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CN111400922A - Method for calculating stress-strain behavior of unidirectional ceramic matrix composite material by adding and removing random strain - Google Patents

Method for calculating stress-strain behavior of unidirectional ceramic matrix composite material by adding and removing random strain Download PDF

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CN111400922A
CN111400922A CN202010210593.XA CN202010210593A CN111400922A CN 111400922 A CN111400922 A CN 111400922A CN 202010210593 A CN202010210593 A CN 202010210593A CN 111400922 A CN111400922 A CN 111400922A
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stress
slip
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CN111400922B (en
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宋迎东
韩笑
高希光
韩栋
张盛
于国强
董洪年
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Nanjing University of Aeronautics and Astronautics
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Abstract

本发明提供了一种单向陶瓷基复合材料任意应变加卸载应力应变行为计算方法，该方法考虑了基体开裂、界面脱粘、滑移区的覆盖等细观损伤机制，以及界面是否完全脱粘和完全滑移等不同界面状态，给出了基体裂纹间距直接计算方法。滑移区覆盖的判断方法更加简洁高效，可快速建立应力和应变之间的方程关系，将应力的求解转换为方程的求解。实现了单向陶瓷基复合材料在任意应变加卸载作用下应力应变行为快速预测，为陶瓷基复合材料动力学研究提供基础。

Figure 202010210593

The invention provides a method for calculating the stress-strain behavior of a unidirectional ceramic matrix composite material with arbitrary strain loading and unloading. For different interface states such as full slip and complete slip, the direct calculation method of matrix crack spacing is given. The method of judging slip zone coverage is more concise and efficient, and the equation relationship between stress and strain can be quickly established, and the solution of stress can be converted into the solution of equation. The rapid prediction of stress-strain behavior of unidirectional ceramic matrix composites under arbitrary strain loading and unloading is realized, which provides a basis for the dynamic research of ceramic matrix composites.

Figure 202010210593

Description

Method for calculating stress-strain behavior of unidirectional ceramic matrix composite material by adding and removing random strain
Technical Field
The invention belongs to the field of prediction of stress-strain behaviors of composite materials, and particularly relates to a method for calculating any strain loading and unloading stress-strain behaviors of a one-way ceramic matrix composite material.
Background
The ceramic matrix composite material has excellent performances of high specific strength, high specific modulus, corrosion resistance, high temperature resistance and the like, and becomes an important advanced material of a heat end part in the fields of aerospace, ships, automobiles and the like. In order to apply the ceramic matrix composite material to a practical structure better and more safely, the stress-strain behavior (i.e., the constitutive behavior) needs to be studied in the design stage. In the actual service environment, the ceramic matrix composite material is inevitably subjected to the action of irregular variable amplitude load. The ceramic matrix composite material is different from a metal material, complex microscopic damage failure occurs under the action of load, and the stress-strain behavior of the material shows obvious nonlinearity. In addition, in the dynamic calculation process, the stress needs to be calculated by an arbitrary displacement loading and unloading curve (namely, a strain loading and unloading curve), so that a constitutive model of the material is further obtained. Therefore, it is necessary to develop a calculation method for any strain plus unload stress strain behavior of the ceramic matrix composite.
In the prior art, patent CN104866690B, "prediction method of stress-strain behavior of any loading and unloading of unidirectional ceramic matrix composite", proposes a method for calculating strain based on any stress loading and unloading of a shear-lag model, and due to the nonlinearity of ceramic matrix composite, the method cannot be directly used for calculating stress based on any strain loading and unloading curve. In addition, patent CN109815643A, "prediction method of arbitrary strain loading and unloading constitutive relation of unidirectional ceramic matrix composite" provides a method for calculating stress of arbitrary strain loading and unloading based on shear hysteresis model, which needs to perform piecewise function integration in equation solution, and consumes long time. Meanwhile, the calculation of the crack spacing of the substrate, which is one of important parameters, is based on stress rather than strain calculation, so that the application of the method has certain difficulty. Furthermore, this method is not fully considered for the conditions that occur in any strain loading and unloading. Therefore, it is necessary to provide a simple and effective calculation method capable of rapidly predicting the stress-strain behavior of the ceramic matrix composite material under any strain loading and unloading.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a method for calculating the strain behavior of any strain and stress of unloading of a unidirectional ceramic matrix composite.
In order to achieve the purpose, the invention adopts the following technical scheme:
the method for calculating the stress-strain behavior of any strain loading and unloading of the unidirectional ceramic matrix composite is characterized by comprising the following steps of:
step 1: judging whether the ceramic matrix composite generates matrix cracks, if not, considering that the material is not damaged, and calculating the stress under the current strain load by adopting a mixing ratio formula; if the matrix crack is generated, calculating the stress by adopting the following steps;
step 2: calculating critical strain corresponding to complete debonding of the interface when the composite material is loaded;
and step 3: calculating critical strain corresponding to complete slippage of an interface when the composite material is unloaded;
and 4, step 4: and (3) obtaining the stress of the composite material through an equation of stress increment by combining the critical strain calculated in the step (2) and the critical strain calculated in the step (3) based on a shear-lag model.
