CN107688686A - A kind of laminate structure crash simulation method based on elastic brittle damage pattern - Google Patents
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Abstract
The invention discloses a kind of laminate structure crash simulation method based on elastic brittle damage pattern, it is related to composite material laminated board analysis technical field.The laminate structure crash simulation method based on elastic brittle damage pattern comprises the steps of：Step 1, laminate through-thickness is divided into some solid element bodies, using each cell cube as analysis object, the damage characteristic of single layer composite is described using symmetrical second order damage tensor；The damage mode of material point is introduced into numerical computations, the concept of equivalent displacement is introduced, using Damage Evolution model of the nonlinear Evolution rule of exponential form as material point；According to the equivalent displacement concept and nonlinear Evolution rule introduced in step 2, damage variable Evolution is calculated；Step 4, calculate the damage stiffness of single layer cell body；Step 5, build fibre reinforced composites constitutive relation of the laminate containing damage.The advantage of the invention is that：Avoid the dependence to experiment and researcher's experience.
Description
Technical Field
The invention relates to the technical field of composite material laminated plate structure progressive damage analysis, in particular to a laminated plate structure damage simulation method based on an elastic brittle damage mode.
Background
With the great application of composite material structures in the fields of aviation, aerospace, navigation, construction and the like, the application of the composite material structures is increasingly complicated. Taking aviation as an example, the development trend of international civil aircraft manufacturers for applying composite materials to civil aircraft structures can show that the application parts of the composite materials are gradually transited from secondary load-bearing structures to large-scale main load-bearing structures no matter whether the companies are Boeing companies or air passenger companies. Compared with a secondary bearing structure, the main bearing structure bears a large load, the thickness of parts of the main bearing structure is much thicker than that of the secondary bearing structure, and the interaction between the composite laminated plates cannot be ignored. At the same time, most fiber reinforced composite laminates exhibit brittle failure, no significant plastic deformation before complete failure, and can continue to bear greater loads after failure of the first layer. These characteristics of composite laminates determine the importance of developing an effective composite laminate slab strength analysis method.
In order to predict the damage propagation and ultimate strength of the fiber reinforced composite laminates, researchers typically employ methods that perform corresponding stiffness degradation on material performance parameters after the material is damaged according to corresponding failure criteria. However, in the early progressive damage analysis model, a two-dimensional finite element method based on the classical laminate theory is adopted for stress solution, and the three-dimensional effect of the laminate and the interaction between layers cannot be well reflected. Later, researchers built progressive damage analysis models for three-dimensional composite laminates and used three-dimensional finite elements for stress solution. After the stress is solved, the damage state of the material integration point is judged according to a reasonable failure criterion. The damage analysis in the existing progressive damage analysis model mainly adopts failure criteria which can distinguish failure modes. When the material integration point is damaged, the material property needs to be degraded. The material property degradation scheme in most composite material structure progressive damage analysis is to directly reduce the material elastic constant according to a material failure mode, and because the reduction coefficient is mostly empirical data, the material property degradation scheme needs to be determined by combining specific materials and structures and through abundant experiments of a large number of experiments and researchers, and the inappropriate reduction coefficient may cause the singularity of a stiffness matrix.
Disclosure of Invention
The invention aims to provide a laminate structure damage simulation method based on an elastic brittle damage mode, so as to solve or at least reduce at least one problem existing in the background technology.
The progressive damage analysis method is a calculation method based on the assumption that the material containing the damage can be continuously loaded according to the property after the performance degradation of the material, and the external load is applied by adopting a method of increasing step by step. In each load increment step, firstly, assuming that the material state is unchanged, establishing a finite element balance equation for the whole composite laminated board and solving the nonlinear equation system to obtain displacement convergence. And then, calculating the stress/strain state of each material integration point according to the obtained displacement convergence solution, substituting the obtained stress/strain into a corresponding material failure criterion, and judging whether the material integration points fail. And if the material point fails, calculating the damage state according to the corresponding damage evolution rule, and degrading the material performance. At this time, the convergence solution of the previously solved nonlinear equation no longer satisfies the balance equation of the whole structure, so that the current load state needs to be kept unchanged, and the finite element balance equation is reestablished according to the degraded material property. And repeating the whole analysis process of stress solving, failure criterion judgment, damage evolution and material performance degradation until new damage does not occur in the whole composite material structure. And increasing the load delta P, entering the next load increment step Pn +1, and repeating the previous solving process until the whole composite laminated board structure finally fails.
