CN111400947B - Method for predicting compression modulus and strength of plane orthogonal woven composite material - Google Patents
Method for predicting compression modulus and strength of plane orthogonal woven composite material Download PDFInfo
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Abstract
A method for predicting the compression modulus and the compression strength of a plane orthogonal braided composite material by considering geometric nonlinearity comprises five steps: selecting a representative unit cell according to the weaving form of a fiber bundle; step two, establishing a mesomechanics model of the fabric in the unit cell under the action of a unidirectional compression load, and solving the deformation of the fabric in the unit cell by using an energy method; step three, calculating the compression modulus of the plane orthogonal woven composite material according to a mixing law; step four, calculating the stress and the strain of the plane woven composite material, and drawing a stress-strain curve; and step five, calculating the compressive strength of the plane orthogonal braided composite material according to the failure strength of the component materials (the fiber bundles and the matrix). The method is convenient and efficient, and the compression modulus and the strength of the plane orthogonal woven composite material can be conveniently and accurately predicted only by determining the weaving form and a small number of component material parameters.
Description
Technical Field
The invention provides a method for predicting the compressive modulus and strength of a plane orthogonal braided composite material by considering geometric nonlinearity, and belongs to the field of composite material design.
Background
The plane-woven composite material has the advantages of good economy, excellent overall performance, impact resistance, high specific strength and specific stiffness and the like as a novel light composite material, and compared with the composite material laminated plate reinforced by unidirectional fibers, the plane-woven composite material laminated plate has great potential in improving interlaminar strength, in-layer strength, damage tolerance and the like. Therefore, the plane woven composite material is widely applied to the aerospace field and the automobile field. The experimental means directly measures the compression modulus and the strength of the plane braided composite material, so that the cost is high, and the test process is easily influenced by many accidental factors; the finite element numerical simulation method needs to establish a complex finite element model, has complex calculation and low calculation efficiency, and is difficult to ensure the calculation precision; therefore, based on the minimum complementary energy principle, the change of the geometric shape of the fiber bundle in the compression deformation process of the planar braided composite material is considered by combining a bending beam model of an ideal fiber bundle and microscopic mechanical analysis, the compressive stress-strain relation of the planar braided composite material, which is considered by geometric nonlinearity, is established by accumulating deformation increment and load increment in the loading process, and on the basis, a fiber buckling theory, a transverse tensile crack theory and an interlayer resin failure theory are introduced to obtain the analytical expression of the compressive modulus and the compressive strength of the planar braided composite material. The macroscopic compression modulus and the compression strength of the plane orthogonal braided composite material can be rapidly and accurately predicted only by a small amount of component material performance parameters, and the method has important academic value and wide engineering application prospect.
Disclosure of Invention
The invention establishes a method for predicting the compressive modulus and the strength of a plane orthogonal braided composite material by considering geometric nonlinearity, and the method has the advantages of simple and convenient calculation, high precision and the like, and the technical scheme is as follows:
step one, according to the orthogonal weaving mode (such as periodicity, repeatability and the like) of the symmetry plane of the fiber bundle, selecting proper periodic units as representative volume elements, thereby determining cell units of the fiber bundle, and defining the orthogonal weaving geometry and size in the cell units.
As shown in fig. 1, coordinate axes x and z represent the direction of the undulation of the fiber bundle and the out-of-plane thickness direction, respectively. To establish a mechanical analysis model, the following assumptions were made for the fabric unit cell structure:
(1) the cross section of the fiber bundle is simplified into a rectangle with a semicircular shape at the left end and a semicircular shape at the right end, wherein the width is a, and the thickness is b.
(2) During compression deformation, the fiber bundle is idealized as a curved beam, and the neutral axis path of the fiber bundle is represented by a trigonometric function.
According to the assumption (1), the cross-sectional area and the moment of inertia of the yarn are respectively
Wherein (j-1, 2), j-1 represents a warp fiber bundle, and j-2 represents a weft fiber bundle.
According to the assumption (2), the neutral axis path of the wavy fiber bundle is represented as
The tangent value of the tangential angle theta at any section of the wavy fiber bundle is
Thus, the longitudinal cross-sectional area of the fiber bundle can be expressed as
In fact, in the process of actually bearing unidirectional compression load, the paths of warp and weft fiber bundles change along with the change of load (as shown in fig. 3), and the change of the geometric shape of the fiber bundles can further influence the compression performance, so that a mesoscopic mechanical model considering the change of the geometric parameters (namely geometric nonlinearity) of the fiber bundles is proposed: the process of starting compression and breaking of the plane orthogonal braided composite material is divided into n steps, and each step applies a compression load delta P to the unidirectional fiber bundles1The length and the height of the warp and weft fiber bundle fluctuation are changed, and the next step is to apply the compression load delta P again on the basis of the new geometric shape1Performing n times of iterative calculation until the compressive strength X of the plane orthogonal braided composite material is reachedc(X is given in step 3 herein)cThe prediction method of (1).
From equations (1) to (4), the fiber bundle volume fraction V can be determined using only the initial geometry (i.e., i ═ 1)t,
Wherein the warp fiber bundles are denoted by subscript 1, the weft fiber bundles are denoted by 2, the individual layer thickness is H, and the volume fraction of the fibers in the fiber bundlesVf1Can be expressed as a number of times as,
Vf1＝Vf/Vt (7)
wherein, VfIs the fiber volume fraction.