In order to optimize the technical scheme, the specific measures adopted further comprise:
further, in the step 1, the maximum strain load including the current and the previous maximum strain loads is obtainedi_max(ii) a If no damage occurs inside the material, the strain corresponds to a stress of:
σi_max＝i_maxEc(1)
wherein E iscIs the modulus of elasticity of the composite material; the mean matrix crack spacing was initially calculated by substituting equation (1) into the Weibull distribution equation shown below:
Figure BDA0002422091790000021
wherein L is the average matrix crack spacing,Lsatis the saturated spacing of cracks in the matrix, sigma0And m is a statistical parameter, L is mixed with the length L of the composite materialcmpComparing, if the average matrix crack spacing is larger than the length of the composite material, no matrix crack is generated, the material is not damaged, and the stress under the current strain is calculated by adopting the formula shown as follows:
σi＝iEc(3)
wherein,ifor the current strain, σiThe stress corresponding to the current strain; otherwise, if the material is damaged, the stress of the material is calculated by adopting the following steps.
Further, in the step 2, based on a shear model, in the loading process, the load reaches a certain magnitude, the composite material interface is completely debonded, and the average matrix crack distance is kept unchanged; at the very complete debonding of the composite, there is a relationship between stress and interfacial debond length as shown below:
Figure BDA0002422091790000022
wherein,
Figure BDA0002422091790000023
critical stress corresponding to just complete debonding of the composite material, Ld_crTan theta is the critical debond length, tan theta, corresponding to when the interface is just fully debonded1Is expressed as
Figure BDA0002422091790000031
C1Is expressed as
Figure BDA0002422091790000032
vfIs the fiber volume fraction, EfIs the modulus of elasticity of the fiber, EcIs the equivalent elastic modulus, tau, of the composite materialiIs the interfacial shear stress of the composite material, rfIs the radius of the fiber, σf_thIs the fiber thermal residual stress; at this point, the critical strain at which the composite just completely debonds
Figure BDA0002422091790000033
Expressed as:
Figure BDA0002422091790000034
further, the interfacial debond length and the average matrix crack spacing satisfy the following equation:
Lcr/2＝Ld_cr(6)
wherein, LcrCombining the formulas (2), (4) and (6) to obtain the critical strain corresponding to the complete debonding of the interface by a numerical solving method, wherein the critical strain is the actual saturated matrix crack spacing corresponding to the complete debonding of the interface
Figure BDA0002422091790000035
Critical stress
Figure BDA0002422091790000036
And actual saturated matrix crack spacing Lcr；
When the current loading strain satisfies equation (7):
Figure BDA0002422091790000037
the average matrix crack spacing L need to be recalculated.
Further, in the step 3, if the peak load of a loading process before the current strain unloading process is the historical maximum strain load, before the current strain unloading process is calculated, the critical unloading strain corresponding to the complete slippage of the interface is calculated; after the last strain loading process is finished, assuming that n pairs of slip regions exist, the stress increment required by complete slip of the interface is as follows:
Figure BDA0002422091790000038
wherein,
Figure BDA0002422091790000039
is the length of the kth forward slip region,
Figure BDA00024220917900000310
the critical strain corresponding to the complete slippage of the interface is obtained by the stress increment required by the complete slippage of the corresponding interface state at the end of the previous strain loading cycle
Figure BDA00024220917900000311
Comprises the following steps:
Figure BDA00024220917900000312
wherein σi-1Loading the load at the end of the cycle for the previous step; during unloading, when the unloading strain is less than
Figure BDA00024220917900000313
When the interface is completely slipped, otherwise, the interface is partially slipped.
Further, in the step 4, based on the shear model, the following is performed:
for any loading process, a material generates a new slip region, gradually covers a reverse slip region, skips over the forward slip region when encountering the forward slip region, and continuously covers the reverse slip region until the loading process is completed, and finally the end point of the newly generated forward slip region falls into the reverse slip region; assuming that the number of the slip zone is gradually reduced from the matrix crack of the material to the bonding zone (i.e. gradually reduced from left to right), at this time, the loading strain is greater than the strain corresponding to the forward slip zone on the left side of the reverse slip zone and is less than the strain corresponding to the forward slip zone on the right side; if the forward slip region completely covers the reverse slip region, the material debonding length and the slip length are increased until the material is completely debonded; when the material is completely debonded, the debonding length and the sliding length of the material are not increased any more;
for the unloading process, the material generates a reverse slip region and gradually covers the forward slip region, when the material encounters the reverse slip region, the material skips the reverse slip region and continuously covers the forward slip region until the unloading process is completed, and finally the terminal point of the newly generated reverse slip region falls into the forward slip region; at the moment, the unloading strain is smaller than the strain corresponding to the left reverse slip zone of the forward slip zone and larger than the strain corresponding to the right reverse slip zone; if the reverse slip zone completely covers the forward slip zone, the reverse slip zone will not continue to be generated because the compression process will not cause the material to generate new debonding and slip zones.