In order to achieve the purpose, the invention adopts the technical scheme that: the method for simulating the structural damage of the laminated board based on the elastic brittle damage mode comprises the following steps:
dividing a laminated board into a plurality of entity unit bodies along the thickness direction, and describing damage characteristics of a single-layer composite material by adopting a symmetrical second-order damage tensor by taking each unit body as an analysis object;
step two, considering a typical representative domain of the material integration point, introducing a damage mode of the material integration point into numerical calculation, and introducing a concept of equivalent displacement; according to the damage type of the elastic brittle material, combining the elastic brittle failure characteristics of the composite material, and adopting an exponential nonlinear evolution law as a damage evolution model of the material integration point;
step three, calculating a damage variable evolution rule of the typical representative domain according to the equivalent displacement concept and the nonlinear evolution rule introduced in the step two;
fourthly, calculating the damage rigidity of the single-layer unit body;
and fifthly, constructing the constitutive relation of the damaged fiber reinforced composite material of the laminated plate, namely that the stress of the laminated plate is equal to the damage rigidity of the unit body multiplied by the strain of the unit body.
Preferably, defining the direction 1 as a direction parallel to the fibers in a plane of the fiber-reinforced composite material, the direction 2 as a direction perpendicular to the fibers in the plane, and the direction 3 as a normal direction out of the plane, the second-order damage tensor is expressed by D:
wherein d is1、d2、d3The damage tensor eigenvalue is a damage tensor eigenvalue which respectively represents the damage degree of three material main directions of the unit body and a damage tensor eigenvalue diA value of between 0 and 1, di0 means no damage to the ith material direction; di1 means that the material in the ith direction completely breaks in the main direction and is no longer loaded.
Preferably, the equivalent displacement is:
wherein,represents the equivalent displacement,. epsilonijRepresenting the strain at the point of integration of the material, LcA characteristic length of a representative domain representative of the material integration point;
the nonlinear evolution law of the exponential form is as follows:
wherein f isNrNIs a softening function of the constitutive relation of the material, AMThe softening coefficient.
Preferably, the calculation of the evolution law of the damage variable of the typical representative domain specifically includes:
d1＝d2＝d3＝0 if F1＜1∩F2＜1∩F3＜1
wherein, F1、F2、F3Obtaining a strength coefficient according to a failure criterion; g1c、G2c、G3cThe fracture energy dissipation rates of the three material main directions of the unidirectional plate are respectively.
Preferably, the specific calculation method of the damage stiffness of the single-layer unit body comprises the following steps:
wherein,
C11＝E1(1-v23v32)γ C22＝E2(1-v13v31)γ C33＝E3(1-v12v21)γ
C44＝G12C55＝G13C66＝G23
C12＝E1(v21+v31v23)γ C13＝E1(v31+v21v32)γ C23＝E2(v32+v31v12)γ
γ＝1/(1-v12v21-v23v32-v13v31-2v21v32v13)
E1,E2,E3tensile and compressive moduli, G, of the single-layer board in three directions 1,2,312Shear modulus, G, for the 1-2 plane of a single ply13Shear modulus, G, for the 1-3 plane of a single ply23Shear modulus at the 2-3 plane of the single layer sheet; v. of12、v21Poisson ratio, v, for a single sheet 1-2 plane13、v31Poisson ratio, v, for a single sheet in the 1-3 plane23、32Is the Poisson's ratio of 2-3 planes of single-layer plates.
