and are the incremental components of shear strain on the 2-3 plane, 1-3 plane and 1-2 plane respectively; and Along Figure 3 The plastic normal strain increment components in directions 1, 2, and 3 are shown. and are the plastic shear strain increment components on the 2-3 plane, 1-3 plane and 1-2 plane respectively; and are the damage variable components along the axial, tangential and normal directions, respectively. and are the damage variable components on the axial-tangential plane, axial-normal plane and tangential-normal plane, respectively. and are the strain components of the subcell (q) in the local coordinate system and A univariate function of ;

[0145] Step 10: From the matrix plastic strain increment Δε pl(q) and interface damage variable d (q) , find the tangent modulus matrix of the daughter cell (q).

[0146] The tangent modulus matrix for the matrix subcell (q) is Depend on

[0147]

[0148] Given. Among them: is the elastic modulus matrix of the daughter cell (q), Δε pl(q) is the plastic strain increment of the matrix, Δε (q) is the strain increment, s (q) is the deflection tensor of stress.

[0149] The tangent modulus matrix for the interface subcell (q) is Depend on

[0150]

[0151] Given. and are the subcells (q) located at the interface in the rectangular coordinate y (q)1 ,y (q)2 ,y (q)3 Stress under and strain Transform to the component array in local curvilinear coordinates a, t, n. The subscripts 11, 22, 33, 23, 13, and 12 have the same meanings as in step 9. T 1 (q) , is the rectangular coordinate y (q)1 ,y (q)2 ,y (q)3The transformation matrix to the local curve coordinates a, t, n is as follows: Combination Figure 2 , the expression of the transformation matrix is:

[0152]

[0153]

[0154] in: is the interface elastic modulus matrix in the local curvilinear coordinate system. Since the Poisson’s ratio of the interface subcell is 0, is a diagonal matrix. I 6×6 is the 6th-order identity matrix.

[0155] is the interface damage variable matrix, and its derivative of the strain component is:

[0156]

[0157] in: is the diagonal matrix of subcell strain in the local curvilinear coordinate system;

[0158] Step 11: Repeat steps 6 to 10 until the tangent moduli of all subcells converge. The convergence criterion here uses a combination of absolute and relative criteria. For the difference between the tangent modulus matrix before and after the first iteration ||ΔC (q) || first ≥50MPa, if the difference in elastic modulus matrix before and after subsequent iterations ||ΔC (q) || is ||ΔC (q) || first 1%, it can be determined that the load step has converged; for ||ΔC (q) || first <50MPa, if the difference between the tangent modulus matrix before and after the subsequent iteration ||ΔC (q) ||<0.01MPa, the load step can be judged to be converged;

[0159] Step 12: Output the microscopic stress field, equivalent plastic strain field, interface damage variable field, macroscopic tangent modulus matrix, macroscopic stress and macroscopic strain of the composite unit cell under the load step. Figure 6 SiC shown f Macroscopic stress analysis of unit cell of / Ti composites and 498MPa, the stress component of the daughter cell (q) Equivalent stress of daughter cell (q) Equivalent plastic strain of daughter cell (q) The damage variable of the interface of the subcell (q) along the normal direction The damage variable along the tangential direction of the interface between the cell and the subcell (q) exist and SiC at 498MPa f The macroscopic tangent modulus matrix of / Ti composite materials is shown in the table below;

[0160]

[0161]

[0162] Step 13: Plot a series of macroscopic stresses and macroscopic strains into a stress-strain curve to obtain the stress-strain response of the composite material under the loading process. Figure 7 The figure shows the transverse stress-strain response. f The / Ti composite material has residual thermal stress before loading, and there is deformation compared to the state before cooling, so the macroscopic strain component Initially non-zero, when drawing Figure 7 Time will Pan to the origin.