Further, in the step 4, it is assumed that the slip zones occur in pairs and the forward slip zone is in front; before the current load cycle, assuming that there are already n pairs of slip zones, for the slip zone that is missing, assuming it is present and of length 0; at the time of initial loading, 0 pair of slippage areas exist;
based on the above settings, when the current strain load is in the loading or unloading process, the stress-strain behavior is calculated as follows:
1) in the loading process, if the current loading strain meets the following relationship:
i≥i_max(10)
the loading process covers all the slip zones, and only one pair of non-zero forward slip zones and a reverse slip zone with the length of 0 exist under the current loading strain load; if the composite material is completely debonded before the current load, the corresponding stress under the current loading strain is:
Figure BDA0002422091790000041
at this time, the length of the first forward slip region is:
Figure BDA0002422091790000042
if the composite material is partially debonded under the current load, the current strain is determinediCurrent stress σiAverage matrix crack spacing L and interfacial debond length LdSubstituting in equations (4) and (5), respectively
Figure BDA0002422091790000043
LcrAnd Ld_crAnd combined with equation (2) with stress σiSolving the equation (5) for unknown number by numerical method to obtain the current loading stress sigmaiAverage matrix crack spacing L and debond length Ld；
At this time, the length of the first forward slip region
Figure BDA0002422091790000051
Comprises the following steps:
Figure BDA0002422091790000052
recording the strain generated in the first forward slip region
Figure BDA0002422091790000053
And strain at the time of first reverse slip region
Figure BDA0002422091790000054
2) In the loading process, if the current loading strain meets the following relationship:
i＜i_max(14)
the current strain loading process will not completely cover all of the slip zones already present and the average matrix crack spacing L remains constant, and the forward slip zone produced by the current loading process will fall within the reverse slip zone satisfying the relationship shown below:
Figure BDA0002422091790000055
wherein j is the number of the reverse slip region meeting the requirement,
Figure BDA0002422091790000056
and
Figure BDA0002422091790000057
the strain generated in the j & ltth & gt and j +1 & ltth & gt forward slip zones respectively;at the moment, the nth to j +1 pairs of slippage areas are covered, a jth reverse slippage area, a jth +1 forward slippage area and a jth +1 reverse slippage area with the length of 0 are newly generated, and the length of the front j-1 pair of slippage areas is not changed;
updating the jth reverse slip zone length as:
Figure BDA0002422091790000058
wherein,
Figure BDA0002422091790000059
is the length of the kth reverse slip zone,
Figure BDA00024220917900000510
the length of the reverse slip zone covered under the current strain load increment;
the length of the j +1 th positive slip region is as follows:
Figure BDA00024220917900000511
stress increment delta sigma corresponding to current loading strainiLength of the covered reverse slip region
Figure BDA00024220917900000512
Satisfies the following relationship:
Figure BDA00024220917900000513
the current loading strain and stress delta satisfy the relationship shown below:
Figure BDA00024220917900000514
wherein,
Figure BDA00024220917900000515
showing the actual stress of the fiber in the kth pair of slip regions and the coordinate axisThe difference between the area and the area formed by the fiber stress and the area formed by the coordinate axis when the composite material is not damaged is assumed, wherein the coordinate axis x represents the distance from one point on the composite material to the crack plane, and the coordinate axis y represents the fiber stress of the point; only when the length of the slip region changes,
Figure BDA0002422091790000061
the change will occur; when there is no change in the slip region,
Figure BDA0002422091790000062
remain unchanged during the loading and unloading processes,
Figure BDA0002422091790000063
the expression is as follows:
Figure BDA0002422091790000064
wherein,
Figure BDA0002422091790000065
the difference value of the peak fiber stress in the kth pair of slippage area and the fiber stress in the bonding area is expressed as follows:
Figure BDA0002422091790000066
equations (16) to (18) and equations (20) to (21) are substituted into equation (19) to produce Δ σiTo solve the object, equation (19) relates to Δ σiThe stress increment delta sigma under the current strain is obtained by solving the unitary quadratic equationi(ii) a There are 2 solutions to the quadratic equation of unity, find the current load σi＝(σi-1+Δσi) The stress generated by the j +1 th and j forward slip regions refers to the stress corresponding to the slip region before the current load calculation; substituting the solving result into formulas (16) to (18) to obtain the slip distinguishing distribution and the stress under the current strain load; recording the j +1 forward slip regionStrain generated
Figure BDA0002422091790000067
And strain generated in the j +1 th reverse slip region
Figure BDA0002422091790000068
3) In the unloading process, if the current unloading strain meets the following relationship:
Figure BDA0002422091790000069
the unloading process will not completely cover all slip zones already present and the average matrix crack spacing remains unchanged; the reverse slip zone resulting from the current unloading process will fall within the forward slip zone satisfying the following relationship:
Figure BDA00024220917900000610
wherein,
Figure BDA00024220917900000611
and
Figure BDA00024220917900000612
respectively corresponding strain generated in the j & ltth & gt and j & lt-1 & gt reverse slip zones; at the moment, n to j +1 pairs of slippage areas are covered, a jth reverse slippage area and a jth forward slippage area are newly generated, and the length of the front j-1 pair of slippage areas is kept unchanged; the jth reverse slip zone length is:
Figure BDA00024220917900000613
wherein,
Figure BDA00024220917900000614
for the length of the forward slip zone covered under the current unloading strain load increment, the length of the jth forward slip zone is as follows:
Figure BDA0002422091790000071
stress increment corresponding to current unloading strain increment and covered forward slip region length
Figure BDA00024220917900000711
Satisfies the following relationship:
Figure BDA0002422091790000073
the current unload strain and stress delta satisfy the relationship shown below:
Figure BDA0002422091790000074
equations (20) to (21) and (24) to (26) are substituted into equation (27) by Δ σiTo solve the object, equation (27) is for Δ σiHas 2 solutions to find the current load sigmai＝(σi-1+Δσi) The stress generated by the jth reverse slip region is less than or equal to the stress generated by the jth reverse slip region and is greater than the stress generated by the jth-1 reverse slip region; substituting the solving result into formulas (24) - (26) to obtain the distribution and stress of the slip region under the current strain load; recording the strain generated in the jth reverse slip region
Figure BDA0002422091790000075
4) When unloaded, the unload strain satisfies the relationship shown below:
Figure BDA0002422091790000076
if the composite material completely slips in the reverse direction, the unloading process covers all existing slip zones, and only one reverse slip zone with the length being non-zero and one forward slip zone with the length being 0 exist under the current load; again because the compression process does not cause the material to create new debonded areas and slip areas, this reverse slip zone is equal to the total length of all slip zones already present:
Figure BDA0002422091790000077
the stress under strain load at this time was calculated by the following formula:
Figure BDA0002422091790000078
recording the strain when the 1 st forward slip region is generated
Figure BDA0002422091790000079
And strain at the time of generation of 1 st reverse slip region
Figure BDA00024220917900000710
The invention has the beneficial effects that:
1. the invention provides a prediction method of any strain loading and unloading stress strain behavior of a unidirectional ceramic matrix composite, which considers mesoscopic damage mechanisms such as matrix cracking, interface debonding, covering of a sliding area and the like, and different interface states such as whether an interface is completely debonded and completely slid or not, and provides a direct calculation method of matrix crack spacing;
2. the method for judging the coverage of the slippage area is simpler and more efficient, the equation relation between stress and strain can be quickly established, the solution of the stress is converted into the solution of the equation, the stress-strain behavior of the unidirectional ceramic matrix composite under the action of any strain loading and unloading is quickly predicted, and a foundation is provided for the dynamics research of the ceramic matrix composite.