Preferably, in the third step, a three-dimensional Hashin failure criterion is adopted as a damage criterion, and specifically:
wherein the subscripts 1,2,3 represent the three orthogonal directions of the unit cell, respectively,andrespectively tensile and compressive ultimate strains in three directions,is the in-plane shear limit strain of the composite single-layer board,andin order to limit the strain at the interlaminar shear, wherein, CiiAre diagonal elements in the elastic matrix of the material.
The invention has the beneficial effects that:
according to the method, based on the elastic brittle damage mechanics theory, the damage state of a material point is represented by introducing the damage variable corresponding to the material damage mode, the rigidity matrix of the material is continuously degraded according to the damage variable, the dependence on the experiment and the experience of a researcher is avoided, and the method can successfully predict the whole process from the damage initiation, the expansion to the final failure of the laminated thick plate made of the composite material and the ultimate strength.
Drawings
Fig. 1 is a flowchart of a method for simulating a laminate structure failure based on a brittle fracture mode according to an embodiment of the present invention.
FIG. 2 is a schematic diagram of the distribution of elementary volume lesions expressed using the lesion tensor eigenvalues in the present invention.
FIG. 3 is a schematic representation of the type of elasto-brittle damage.
Detailed Description
In order to make the implementation objects, technical solutions and advantages of the present invention clearer, the technical solutions in the embodiments of the present invention will be described in more detail below with reference to the accompanying drawings in the embodiments of the present invention. In the drawings, the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The described embodiments are only some, but not all embodiments of the invention. The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention. Embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
In the description of the present invention, it is to be understood that the terms "center", "longitudinal", "lateral", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc., indicate orientations or positional relationships based on those shown in the drawings, and are used merely for convenience in describing the present invention and for simplifying the description, but do not indicate or imply that the device or element being referred to must have a particular orientation, be constructed and operated in a particular orientation, and thus, should not be construed as limiting the scope of the present invention.
As shown in fig. 1, fig. 2 and fig. 3, a method for simulating the damage of a laminate structure based on a brittle fracture mode comprises the following steps:
step one, dividing the laminated board into a plurality of entity unit bodies along the thickness direction, taking each unit body as an analysis object, and describing the damage characteristics of the single-layer composite material by adopting a symmetrical second-order damage tensor D which is specifically a symmetrical second-order damage tensor D because the damage main direction of the fiber reinforced composite material containing the orthotropic damage is the same as the material main direction
Wherein d is1、d2、d3For the damage tensor eigenvalue, the damage degrees of the three material main directions of the unit body are respectively represented, namely the reduction proportion of the effective bearing area in each direction perpendicular to the unit body can be quantized to describe the distribution density of the microcracks in the fiber direction, the transverse direction of the matrix and the thickness direction. Eigenvalue of damage tensor diA value of between 0 and 1, di0 means no damage to the ith material direction; di1 means that the material main direction of the ith direction is completeBreaking and no longer bearing; diLarger values indicate more severe damage.
It can be understood that the number of the unit bodies can be determined according to the actual loading condition, and when the laminated board bears the in-plane load, a few unit bodies can be divided, so that the analysis efficiency is improved; when the laminated board bears out-of-plane load, the number of the divided unit bodies needs to be increased so as to improve the analysis precision.
It is understood that each unit cell may be a single layer of 1 plate or a plurality of single layers of plates.
The material properties in the progressive damage and damage process of the composite material structure are described by a damage and degradation model, and can be generally divided into three categories, as shown in fig. 3: the first type is that the material point is destroyed immediately after meeting the failure criterion and no longer continues to bear the load, i.e. is unloaded instantaneously; the second is to keep the stress carried constant after the failure criterion is met at the material point; the third type is that the bearing capacity is gradually lost after the material point meets the failure criterion, the condition is divided into two paths of gradually losing the bearing capacity, namely linear gradually unloading and nonlinear gradually unloading, and the evolution of the damage variable in the continuous medium damage mechanical analysis model is generally the third type of condition. The method adopts an exponential nonlinear evolution law as a damage evolution model of the material points.
