CN108446415B - Robust design method suitable for coating particle composite material considering geometric random distribution - Google Patents

Robust design method suitable for coating particle composite material considering geometric random distribution Download PDF

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CN108446415B
CN108446415B CN201711399125.6A CN201711399125A CN108446415B CN 108446415 B CN108446415 B CN 108446415B CN 201711399125 A CN201711399125 A CN 201711399125A CN 108446415 B CN108446415 B CN 108446415B
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coating
random
particles
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particle composite
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CN108446415A (en
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文聘
叶红玲
刘东来
杨庆生
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Beijing University of Technology
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
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    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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Abstract

The invention provides a robust design method suitable for coating particle composite materials considering geometric random distribution, so as to ensure the reliability of the composite materials in the manufacturing and preparation process. The method mainly comprises the following steps: (1) selecting a microstructure model of the coating material; (2) introducing random variables to establish a calculation model considering geometric random distribution; (3) obtaining Young modulus, shear modulus and strength index of the material by multi-scale analysis, and performing parameter sensitivity analysis by adopting a response surface method; (4) a numerical simulation of a specific coated particle composite robust design. The invention provides a reliable theoretical basis for the development of the coating particle composite material in engineering practice, and can save the design cost.

Description

Robust design method suitable for coating particle composite material considering geometric random distribution
Technical Field
The invention belongs to the field related to design optimization of particle coating composite materials, and particularly relates to a robust design method suitable for coating particle composite materials considering geometric random distribution.
Background
Spherical particle coating materials are common composite materials in engineering and have important application in various fields. The coating mainly has the following two functions: firstly, the strength and other mechanical properties of the raw materials are improved as a reinforcing item; and the second is used as a protective layer to prevent the material from generating chemical reaction and losing efficacy. Particle coating materials theoretically have good properties, but in practice face a number of problems, since the production process has a number of uncertainties such as temperature, processing errors, etc. which cause the material to produce a random distribution of particle coatings during production. This randomness is manifested in two ways, namely the non-uniformity of the coating, but the randomness of the spatial position of the particles throughout the coating. The optimization design of particle composite materials from the point of microstructure is a popular problem in the field of composite materials at present. The robustness design mainly researches the response problem of the system function to parameter variation, and reduces the sensitivity of the system function to the uncontrollable design parameter variation by adjusting the easily controlled design parameters. In the preparation of the particle coating composite material, random variables are selected to establish a mathematical model due to the randomness of the distribution of particles and coatings caused by various uncertain factors, and parameter sensitivity analysis is carried out according to the robustness principle. Therefore, the invention provides a design method of a randomly distributed coating particle composite material based on mathematical methods such as statistics, perturbation analysis, random homogenization theory and the like.
Disclosure of Invention
The invention provides a robust design method suitable for coating particle composite material considering geometric random distribution based on theories of mathematical statistics, perturbation analysis, random homogenization and the like, and adopts the following technical scheme to realize the aim, wherein the method comprises the following steps:
firstly, carrying out parametric modeling on micro-scale geometric fluctuation;
secondly, introducing random variables to establish a calculation model considering weak fluctuation effect;
thirdly, carrying out multi-scale analysis to obtain Young modulus, shear modulus and strength indexes of the material, and carrying out parameter sensitivity analysis by a response surface method;
and fourthly, simulating the numerical value of the coating particle composite material robust design.
Preferably, in said first step, in order to characterize the ripple effect during the preparation process, the following function is introduced to represent the microscopic geometry X:
X＝X(R,VP,AP(m),h,Vc,Ac(q)) (1)
wherein R is the particle radius, VpIs the volume fraction of the particles, AP(m) is a system motion parameter, m is a related particle morphology parameter, h is the average coating thickness, VCIs the volume fraction of the coating, AC(q) is a coating morphology parameter related to the eccentricity h of the particle coating. Generating a jth microstructure X by controlling parameters within parenthesesjThe total number J is 1 … Jtot。
Preferably, random variables are introduced in the second step to establish a calculation model considering weak fluctuation effect, and the modeling process is completed in three aspects: 1) establishing a regular distribution model, wherein all particles are overlapped with the centroid of the coating and are regularly arranged in the matrix, and the radius of the particles and the thickness of the coating are random numbers; 2) moving particles in the coating, and representing the disorder state in the material by using the centroid shift rate (eccentricity) q of the particles and the coating, wherein the q value is a normal distribution random number; 3) the particle coating is controlled by randomly generated motion vectors, which are moved throughout the matrix, the motion of the particles being expressed as the number m of particles moved.