Drawings
FIG. 1 is a graphical representation of an arbitrary strain-plus-unload curve.
FIG. 2 is a graph of a ceramic matrix composite slip zone profile and a fiber stress profile.
FIG. 3 is a predicted stress-strain curve of the ceramic matrix composite under the strain-load curve of FIG. 1.
Detailed Description
The present invention will now be described in further detail with reference to the accompanying drawings.
Table 1 is a table of parameters for a unidirectional ceramic matrix composite.
TABLE 1 Material parameters
Figure BDA0002422091790000081
The method for calculating the stress-strain behavior of any strain loading and unloading of the unidirectional ceramic matrix composite specifically comprises the following steps:
step 1: and judging whether the ceramic matrix composite generates matrix cracks, if not, determining that the material is not damaged, and calculating the stress under the current strain load by adopting a mixing ratio formula. If a matrix crack occurs, the stress is calculated by the following procedure.
Obtaining a maximum pre-strain load including the current andi_max. If no damage occurs inside the material, the strain corresponds to a stress of:
σi_max＝i_maxEc(1)
wherein E iscIs the modulus of elasticity of the composite material. The mean matrix crack spacing was initially calculated by substituting equation (1) into the Weibull distribution equation shown below:
Figure BDA0002422091790000091
wherein L is the average matrix crack spacing, LsatIs the saturated spacing of cracks in the matrix, sigma0And m is a statistical parameter, L is compared with the length L of the composite materialcmpA comparison is made. If the average matrix crack spacing is larger than the length of the composite material, no matrix crack is generated, the material is not damaged, and the stress under the current strain is calculated by adopting the formula shown as follows:
σi＝iEc(3)
wherein,ifor the current strain, σiCorresponding to the current strainAnd (4) stress. Otherwise, if the material is damaged, the stress of the material is calculated by adopting the following steps.
FIG. 1 is a schematic view of an arbitrary strain-plus-unload curve. The present example is illustrated with the last unloading process as the current strain load cycle. The composite material has already developed matrix cracks prior to the current strain relief cycle, and therefore the calculation of the stress-strain curve for the current strain relief cycle requires the following steps.
Step 2: and calculating the critical strain corresponding to complete debonding of the interface when the composite material is loaded. Based on a shear-lag model, in the loading process, the load reaches a certain magnitude, the composite material interface is completely debonded, and the average matrix crack spacing is kept unchanged. At the very complete debonding of the composite, there is a relationship between stress and interfacial debond length as shown below:
Figure BDA0002422091790000092
wherein,
Figure BDA0002422091790000093
critical stress corresponding to just complete debonding of the composite material, Ld_crTan theta is the critical debond length, tan theta, corresponding to when the interface is just fully debonded1Is expressed as
Figure BDA0002422091790000094
C1Is expressed as
Figure BDA0002422091790000095
vfIs the fiber volume fraction, EfIs the modulus of elasticity of the fiber, EcIs the equivalent elastic modulus, tau, of the composite materialiIs the interfacial shear stress of the composite material, rfIs the radius of the fiber, σf_thIs the fiber thermal residual stress. At this point, the strain at which the composite just completely debonded
Figure BDA0002422091790000096
Can be expressed as:
Figure BDA0002422091790000097
further, the interfacial debond length and the average matrix crack spacing satisfy the following equation:
Lcr/2＝Ld_cr(6)
wherein, LcrThe critical strain corresponding to the complete debonding of the interface can be obtained by combining the formulas (2), (4) and (6) and a numerical solution method for the actual saturated matrix crack spacing corresponding to the complete debonding of the interface
Figure BDA0002422091790000101
Critical stress
Figure BDA0002422091790000102
And actual saturated matrix crack spacing Lcr. Wherein the matrix crack spacing only needs to be recalculated if the current loading strain satisfies equation (7).
Figure BDA0002422091790000103
In this example, complete debonding of the composite corresponds to a strain of 0.0057. In addition, the unloading process does not need to recalculate the crack spacing of the matrix, and the step is skipped. After the loading of the previous step is finished, the average matrix crack spacing is 1502 mu m.
And step 3: and if the peak load of a loading process before the current strain unloading process is the historical maximum strain load, calculating the critical unloading strain corresponding to the complete slippage of the interface before the calculation of the current strain unloading process. After the last strain loading process is finished, assuming that n pairs of slip regions exist, the stress increment required by complete slip of the interface is as follows:
Figure BDA0002422091790000104
wherein,
Figure BDA0002422091790000105
is the length of the kth forward slip region,
Figure BDA0002422091790000106
the stress increment required for complete slippage of the corresponding interface state at the end of the previous strain loading cycle. The critical strain corresponding to complete interface slip is:
Figure BDA0002422091790000107
wherein σi-1And loading the load at the end of the cycle process for the previous step. During unloading, when the unloading strain is less than
Figure BDA0002422091790000108
When the interface is completely slipped, otherwise, the interface is partially slipped.