Step two, considering a typical representative domain of the material integration points, namely a unit body represented by the material integration points, introducing a damage mode of the material integration points into numerical calculation, and introducing a concept of equivalent displacement;
wherein,represents the equivalent displacement,. epsilonijRepresenting the strain at the point of integration of the material, LcCharacteristic length of a typical representative field representing an integral point of a material, i.e. a unit volumeLength of (d).
According to the damage type of the elastic brittle material and the elastic brittle failure characteristics of the composite material, the following nonlinear evolution law in an exponential form is adopted as a damage evolution model of the material integration point:
wherein f isNrNIs a softening function of the constitutive relation of the material, AMThe softening coefficient.
Step three, according to the equivalent displacement introduced in the step twoCalculating a damage variable evolution rule of a typical representative domain according to the concept and the nonlinear evolution rule;
d1＝d2＝d3＝0 if F1＜1∩F2＜1∩F3＜1
wherein, F1、F2、F3Obtaining a strength coefficient according to a failure criterion; g1c、G2c、G3cThe fracture energy dissipation rates of the three material main directions of the unidirectional body respectively.
In this embodiment, a three-dimensional Hash i n failure criterion is adopted as a damage criterion, and the specific calculation method of the strength coefficient is as follows:
wherein the subscripts 1,2,3 represent the three orthogonal directions of the unit cell, respectively,andrespectively tensile and compressive ultimate strains in three directions,is the in-plane shear limit strain of the composite single-layer board,andin order to limit the strain at the interlaminar shear, wherein, CiiAre diagonal elements in the elastic matrix of the material.
Fourthly, calculating the damage rigidity of the single-layer unit body;
effective stress of unit bodyIs defined as the unit effective area on the cross sectionThe force experienced, the apparent stress σ, is defined as the force experienced by the undamaged material cross-sectional area a, i.e., the typical Cauchy stress. Effective area due to material damageSmaller than the apparent area a, so the effective stress is quantitatively greater than the apparent stress. For an isotropic body, due toThereby removing effective stress of the damaged unit bodyRelationship to apparent stress σ:
d represents the damage degree of the main direction of the unit body material, namely the reduction proportion of the effective bearing area in each direction vertical to the unit body;
for anisotropic damage, formulaThe effective stress defined may be generalized as:
wherein M (D) is a fourth order symmetric tensor.
When taking the material principal direction coordinate system, the matrix form of m (d) can be expressed as:
the column vector formats of the Cauchy stress tensor σ and the strain tensor ε are respectively:
σ＝[σ11,σ22,σ33,σ12,σ13,σ23]T
ε＝[ε11,ε22,ε33,σ12,ε13,ε23]T
then formulaThe form of each component of the effective stress defined in (1) in the main direction coordinate system of the material is as follows:
as can be seen from the assumption of energy equivalence, the elastic strain energy density expressed by the effective stress is the same as the elastic strain energy density expressed by the apparent stress, that is, there are:
will be provided withSubstituted typeIn (1),due to the fact thatε and M (D)1Are all symmetrical tensors, then:
then each effective strain component is:
the nondestructive orthotropic single-layer composite material has the constitutive relation matrix form as follows:
σ＝[C]ε
in the formula,
in the formula,
the elastic strain energy density expressed in terms of apparent stress is:
the elastic strain energy density expressed in terms of effective stress is:
expressing the effective stressSubstituted typeThe following can be obtained:
comparison typeAnd formulaIt is possible to obtain:
Cd＝M-1:C:M-T
definition CdThe specific matrix expression is:
wherein, CdThe non-zero term in (a) can be given by (without repeating the corner mark):
wherein,
C11＝E1(1-v23v32)γ C22＝E2(1-v13v31)γ C33＝E3(1-v12v21)γ
C44＝G12C55＝G13C66＝G23
C12＝E1(v21+v31v23)γ C13＝E1(v31+v21v32)γ C23＝E2(v32+v31v12)γ
γ＝1-/(1-v12v21-v23v32-v13v31-2v21v32v13)
E1,E2,E3tensile and compressive moduli, G, of the single-layer board in three directions 1,2,312Shear modulus, G, for the 1-2 plane of a single ply13Shear modulus, G, for the 1-3 plane of a single ply23Shear modulus at the 2-3 plane of the single layer sheet; v. of12、v21Poisson ratio, v, for a single sheet 1-2 plane13、v31Poisson ratio, v, for a single sheet in the 1-3 plane23、v32Is the Poisson's ratio of 2-3 planes of single-layer plates.