Therefore, the length and height of the warp fiber bundle fluctuation of step i +1 are respectively,
L1,i+1＝L1,i-ΔL1,i (8)
h1,i+1＝h1,i+Δh1,i (9)
therefore, the length and height of the weft fiber bundle fluctuation of the (i + 1) th step are respectively,
wherein i represents the ith step (i ═ 1,2,3.. n), and L1,i，h1,i，L2,iAnd h2,iRespectively the length and height, Delta L, of the warp and weft fiber bundle fluctuation in the ith step1,i,Δh1,i,ΔL2,iAnd Δ h2,iRespectively displacement of the warp and weft fiber bundles in the ith step along the x direction and the z direction.
And step two, applying unidirectional compression load to the fiber bundles in the cell unit in the step one, and simultaneously performing stress analysis on the fiber bundles of the woven fabric in the cell unit, thereby establishing a mesomechanics model of the planar orthogonal woven composite cell unit fiber woven fabric, determining the total strain residual energy of the fiber fabric in the cell unit, and solving the deformation of the cell unit fiber woven fabric by using a minimum residual energy principle and a unit load method.
The stress of the longitudinal and latitudinal fiber bundles is shown in fig. 4, and based on microscopic mechanical analysis, the compression load delta P is obtained in the step i1Under the action of,. DELTA.QiIs the interaction force between the warp and weft fiber bundles in the ith step, delta M1,iAnd Δ M2,iRespectively the restraining moment of the warp and weft fiber bundles in the ith step. The internal force delta F on any section of the ith step of warp fiber bundle1,i(x) Is composed of
Delta M for bending moment of any cross section of warp fiber bundle in the ith step1,i(x) Is shown as
Similarly, the internal force and the bending moment on any section of the weft-wise fiber bundle in the ith step are respectively
In formulae (12) to (15), Δ M1,i、ΔM2,iAnd Δ QiFor the unknowns, the three unknowns and the compression load Δ P are calculated using the minimum energy remaining principle1In the context of (a) or (b),
strain energy of warp fiber bundle in step i
Can be expressed as
In the formula, E1The modulus of elasticity in the longitudinal direction of the fiber bundle was obtained by the law of mixing.
Similarly, strain energy of weft fiber bundle in step i
Can be expressed as
By substituting equations (12) to (15) into equations (16) and (17), the total residual energy of the warp and weft fiber bundles in step i can be obtained. Total residual energy of i-step warp fiber bundle
Is composed of
Total strain energy of weft fiber bundle of step i
Is composed of
Wherein, the intermediate variable S in the ith stepm,i(m ═ 1,2,3.. 16) is listed in appendix a. Therefore, the total residual energy of the ith step cell can be obtained from the formulas (18) and (19)
By using the principle of minimum complementary energy for the unit cells, Δ M can be determined separately1,i,ΔM2,iAnd Δ Qi，
ΔM1,i、ΔM2,i、ΔQiAnd Δ P1Is expressed as
By solving the system of equations (22) using the Clamer's law, M can be expressed by P, respectively1,i，M2,iAnd Qi：
ΔM1,i＝W1,iΔP1 (23)
ΔM2,i＝W2,iΔP1 (24)
ΔQi＝W3,iΔP1 (25)
In the formula, the intermediate variable W in the ith stepk,i(k ═ 1,2,3) is listed in appendix B.
Step i in compression load Δ P1Internal force Δ QiAnd bending moment Δ M1,iUnder the action of the unit load method, the relative deformation quantity Delta L of the warp fiber bundle along the x direction is obtained1,i：
Deformation delta h of warp direction fiber bundle along z direction at wave crest or wave trough1,i：
Relative deformation amount Delta L of weft fiber bundles along x direction2,i：
The deformation delta h of the crest or the trough of the weft fiber bundle along the z direction2,i：
And step three, calculating the compression modulus of the plane orthogonal braided composite material according to the deformation continuous condition and the mixing law.
Strain increment delta of step iεiCan be expressed as
The stress increment of the longitudinal section of the weft fiber bundle in the compressive load direction can be expressed according to the deformation continuous condition
Δσ2,i＝E2Δεi (31)
Compression load delta N borne by longitudinal section of ith weft fiber bundle1Is composed of
Δ P can be obtained by substituting expressions (26), (30) and (31) into expression (32)1And Δ N1The relational expression of (1):
therefore, the warp compression modulus E of the ith-step plane-woven fiber bundle cell structuretc,iCan be expressed as
Using the well-known law of mixture, the warp-wise compressive modulus E of the i-th orthogonal plane braided composite materialc,iCan be prepared from resin elastic modulus EmAnd compressive modulus E of plane woven fiber bundle cell structuretc,iTo obtain
And step four, calculating the stress and the strain of the plane braided composite material according to a stress-strain constitutive equation, simultaneously drawing a stress-strain curve, and fitting the stress-strain curve according to least square normativity.
The warp-wise compressive stress increment of the i-th step plane braided composite material can be expressed as
Delta epsilon is determined stepwise by equations (30) and (36)i(i ═ 1,2,3,. n) and Δ σi(i ═ 1,2,3.. n). Therefore, Δ ∈ is calculated by accumulationiAnd Δ σiThe relationship between compressive stress and strain in a large deformation range can be obtained:
by fitting data points [ epsilon ]e,σe]( e 1,2,3.. n.) stress-strain curves of the braided composite material under compressive loading can be obtained for orthogonal planes up to σeThe compressive strength of the orthogonal plane braided composite material is reached. The resulting data point [ epsilon ]e,σe](E1, 2,3.. n) by using least square method to carry out linear fitting, the compressive modulus E of the warp direction of the orthogonal plane braided composite material can be obtained respectivelyc：
In the formula (I), the compound is shown in the specification,
and
the average stress and the average strain experienced by the PWF composite material are respectively.