In this example, step 3 need not be performed because the peak load of the loading process prior to the current load is not the historical maximum strain load. During the previous unload cycle calculation, the strain corresponding to complete interface slip was-0.0013.
And 4, step 4: based on a shear-lag model, for any loading process, a material generates a new slip region and gradually covers a reverse slip region, when a forward slip region is encountered, the forward slip region is skipped, the reverse slip region is continuously covered until the loading process is completed, and finally the end point of the newly generated forward slip region falls into the reverse slip region. At this time, the loading strain is greater than the strain corresponding to the left forward slip region of the reverse slip region and less than the strain corresponding to the right forward slip region (assuming that the number of the slip regions gradually decreases from the crack of the substrate of the material to the bonding region, i.e. gradually decreases from left to right). If the forward slip zone completely covers the reverse slip zone, the material debonding length and the slip length will increase until the material is completely debonded. When fully debonded, the debonded length and the slip length of the material will not increase. In the unloading process, the material generates a reverse slip region and gradually covers the forward slip region, when the material encounters the reverse slip region, the material skips the reverse slip region and continuously covers the forward slip region until the unloading process is completed, and finally the terminal point of the newly generated reverse slip region falls into the forward slip region. At this time, the unloading strain is smaller than the strain corresponding to the left reverse slip region of the forward slip region and larger than the strain corresponding to the right reverse slip region. If the reverse slip zone completely covers the forward slip zone, the reverse slip zone will not continue to be generated because the compression process will not cause the material to generate new debonding and slip zones.
It is assumed that the slip zones occur in pairs with the forward slip zone preceding. Before the current load cycle, it is assumed that there are already n pairs of slip zones, and for the slip zone that is missing, it is assumed that it is present and of length 0. At initial loading, there are 0 pairs of slip zones.
Based on the above discussion, the current strain load is discussed below as a calculation of stress during loading or unloading.
1) In the loading process, if the current loading strain meets the following relationship:
i≥i_max(10)
the loading process covers all the slip zones, and only one pair of non-zero forward slip zones and a reverse slip zone with the length of 0 exist under the current loading strain load; if the composite material is completely debonded before the current load, the corresponding stress under the current loading strain is:
Figure BDA0002422091790000111
at this time, the length of the first forward slip region is:
Figure BDA0002422091790000112
if the composite material is partially debonded under the current load, the current strain is determinediCurrent stress σi(unknown parameters), average matrix crack spacing L and interfacial debond length LdSubstituting in equations (4) and (5), respectively
Figure BDA0002422091790000113
LcrAnd Ld_crWill combine equation (2) with stress σiSolving the equation (5) for unknown number by numerical method to obtain the current loading stress sigmaiAverage matrix crack spacing L and debond length Ld。
At this time, the length of the first slip zone is:
Figure BDA0002422091790000114
recording the strain generated in the first forward slip region
Figure BDA0002422091790000115
And strain at the time of first reverse slip region
Figure BDA0002422091790000116
2) In the loading process, if the current loading strain meets the following relationship:
i＜i_max(14)
the current strain loading process will not completely cover all of the slip zones already present and the average matrix crack spacing L remains the same.
Figure BDA0002422091790000121
j is the reverse glide zone number that meets the requirements, where,
Figure BDA0002422091790000122
and
Figure BDA0002422091790000123
the strains at the time of generation of the j-th and j + 1-th forward slip regions, respectively. At this time, the n to j +1 pairs of slip regions are covered, and a new j reverse slip region, a j +1 forward slip region and a length of 0 are generatedThe front j-1 pair of slip zones of the (j + 1) th reverse slip zone have no change in length.
The jth reverse slip zone length is:
Figure BDA0002422091790000124
wherein,
Figure BDA0002422091790000125
is the length of the kth reverse slip zone,
Figure BDA0002422091790000126
is the length of the reverse slip zone covered at the current strain load increment.