And in the numerical calculation process, accumulating the damage amount in the third step along with the increment of the load, and substituting the damage amount into the damage rigidity matrix of the single-layer unit body to further obtain the damaged elastic matrix of the whole laminated board in the instant state. And continuously looping iteration to finally reach a complete breakage state which typically represents that the damage amount of the domain is accumulated to be 1. Thereby realizing the numerical simulation of the damage process of the composite material laminated thick plate structure.
Finally, it should be pointed out that: the above examples are only for illustrating the technical solutions of the present invention, and are not limited thereto. Although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.
Claims (6)
1. A method for simulating the structural damage of a laminated board based on an elastic brittle damage mode is characterized by comprising the following steps:
dividing a laminated board into a plurality of entity unit bodies along the thickness direction, and describing damage characteristics of a single-layer composite material by adopting a symmetrical second-order damage tensor by taking each unit body as an analysis object;
step two, considering a typical representative domain of the material integration point, introducing a damage mode of the material integration point into numerical calculation, and introducing a concept of equivalent displacement; according to the damage type of the elastic brittle material, combining the elastic brittle failure characteristics of the composite material, and adopting an exponential nonlinear evolution law as a damage evolution model of the material integration point;
step three, calculating a damage variable evolution rule of the typical representative domain according to the equivalent displacement concept and the nonlinear evolution rule introduced in the step two;
fourthly, calculating the damage rigidity of the single-layer unit body;
and fifthly, constructing the constitutive relation of the damaged fiber reinforced composite material of the laminated plate, namely that the stress of the laminated plate is equal to the damage rigidity of the unit body multiplied by the strain of the unit body.
2. The method for simulating structural failure of a laminate based on brittle elastic damage mode according to claim 1, wherein the direction 1 is defined as the direction parallel to the fibers in the plane of the fiber-reinforced composite, the direction 2 is the direction perpendicular to the fibers in the plane, the direction 3 is the out-of-plane normal direction, and the second-order damage tensor is expressed as D:
wherein d is1、d2、d3The damage tensor eigenvalue is a damage tensor eigenvalue which respectively represents the damage degree of three material main directions of the unit body and a damage tensor eigenvalue diA value of between 0 and 1, di0 means no damage to the ith material direction; di1 means that the material in the ith direction completely breaks in the main direction and is no longer loaded.
3. The method for simulating structural failure of a laminate based on brittle elastic damage mode according to claim 2, wherein the equivalent displacement is:
<mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>&delta;</mi> <mrow> <mi>e</mi> <mi>q</mi> <mo>,</mo> <mi>i</mi> <mi>j</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <msub> <mi>&epsiv;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>L</mi> <mi>c</mi> </msup> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
wherein,represents the equivalent displacement,. epsilonijRepresenting the strain at the point of integration of the material, LcA characteristic length of a representative domain representative of the material integration point;
the nonlinear evolution law of the exponential form is as follows:
wherein f isNrNIs a softening function of the constitutive relation of the material, AMThe softening coefficient.