In the same way, the latitudinal compression modulus of the orthogonal plane woven composite material can be obtained, the prediction method of the orthogonal plane woven composite material is consistent with the longitudinal direction, and only the initial geometric parameters need to be changed.
And step five, according to the failure strength of component materials (fiber bundles and resin), adopting a fiber buckling theory and a transverse tensile crack theory as failure criteria of the fiber bundles, adopting a maximum compressive stress criterion as the failure criteria of the interlayer resin, and calculating the compressive strength of the plane orthogonal woven composite material based on the deformation continuous condition to obtain an analytic solution.
There are two main reasons for compression failure of flat woven composites: (1) because the fiber bundles are the major part of the compressive load, the compressive failure of the composite may be caused by a failure of the fiber bundles to compress; (2) another reason is that the resin between the fiber bundle layers reaches compressive strength, the resin breaks, loses lateral support to the fiber bundle, and the fiber bundle fails.
For the first reason, the compressive strength of the orthogonal plane woven composite material is predicted by adopting a fiber buckling theory and a fiber bundle transverse tension crack theory. There are two types of buckling of fibers in resin: the fibers are buckled in opposite directions to form a tensile pattern and the fibers are buckled in the same direction to form a shear pattern.
For the tensile pattern formed by reversely buckling the fibers, the resin alternately generates tension-compression deformation perpendicular to the fibers, as shown in FIG. 5, and the compressive strength X of the fiber bundle is predicted by the tensile patterny1Is composed of
In the formula, Vf1Is a fiber inVolume fraction in fiber bundle, EfIs the modulus of elasticity of the fiber.
For fiber co-buckling to form a shear pattern, the resin undergoes shear deformation, as shown in FIG. 5, using the shear pattern to predict the fiber bundle compressive strength Xy2Is composed of
In the formula, GmIs the shear modulus of the resin.
When the unidirectional tape composite material is compressed longitudinally, debonding and cracking along the fiber direction occur firstly, and finally transverse tension fracture is formed to cause damage, as shown in figure 6, the fiber bundle compression strength X predicted by adopting the transverse fracture theoryy3Is composed of
In the formula, vfIs the Poisson ratio, v, of the fibresmIs the poisson's ratio of the resin. EpsilonmuIs the transverse strain of resin failure.
Based on the above two fiber bundle failure theories, the minimum value should be taken to determine the compressive strength X of the fiber bundley：
Xy＝min(Xy1,Xy2,Xy3) (45)
According to mesoscopic mechanical analysis, because the warp fiber bundles are of a symmetrical structure, the maximum positive stress at the wave crest and the wave trough of the nth step is equal, and only the maximum positive stress sigma of the wave trough needs to be consideredmax，
In the formula, z00 denotes the position of maximum positive stress of the trough.
By substituting equation (23) into equation (46), the maximum positive stress at the trough of the warp fiber bundle can be obtained
When the maximum positive stress at the trough reaches XyWhen the orthogonal plane woven composite material fails, the critical external compression load applied to the warp fiber bundles can be obtained by substituting equation (45) into equation (47):
the compression load of the longitudinal section of the weft fiber bundle due to deformation coordination can be obtained by the formula (33):
in the plane woven composite compression test, the calculation method of the warp direction compression strength of the plane woven composite is to divide the failure compression load by the initial geometric cross-sectional area, so that the initial geometric shape is adopted for predicting the warp direction compression strength by adopting the analytical model, namely i is 1. Thus, the warp-wise compressive strength X of the flat woven composite predicted by fiber bundle failurec1Can be expressed as
Wherein H is the monolayer thickness of the PWF composite.
Resin failure between fiber bundle layers in a flat woven composite material As shown in FIG. 7, the warp-wise compression modulus E of the flat woven fiber bundle cell structure increases with increasing compression load in consideration of the change in the geometrical parameters of the materialtc,iGradually reduced, but the warp-wise average strain of the cell body structure of the plane weaving fiber bundle is always equal to the strain of the resin
In the formula, σtc,nAnd σm,nRespectively representing the mean stress of the warp direction of the n-th step plane woven fiber bundle cell body structure and the stress of the resin
According to the law of mixing of stresses,
σn＝σtc,nVt+σm,n(1-Vt) (52)
substitution of formula (51) into formula (52) can give
Stress σ of outer layer resin when step nm,nTo reach XmcWhen the plane woven composite fails, i.e. sigma is about tom,n＝XmcSubstitution into equation (53) can obtain the warp direction compressive strength X of the flat woven composite material predicted from the resin failure between fiber bundle layersc2：
In the same way, the weft compression strength of the plane woven composite material can be obtained.
The invention relates to a method for predicting the compression modulus and strength of a plane orthogonal braided composite material by taking geometric nonlinearity into consideration, which is characterized in that the compression modulus and strength of the plane orthogonal braided composite material can be conveniently and quickly predicted by taking the geometric nonlinearity into consideration and according to the braided geometric shape and size and the performance parameters of fiber bundles and resin.
Drawings
Fig. 1 is a braided shape of a planar orthogonal braided composite material.
FIG. 2 is an idealized soma unit.
Fig. 3 is a fiber bundle compression process.
FIG. 4 is an idealized cross-section and internal forces for warp and weft fiber bundles.
