CN107695535A - A kind of hard brittle material laser cutting householder method, model creation method and system - Google Patents
A kind of hard brittle material laser cutting householder method, model creation method and system Download PDFInfo
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- CN107695535A CN107695535A CN201710942124.5A CN201710942124A CN107695535A CN 107695535 A CN107695535 A CN 107695535A CN 201710942124 A CN201710942124 A CN 201710942124A CN 107695535 A CN107695535 A CN 107695535A
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- 239000000463 material Substances 0.000 title claims abstract description 130
- 238000000034 method Methods 0.000 title claims abstract description 97
- 238000003698 laser cutting Methods 0.000 title claims abstract description 45
- 239000000446 fuel Substances 0.000 claims abstract description 80
- 230000008646 thermal stress Effects 0.000 claims abstract description 74
- 238000009826 distribution Methods 0.000 claims abstract description 52
- 238000000556 factor analysis Methods 0.000 claims abstract description 49
- 238000005520 cutting process Methods 0.000 claims abstract description 41
- 238000005315 distribution function Methods 0.000 claims description 5
- 238000004458 analytical method Methods 0.000 claims description 4
- 230000003247 decreasing effect Effects 0.000 abstract description 4
- 229910052594 sapphire Inorganic materials 0.000 description 28
- 239000010980 sapphire Substances 0.000 description 28
- 230000006870 function Effects 0.000 description 22
- 238000005516 engineering process Methods 0.000 description 4
- 230000009466 transformation Effects 0.000 description 3
- 238000010586 diagram Methods 0.000 description 2
- 239000010437 gem Substances 0.000 description 2
- 229910001751 gemstone Inorganic materials 0.000 description 2
- 239000011521 glass Substances 0.000 description 2
- 230000035882 stress Effects 0.000 description 2
- 241000208340 Araliaceae Species 0.000 description 1
- 230000005483 Hooke's law Effects 0.000 description 1
- 244000283207 Indigofera tinctoria Species 0.000 description 1
- 235000005035 Panax pseudoginseng ssp. pseudoginseng Nutrition 0.000 description 1
- 235000003140 Panax quinquefolius Nutrition 0.000 description 1
- 241000219000 Populus Species 0.000 description 1
- 238000010521 absorption reaction Methods 0.000 description 1
- 230000006835 compression Effects 0.000 description 1
- 238000007906 compression Methods 0.000 description 1
- 238000010276 construction Methods 0.000 description 1
- 238000001816 cooling Methods 0.000 description 1
- 239000013078 crystal Substances 0.000 description 1
- 230000007547 defect Effects 0.000 description 1
- 238000001514 detection method Methods 0.000 description 1
- 230000000694 effects Effects 0.000 description 1
- 238000002309 gasification Methods 0.000 description 1
- 235000008434 ginseng Nutrition 0.000 description 1
- 230000005484 gravity Effects 0.000 description 1
- 230000008642 heat stress Effects 0.000 description 1
- -1 life Substances 0.000 description 1
- 238000004519 manufacturing process Methods 0.000 description 1
- 230000003287 optical effect Effects 0.000 description 1
- 239000013307 optical fiber Substances 0.000 description 1
- 238000002360 preparation method Methods 0.000 description 1
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
- B23K—SOLDERING OR UNSOLDERING; WELDING; CLADDING OR PLATING BY SOLDERING OR WELDING; CUTTING BY APPLYING HEAT LOCALLY, e.g. FLAME CUTTING; WORKING BY LASER BEAM
- B23K26/00—Working by laser beam, e.g. welding, cutting or boring
- B23K26/36—Removing material
- B23K26/38—Removing material by boring or cutting
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
- B23K—SOLDERING OR UNSOLDERING; WELDING; CLADDING OR PLATING BY SOLDERING OR WELDING; CUTTING BY APPLYING HEAT LOCALLY, e.g. FLAME CUTTING; WORKING BY LASER BEAM
- B23K26/00—Working by laser beam, e.g. welding, cutting or boring
- B23K26/70—Auxiliary operations or equipment
- B23K26/702—Auxiliary equipment
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Abstract
This application discloses a kind of hard brittle material laser cutting householder method, model creation method and system, including determine the equation of heat conduction corresponding to laser cutting flat board process；Using the boundary condition and the equation of heat conduction of flat board, three-dimensional temperature field expression formula of the flat board when being cut inside it is determined；Based on three-dimensional temperature field expression formula, using Airy thermal stress function methods, flat board corresponding fuel factor analysis model when cut is determined；Fuel factor distribution map corresponding with fuel factor analysis model is drawn, and fuel factor distribution map is sent to default cutting auxiliary terminal display screen and shown.The application enables staff by checking the fuel factor distribution map for cutting auxiliary terminal display screen and showing, thermal stress during precognition laser cutting hard brittle material and the concrete condition of thermal strain in advance, thus, it is possible to the actual cutting operation of ancillary staff, it is greatly decreased time-consuming during Laser Processing hard brittle material, and is advantageous to be lifted the precision of material cutting.
