CN105930579B - Residual Stiffness prediction technique after a kind of oxidation of control of two-dimensional braided ceramic matric composite - Google Patents
Residual Stiffness prediction technique after a kind of oxidation of control of two-dimensional braided ceramic matric composite Download PDFInfo
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- 238000007254 oxidation reaction Methods 0.000 title claims abstract description 67
- 230000003647 oxidation Effects 0.000 title claims abstract description 63
- 239000002131 composite material Substances 0.000 title claims abstract description 40
- 238000000034 method Methods 0.000 title claims abstract description 18
- 239000000919 ceramic Substances 0.000 title abstract 4
- HPNSNYBUADCFDR-UHFFFAOYSA-N chromafenozide Chemical compound CC1=CC(C)=CC(C(=O)N(NC(=O)C=2C(=C3CCCOC3=CC=2)C)C(C)(C)C)=C1 HPNSNYBUADCFDR-UHFFFAOYSA-N 0.000 title abstract 4
- 239000000835 fiber Substances 0.000 claims abstract description 44
- 239000000463 material Substances 0.000 claims abstract description 23
- 230000000737 periodic effect Effects 0.000 claims abstract description 16
- 239000011153 ceramic matrix composite Substances 0.000 claims description 36
- 238000006073 displacement reaction Methods 0.000 claims description 22
- OKTJSMMVPCPJKN-UHFFFAOYSA-N Carbon Chemical compound [C] OKTJSMMVPCPJKN-UHFFFAOYSA-N 0.000 claims description 11
- 229910052799 carbon Inorganic materials 0.000 claims description 11
- 230000015556 catabolic process Effects 0.000 claims description 10
- 238000006731 degradation reaction Methods 0.000 claims description 10
- 238000006243 chemical reaction Methods 0.000 claims description 6
- 230000008859 change Effects 0.000 claims description 5
- 229920000049 Carbon (fiber) Polymers 0.000 claims description 4
- 239000004917 carbon fiber Substances 0.000 claims description 4
- 230000004913 activation Effects 0.000 claims description 3
- QVGXLLKOCUKJST-UHFFFAOYSA-N atomic oxygen Chemical compound [O] QVGXLLKOCUKJST-UHFFFAOYSA-N 0.000 claims description 3
- 239000011248 coating agent Substances 0.000 claims description 3
- 238000000576 coating method Methods 0.000 claims description 3
- 238000005336 cracking Methods 0.000 claims description 3
- 239000007789 gas Substances 0.000 claims description 3
- VNWKTOKETHGBQD-UHFFFAOYSA-N methane Chemical compound C VNWKTOKETHGBQD-UHFFFAOYSA-N 0.000 claims description 3
- 229910052760 oxygen Inorganic materials 0.000 claims description 3
- 239000001301 oxygen Substances 0.000 claims description 3
- 230000035515 penetration Effects 0.000 claims description 3
- 239000000758 substrate Substances 0.000 claims description 2
- 238000002474 experimental method Methods 0.000 abstract description 3
- 238000012360 testing method Methods 0.000 abstract description 2
- 230000001590 oxidative effect Effects 0.000 abstract 1
- 239000011204 carbon fibre-reinforced silicon carbide Substances 0.000 description 12
- 238000005034 decoration Methods 0.000 description 2
- 230000007547 defect Effects 0.000 description 2
- 238000010586 diagram Methods 0.000 description 2
- 238000012986 modification Methods 0.000 description 2
- 230000004048 modification Effects 0.000 description 2
- 238000005452 bending Methods 0.000 description 1
- 150000001722 carbon compounds Chemical class 0.000 description 1
- 230000007797 corrosion Effects 0.000 description 1
- 238000005260 corrosion Methods 0.000 description 1
- 230000008878 coupling Effects 0.000 description 1
- 238000010168 coupling process Methods 0.000 description 1
- 238000005859 coupling reaction Methods 0.000 description 1
- 230000005489 elastic deformation Effects 0.000 description 1
- 230000007613 environmental effect Effects 0.000 description 1
- 230000010354 integration Effects 0.000 description 1
- 239000011159 matrix material Substances 0.000 description 1
- 239000011148 porous material Substances 0.000 description 1
- 230000008569 process Effects 0.000 description 1
- 238000011160 research Methods 0.000 description 1
- 238000001878 scanning electron micrograph Methods 0.000 description 1
- 238000010008 shearing Methods 0.000 description 1
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Abstract
The present invention relates to Residual Stiffness prediction techniques after a kind of oxidation of control of two-dimensional braided ceramic matric composite.Rigidity refers to the ability that material resists flexible deformation in stress.By the rigidity of analysis of material each component, that is, it can determine the Stress distribution of material internal.Therefore the present invention provides a kind of methods of Residual Stiffness after energy Accurate Prediction control of two-dimensional braided ceramic matric composite oxidation.The kinetic model for considering fiber oxidation is proposed, establishes the micro-scale model and control of two-dimensional braided ceramic matric composite mesoscale model for considering fiber oxidation on this basis.Residual Stiffness using FInite Element, by applying periodic boundary condition, after calculating material oxidation.The present invention can accurately predict the material and not need to spend a large amount of human and material resources to go to test by experiment in the Residual Stiffness in different oxidization times, different oxidizing temperatures section, therefore save a large amount of experimentation cost.