FIG. 5 is a theoretical representation of fiber buckling. Wherein fig. 5(a) is a schematic diagram of a stretching type fiber buckling theory, and fig. 5(b) is a schematic diagram of a shearing type fiber buckling theory.
Fig. 6 is a schematic diagram of transverse spalling theory.
FIG. 7 is a schematic illustration of interlayer resin compression failure.
FIG. 8 is a flow chart for predicting compressive modulus and strength of a planar orthogonal braided composite material.
The symbols in the figures are as follows:
in fig. 1 a and b are the cross-sectional width and thickness of the fiber bundle, 1. undulation of the fiber bundle, 2. interlayer resin, respectively.
A in FIG. 21And b1Respectively the width and thickness of the cross section of the warp-wise fibre bundle, a2And b2Width and thickness of cross-section of weft-wise fibre bundle, L1,iAnd L2,iThe warp and weft fiber bundle fluctuation length in the ith step is respectively, H is the single-layer thickness of the composite material woven on the plane, 3 is the warp fiber bundle, and 4 is the weft fiber bundle.
Δ P in FIG. 31For the compressive load applied to the warp fiber bundles, 5 weft fiber bundles, 6 warp fiber bundles.
Δ M in FIG. 41,iBending moment, Δ M, of the warp fiber bundle in step i2,iBending moment, Δ Q, of the weft-wise fibre bundleiAnd (3) the interaction force of the warp and weft fiber bundles in the ith step, theta is the tangential angle of any section of the center line of the fiber bundles, 7 degrees of warp fiber bundles and 8 degrees of weft fiber bundles.
In FIG. 5, 9. fibers are bent and 10. resin.
In fig. 6, 11, transverse tensile failure, 12, fibers, 13, resin.
In fig. 7, 14 wave fiber bundles, 15 interlaminar resin compression failure.
The specific implementation mode is as follows:
step one, according to the orthogonal weaving mode (such as periodicity, repeatability and the like) of the symmetry plane of the fiber bundle, selecting proper periodic units as representative volume elements, thereby determining cell units of the fiber bundle, and defining the orthogonal weaving geometry and size in the cell units.
As shown in fig. 1, coordinate axes x and z represent the direction of the undulation of the fiber bundle and the out-of-plane thickness direction, respectively. To establish a mechanical analysis model, the following assumptions were made for the fabric unit cell structure:
(3) the cross section of the fiber bundle is simplified into a rectangle with a semicircular shape at the left end and a semicircular shape at the right end, wherein the width is a, and the thickness is b.
(4) During compression deformation, the fiber bundle is idealized as a curved beam, and the neutral axis path of the fiber bundle is represented by a trigonometric function.
According to the assumption (1), the cross-sectional area and the moment of inertia of the yarn are respectively
Wherein (j-1, 2), j-1 represents warp fiber bundle, and j-1 represents weft fiber bundle.
According to the assumption (2), the neutral axis path of the wavy fiber bundle is represented as
The tangent value of the tangential angle theta at any section of the wavy fiber bundle is
Thus, the longitudinal cross-sectional area of the fiber bundle can be expressed as
In fact, in the process of actually bearing unidirectional compression load, the paths of warp and weft fiber bundles are changed along with the change of load in the process of weaving the composite material in the plane orthogonal modeChanges occur (as shown in fig. 3), which in turn further affect the compression performance, and therefore, a mesomechanics model is proposed herein that takes into account changes in the fiber bundle geometric parameters (i.e., geometric non-linearities): the process of starting compression and breaking of the plane orthogonal braided composite material is divided into n steps, and each step applies a compression load delta P to the unidirectional fiber bundles1The length and the height of the warp and weft fiber bundle fluctuation are changed, and the next step is to apply the compression load delta P again on the basis of the new geometric shape1Performing n times of iterative calculation until the compressive strength X of the plane orthogonal braided composite material is reachedc(X is given in step 3 herein)cThe prediction method of (1).
From equations (1) to (4), the fiber bundle volume fraction V can be determined using only the initial geometry (i.e., i ═ 1)t,
Wherein the warp fiber bundles are indicated by subscript 1, the weft fiber bundles are indicated by 2, the single layer thickness is H, and the volume fraction V of the fibers in the fiber bundlesf1Can be expressed as a number of times as,
Vf1＝Vf/Vt (7)
wherein, VfIs the fiber volume fraction.
Therefore, the length and height of the warp fiber bundle fluctuation of step i +1 are respectively,
L1,i+1＝L1,i-ΔL1,i (8)
h1,i+1＝h1,i+Δh1,i (9)
therefore, the length and height of the weft fiber bundle fluctuation of the (i + 1) th step are respectively,
L2,i+1＝L2,i-ΔL2,i (9)
h2,i+1＝h2,i+Δh2,i (10)
wherein i represents the ith step (i ═ 1,2,3.. n), and L1,i，h1,i，L2,iAnd h2,iRespectively the length and height, Delta L, of the warp and weft fiber bundle fluctuation in the ith step1,i,Δh1,i,ΔL2,iAnd Δ h2,iRespectively displacement of the warp and weft fiber bundles in the ith step along the x direction and the z direction.
And step two, applying unidirectional compression load to the fiber bundles in the cell unit in the step one, and simultaneously performing stress analysis on the fiber bundles of the woven fabric in the cell unit, thereby establishing a mesomechanics model of the planar orthogonal woven composite cell unit fiber woven fabric, determining the total strain residual energy of the fiber fabric in the cell unit, and solving the deformation of the cell unit fiber woven fabric by using a minimum residual energy principle and a unit load method.
