| CPC G16C 60/00 (2019.02) [G06F 30/23 (2020.01); G16C 10/00 (2019.02); G16C 20/80 (2019.02); G06F 2111/10 (2020.01); G06F 2111/14 (2020.01); G06F 2113/26 (2020.01)] | 3 Claims |
|
1. A multi-scale simulation method for mechanical behavior of a multiphase composite material, comprising the following steps:
(1) first-principles calculation under nano scale:
conducting first-principles calculation under nano scale to respectively obtain elastic properties of metal phase and ceramic phase of the multiphase composite material at temperature 0 K and an energy corresponding to a crystal structure of different volumes;
calculation scheme is as follows:
1.1 calculating Young's modulus, bulk modulus, shear modulus and Poisson's ratio of metal and ceramic phases of the multiphase composite material under temperature 0 K;
firstly, structural relaxation of a single crystal structure of metal and ceramic phase materials is carried out respectively; elastic constant C as matrix element of a stiffness matrix is calculated by using the crystal structure obtained after the structural relaxation;
 wherein σi εi cij (i,j=1,2,3,4,5,6) represent stress, strain and elastic constants of single crystal material respectively; then the calculated elastic constants are converted into Young's modulus, bulk modulus, shear modulus and Poisson's ratio by using Voigt-Reuss-Hill approximation;
B=(Bv+Br)/2 (2)
G=(Gv+Gr)/2 (3)
E=9B·G/(3·B+G) (4)
υ=0.5·(3B−2G)/(3B+G) (5)
wherein B represents bulks modulus, G represents shear modulus, E represents Young's modulus and V represents Poisson's ratio; Poisson's ratio is approximately unchanged with temperature; the bulks modulus, the shear modulus and the Young's modulus will change with the change of temperature; in formula (2) and (3), subscript v and r represent the approximate calculation methods of Voigt and Reuss respectively;
Bv=(c11+c22+c33)/9+2(c12+c23+c13)/9 (6)
Gv=(c11+c22+c33−c12−c13−c23)/15+(c44+c55+c66)/5 (7)
Br=1/(s11+s22+s33+2s12+2s13+2s23) (8)
Gr=15/(4s11+4s22+4s33−4s12−4s13−4s23+3s44+3s55+3s6) (9)
wherein Sij (i,j=1,2,3,4,5,6) represents matrix element of a compliance matrix S, which is compliance coefficient of the material; the compliance matrix S and a stiffness matrix C are inverse to each other;
1.2 calculating volume-energy relations of the metal and ceramic phases in the multiphase composite material at temperature 0 K;
the crystal structure obtained after structural relaxation of metal and ceramic phases in step 1.1 is shrunk or expanded in a range of −10% to +10%, and energies corresponding to different cell volumes are calculated;
(2) thermodynamic calculation of mesoscopic scale:
conducting thermodynamic calculation of mesoscopic scale, based on quasi-harmonic Debye model, to obtain elastic properties and thermal expansion coefficients of metal phase and ceramic phase in the multiphase composite material at different temperatures respectively; calculation scheme is as follows:
2.1 calculating the Young's modulus of metal and ceramic phase in the multiphase composite material at different temperatures:
non-equilibrium Gibbs free energy can be described as:
G*=E(V)+pV+Avib(θ,T) (10)
wherein, E(V) represents energies corresponding to the crystal structure with different volumes in step 1.2, P and V are external pressure and cell volume respectively, T represents temperature, Avib represents vibrational Helmholtz free energy, θ represents Debye temperature; vibrational Helmholtz free energy Avib can be expressed as:
 wherein, D(θ/T) represents the Debye function,
 n represents the number of atoms in each cell, kn, represents Boltzmann constant, and the Debye temperature can be expressed as:
 wherein, M represents cell mass, v represents the Poisson's ratio, h represents reduced Planck constant, and f(v) represents the Poisson's ratio function
 Bs represents adiabatic elastic modulus, which can be obtained by using volume-energy relationship data obtained in step 1.2, a calculation method is as follows:
 for a given (p,T), non-equilibrium Gibbs free energy minimizes the volume:
 according to equation (14), thermodynamic state of system can be obtained, an universal equation of system state can be obtained, and the relationship between p, T and V can be determined; further, the bulk modulus of a certain temperature can be expressed as:
 temperature effect can be introduced into the bulk modulus by calculating formula (15), and the bulk modulus BT at a specific temperature can be obtained; by formula (16):
ET=3BT(1−υ) (16)