The length of the j +1 th positive slip region is as follows:
Figure BDA0002422091790000127
stress increment delta sigma corresponding to current loading strainiLength of the covered reverse slip region
Figure BDA0002422091790000128
Satisfies the following relationship:
Figure BDA0002422091790000129
the current loading strain and stress delta satisfy the relationship shown below:
Figure BDA00024220917900001210
wherein,
Figure BDA00024220917900001211
the difference value between the area enclosed by the fiber actual stress and the coordinate axis in the kth pair of slippage areas and the area enclosed by the fiber stress and the coordinate axis when the composite material is not damaged is shown (the coordinate axis x represents the previous one on the composite material)The distance of the point from the crack plane, the coordinate axis y representing the fiber stress at that point); only when the length of the slip region changes,
Figure BDA00024220917900001212
the change will occur; when there is no change in the slip region,
Figure BDA00024220917900001213
remain unchanged during the loading and unloading processes,
Figure BDA00024220917900001214
the expression is as follows:
Figure BDA00024220917900001215
wherein,
Figure BDA00024220917900001216
the difference value of the peak fiber stress in the kth pair of slippage area and the fiber stress in the bonding area is expressed as follows:
Figure BDA00024220917900001217
equations (16) to (18) and equations (20) to (21) are substituted into equation (19) to produce Δ σiTo solve the object, equation (19) relates to Δ σiThe stress increment delta sigma under the current strain is obtained by solving the unitary quadratic equationi(ii) a There are 2 solutions to the quadratic equation of unity, find the current load σi＝(σi-1+Δσi) The stress generated by the j +1 th and j forward slip regions refers to the stress corresponding to the slip region before the current load calculation; substituting the solving result into formulas (16) to (18) to obtain the slip distinguishing distribution and the stress under the current strain load; recording the strain generated in the j +1 forward slip region
Figure BDA0002422091790000131
And strain generated in the j +1 th reverse slip region
Figure BDA0002422091790000132
3) In the unloading process, if the current unloading strain meets the following relationship:
Figure BDA0002422091790000133
the unloading process will not completely cover all slip zones already present and the average matrix crack spacing remains unchanged. The reverse slip zone resulting from the current unloading process will fall within the forward slip zone satisfying the following relationship:
Figure BDA0002422091790000134
wherein,
Figure BDA0002422091790000135
and
Figure BDA0002422091790000136
corresponding strains when the j-th and j-1-th reverse slip regions are generated respectively. At the moment, n to j +1 pairs of slip zones are covered, a new jth reverse slip zone and a jth forward slip zone are generated, and the length of the front j-1 pair of slip zones is kept unchanged. The jth reverse slip zone length is:
Figure BDA0002422091790000137
wherein,
Figure BDA0002422091790000138
for the length of the forward slip zone covered under the current unloading strain load increment, the length of the jth forward slip zone is as follows:
Figure BDA0002422091790000139
stress increment corresponding to current unloading strain increment and covered forward slip region length
Figure BDA00024220917900001310
Satisfies the following relationship:
Figure BDA00024220917900001311
the current unload strain and stress delta satisfy the relationship shown below:
Figure BDA0002422091790000141
equations (20) to (21) and (24) to (26) are substituted into equation (27) by Δ σiTo solve the object, equation (27) is for Δ σiHas 2 solutions to find the current load sigmai＝(σi-1+Δσi) The stress generated by the jth reverse slip region is less than or equal to the stress generated by the jth reverse slip region and is greater than the stress generated by the jth-1 reverse slip region; substituting the solving result into formulas (24) - (26) to obtain the distribution and stress of the slip region under the current strain load; recording the strain generated in the jth reverse slip region
Figure BDA0002422091790000142
4) When unloaded, the unload strain satisfies the relationship shown below:
Figure BDA0002422091790000143
if the composite material slips completely in the reverse direction, the unloading process will cover all slip zones that already exist, and only one reverse slip zone with a length of non-zero and one forward slip zone with a length of 0 will exist under the current load. Again because the compression process does not cause the material to create new debonded areas and slip areas, this reverse slip zone is equal to the total length of all slip zones already present:
Figure BDA0002422091790000144
the stress under strain load at this time was calculated by the following formula:
Figure BDA0002422091790000145
recording the variation of the 1 st forward slip region
Figure BDA0002422091790000146
And strain at the time of generation of 1 st reverse slip region
Figure BDA0002422091790000147
In this example, the strain during the current unloading process is not less than the strain corresponding to the complete slip of the interface, so the case 3) in step 4 is performed during the whole unloading process. After the current unload strain load cycle is calculated, the composite material slip zone distribution and the fiber stress distribution are shown in fig. 2, and the slip zone length is shown in table 2. From which it can be found that there are 2 pairs of slip zones. Under any of the strain-plus-unload curves in fig. 1, the stress-strain prediction curve of the material is shown in fig. 3.
TABLE 2 Length distribution of glide region
Number of slipping zone Forward slip zone length/um Reverse slip zone length/um
1 9.08 44.94
2 5.59 49.83
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.

Claims (6)

1. The method for calculating the stress-strain behavior of any strain loading and unloading of the unidirectional ceramic matrix composite is characterized by comprising the following steps of:
step 1: judging whether the ceramic matrix composite generates matrix cracks, if not, considering that the material is not damaged, and calculating the stress under the current strain load by adopting a mixing ratio formula; if the matrix crack is generated, calculating the stress by adopting the following steps;
step 2: calculating critical strain corresponding to complete debonding of the interface when the composite material is loaded;
and step 3: calculating critical strain corresponding to complete slippage of an interface when the composite material is unloaded;
and 4, step 4: and (3) obtaining the stress of the composite material through an equation of stress increment by combining the critical strain calculated in the step (2) and the critical strain calculated in the step (3) based on a shear-lag model.
2. The method for calculating the random strain plus unload stress-strain behavior of the unidirectional ceramic matrix composite according to claim 1, wherein: in the step 1, the current and previous maximum strain loads are obtainedi_max(ii) a If no damage occurs inside the material, the strain corresponds to a stress of:
σi_max＝i_maxEc(1)
wherein,Ecis the modulus of elasticity of the composite material; the mean matrix crack spacing was initially calculated by substituting equation (1) into the Weibull distribution equation shown below:
Figure FDA0002422091780000011
wherein L is the average matrix crack spacing, LsatIs the saturated spacing of cracks in the matrix, sigma0And m is a statistical parameter, L is mixed with the length L of the composite materialcmpComparing, if the average matrix crack spacing is larger than the length of the composite material, no matrix crack is generated, the material is not damaged, and the stress under the current strain is calculated by adopting the formula shown as follows:
σi＝iEc(3)
wherein,ifor the current strain, σiThe stress corresponding to the current strain; otherwise, if the material is damaged, the stress of the material is calculated by adopting the following steps.