The stress of the longitudinal and latitudinal fiber bundles is shown in fig. 4, and based on microscopic mechanical analysis, the compression load delta P is obtained in the step i1Under the action of,. DELTA.QiIs the interaction force between the warp and weft fiber bundles in the ith step, delta M1,iAnd Δ M2,iRespectively the restraining moment of the warp and weft fiber bundles in the ith step. The internal force delta F on any section of the ith step of warp fiber bundle1,i(x) Is composed of
Delta M for bending moment of any cross section of warp fiber bundle in the ith step1,i(x) Is shown as
Similarly, the internal force and the bending moment on any section of the weft-wise fiber bundle in the ith step are respectively
In formulae (12) to (15), Δ M1,i、ΔM2,iAnd Δ QiFor the unknowns, the three unknowns and the compression load Δ P are calculated using the minimum energy remaining principle1In the context of (a) or (b),
strain energy of warp fiber bundle in step i
Can be expressed as
In the formula, E1The modulus of elasticity in the longitudinal direction of the fiber bundle was obtained by the law of mixing.
Similarly, strain energy of the i-th step weft fiber bundle
Can be expressed as
By substituting equations (12) to (15) into equations (16) and (17), the total residual energy of the warp and weft fiber bundles in step i can be obtained. Total residual energy of i-step warp fiber bundle
Is composed of
Total strain energy of weft fiber bundle of step i
Is composed of
Wherein, the intermediate variable S in the ith stepm,i(m ═ 1,2,3.. 16) is listed in appendix a. Therefore, the total residual energy of the ith step cell can be obtained from the formulas (18) and (19)
By using the principle of minimum complementary energy for the unit cells, Δ M can be determined separately1,i,ΔM2,iAnd Δ Qi，
ΔM1,i、ΔM2,i、ΔQiAnd Δ P1Is expressed as
By solving the system of equations (22) using the kramer's law, M can be expressed as P, respectively1,i，M2,iAnd Qi：
ΔM1,i＝W1,iΔP1 (23)
ΔM2,i＝W2,iΔP1 (24)
ΔQi＝W3,iΔP1 (25)
In the formula, the intermediate variable W in the ith stepk,i(k ═ 1,2,3) is listed in appendix B.
Step i in compression load Δ P1Internal force Δ QiAnd bending moment Δ M1,iUnder the action of the unit load method, the relative deformation quantity Delta L of the warp fiber bundle along the x direction is obtained1,i：
Deformation delta h of warp direction fiber bundle along z direction at wave crest or wave trough1,i：
Relative deformation amount Delta L of weft fiber bundles along x direction2,i：
The deformation delta h of the crest or the trough of the weft fiber bundle along the z direction2,i：
And step three, calculating the compression modulus of the plane orthogonal braided composite material according to the deformation continuous condition and the mixing law.
Strain increment delta epsilon of step iiCan be expressed as
The stress increment of the longitudinal section of the weft fiber bundle in the compressive load direction can be expressed according to the deformation continuous condition
Δσ2,i＝E2Δεi (31)
Compression load delta N borne by longitudinal section of ith weft fiber bundle1Is composed of
Δ P can be obtained by substituting expressions (26), (30) and (31) into expression (32)1And Δ N1Table of relationshipsThe expression is as follows:
therefore, the warp compression modulus E of the ith-step plane-woven fiber bundle cell structuretc,iCan be expressed as
Using the well-known law of mixture, the warp-wise compressive modulus E of the i-th orthogonal plane braided composite materialc,iCan be prepared from resin elastic modulus EmAnd compressive modulus E of plane woven fiber bundle cell structuretc,iTo obtain
And step four, calculating the stress and the strain of the plane braided composite material according to a stress-strain constitutive equation, simultaneously drawing a stress-strain curve, and fitting the stress-strain curve according to least square normativity.
The warp-wise compressive stress increment of the i-th step plane braided composite material can be expressed as
Delta epsilon is determined stepwise by equations (30) and (36)i(i ═ 1,2,3,. n) and Δ σi(i ═ 1,2,3.. n). Therefore, Δ ∈ is calculated by accumulationiAnd Δ σiThe relation between the compressive stress and the strain in a large deformation range can be obtained:
by fitting data points [ epsilon ]e,σe]( e 1,2,3.. n.) stress-strain curves of the braided composite material under compressive loading can be obtained for orthogonal planes up to σeThe compressive strength of the orthogonal plane braided composite material is reached. The resulting data point [ epsilon ]e,σe](E1, 2,3.. n) by using least square method to carry out linear fitting, the compressive modulus E of the warp direction of the orthogonal plane braided composite material can be obtained respectivelyc：
In the formula (I), the compound is shown in the specification,
and
the average stress and the average strain to which the PWF composite is subjected are the same.
In the same way, the latitudinal compression modulus of the orthogonal plane woven composite material can be obtained, the prediction method of the orthogonal plane woven composite material is consistent with the longitudinal direction, and only the initial geometric parameters need to be changed.
And step five, according to the failure strength of component materials (fiber bundles and resin), adopting a fiber buckling theory and a transverse tensile crack theory as failure criteria of the fiber bundles, adopting a maximum compressive stress criterion as the failure criteria of the interlayer resin, and calculating the unidirectional compressive strength of the plane orthogonal braided composite material based on the deformation continuous condition to obtain an analytic solution.
There are two main reasons for compression failure of flat woven composites: (1) because the fiber bundles are the major part of the compressive load, the compressive failure of the composite may be caused by a failure of the fiber bundles to compress; (2) another reason is that the resin between the fiber bundle layers reaches compressive strength, the resin breaks, loses lateral support to the fiber bundle, and the fiber bundle fails.