the Young's modulus at different temperatures can be obtained; the Young's modulus ET at temperature 0 K calculated by formula (16) and the Young's modulus E at temperature 0 K calculated in step 1.1 has a difference ΔE=ET,0−E; this difference is used as the zero correction term of Young's modulus of other arbitrary temperatures calculated by formula (16), and the Young's modulus of any temperature is calculated accordingly E(T)=ET−ΔE;
then the shear modulus G(T) at a particular temperature is calculated by
 2.2 calculating the thermal expansion coefficients of metal and ceramic phases at different temperatures:
heat capacity at constant pressure Cv and Gruneisen parameters γ can be described as:
 based on formula (15), (17) and (18), the coefficient of thermal expansion α(T) can be obtained:
 therefore, the temperature effect can be introduced into the coefficient of thermal expansion to calculate the coefficient of thermal expansion at different temperatures;
(3) molecular dynamics simulation under micro scale:
molecular dynamics simulation under micro scale can obtain the plastic properties of metal phase in the multiphase composites at a specific temperature (that is, different temperatures can be taken) and a specific grain size, plasticity of hard and brittle ceramic phase is negligible compared with that of metal phase;
firstly, a single crystal model of metal phase in the multiphase composite is obtained, and then a polycrystalline model with different grain sizes within a relatively narrow range is constructed by Vironoi method; compression simulation is carried out for polycrystalline models with different grain sizes in a relatively narrow range at a specific temperature, and stress-strain curves are drawn; yield strength and hardening coefficient can be read from the stress-strain curves; it is assumed that the hardening coefficient is independent of grain size; in order to reduce the error, hardening coefficient of polycrystalline models of all grain sizes can be expressed as an average of hardening coefficient of several different small-grain sizes;
for the yield strength σs of any model with different grain size, it can be obtained by fitting the yield strength data of polycrystalline model in a relatively narrow range read from the stress-strain curve by using the Hall-Petch relation;
σs−=σ0+kd−1/2 (20)
wherein, σs represents the yield strength, σ0 represents lattice friction resistance when a single dislocation moves, k is a constant, and d represents desirable arbitrary grain size;
in the fitting process, a calculation result of Peirls-Nabarro stress τp is adopted, σ0≈τp;
a value of Peirls-Nabarro stress of the metal phase main slip system is used as the Peirls-Nabarro stress, and a calculation formula is as follows:
 wherein, v represents Poisson's ratio, and takes the calculated result in step 1.1; a represents plane spacing of the sliding plane, and b represents atomic spacing in the sliding direction, which can be calculated by using the relaxed crystal structure in step 1.1 through the geometric relationship; G(T) is the shear modulus at a specific temperature, and calculated results in step 2.1 are taken;
(4) building three-dimensional finite element model of real microstructure of multiphase composites:
a dual beam focused ion beam system is used to obtain composite materials cuboids with micron side length; after image stack is obtained through focused ion beam experiment, an image processing is carried out, the image processing is as follows:
first, FIB Stack Wizard module is applied to align and crop the image, as well as correct the shearing and shadow of the image, then median filter and edge-preserving smoothing Gaussian filter are used for denoising, finally, the image is binarized by threshold segmentation to distinguish the metal phase and ceramic phase; therefore, composite microstructure images are obtained which can be used to construct a geometric model in finite element model; then constructing the geometric model;
(5) simulating the mechanical behavior of multiphase composites by using finite element method
according to the Poisson's ratio obtained from step (1), Young's modulus and thermal expansion coefficient of metal and ceramic phase at particular temperature obtained in step (2), yield strength and hardening coefficient of metal phase with particular grain size at specific temperature obtained in step (3), a parameter model in finite element model of the ceramic and metal phase at different temperature is built;
reading the geometric model in finite element model of composite material structure constructed in step (4); endowing the parameter model of ceramic phase and metal phase with different temperature to the geometric model of composite material microstructure for calculation.
|