Preferably, in the third step, a random homogenization method is adopted to perform reliability analysis on the multi-scale macroscopic mechanical property. Assuming that a selected material elastic constitutive matrix [ D ] contains a random variable alpha and obeys normal distribution, adopting first-order perturbation expansion as follows:
[D]≈[D]0+[D]1α (2)
in the formula [ D]0Is [ D ]]Zero order term of [ D ]]1Is [ D ]]The first order term of (a). Material macroscopic equivalent elastic matrix
Figure BDA0001519088620000021
The first order perturbation assumption is also exploited as follows:
Figure BDA0001519088620000022
in the formula XjThe reference numerals for the microstructure are used,
Figure BDA0001519088620000023
is a model XjThe macroelastic matrix of (a).
Figure BDA0001519088620000024
In the form of a zero-order term,
Figure BDA0001519088620000025
is a first order term. Since the random variable α is assumed to be a normally distributed variable whose expected value is zero, the macroscopic equivalent elastic matrix
Figure BDA0001519088620000026
Average value of (2) Exp [ deg. ]]Variance Var [ phi ], [ phi ]]Defined by the integral as follows:
Figure BDA0001519088620000027
Figure BDA0001519088620000028
introducing a correction factor β on the basis of equation (3) yields:
Figure BDA0001519088620000029
applying the above theory to the geometric fluctuations of the microstructure, one can derive:
Figure BDA00015190886200000210
in the formula Pr(Xj) Representing a microscopic model XjThe probability of occurrence. Transforming the obtained macroscopic equivalent elastic matrix to obtain the engineering constant Young modulus EHShear modulus GHAnd strength index microscopic stress concentration coefficient Vth。
Figure BDA0001519088620000031
Figure BDA0001519088620000032
Figure BDA0001519088620000033
Figure BDA0001519088620000034
Figure BDA0001519088620000035
Figure BDA0001519088620000036
Figure BDA00015190886200000310
Figure BDA0001519088620000037
Figure BDA0001519088620000038
Figure BDA0001519088620000039
Figure BDA0001519088620000041
Figure BDA0001519088620000042
Preferably, in the fourth step, the influence of geometric fluctuation is contrastively analyzed, 24 groups of calculation models are sampled, a response surface method is applied to fitting, a response surface method is adopted to obtain a microscopic stress concentration coefficient-Mises stress threshold curve, and the young modulus E is drawnHShear modulus GHAnd strength index microscopic stress concentration coefficient VthWhile the response surface of
And carrying out parameter sensitivity analysis on the Young modulus, the shear modulus and the microscopic stress concentration coefficient to obtain a robustness design.
Drawings
FIG. 1 is a particle coating microstructure model;
FIG. 2 is a schematic diagram of a rule model;
FIG. 3 is a graph I of a stochastic model incorporating eccentricity q;
FIG. 4 is a location randomization model II;
FIG. 5 is a computational model flow diagram;
FIG. 6 is a graph of numerical simulation results;
FIG. 7 is a stress concentration factor versus Mises stress threshold curve;
FIG. 8 is a Young's modulus response surface;
FIG. 9 is a shear modulus response surface;
FIG. 10 is a stress concentration coefficient response surface;
fig. 11 is a robustness plan.
Detailed description of the preferred embodiment
The invention provides a coating particle composite material robust design method considering geometric fluctuation effect based on theories of mathematical statistics, perturbation analysis, random homogenization and the like, and the specific implementation steps are as follows:
the first step is as follows: selecting a microstructure of an optimized coating material scheme for user input of microstructure parameters XjThe selected model is shown in fig. 1.
The second step is that: the wave model is established as shown in fig. 2, the modeling process is shown in fig. 3, and the method is completed by the following three steps:
(1) establishing a regular distribution model, wherein all particles are overlapped with the centroid of the coating and are regularly arranged in the matrix, the radius of the particles and the thickness of the coating are random numbers, and the random numbers are generated by a uniform distribution characteristic random number generator and meet normal distribution;
(2) moving particles in the coating, and representing the disorder state in the material by using the centroid shift rate (eccentricity) q of the particles and the coating, wherein the q value is a normal distribution random number;
(3) the particle coating is controlled by randomly generated motion vectors, which are moved throughout the matrix, the motion of the particles being expressed as the number m of particles moved.
The third step: and (3) giving theoretical basis of multi-scale analysis method and parameter sensitivity analysis considering the fluctuation effect particle coating model. The random homogenization method is based on a first-order perturbation expansion theory, and represents the uncertainty of macroscopic mechanical property by introducing a random variable alpha.
The comparative analysis of the model containing geometric fluctuation and the model containing no geometric fluctuation is shown in FIG. 4; the 24 sets of calculation models were performed, and the parameters of the 24 sets of calculation models are shown in the following table.
Figure BDA0001519088620000051
And obtaining a microscopic stress concentration coefficient-Mises stress threshold curve and Young modulus EHShear modulus GHAnd strength index microscopic stress concentration coefficient VthAs shown in fig. 6-8. Each regression fit equation is shown below:
Figure BDA0001519088620000052
Figure BDA0001519088620000053
Figure BDA0001519088620000054
4. calculating the derivative of the design variable based on the above response surface formula
Figure BDA0001519088620000059
And
Figure BDA0001519088620000056
the sensitivity analysis is performed to obtain a robust design with a smaller sensitivity coefficient as shown in fig. 11.
Figure BDA0001519088620000057
Figure BDA0001519088620000058