Description
Technical Field
The invention relates to a method for predicting residual stiffness of a two-dimensional woven ceramic matrix composite after oxidation.
Background
The two-dimensional woven ceramic matrix composite has excellent performances of high specific strength, high specific modulus, high temperature resistance, corrosion resistance, low density and the like, and has wide requirements on high-temperature protection systems of aerospace aircrafts. In the using process of the material, due to the influence of high-temperature environmental factors, oxidation damage is gradually generated, the mechanical property of the material is reduced, and the service life and the safety of engineering components are further seriously influenced. The rigidity refers to the capability of the material to resist elastic deformation when stressed, and the research on the residual rigidity of the two-dimensional plain weave ceramic matrix composite after oxidation has important significance for the application of the material.
The application range of the unidirectional ceramic matrix composite is limited due to the defects of weak mechanical property in the non-fiber direction and the like. The two-dimensional woven structure ceramic matrix composite overcomes the defects of a one-way composite, and meanwhile, the integration of fiber bundles in the thickness direction is higher, the shearing strength between material layers is increased, the layering phenomenon is reduced, and the impact resistance and the bending fatigue resistance of the composite are improved, so that the application range of the ceramic matrix composite is greatly expanded.
However, because the two-dimensional woven ceramic matrix composite is a novel structural material, no efficient method for predicting the residual stiffness after oxidation exists at home and abroad, and a patent of the invention is not disclosed. The oxidation damage and rigidity model of the 2D-C/SiC composite material [ J ] the composite material science, 2009,26(3):175-181.) is tested by an experimental method to test the residual rigidity of the 2D C/SiC composite material in an environment of 700 ℃, a calculation formula is established based on the change of a microscopic structure, and the calculated value is in accordance with the experimental value. However, a large amount of experimental funds are consumed through an experimental mode, and the proposed calculation model can only calculate the residual stiffness at discrete specific temperatures.
At present, how to accurately predict the residual stiffness of the two-dimensional woven ceramic matrix composite after oxidation is an important and difficult-to-solve problem in the technical field.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the prior art, a prediction method capable of effectively predicting residual stiffness of the two-dimensional woven ceramic matrix composite after oxidation is provided.
The technical scheme is as follows: a method for predicting residual stiffness of a two-dimensional woven ceramic matrix composite after oxidation comprises the following steps:
(1) establishing an oxidation kinetic model based on a mass loss rate theory and a fiber degradation rule hypothesis;
(2) establishing an oxidized micro-scale unit cell model by adopting finite element software based on an oxidation kinetic model;
(3) applying periodic boundary conditions of the microscopic-scale unit cell model;
(4) calculating elastic parameters of the unit cell model in 6 directions;
(5) establishing a two-dimensional woven ceramic matrix composite material unit cell model by adopting finite element software;
(6) taking the elastic parameters in 6 directions of the oxidized micro-scale unit cell model obtained by calculation as basic attributes of the yarns, and bringing the basic attributes into a two-dimensional woven ceramic matrix composite unit cell model;
(7) applying periodic boundary conditions of the two-dimensional woven ceramic matrix composite unit cell model;
(8) and calculating to obtain the axial residual elastic modulus of the two-dimensional woven ceramic matrix composite.