4. The method for simulating the structural damage of the laminated board based on the elastic brittle damage mode as claimed in claim 3, wherein the calculation of the damage variable evolution law of the typical representative domain is specifically as follows:
d1＝d2＝d3＝0 if F1＜1∩F2＜1∩F3＜1
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <msubsup> <mi>&sigma;</mi> <mn>11</mn> <mi>f</mi> </msubsup> <msubsup> <mi>&delta;</mi> <mrow> <mi>e</mi> <mi>q</mi> <mo>,</mo> <mn>11</mn> </mrow> <mi>f</mi> </msubsup> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>/</mo> <msub> <mi>G</mi> <mrow> <mn>1</mn> <mi>c</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mi>f</mi> </mrow> </mtd> <mtd> <mrow> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <msubsup> <mi>&sigma;</mi> <mn>22</mn> <mi>f</mi> </msubsup> <msubsup> <mi>&delta;</mi> <mrow> <mi>e</mi> <mi>q</mi> <mo>,</mo> <mn>22</mn> </mrow> <mi>f</mi> </msubsup> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>/</mo> <msub> <mi>G</mi> <mrow> <mn>2</mn> <mi>c</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mi>f</mi> </mrow> </mtd> <mtd> <mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>d</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <msubsup> <mi>&sigma;</mi> <mn>33</mn> <mi>f</mi> </msubsup> <msubsup> <mi>&delta;</mi> <mrow> <mi>e</mi> <mi>q</mi> <mo>,</mo> <mn>33</mn> </mrow> <mi>f</mi> </msubsup> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mn>3</mn> </msub> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>/</mo> <msub> <mi>G</mi> <mrow> <mn>3</mn> <mi>c</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>F</mi> <mn>3</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mi>f</mi> </mrow> </mtd> <mtd> <mrow> <msub> <mi>F</mi> <mn>3</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced>
wherein, F1、F2、F3Obtaining a strength coefficient according to a failure criterion; g1c、G2c、G3cThe fracture energy dissipation rates of the three material main directions of the unidirectional plate are respectively.
5. The method for simulating the structural damage of the laminated board based on the elastic brittle damage mode as claimed in claim 4, wherein the specific calculation method of the damage rigidity of the single-layer unit body is as follows:
<mrow> <msup> <mi>C</mi> <mi>d</mi> </msup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>C</mi> <mn>11</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>C</mi> <mn>12</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>C</mi> <mn>13</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>C</mi> <mn>21</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>C</mi> <mn>22</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>C</mi> <mn>23</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>C</mi> <mn>31</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>C</mi> <mn>32</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>C</mi> <mn>33</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msubsup> <mi>C</mi> <mn>44</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msubsup> <mi>C</mi> <mn>55</mn> <mi>d</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msubsup> <mi>C</mi> <mn>66</mn> <mi>d</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
wherein,
C11＝E1(1-v23v32)γ C22＝E2(1-ν13ν31)γ C33＝E3(1-ν12ν21)γ
C44＝G12C55＝G13C66＝G23
C12＝E1(v21+v31v23)γ C13＝E1(v31+v21v32)γ C23＝E2(v32+v31v12)γ
γ＝1/(1v12v21v23v32v13v312v21v32v13)
E1,E2,E3tensile and compressive moduli, G, of the single-layer board in three directions 1,2,312Shear modulus, G, for the 1-2 plane of a single ply13Shear modulus, G, for the 1-3 plane of a single ply23Shear modulus at the 2-3 plane of the single layer sheet; v. of12、v21Poisson ratio, v, for a single sheet 1-2 plane13、v31Poisson ratio, v, for a single sheet in the 1-3 plane23、v32Is the Poisson's ratio of 2-3 planes of single-layer plates.