Description
Technical field
The present invention relates to laser cutting technique field, more particularly to a kind of hard brittle material laser cutting householder method, mould
Type creation method and system.
Background technology
Present consumer electronics such as mobile phone pours in the hard brittle material such as life, glass, sapphire of people more and more
Used on a large scale as display screen, Home key, cell-phone camera head protection form etc., because traditional processing is using machinery
Mode (such as break bar cutting) directly contacts processing, it is easy to causes to cause glass, indigo plant because of direct active force in process
The hard brittle materials such as jewel produce defect and amplified, and can finally cause this hard brittle material to rupture, thus develop laser now
Untouchable processing, substantially much Laser Processing manufacturer is using laser cutting hard brittle material at present, due to swashing in design
During light processing technology solution, staff can not predict the thermal stress in cutting hard brittle material and the tool of thermal strain in advance
Body situation, so as to cause staff to be essentially all to cut hard fragility material by being constantly changing laser technical parameterses come trial cut
Material, then observes laser processing workpiece cross-section morphology by detection device, very time-consuming.
In summary as can be seen that how to reduce Laser Processing hard brittle material during it is time-consuming be at present it is urgently to be resolved hurrily
The problem of.
The content of the invention
In view of this, it is an object of the invention to provide a kind of hard brittle material laser cutting householder method, model creation
Method and system, it can be greatly decreased time-consuming during Laser Processing hard brittle material.Its concrete scheme is as follows：
A kind of hard brittle material is cut by laser householder method, and the householder method is to by hard brittle material system using laser
Standby flat board carries out actually cutting the method completed before, including：
Determine the equation of heat conduction corresponding to the laser cutting flat board process；Wherein, the laser is symmetrical for circle
The laser of Gaussian, the hard brittle material are orthotropic material；
Boundary condition and the equation of heat conduction using the flat board, determine the flat board when cut in it
The three-dimensional temperature field expression formula in portion；
Based on the three-dimensional temperature field expression formula, using Airy thermal stress function methods, determine that the flat board is being cut
When corresponding fuel factor analysis model；Wherein, the fuel factor analysis model includes thermal stress analytic solutions and thermal strain analytic solutions；
Draw corresponding with fuel factor analysis model fuel factor distribution map, and by the fuel factor distribution map send to
Default cutting auxiliary terminal display screen is shown；Wherein, the fuel factor distribution map includes thermal stress distribution figure and heat should
Become distribution map.
Optionally, the equation of heat conduction, it is specially：
In formula, θ=T-TaRepresent plate material temperature T-phase for air themperature TaTemperature rise；kx、kyAnd kzRespectively institute
State the thermal conductivity factor of hard brittle material along x-axis on the flat board, y-axis, three directions of z-axis component；qV(x, y, z) is described
The thermal source distribution function of hard brittle material.
Optionally, the three-dimensional temperature field expression formula, it is specially：
Optionally, the thermal stress analytic solutions, it is specially：
Optionally, the process of the flat board corresponding thermal strain analytic solutions when being cut is determined, including：
The thermal stress analytic solutions are inputted to the relational expression between default thermal strain and thermal stress, obtaining the heat should
Become analytic solutions；
Wherein, the relational expression between the thermal strain and thermal stress, it is specially：
The invention also discloses one kind to be cut by laser hard brittle material fuel factor analysis model creation method, the establishment side
Method is that the flat board prepared by hard brittle material is carried out actually cutting the method completed before using laser, including：
Determine the equation of heat conduction corresponding to the laser cutting flat board process；Wherein, the laser is symmetrical for circle
The laser of Gaussian, the hard brittle material are orthotropic material；
Boundary condition and the equation of heat conduction using the flat board, determine the flat board when cut in it
The three-dimensional temperature field expression formula in portion；
Based on the three-dimensional temperature field expression formula, using Airy thermal stress function methods, determine that the flat board is being cut
When corresponding fuel factor analysis model；Wherein, the fuel factor analysis model includes thermal stress analytic solutions and thermal strain analytic solutions.