As a preferable embodiment of the present invention, in the step (1), the mass loss rate is theoretically divided into two temperature ranges:
1) when the temperature is between 400 and 700 ℃, the formula is as follows:
wherein λ isrIs the mass loss rate of the composite material, W is the mass of the composite material, Δ W is the mass change of the material, K0Is a constant that is related to the rate of oxidation,is the volume fraction of oxygen, P is atmospheric pressure, McIs the molar mass of the carbon fiber, R is the gas constant, T is the ambient temperature, ErIs oxidation reaction activation energy, t is oxidation time, SeffIs the effective reaction area of the carbon; wherein S iseffμ is the reaction effective coefficient of carbon;
2) when the temperature is between 700 and 900 ℃, the formula is as follows:
wherein N iscIs the molar density of carbon, λ is a constant related to the initial state, TcIs the cracking temperature of the substrate, LcIs the coating thickness;
the fiber degradation law assumes: assuming that the fiber degrades at high temperature with a circular law, the formula is as follows:
where δ is the oxidation length of the fiber, ρfAnd ρcRespectively, the density of the fiber and the composite material, L is the length of the composite material, H is the height of the composite material, N is the amount of carbon, N is the amount of carbonfIs the number of fibers per unit area;
subjecting the composite material to a mass loss ratio lambdarSubstituting into the calculation equation (3) for the oxidation length δ of the fiber yields:
1) when the temperature is between 400 and 700 ℃:
2) when the temperature is between 700 and 900 ℃:
the remaining radii of the oxidized fibers thus obtained are:
1) when the temperature is between 400 and 700 ℃, the formula is as follows:
2) when the temperature is between 700 and 900 ℃, the formula is as follows:
wherein rf is the residual radius of the fiber after oxidation, rf0Is the initial radius of the fiber when not oxidized.
In the preferred embodiment of the present invention, in the step (2), the oxidation is assumed to be a uniform penetration oxidation, and the radius of the fiber after oxidation is equal.
In the step (3), the boundary condition is applied to satisfy the continuity of the displacement and the consistency of the stress distribution in two opposite planes of the model.
In a preferred embodiment of the present invention, in the step (4), the elastic parameters in the 6 directions include elastic moduli E in the x, y, and z directionsx、Ey、EzShear modulus G in xy, xz, yz directionsxy、Gxz、GyzAnd poisson's ratio vxy、vxz、vyz。
As a preferred embodiment of the present invention, in the step (7), the applying of the periodic boundary condition is:
wherein Z + and Z-respectively represent two opposite boundary surfaces perpendicular to the Z-axis,is at the same timeThe displacement on the surface of the Z + boundary,for displacement on the Z-boundary surface, xi Z+Is the displacement of a node on the Z + surface, xi Z-Is the amount of displacement of the node on the Z-surface,for periodic part of the displacement on the boundary surface, θiIs the average strain tensor of the periodic structure.
Has the advantages that: the method for predicting the residual stiffness of the two-dimensional woven ceramic matrix composite after oxidation, provided by the invention, provides a dynamic model considering fiber oxidation based on a mass loss rate model and a fiber degradation rule hypothesis. Based on an oxidation kinetic model, a micro-scale model considering fiber oxidation and a two-dimensional woven ceramic matrix composite material single-cell scale model are established by adopting a finite element method, and the residual stiffness of the material is predicted. The prediction model provided by the invention fully considers the degradation rule of the fiber along with the oxidation time and temperature, so that the residual stiffness of the two-dimensional woven ceramic matrix composite material after oxidation can be accurately predicted, and a large amount of experiment cost is saved.
Drawings
FIG. 1 is a scanning electron micrograph of carbon fiber oxidation near the crack tip;
FIG. 2 is a two-dimensional planar model of a ceramic matrix composite;
FIG. 3 is a schematic view of a unidirectional ceramic matrix composite oxidation;
FIG. 4 is a microscopic model after oxidation;
FIG. 5 is a schematic representation of the microscopic model boundary conditions after oxidation;
FIG. 6 is a two-dimensional plain woven ceramic matrix composite unit cell model;
FIG. 7 is a schematic diagram of boundary conditions of a two-dimensional plain woven ceramic matrix composite cell model;
FIG. 8 is a detailed flow diagram of the predictive model;
FIG. 9 is a comparison graph of predicted values of residual modulus of elasticity in the axial direction of a two-dimensional plain weave C/SiC composite material at 700 ℃ in an air environment and experimental values;
FIG. 10 is a comparison graph of predicted values of residual modulus of elasticity in the axial direction of a two-dimensional plain weave C/SiC composite material at 800 ℃ in an air environment with experimental values;
FIG. 11 is a comparison graph of predicted values of residual modulus of elasticity in the axial direction of a two-dimensional plain weave C/SiC composite material at 850 ℃ in an air environment and experimental values;
FIG. 12 is a graph comparing a predicted value of the residual modulus of elasticity in the axial direction of a two-dimensional plain-woven C/SiC composite material with an experimental value at 900 ℃ in an air atmosphere.