For the first reason, the compressive strength of the orthogonal plane woven composite material is predicted by adopting a fiber buckling theory and a fiber bundle transverse tension crack theory. There are two types of buckling of fibers in resin: the fibers are buckled in opposite directions to form a tensile pattern and the fibers are buckled in the same direction to form a shear pattern.
For the tensile pattern formed by reversely buckling the fibers, the resin alternately generates tension-compression deformation perpendicular to the fibers, as shown in FIG. 5, and the compressive strength X of the fiber bundle is predicted by the tensile patterny1Is composed of
In the formula, Vf1The volume fraction of the fibres in the fibre bundle, EfIs the modulus of elasticity of the fiber.
For fiber co-buckling to form a shear pattern, the resin undergoes shear deformation, as shown in FIG. 5, using the shear pattern to predict the fiber bundle compressive strength Xy2Is composed of
In the formula, GmIs the shear modulus of the resin.
When the unidirectional tape composite material is compressed longitudinally, debonding and cracking along the fiber direction occur firstly, and finally transverse tension fracture is formed to cause damage, as shown in figure 6, the fiber bundle compression strength X predicted by adopting the transverse fracture theoryy3Is composed of
In the formula, vfIs the Poisson ratio, v, of the fibresmIs the poisson's ratio of the resin. EpsilonmuIs the transverse strain of resin failure.
Based on the above two fiber bundle failure theories, the minimum value should be taken to determine the compressive strength X of the fiber bundley：
Xy＝min(Xy1,Xy2,Xy3) (45)
According to mesoscopic mechanical analysis, because the warp fiber bundles are of a symmetrical structure, the maximum positive stress at the wave crest and the wave trough of the nth step is equal, and only the maximum positive stress sigma of the wave trough needs to be consideredmax，
In the formula, z00 denotes the position of maximum positive stress of the trough.
By substituting equation (23) into equation (46), the maximum normal stress at the trough of the warp direction fiber bundle can be obtained
When the maximum positive stress at the wave trough reaches XyWhen the orthogonal plane woven composite material fails, the critical external compression load applied to the warp fiber bundles can be obtained by substituting formula (45) into formula (47):
the compression load of the longitudinal section of the weft fiber bundle due to deformation coordination can be obtained by the formula (33):
in the plane woven composite compression test, the calculation method of the warp-direction compression strength of the plane woven composite is to divide the failure compression load by the initial geometric cross-sectional area, so that the initial geometric shape is adopted for predicting the warp-direction compression strength by adopting the analytical model, namely i is 1. Thus, the warp-wise compressive strength X of the flat woven composite predicted by fiber bundle failurec1Can be expressed as
Wherein H is the monolayer thickness of the PWF composite.
Resin failure between fiber bundle layers in a flat woven composite material As shown in FIG. 7, the warp-wise compression modulus E of the flat woven fiber bundle cell structure increases with increasing compression load in consideration of the change in the geometrical parameters of the materialtc,iGradually reduced, but the warp-wise average strain of the cell body structure of the plane weaving fiber bundle is always equal to the strain of the resin
In the formula, σtc,nAnd σm,nRespectively representing the mean stress of the warp direction of the n-th step plane woven fiber bundle cell body structure and the stress of the resin
According to the law of mixing of stresses,
σn＝σtc,nVt+σm,n(1-Vt) (52)
substitution of formula (51) into formula (52) can give
Stress σ of outer layer resin when step nm,nTo reach XmcWhen the plane woven composite fails, i.e. sigma is about tom,n＝XmcSubstitution intoIn the formula (53), the warp-wise compressive strength X of the flat woven composite predicted from the resin failure between fiber bundle layers can be obtainedc2：
In the same way, the weft compression strength of the plane woven composite material can be obtained.
The invention relates to a method for predicting the compression modulus and strength of a plane orthogonal braided composite material by taking geometric nonlinearity into consideration, which is characterized in that the compression modulus and strength of the plane orthogonal braided composite material can be conveniently and quickly predicted by taking the geometric nonlinearity into consideration and according to the braided geometric shape and size and the performance parameters of fiber bundles and resin.