6. The method of simulating laminate structural failure based on brittle elastic damage mode according to claim 5, characterized by: in the third step, a three-dimensional Hashin failure criterion is adopted as a damage criterion, and the method specifically comprises the following steps:
<mrow> <msubsup> <mi>F</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>11</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mn>11</mn> <mrow> <mi>f</mi> <mo>,</mo> <mi>t</mi> </mrow> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>12</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mn>12</mn> <mi>f</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>13</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mn>13</mn> <mi>f</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&GreaterEqual;</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>&sigma;</mi> <mn>11</mn> </msub> <mo>></mo> <mn>0</mn> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>11</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mn>11</mn> <mrow> <mi>f</mi> <mo>,</mo> <mi>c</mi> </mrow> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&GreaterEqual;</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>&sigma;</mi> <mn>11</mn> </msub> <mo><</mo> <mn>0</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
<mrow> <msubsup> <mi>F</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mfrac> <mrow> <mo>(</mo> <msub> <mi>&epsiv;</mi> <mn>22</mn> </msub> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> <mo>)</mo> </mrow> <mrow> <msubsup> <mi>&epsiv;</mi> <mn>22</mn> <mrow> <mi>f</mi> <mo>,</mo> <mi>t</mi> </mrow> </msubsup> <msubsup> <mi>&epsiv;</mi> <mn>33</mn> <mrow> <mi>f</mi> <mo>,</mo> <mi>t</mi> </mrow> </msubsup> </mrow> </mfrac> <mn>2</mn> </msup> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&epsiv;</mi> <mn>22</mn> </msub> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>&epsiv;</mi> <mn>23</mn> <mi>f</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>12</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mn>12</mn> <mi>f</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>13</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mn>13</mn> <mi>f</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <msubsup> <mi>&epsiv;</mi> <mn>23</mn> <mn>2</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>&epsiv;</mi> <mn>23</mn> <mi>f</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>&GreaterEqual;</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>&epsiv;</mi> <mn>22</mn> </msub> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> <mo>></mo> <mn>0</mn> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>&epsiv;</mi> <mn>22</mn> </msub> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <msubsup> <mi>&epsiv;</mi> <mn>22</mn> <mrow> <mi>f</mi> <mo>,</mo> <mi>c</mi> </mrow> </msubsup> <msubsup> <mi>&epsiv;</mi> <mn>33</mn> <mrow> <mi>f</mi> <mo>,</mo> <mi>c</mi> </mrow> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&epsiv;</mi> <mn>22</mn> </msub> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> </mrow> <msubsup> <mi>&epsiv;</mi> <mn>22</mn> <mrow> <mi>f</mi> <mo>,</mo> <mi>c</mi> </mrow> </msubsup> </mfrac> <mrow> <mo>(</mo> <mfrac> <msubsup> <mi>&epsiv;</mi> <mn>22</mn> <mrow> <mi>f</mi> <mo>,</mo> <mi>c</mi> </mrow> </msubsup> <mrow> <mn>2</mn> <msubsup> <mi>&epsiv;</mi> <mn>12</mn> <mi>f</mi> </msubsup> </mrow> </mfrac> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&epsiv;</mi> <mn>22</mn> </msub> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>&epsiv;</mi> <mn>23</mn> <mi>f</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>12</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mrow> <mi>1</mi> <mn>2</mn> </mrow> <mi>f</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>13</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mrow> <mi>1</mi> <mn>3</mn> </mrow> <mi>f</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <msubsup> <mi>&epsiv;</mi> <mn>23</mn> <mn>2</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>&epsiv;</mi> <mn>23</mn> <mi>f</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>&GreaterEqual;</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>&epsiv;</mi> <mn>22</mn> </msub> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> <mo>></mo> <mn>0</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
<mrow> <msubsup> <mi>F</mi> <mn>3</mn> <mn>2</mn> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mi>33</mi> <mrow> <mi>f</mi> <mo>,</mo> <mi>t</mi> </mrow> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>13</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mi>13</mi> <mi>f</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>23</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mi>23</mi> <mi>f</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&GreaterEqual;</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> <mo>></mo> <mn>0</mn> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mi>33</mi> <mrow> <mi>f</mi> <mo>,</mo> <mi>c</mi> </mrow> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>13</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mi>13</mi> <mi>f</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>&epsiv;</mi> <mn>23</mn> </msub> <msubsup> <mi>&epsiv;</mi> <mi>23</mi> <mi>f</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&GreaterEqual;</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>&epsiv;</mi> <mn>33</mn> </msub> <mo><</mo> <mn>0</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
wherein the subscripts 1,2,3 represent the three orthogonal directions of the unit cell, respectively,and(i is 1,2,3) is the tensile and compressive limit strain in three directions respectively,is the in-plane shear limit strain of the composite single-layer board,andin order to limit the strain at the interlaminar shear, wherein, CiiAre diagonal elements in the elastic matrix of the material.
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