Claims (4)

1. A robust design method for a coated particle composite material considering geometric random distribution, comprising the steps of:
firstly, carrying out parametric modeling on micro-scale geometric fluctuation;
secondly, introducing random variables to establish a calculation model considering weak fluctuation effect;
thirdly, carrying out multi-scale analysis to obtain Young modulus, shear modulus and strength indexes of the material, and carrying out parameter sensitivity analysis by a response surface method;
fourthly, coating particle composite material robust design: the method comprises the steps of carrying out comparative analysis on the influence of geometric fluctuation, sampling 24 groups of calculation models, fitting by using a response surface method, obtaining a microscopic stress concentration coefficient-Mises stress threshold value curve by using the response surface method, drawing response surfaces of the Young modulus, the shear modulus and the strength index microscopic stress concentration coefficient, carrying out derivation based on the response surfaces, and carrying out parameter sensitive analysis on the Young modulus, the shear modulus and the microscopic stress concentration coefficient to obtain a robustness design.
2. A robust design method considering geometric random distribution suitable for coating particle composites as claimed in claim 1 characterized by that in said first step, in order to characterize the ripple effect during the preparation, the following function is introduced to represent the microscopic geometry X:
X＝X(R,VP,AP(m),h,Vc,Ac(q)) (1)
wherein R is the particle radius, VpIs the volume fraction of the particles, AP(m) is a system motion parameter, m is a related particle morphology parameter, h is the average coating thickness, VCIs the volume fraction of the coating, AC(q) is a coating morphology parameter related to the eccentricity h of the particle coating; by controlling parameters within bracketsGenerating the jth microstructure XjThe total number J is 1 … Jtot。
3. The robust design method for the coated particle composite material considering the geometric random distribution as claimed in claim 2, wherein the random variables are introduced in the second step to establish a calculation model considering the weak fluctuation effect, and the modeling process is completed in three aspects: 1) establishing a regular distribution model, wherein all particles are overlapped with the centroid of the coating and are regularly arranged in the matrix, and the radius of the particles and the thickness of the coating are random numbers; 2) moving particles in the coating, and representing the disorder state in the material by using the centroid shift rate q of the particles and the coating, wherein the q value is a normal distribution random number; 3) the particle coating is controlled by randomly generated motion vectors, which are moved throughout the matrix, the motion of the particles being expressed as the number m of particles moved.
4. The robust design method considering geometric random distribution for the coated particle composite material as claimed in claim 3, wherein in the third step, reliability analysis of multi-scale macro-mechanical properties is performed using a random homogenization method; assuming that a selected material elastic constitutive matrix [ D ] contains a random variable alpha and obeys normal distribution, adopting first-order perturbation expansion as follows:
[D]≈[D]0+[D]1α (2)
in the formula [ D]0Is [ D ]]Zero order term of [ D ]]1Is [ D ]]The first order term of (1); material macroscopic equivalent elastic matrix
Figure FDA0003124637720000021
The first order perturbation assumption is also exploited as follows:
Figure FDA0003124637720000022
in the formula XjThe reference numerals for the microstructure are used,
Figure FDA0003124637720000023
is a model XjA macro-elastic matrix of (a);
Figure FDA0003124637720000024
in the form of a zero-order term,
Figure FDA0003124637720000025
for the first order term, the macroscopic equivalent elastic matrix is assumed to be a normally distributed variable with an expected value of zero
Figure FDA0003124637720000026
Average value of (2) Exp [ deg. ]]Variance Var [ phi ], [ phi ]]Defined by the integral as follows:
Figure FDA0003124637720000027
Figure FDA0003124637720000031
introducing a correction factor β on the basis of equation (3) yields:
Figure FDA0003124637720000032
applying the above theory to the geometric fluctuations of the microstructure, one can derive:
Figure FDA0003124637720000033
in the formula, Pr (X)j) Representing a microscopic model XjThe probability of occurrence; transforming the obtained macroscopic equivalent elastic matrix to obtain the engineering constant Young modulus EHShear modulus GHAnd strength index microscopic stress concentration coefficient Vth；
Figure FDA0003124637720000034
Figure FDA0003124637720000035
Figure FDA0003124637720000036
Figure FDA0003124637720000037
Figure FDA0003124637720000038
Figure FDA0003124637720000039
Figure FDA00031246377200000310
Figure FDA00031246377200000311
Figure FDA00031246377200000312
Figure FDA0003124637720000041
Figure FDA0003124637720000042
Figure FDA0003124637720000043
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