Optionally, the thermal stress analytic solutions, it is specially：
Optionally, the process of the flat board corresponding thermal strain analytic solutions when being cut is determined, including：
The thermal stress analytic solutions are inputted to the relational expression between default thermal strain and thermal stress, obtaining the heat should
Become analytic solutions；
Wherein, the relational expression between the thermal strain and thermal stress, it is specially：
The present invention accordingly discloses a kind of hard brittle material laser cutting accessory system, the course of work of the accessory system
To carry out actually cutting the process completed before to the flat board prepared by hard brittle material using laser；The accessory system bag
Include：
Equation of heat conduction determining module, for determining the heat transfer side corresponding to the laser cutting flat board process
Journey；Wherein, the laser is the laser of circle symmetrical Gaussian type, and the hard brittle material is orthotropic material；
Temperature field expression formula determining module, boundary condition and the equation of heat conduction for the utilization flat board, really
Make three-dimensional temperature field expression formula of the flat board when being cut inside it；
Fuel factor analysis model determining module, for based on the three-dimensional temperature field expression formula, utilizing Airy thermal stress letters
Number methods, determine the flat board corresponding fuel factor analysis model when cut；Wherein, the fuel factor analysis model includes
Thermal stress analytic solutions and thermal strain analytic solutions；
Fuel factor distribution map drafting module, for drawing fuel factor distribution map corresponding with the fuel factor analysis model,
And the fuel factor distribution map is sent to default cutting auxiliary terminal display screen and shown；Wherein, the fuel factor point
Butut includes thermal stress distribution figure and thermal strain distribution map.
The present invention further correspondingly discloses a kind of laser cutting hard brittle material fuel factor analysis model and creates system, the wound
The course of work for building system is that the flat board prepared by hard brittle material is carried out actually cutting the process completed before using laser；
The establishment system includes：
Equation of heat conduction determining module, for determining the heat transfer side corresponding to the laser cutting flat board process
Journey；Wherein, the laser is the laser of circle symmetrical Gaussian type, and the hard brittle material is orthotropic material；
Temperature field expression formula determining module, boundary condition and the equation of heat conduction for the utilization flat board, really
Make three-dimensional temperature field expression formula of the flat board when being cut inside it；
Fuel factor analysis model determining module, for based on the three-dimensional temperature field expression formula, utilizing Airy thermal stress letters
Number methods, determine the flat board corresponding fuel factor analysis model when cut；Wherein, the fuel factor analysis model includes
Thermal stress analytic solutions and thermal strain analytic solutions.
It can be seen that the application first with the flat board prepared by hard brittle material boundary condition and laser cutting by hard fragility
The equation of heat conduction corresponding to flat board process prepared by material, determine three-dimensional temperature field table of the flat board when being cut inside it
Up to formula, then on this basis, using Airy thermal stress function methods, the flat board corresponding fuel factor when cut is determined
Analysis model, and corresponding fuel factor distribution map is drawn, finally fuel factor distribution map is sent to default cutting auxiliary terminal
Display screen is shown, so that staff can be by checking that the fuel factor that cutting auxiliary terminal display screen is shown is distributed
Figure, thermal stress when being cut by laser hard brittle material and the concrete condition of thermal strain is predicted in advance, thus, it is possible to back work people
The actual cutting operation of member, is greatly decreased time-consuming during Laser Processing hard brittle material, and is advantageous to lift material cutting
Precision.
Brief description of the drawings
In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing
There is the required accompanying drawing used in technology description to be briefly described, it should be apparent that, drawings in the following description are only this
The embodiment of invention, for those of ordinary skill in the art, on the premise of not paying creative work, can also basis
The accompanying drawing of offer obtains other accompanying drawings.
Fig. 1 is that a kind of hard brittle material is cut by laser aided process flow sheet figure disclosed in the embodiment of the present application；
Fig. 2 is laser cutting sapphire flat board process schematic；
Fig. 3 is anisotropy plate material schematic diagram；
The temperature profile of laser entrance face when Fig. 4 is laser cutting anisotropy sapphire plate material；
Fig. 5 is the thermal stress σ of sapphire flat boardxxDistribution of contours figure；
Fig. 6 is the thermal stress σ of sapphire flat boardyyDistribution of contours figure；
Fig. 7 is the thermal stress σ of sapphire flat boardxyDistribution of contours figure；
Fig. 8 is the thermal strain ε of sapphire flat boardxxDistribution of contours figure；
Fig. 9 is the thermal strain ε of sapphire flat boardyyDistribution of contours figure；
Figure 10 is the thermal strain ε of sapphire flat boardxyDistribution of contours figure；
Figure 11 is a kind of laser cutting hard brittle material fuel factor analysis model creation method disclosed in the embodiment of the present application
Flow chart；
Figure 12 is that a kind of hard brittle material is cut by laser accessory system structural representation disclosed in the embodiment of the present application；
Figure 13 is that a kind of laser cutting hard brittle material fuel factor analysis model creates system disclosed in the embodiment of the present application
Structural representation.