Detailed Description
The invention is further explained below with reference to the drawings.
In this embodiment, a two-dimensional plain woven C/SiC composite material is taken as an example, and the residual stiffness after oxidation of the material in the range of 700 ℃ to 900 ℃ is predicted, wherein the material performance parameters are shown in table 1.
TABLE 1
As shown in fig. 8, the method comprises the following specific steps:
(1) and establishing an oxidation kinetic model based on a mass loss rate theory and a fiber degradation rule hypothesis. Wherein, the mass loss rate theory is divided into two temperature intervals:
1) when the temperature is between 400 and 700 ℃, the mass loss rate formula of the composite material is as follows:
wherein λ isrIs the mass loss rate of the composite; w is the mass of the composite; Δ W is the mass change in the mass loss rate of the composite material; k0Is a constant related to the oxidation rate;is the volume fraction of oxygen, taken here as 20.95%; p is atmospheric pressure, taken here at 101.325 KPa; mcIs the molar mass of the carbon fibers, here taken to be 12X 103kg/mol; r is a gas constant, here taken as 8.3145J/(mol. K); t is ambient temperature; erIs oxidation reaction activation energy; t is the oxidation time; seffIs the effective reaction area of carbon, SeffRelated to the mass of the sample, denoted Seffμ is the effective coefficient of reaction of carbon, and μ depends on the microcrack area and pore cross-sectional area of the sample and the sample density, and can be determined experimentally.
2) When the temperature is between 700 and 900 ℃, the mass loss rate formula of the composite material is as follows:
wherein N iscIs the molar density of carbon; λ is a constant related to the initial state; t iscThe cracking temperature of the matrix is 1030 ℃; l iscIs the coating thickness.
The fiber degradation law is assumed: according to the scanning electron microscope photograph shown in FIG. 1 and the two-dimensional planar model of the ceramic matrix composite shown in FIG. 2, it is assumed that the fibers are in a circular shape at high temperaturePerforming degradation, wherein the oxidation length of the fiber is the length of the BD segment in the figure as shown in figure 3; r is0i.e., the distance of OD from the center O of the fitted oxidized region to the unoxidized surface of the fiber, r' is the distance of OC and is the radius of the fitted oxidized region, and α is r0And r' is included angle. The fiber degradation law formula is as follows:
wherein δ is the oxidation length of the fiber; rhofAnd ρcThe densities of the fiber and the composite material are respectively expressed; l is the length of the composite and H is the height of the composite, as shown in FIG. 2; n is the amount of carbon species; n is a radical offIs the number of fibers per unit area.
The mass loss rate lambda of the composite materialrThe calculation equation (3) introduced into the oxidation length δ of the fiber yields:
1) when the temperature is between 400 and 700 ℃:
2) when the temperature is between 700 and 900 ℃:
initial radius if fiber not oxidized is rf0The residual radius of the oxidized fiber is rf, and the relationship is rf0δ residual radius of the fiber after oxidation:
1) when the temperature is between 400 and 700 ℃, the formula is as follows:
2) when the temperature is between 700 and 900 ℃, the formula is as follows:
(2) based on the oxidation kinetics model in the step (1), namely the equations (6) and (7), ANSYS software is adopted to establish a micro-scale unit cell model after oxidation, as shown in FIG. 4. Wherein, assuming that the oxidation is uniform penetration oxidation, the radius of the fiber is equal after oxidation.
(3) Periodic boundary conditions of the microscopic-scale unit cell model are applied: the applied boundary conditions satisfy the continuity of displacement and the consistency of the stress distribution in two opposite planes of the unit cell model at the microscopic scale, as shown in fig. 5; the applied periodic boundary conditions are shown in table 2.