3. The method for calculating the random strain plus unload stress-strain behavior of the unidirectional ceramic matrix composite according to claim 2, wherein: in the step 2, based on a shear model, in the loading process, the load reaches a certain size, the composite material interface is completely debonded, and the average matrix crack distance is kept unchanged; at the very complete debonding of the composite, there is a relationship between stress and interfacial debond length as shown below:
Figure FDA0002422091780000021
wherein,
Figure FDA0002422091780000022
critical stress corresponding to just complete debonding of the composite material, Ld_crTan theta is the critical debond length, tan theta, corresponding to when the interface is just fully debonded1Is expressed as
Figure FDA0002422091780000023
C1Is expressed as
Figure FDA0002422091780000024
vfIs the fiber volume fraction, EfIs the modulus of elasticity of the fiber, EcIs the equivalent elastic modulus, tau, of the composite materialiIs the interfacial shear stress of the composite material, rfIs the radius of the fiber, σf_thIs the fiber thermal residual stress; at this point, the critical strain at which the composite just completely debonds
Figure FDA0002422091780000025
Expressed as:
Figure FDA0002422091780000026
further, the interfacial debond length and the average matrix crack spacing satisfy the following equation:
Lcr/2＝Ld_cr(6)
wherein, LcrCombining the formulas (2), (4) and (6) to obtain the critical strain corresponding to the complete debonding of the interface by a numerical solving method, wherein the critical strain is the actual saturated matrix crack spacing corresponding to the complete debonding of the interface
Figure FDA00024220917800000212
Critical stress
Figure FDA00024220917800000213
And actual saturated matrix crack spacing Lcr；
When the current loading strain satisfies equation (7):
Figure FDA0002422091780000027
the average matrix crack spacing L need to be recalculated.
4. The method for calculating the random strain plus unload stress-strain behavior of the unidirectional ceramic matrix composite according to claim 3, wherein: in the step 3, if the peak load of a loading process before the current strain unloading process is the historical maximum strain load, the critical unloading strain corresponding to the complete slippage of the interface is calculated before the current strain unloading process is calculated; after the last strain loading process is finished, assuming that n pairs of slip regions exist, the stress increment required by complete slip of the interface is as follows:
Figure FDA0002422091780000028
wherein,
Figure FDA0002422091780000029
is the length of the kth forward slip region,
Figure FDA00024220917800000210
the critical strain corresponding to the complete slippage of the interface is obtained by the stress increment required by the complete slippage of the corresponding interface state at the end of the previous strain loading cycle
Figure FDA00024220917800000211
Comprises the following steps:
Figure FDA0002422091780000031
wherein σi-1Loading the load at the end of the cycle for the previous step; during unloading, when the unloading strain is less than
Figure FDA0002422091780000033
When the interface is completely slipped, otherwise, the interface is partially slipped.
5. The method for calculating the random strain plus unload stress-strain behavior of the unidirectional ceramic matrix composite according to claim 4, wherein: in the step 4, based on the shear model, the following steps are performed:
for any loading process, a material generates a new slip region, gradually covers a reverse slip region, skips over the forward slip region when encountering the forward slip region, and continuously covers the reverse slip region until the loading process is completed, and finally the end point of the newly generated forward slip region falls into the reverse slip region; assuming that the number of the slip zone is gradually reduced from the matrix crack of the material to the bonding zone (i.e. gradually reduced from left to right), at this time, the loading strain is greater than the strain corresponding to the forward slip zone on the left side of the reverse slip zone and is less than the strain corresponding to the forward slip zone on the right side; if the forward slip region completely covers the reverse slip region, the material debonding length and the slip length are increased until the material is completely debonded; when the material is completely debonded, the debonding length and the sliding length of the material are not increased any more;
for the unloading process, the material generates a reverse slip region and gradually covers the forward slip region, when the material encounters the reverse slip region, the material skips the reverse slip region and continuously covers the forward slip region until the unloading process is completed, and finally the terminal point of the newly generated reverse slip region falls into the forward slip region; at the moment, the unloading strain is smaller than the strain corresponding to the left reverse slip zone of the forward slip zone and larger than the strain corresponding to the right reverse slip zone; if the reverse slip zone completely covers the forward slip zone, the reverse slip zone will not continue to be generated because the compression process will not cause the material to generate new debonding and slip zones.