[0163] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A constitutive model establishment and numerical calculation method for SiC fiber reinforced titanium-based composite materials, characterized in that: The following steps are involved: Step 1: According to SiC fiber reinforced titanium matrix composite material SiC f / Ti volume fraction and interface thickness, establish SiC f / Ti composite unit cell model, and assign mechanical property parameters of each component of the composite material in the unit cell model; Step 2: Divide the unit cell model described in step 1 into n q arbitrary hexahedral subcells, numbered from 1 to n q , any one subcell is called subcell (q); subcell (q) is transformed into a standard cube subcell through isoparametric transformation, and the corresponding relationship between the coordinates of subcell (q) and the coordinates of the standard cube subcell is established; Step 3: Set the displacement increment of the subcell (q) The expression, superscript (q) is the subcell number, subscript i is the coordinate component symbol, which can be 1, 2, or 3; displacement increment The macroscopic displacement increment divided into subcells (q) and the microscopic displacement increment of the daughter cell (q) The sum of the two terms, Step 4: Introduce the average displacement increment of the daughter cell (q) on the outer surface (p) and average surface force increment Where: the superscript (q,p) represents the outer surface (p) of the subcell (q), and then the incremental form local stiffness equation of the subcell (q) is established; Step 5: According to the continuity of displacement and surface force, the local stiffness equation of each subcell is assembled into the global stiffness equation of the composite unit cell; Step 6: Solve the global stiffness equation of the composite unit cell to obtain the tensor increment for each strain concentration and Step 7: From the subcell volume V (q) , strain concentration tensor increment and and the subcell tangent modulus matrix Get the macroscopic tangent modulus matrix of the composite material Macroscopic equivalent thermal expansion coefficient α H and homogenized constitutive equations; Step 8: Set the macroscopic stress increment of the composite unit cell at each load step and the temperature increment △(δT), the macroscopic strain increment is calculated according to the homogenized constitutive equation at a certain load step Step 9: Increment the macroscopic strain Substitute into the overall stiffness equation and solve it to obtain the stress increment △σ of the subcell (q) (q) , strain increment △ε (q) , matrix plastic strain increment △ε pl(q) and interface damage variable d (q) ; Step 10: According to the plastic strain increment △ε of the matrix pl(q) and interface damage variable d (q) , update the tangent modulus matrix of the subcell (q) Global stiffness equation and macroscopic tangent modulus matrix Step 11: Repeat steps 6 to 10 until the tangent modulus matrix convergence; Step 12: Output the microscopic field and macroscopic tangent modulus matrix of the composite unit cell under this load step Macro stress and macroscopic strain Step 13: The macroscopic tangent modulus matrix obtained from the previous step Used for mechanical analysis of composite structural parts; a series of macroscopic stresses obtained from the loading history Macro strain Obtain the stress-strain response of the composite material under a certain loading process; In step 4: the average displacement increment of the daughter cell (q) on the outer surface (p) for: in: is the microscopic displacement increment of the daughter cell (q) on the outer surface (p), ξ m , n is the standard cube coordinate component, the subscripts i, m, n are the coordinate component symbols, and the values ​​are 1, 2, and 3; Average surface force increment for: in: is the increment of the surface force of the daughter cell (q) on the outer surface (p), The matrix is ​​represented as The subscripts i, m, and n are all coordinate component symbols, and are 1, 2, and 3; The local stiffness equation in incremental form is: in: N (q,p) is the normal vector matrix, containing the outer normal vector n of the subcell (q) on the outer surface (p) (q,p) ; is the tangent modulus matrix; is the macroscopic strain increment; Γ (q) is the thermal stress array, including the tangent modulus and thermal expansion coefficient of the subcell (q); △(δT) is the temperature change increment; is the tangent local stiffness matrix of the subcell (q) on the outer surface (p); is the average displacement increment of all external surfaces of the subcell (q) The array formed; In step 5: the overall stiffness equation of the composite material is: Where: K t is the overall tangent stiffness matrix; It is an array composed of the average displacement increments of the outer surfaces of all sub-cells on the composite unit cell; The macroscopic strain increment The resulting load matrix, where: matrix D t Contains the difference between the tangent moduli of adjacent subcells; Γ△(δT) is the load array generated by the temperature change increment △(δT), where: the matrix Γ contains the difference between the tangent moduli and thermal expansion coefficients of adjacent subcells; In step 6: Strain concentration tensor increment and The strain increment △ε of the composite subcell is established (q) and macroscopic strain increment The relationship between the temperature increment △(δT) is: In step 7: composite macroscopic tangent modulus matrix and the macroscopic equivalent thermal expansion coefficient α H The expressions are: Where: v q is the volume fraction of the daughter cell (q) in the whole unit cell, is the daughter cell tangent modulus matrix; The homogenized constitutive equation is in: is the macro stress increment.

2. The constitutive model establishment and numerical calculation method of SiC fiber reinforced titanium-based composite material according to claim 1 is characterized in that: In step 1: SiC f The macro coordinate system of the unit cell model of the / Ti composite material is O-x1x2x3, where: O is the origin, and x1, x2, and x3 are called macro coordinate components.

3. The constitutive model establishment and numerical calculation method of SiC fiber reinforced titanium-based composite material according to claim 1, characterized in that: In step 2: the coordinate system of any hexahedral subcell (q) is O (q) -y (q)1 y (q)2 y (q)3 , where: (q) is the origin, y (q)1 ,y (q)2 ,y (q)3 is called the microscopic coordinate component; the coordinate system of the standard cubic subcell is O s -ξ1ξ2ξ3, where: O s is the origin, ξ1, ξ2, ξ3 are standard cubic subcells, and their range is -1≤ξ1,ξ2,ξ3≤1; the macroscopic coordinate components x1,x2,x3 and the microscopic coordinate components y (q)1 ,y (q)2 ,y (q)3 The relationship is in: From point O to O (q) Vector.

4. The constitutive model establishment and numerical calculation method of SiC fiber reinforced titanium-based composite material according to claim 1, characterized in that: In step 3: macro displacement increment The expression is in: is the macroscopic strain increment component, x j is the macroscopic coordinate, the subscripts i and j are the coordinate component symbols, which are 1, 2, and 3 respectively; the microscopic displacement increment The expression is: in: and is the unknown coefficient in the microscopic displacement increment. The subscript i is the coordinate component sign and takes 1, 2, or 3.

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