TABLE 2
| No | S(x-,y,z) | S(x+,y,z) | S(x,y-,z) | S(x,y+,z) | S(x,y,z-) | S(x,y,z+) |
| 1 | ux＝0 | ux＝0.1 | uy＝0 | uy＝const | uz＝0 | uz＝const |
| 2 | ux＝0 | ux＝const | uy＝0 | uy＝0.1 | uz＝0 | uz＝const |
| 3 | ux＝0 | ux＝const | uy＝0 | uy＝const | uz＝0 | uz＝0.1 |
| 4 | ux＝0 | ux＝const | uz＝0 | uz＝0 | uy＝0 | uy＝0.1 |
| 5 | uz＝0 | uz＝0 | uy＝0 | uy＝const | ux＝0 | ux＝0.1 |
| 6 | uy＝0 | uy＝0 | ux＝0 | ux＝0.1 | uz＝0 | uz＝const |
In the table: s (x)-,y,z)、S(x+Y, z) are two planes with minimum and maximum x-direction coordinates, S (x, y)-,z)、S(x,y+Z) are the two planes with the smallest and largest y-direction coordinates, S (x, y, z)-)、S(x,y,z+) The two planes with the smallest and the largest z-direction coordinate are respectively. u. ofx、uy、uzDisplacement constraints in x, y and z directions respectively, const represents displacement coupling of all nodes in a plane.
(4) Calculating elastic parameters of the unit cell model in 6 directions, wherein the elastic parameters in 6 directions comprise elastic moduli E in x, y and z directionsx、Ey、EzShear modulus G in xy, xz, yz directionsxy、Gxz、GyzAnd poisson's ratio vxy、vxz、vyz。
(5) Adopting ANSYS software to establish a two-dimensional woven ceramic matrix composite material single-cell model as shown in figure 6; the dimensional parameters of the model are shown in table 3.
TABLE 3
| hf | a | hb | b |
| 0.056 | 0.4 | 0.02 | 0.2 |
Wherein a is the width of the yarn, b is the space between the yarns in the same direction, hfIs the yarn thickness, hbIs the thickness of the base layer.
(6) Taking the elastic parameters in 6 directions of the oxidized micro-scale unit cell model obtained by calculation as basic attributes of the yarns, and bringing the basic attributes into a two-dimensional woven ceramic matrix composite unit cell model;
(7) applying the periodic boundary conditions of the two-dimensional woven ceramic matrix composite unit cell model, as shown in FIG. 7, the expression is:
wherein Z + and Z-respectively represent two opposite boundary surfaces perpendicular to the Z-axis,for a displacement on the Z + boundary surface,for displacement on the Z-boundary surface, xi Z+Is the displacement of a node on the Z + surface, xi Z-Is the amount of displacement of the node on the Z-surface,for periodic part of the displacement on the boundary surface, θiIs the average strain tensor of the periodic structure.
The relative displacement between the two boundary surfaces Z + and Z-is expressed as:
wherein,indicating the amount of displacement change of the node on both the boundary surfaces Z + and Z-.
(8) According to the formulaELIs the axial elastic modulus, σ, of the materialmIs the mean stress in the axial direction of the unit, epsilonmAnd calculating the axial residual elastic modulus of the two-dimensional plain weave C/SiC composite material for the unit axial average strain.
FIG. 9 shows a comparison of predicted values of residual modulus of elasticity in the axial direction of a two-dimensional plain weave C/SiC composite material with experimental values at an ambient temperature of 700 ℃. FIG. 10 shows a comparison of predicted values of residual modulus of elasticity in the axial direction of a two-dimensional plain weave C/SiC composite material with experimental values at an ambient temperature of 800 ℃. FIG. 11 shows a comparison of predicted values of residual modulus of elasticity in the axial direction of a two-dimensional plain weave C/SiC composite material with experimental values at an ambient temperature of 850 ℃. FIG. 12 shows a comparison of predicted values of residual modulus of elasticity in the axial direction of a two-dimensional plain weave C/SiC composite material with experimental values at an ambient temperature of 900 ℃. Through comparison, the method disclosed by the invention can effectively predict the residual stiffness of the two-dimensional woven ceramic matrix composite after oxidation.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.