6. The method for calculating the random strain plus unload stress-strain behavior of the unidirectional ceramic matrix composite according to claim 4, wherein: in the step 4, the slippage areas are assumed to appear in pairs, and the forward slippage area is in front; before the current load cycle, assuming that there are already n pairs of slip zones, for the slip zone that is missing, assuming it is present and of length 0; at the time of initial loading, 0 pair of slippage areas exist;
based on the above settings, when the current strain load is in the loading or unloading process, the stress-strain behavior is calculated as follows:
1) in the loading process, if the current loading strain meets the following relationship:
i≥i_max(10)
the loading process covers all the slip zones, and only one pair of non-zero forward slip zones and a reverse slip zone with the length of 0 exist under the current loading strain load; if the composite material is completely debonded before the current load, the corresponding stress under the current loading strain is:
Figure FDA0002422091780000032
at this time, the length of the first forward slip region is:
Figure FDA0002422091780000041
if the composite material is partially debonded under the current load, the current strain is determinediCurrent stress σiAverage matrix crack spacing L and interfacial debond length LdSubstituting in equations (4) and (5), respectively
Figure FDA0002422091780000046
LcrAnd Ld_crAnd combined with equation (2) with stress σiSolving the equation (5) for unknown number by numerical method to obtain the current loading stress sigmaiAverage matrix crack spacing L and debond length Ld；
At this time, the length of the first forward slip region
Figure FDA0002422091780000047
Comprises the following steps:
Figure FDA0002422091780000042
recording the strain generated in the first forward slip region
Figure FDA0002422091780000048
And strain at the time of first reverse slip region
Figure FDA0002422091780000049
2) In the loading process, if the current loading strain meets the following relationship:
i＜i_max(14)
the current strain loading process will not completely cover all of the slip zones already present and the average matrix crack spacing L remains constant, and the forward slip zone produced by the current loading process will fall within the reverse slip zone satisfying the relationship shown below:
Figure FDA0002422091780000043
wherein j is the number of the reverse slip region meeting the requirement,
Figure FDA00024220917800000410
and
Figure FDA00024220917800000411
the strain generated in the j & ltth & gt and j +1 & ltth & gt forward slip zones respectively; at the moment, the nth to j +1 pairs of slippage areas are covered, a jth reverse slippage area, a jth +1 forward slippage area and a jth +1 reverse slippage area with the length of 0 are newly generated, and the length of the front j-1 pair of slippage areas is not changed;
updating the jth reverse slip zone length as:
Figure FDA0002422091780000044
wherein,
Figure FDA00024220917800000412
is the length of the kth reverse slip zone,
Figure FDA00024220917800000413
the length of the reverse slip zone covered under the current strain load increment;
the length of the j +1 th positive slip region is as follows:
Figure FDA0002422091780000045
stress increment delta sigma corresponding to current loading strainiLength of the covered reverse slip region
Figure FDA00024220917800000513
Satisfies the following relationship:
Figure FDA0002422091780000051
the current loading strain and stress delta satisfy the relationship shown below:
Figure FDA0002422091780000052
wherein,
Figure FDA0002422091780000053
the difference value between the area enclosed by the fiber actual stress in the kth pair of slippage areas and the coordinate axis and the area enclosed by the fiber stress and the coordinate axis when the composite material is supposed to be not damaged is represented, the coordinate axis x represents the distance from one point on the composite material to the crack plane, and the coordinate axis x less represents the fiber stress of the point; only when the length of the slip region changes,
Figure FDA0002422091780000054
the change will occur; when there is no change in the slip region,
Figure FDA0002422091780000055
remain unchanged during the loading and unloading processes,
Figure FDA0002422091780000056
the expression is as follows:
Figure FDA0002422091780000057
wherein,
Figure FDA0002422091780000058
the difference value of the peak fiber stress in the kth pair of slippage area and the fiber stress in the bonding area is expressed as follows:
Figure FDA0002422091780000059
equations (16) to (18) and equations (20) to (21) are substituted into equation (19) to produce Δ σiTo solve the object, equation (19) relates to Δ σiThe stress increment delta sigma under the current strain is obtained by solving the unitary quadratic equationi(ii) a There are 2 solutions to the quadratic equation of unity, find the current load σi＝(σi-1+Δσi) The stress generated by the j +1 th and j forward slip regions refers to the stress corresponding to the slip region before the current load calculation; substituting the solving result into formulas (16) to (18) to obtain the slip distinguishing distribution and the stress under the current strain load; recording the strain generated in the j +1 forward slip region
Figure FDA00024220917800000510
And strain generated in the j +1 th reverse slip region
Figure FDA00024220917800000511
3) In the unloading process, if the current unloading strain meets the following relationship:
Figure FDA00024220917800000512
the unloading process will not completely cover all slip zones already present and the average matrix crack spacing remains unchanged; the reverse slip zone resulting from the current unloading process will fall within the forward slip zone satisfying the following relationship:
Figure FDA0002422091780000061
wherein,
Figure FDA0002422091780000062
and
Figure FDA0002422091780000063
respectively corresponding strain generated in the j & ltth & gt and j & lt-1 & gt reverse slip zones; at the moment, n to j +1 pairs of slippage areas are covered, a jth reverse slippage area and a jth forward slippage area are newly generated, and the length of the front j-1 pair of slippage areas is kept unchanged; the jth reverse slip zone length is:
Figure FDA0002422091780000064
wherein,
Figure FDA0002422091780000065
for the length of the forward slip zone covered under the current unloading strain load increment, the length of the jth forward slip zone is as follows:
Figure FDA0002422091780000066
stress increment corresponding to current unloading strain increment and covered forward slip region length
Figure FDA0002422091780000067
Satisfies the following relationship:
Figure FDA0002422091780000068
the current unload strain and stress delta satisfy the relationship shown below:
Figure FDA0002422091780000069
equations (20) to (21) and (24) to (26) are substituted into equation (27) by Δ σiTo solve the object, equation (27) is for Δ σiHas 2 solutions to find the current load sigmai＝(σi-1+Δσi) The stress generated by the jth reverse slip region is less than or equal to the stress generated by the jth reverse slip region and is greater than the stress generated by the jth-1 reverse slip region; substituting the solving result into formulas (24) - (26) to obtain the distribution and stress of the slip region under the current strain load; recording the strain generated in the jth reverse slip region
Figure FDA00024220917800000610
4) When unloaded, the unload strain satisfies the relationship shown below:
Figure FDA00024220917800000611
if the composite material completely slips in the reverse direction, the unloading process covers all existing slip zones, and only one reverse slip zone with the length being non-zero and one forward slip zone with the length being 0 exist under the current load; again because the compression process does not cause the material to create new debonded areas and slip areas, this reverse slip zone is equal to the total length of all slip zones already present:
Figure FDA0002422091780000071
the stress under strain load at this time was calculated by the following formula:
Figure FDA0002422091780000072
recording the strain when the 1 st forward slip region is generated
Figure FDA0002422091780000073
And strain at the time of generation of 1 st reverse slip region
Figure FDA0002422091780000074
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