Appendix A
Appendix B
Claims (1)
1. A method for predicting the compressive modulus and strength of a planar orthogonal braided composite material by considering geometric nonlinearity is characterized by comprising the following steps: the method comprises the following specific steps:
selecting a proper periodic unit as a representative volume element according to a symmetrical plane orthogonal weaving mode of a fiber bundle, thereby determining a cell unit of the fiber bundle, and defining the orthogonal weaving geometric shape and size in the cell unit;
coordinate axes x and z represent the fiber bundle fluctuation direction and the out-of-plane thickness direction respectively, and the following assumptions are made for the fabric unit cell structure in order to establish a mechanical analysis model:
(1) the cross section of the fiber bundle is simplified into a rectangle with a semicircular shape at the left end and a semicircular shape at the right end, wherein the width is a, and the thickness is b;
(2) in the compression deformation process, the fiber bundle is idealized into a curved beam, and the neutral axis path of the fiber bundle is represented by a trigonometric function; according to the assumption (1), the cross-sectional area and the moment of inertia of the yarn are respectively
Wherein j 1,2, j 1 represents warp fiber bundle, and j 2 represents weft fiber bundle;
according to the assumption (2), the neutral axis path of the wavy fiber bundle is represented as
The tangent value of the tangential angle theta at any section of the wavy fiber bundle is
Thus, the longitudinal cross-sectional area of the fiber bundle can be expressed as
The process of starting compression and breaking of the plane orthogonal braided composite material is divided into n steps, and each step applies a compression load delta P to the fiber bundle1The length and the height of the warp and weft fiber bundle fluctuation are changed, and the next step is to apply the compression load delta P again on the basis of the new geometric shape1Performing n times of iterative calculation until the compressive strength X of the plane orthogonal braided composite material is reachedc；
From equations (1) to (4), the volume fraction V of the fiber bundle can be obtained by using only the initial geometry, i.e., 1t,
Wherein the warp fiber bundles are indicated by subscript 1, the weft fiber bundles are indicated by 2, the single layer thickness is H, and the volume fraction V of the fibers in the fiber bundlesf1It can be expressed as a number of,
Vf1＝Vf/Vt (7)
wherein, VfIs the fiber volume fraction;
therefore, the length and height of the warp fiber bundle fluctuation of step i +1 are respectively,
L1,i+1＝L1,i-ΔL1,i (8)
h1,i+1＝h1,i+Δh1,i (9)
therefore, the length and height of the weft fiber bundle fluctuation of the (i + 1) th step are respectively,
wherein, i represents the ith step i ═ 1,2,3.. n, L1,i，h1,i，L2,iAnd h2,iRespectively the length and height, Delta L, of the warp and weft fiber bundle fluctuation in the ith step1,i,Δh1,i,ΔL2,iAnd Δ h2,iRespectively displacement of the warp and weft fiber bundles in the ith step along the x direction and the z direction;
step two, applying unidirectional compression load to the fiber bundles in the cell unit in the step one, and simultaneously performing stress analysis on the fiber bundles of the woven fabric in the cell unit, thereby establishing a mesoscopic mechanical model of the fiber woven fabric of the cell unit of the planar orthogonal woven composite material, determining the total strain residual energy of the fiber fabric in the cell unit, and solving the deformation of the fiber woven fabric of the cell unit by using a minimum residual energy principle and a unit load method;
based on the mesomechanics analysis, the compression load delta P is obtained in the step i1Under the action of,. DELTA.QiIs the interaction force between the warp and weft fiber bundles in the ith step, delta M1,iAnd Δ M2,iRespectively the restraining moment of the warp and weft fiber bundles in the ith step, and the internal force delta F on any section of the warp fiber bundles in the ith step1,i(x) Is composed of
Delta M for bending moment of any cross section of warp fiber bundle in the ith step1,i(x) Is shown as
Similarly, the internal force and the bending moment on any section of the weft-wise fiber bundle in the ith step are respectively
In formulae (12) to (15), Δ M1,i、ΔM2,iAnd Δ QiFor the unknowns, the three unknowns and the compression load Δ P are calculated using the minimum energy remaining principle1In the context of (a) or (b),
strain energy of warp fiber bundle in step i
Can be expressed as
In the formula, E1Expressing the longitudinal elastic modulus of the fiber bundle, and obtaining the elastic modulus by using a mixing law;
similarly, strain energy of weft fiber bundle in step i
Can be expressed as
By substituting equations (12) to (15) into equations (16) and (17), the total residual energy of the warp and weft fiber bundles in step i and the total residual energy of the warp fiber bundles in step i can be obtained
Is composed of
Total strain energy of weft fiber bundle of step i
Is composed of
Wherein S ism,iAnd m is 1,2,3.. 16 in the step iAn intermediate variable; therefore, the total residual energy of the ith step cell can be obtained from the formulas (18) and (19)
By using the principle of minimum complementary energy for the unit cells, Δ M can be determined separately1,i,ΔM2,iAnd Δ Qi，
ΔM1,i、ΔM2,i、ΔQiAnd Δ P1Is expressed as
By solving the system of equations (22) using the Clamer's law, M can be expressed by P, respectively1,i，M2,iAnd Qi：
ΔM1,i＝W1,iΔP1 (23)
ΔM2,i＝W2,iΔP1 (24)
ΔQi＝W3,iΔP1 (25)
In the formula, Wk,iK is 1,2 and 3 are intermediate variables in the step i;
step i in compression load Δ P1Internal force Δ QiAnd bending moment Δ M1,iUnder the action of the unit load method, the relative deformation quantity Delta L of the warp fiber bundle along the x direction is obtained1,i：
Deformation delta h of warp direction fiber bundle along z direction at wave crest or wave trough1,i：
Relative deformation amount Delta L of weft fiber bundles along x direction2,i：
The deformation delta h of the crest or the trough of the weft fiber bundle along the z direction2,i：
Calculating the compression modulus of the plane orthogonal braided composite material according to the deformation continuous condition and the mixing law;
strain increment delta epsilon of step iiCan be expressed as
The stress increment of the longitudinal section of the weft fiber bundle in the compressive load direction can be expressed according to the deformation continuous condition
Δσ2,i＝E2Δεi (31)
Compression load delta N borne by longitudinal section of ith weft fiber bundle1Is composed of
Δ P can be obtained by substituting expressions (26), (30) and (31) into expression (32)1And Δ N1The relational expression of (1):
therefore, the warp compression modulus E of the ith-step plane-woven fiber bundle cell structuretc,iCan be expressed as