Claims (5)
1. The method for predicting the residual stiffness of the two-dimensional woven ceramic matrix composite after oxidation is characterized by comprising the following steps of:
(1) establishing an oxidation kinetic model based on a mass loss rate theory and a fiber degradation rule hypothesis;
(2) establishing an oxidized micro-scale unit cell model by adopting finite element software based on an oxidation kinetic model;
(3) applying periodic boundary conditions of the microscopic-scale unit cell model;
(4) calculating elastic parameters of the unit cell model in 6 directions;
(5) establishing a two-dimensional woven ceramic matrix composite material unit cell model by adopting finite element software;
(6) taking the elastic parameters in 6 directions of the oxidized micro-scale unit cell model obtained by calculation as basic attributes of the yarns, and bringing the basic attributes into a two-dimensional woven ceramic matrix composite unit cell model;
(7) applying periodic boundary conditions of the two-dimensional woven ceramic matrix composite unit cell model;
(8) calculating to obtain the axial residual elastic modulus of the two-dimensional woven ceramic matrix composite;
in the step (1), the mass loss rate theory is divided into two temperature intervals:
1) when the temperature is between 400 and 700 ℃, the formula is as follows:
wherein λ isrIs the mass loss rate of the composite material, W is the mass of the composite material, Δ W is the mass change of the material, K0Is a constant that is related to the rate of oxidation,is the volume fraction of oxygen, P is atmospheric pressure, McIs the molar mass of the carbon fiber, R is the gas constant, T is the ambient temperature, ErIs oxidation reaction activation energy, t is oxidation time, SeffIs the effective reaction area of the carbon; wherein S iseffμ is the reaction effective coefficient of carbon;
2) when the temperature is between 700 and 900 ℃, the formula is as follows:
wherein N iscIs the molar density of carbon, λ is a constant related to the initial state, TcIs the cracking temperature of the substrate, LcIs the coating thickness;
the fiber degradation law assumes: assuming that the fiber degrades at high temperature with a circular law, the formula is as follows:
where δ is the oxidation length of the fiber, ρfAnd ρcRespectively, the density of the fiber and the composite material, L is the length of the composite material, H is the height of the composite material, N is the amount of carbon, N is the amount of carbonfIs the number of fibers per unit area;
subjecting the composite material to a mass loss ratio lambdarSubstituting into the calculation equation (3) for the oxidation length δ of the fiber yields:
1) when the temperature is between 400 and 700 ℃:
2) when the temperature is between 700 and 900 ℃:
the remaining radii of the oxidized fibers thus obtained are:
1) when the temperature is between 400 and 700 ℃, the formula is as follows:
2) when the temperature is between 700 and 900 ℃, the formula is as follows:
wherein rf is the residual radius of the fiber after oxidation, rf0Is the initial radius of the fiber when not oxidized.
2. The method for predicting the residual stiffness of the two-dimensional woven ceramic matrix composite after oxidation according to claim 1, wherein: in the step (2), it is assumed that the oxidation is uniform penetration oxidation, and the radius of each part of the oxidized fiber is equal.
3. The method for predicting the residual stiffness of the two-dimensional woven ceramic matrix composite after oxidation according to claim 1, wherein: in the step (3), the applied boundary condition satisfies the continuity of displacement and the consistency of stress distribution in two opposite planes of the model.
4. The method for predicting the residual stiffness of the two-dimensional woven ceramic matrix composite after oxidation according to claim 1, wherein: in the step (4), the elastic parameters in the 6 directions include elastic moduli E in the x, y, and z directionsx、Ey、EzShear modulus G in xy, xz, yz directionsxy、Gxz、GyzAnd poisson's ratio vxy、vxz、vyz。
5. The method for predicting the residual stiffness of the two-dimensional woven ceramic matrix composite after oxidation according to claim 1, wherein: in the step (7), the applying of the periodic boundary condition is as follows:
wherein Z + and Z-respectively represent two opposite boundary surfaces perpendicular to the Z-axis,for a displacement on the Z + boundary surface,for displacement on the Z-boundary surface, xi Z+Is the displacement of a node on the Z + surface, xi Z-Is the amount of displacement of the node on the Z-surface,for periodic part of the displacement on the boundary surface, θiIs the average strain tensor of the periodic structure.
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