Using the law of mixture, the warp-wise compression modulus E of the i-th orthogonal plane braided composite materialc,iCan be prepared from resin elastic modulus EmAnd compressive modulus E of plane woven fiber bundle cell structuretc,iTo obtain
Calculating the stress and the strain of the plane braided composite material according to a stress-strain constitutive equation, simultaneously drawing a stress-strain curve, and fitting the stress-strain curve according to least square normalcy;
the warp-wise compressive stress increment of the i-th step plane braided composite material can be expressed as
Delta epsilon is determined stepwise by equations (30) and (36)i1,2,3,. n and Δ σiN ═ 1,2,3,. n; therefore, Δ ∈ is calculated by accumulationiAnd Δ σiThe relationship between compressive stress and strain in a large deformation range can be obtained:
by fitting data points [ epsilon ]e,σe]N can obtain the stress-strain curve of the braided composite material under the action of compression load and in the normal plane until the stress-strain curve reaches sigmaeUntil the compressive strength of the orthogonal plane woven composite material is reached, the obtained data point [ epsilon ]e,σe]N, and linear fitting is carried out by adopting a least square method, so that the compression modulus E of the warp direction of the orthogonal plane woven composite material can be obtained respectivelyc：
In the formula (I), the compound is shown in the specification,
and
the average stress and the average strain borne by the PWF composite material are respectively;
in the same way, the latitudinal compression modulus of the orthogonal plane woven composite material can be obtained, the prediction method is consistent with the longitudinal direction, and only the initial geometric parameters need to be changed;
step five, according to the failure strength of component materials, adopting a fiber buckling theory and a transverse tensile crack theory as failure criteria of fiber bundles, adopting a maximum compressive stress criterion as the failure criteria of interlayer resin, and calculating the compressive strength of the plane orthogonal woven composite material based on deformation continuous conditions to obtain an analytic solution;
there are two reasons for planar woven composite compression failure: (1) the compressive failure of the composite is caused by a failure of the fiber bundle to compress due to the fiber bundle as part of the compressive load; (2) another reason is that the resin between the fiber bundle layers reaches compressive strength, the resin is destroyed, the lateral support capability of the fiber bundle is lost, and the fiber bundle fails;
for the first reason, the compressive strength of the orthogonal plane woven composite material is predicted by adopting a fiber buckling theory and a fiber bundle transverse tension crack theory, and buckling of fibers in a matrix can be generated in two modes: the fibers are buckled in opposite directions to form a tensile mode and the fibers are buckled in the same direction to form a shearing mode;
for the tensile mode formed by reversely buckling the fibers, the matrix alternately generates tension-compression deformation perpendicular to the fibers, and the compressive strength X of the fiber bundle predicted by the tensile modey1Is composed of
In the formula, Vf1The volume fraction of the fibres in the fibre bundle, EfIs the modulus of elasticity of the fiber;
for fiberCo-directional buckling to form a shear pattern, the matrix undergoing shear deformation, and the predicted fiber bundle compressive strength X using the shear patterny2Is composed of
In the formula, GmIs the shear modulus of the resin;
the unidirectional tape composite material usually generates debonding and cracking along the fiber direction firstly during longitudinal compression, finally forms transverse tension crack to be damaged, and the fiber bundle compression strength X predicted by adopting the transverse cracking theoryy3Is composed of
In the formula, vfIs the Poisson ratio, v, of the fibresmIs the poisson's ratio of the resin; epsilonmuTransverse strain for matrix failure;
based on the above two fiber bundle failure theories, the minimum value should be taken to determine the compressive strength X of the fiber bundley：
Xy＝min(Xy1,Xy2,Xy3) (45)
According to mesoscopic mechanical analysis, because the warp fiber bundles are of a symmetrical structure, the maximum positive stress at the wave crest and the wave trough of the nth step is equal, and only the maximum positive stress sigma of the wave trough needs to be consideredmax，
In the formula, z00 denotes the position of maximum positive stress of the trough;
by substituting equation (23) into equation (46), the maximum positive stress at the trough of the warp fiber bundle can be obtained
When the maximum positive stress at the trough reaches XyWhen the orthogonal plane woven composite material fails, the critical external compression load applied to the warp fiber bundles can be obtained by substituting equation (45) into equation (47):
the compression load of the longitudinal section of the weft fiber bundle due to deformation coordination can be obtained by the formula (33):
in a plane woven composite material compression test, the calculation method of the warp-direction compression strength of the plane woven composite material is that the failure compression load is divided by the initial geometric cross-sectional area, and the warp-direction compression strength is predicted by adopting an initial geometric shape, namely i is 1; thus, the warp-wise compressive strength X of the flat woven composite predicted by fiber bundle failurec1Can be expressed as
Wherein H is the single layer thickness of the PWF composite;
the resin matrix among fiber bundle layers in the plane weaving composite material is damaged, and the warp direction compression modulus E of the cell body structure of the plane weaving fiber bundle is increased along with the increase of the compression load under the condition of considering the change of the geometrical parameters of the materialtc,iGradually reduced, but the warp-wise average strain of the cell body structure of the plane weaving fiber bundle is always equal to the strain of the resin
In the formula, σtc,nAnd σm,nRespectively representing the average stress of the warp direction of the n-step plane weaving fiber bundle cell body structure and the stress of resin;
according to the law of mixing of stresses,
σn＝σtc,nVt+σm,n(1-Vt) (52)
substitution of formula (51) into formula (52) can give
Stress σ of outer layer resin when step nm,nTo reach XmcWhen the plane woven composite material fails, i.e. sigmam,n＝XmcSubstitution into equation (53) can obtain the warp direction compressive strength X of the flat woven composite material predicted from the resin failure between fiber bundle layersc2：
In the same way, the weft compression strength of the plane woven composite material can be obtained.
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