Embodiment
Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is carried out clear, complete
Site preparation describes, it is clear that described embodiment is only part of the embodiment of the present invention, rather than whole embodiments.It is based on
Embodiment in the present invention, those of ordinary skill in the art are obtained every other under the premise of creative work is not made
Embodiment, belong to the scope of protection of the invention.
The embodiment of the present application discloses a kind of hard brittle material laser cutting householder method, and the householder method is to utilize laser
The flat board prepared by hard brittle material is carried out actually cutting the method completed before, shown in Figure 1, the householder method bag
Include：
Step S11：It is determined that the equation of heat conduction corresponding to laser cutting flat board process；Wherein, laser is circle symmetrical Gaussian
The laser of type, hard brittle material are orthotropic material.
In the present embodiment, laser is the laser of circle symmetrical Gaussian type, and hard brittle material is orthotropic material, such as blue
Jewel etc..It is shown in Figure 2, the schematic diagram of laser cutting sapphire flat board is shown in Fig. 2.Specifically, exported by laser
The pulse laser of controlled high repetition frequency, the pulse laser beam is by the optical focusing system built in cutting head by laser focal spot
Focus on the surface of sapphire flat board, form trickle one by one, high-energy-density hot spot.The laser of each high-energy
Body surface is just sputtered an atomic tiny hole by pulse moment, and under the control of the computer, laser cutting head is with being cut
Material carries out relative motion by the figure painted in advance, and being connected together along the micropore one by one of cutting track to form one and cut
Seam, in laser cutting process, coaxial high pressure draught sprays from cutting head with light beam, will melt or the material of gasification is by otch
Bottom blowout, reach that cutting section is smooth, smooth, impulse- free robustness, can thus complete the laser cutting of plate material part.
Step S12：Using the boundary condition and the equation of heat conduction of flat board, determine flat board when cut inside it
Three-dimensional temperature field expression formula.
Step S13：Based on three-dimensional temperature field expression formula, using Airy thermal stress function methods, determine that flat board is being cut
When corresponding fuel factor analysis model；Wherein, fuel factor analysis model includes thermal stress analytic solutions and thermal strain analytic solutions.
In the present embodiment, the process of flat board corresponding thermal strain analytic solutions when being cut is determined, can specifically be included：
Thermal stress analytic solutions are inputted to the relational expression between default thermal strain and thermal stress, obtain thermal strain analytic solutions.
Step S14：Draw corresponding with fuel factor analysis model fuel factor distribution map, and by fuel factor distribution map send to
Default cutting auxiliary terminal display screen is shown；Wherein, fuel factor distribution map includes thermal stress distribution figure and thermal strain point
Butut.
It can be seen that the embodiment of the present application first with the flat board prepared by hard brittle material boundary condition and laser cutting by
The equation of heat conduction corresponding to flat board process prepared by hard brittle material, determine three-dimensional temperature of the flat board when being cut inside it
Field expression formula is spent, then on this basis, using Airy thermal stress function methods, determines that the flat board is corresponding when cut
Fuel factor analysis model, and corresponding fuel factor distribution map is drawn, finally fuel factor distribution map is sent auxiliary to default cutting
Terminal display screen is helped to be shown, so that the thermal effect that staff can show by checking cutting auxiliary terminal display screen
Distribution map is answered, in advance thermal stress during precognition laser cutting hard brittle material and the concrete condition of thermal strain, thus, it is possible to aid in
The actual cutting operation of staff, is greatly decreased time-consuming during Laser Processing hard brittle material, and is advantageous to lift material
Expect the precision of cutting.
The embodiment of the present application analyzes mould to the fuel factor in hard brittle material laser cutting householder method in a upper embodiment
The determination process of type is further described in more detail.It is specific as follows：
Industrial laserses cutting is mainly by hot-working of the laser focal spot of high-energy to cutting material, and heat is by being added
Work material carries out heat transfer, as shown in figure 3, anisotropy is cut the size 2a x 2b x h of plate material, Laser Focusing light
Beam and passes through plate material by the center incident of material, be cut material laser entrance face and exit facet by high voltage co-axial
The forced convertion cooling of gas, 4 sides are because of the fixation by fixture, in order to which letter is in theory deduction, it is assumed that this 4 faces are air pair
Stream.Due to the symmetry of problem, 1/4 model is can use to analyze.The then heat transfer side in orthotropy laser-cut material
Journey can be written as：
In formula, θ=T-TaRepresent plate material temperature T-phase for air themperature TaTemperature rise；kx、kyAnd kzIt is respectively hard
The thermal conductivity factor of fragile material along x-axis on flat board, y-axis, three directions of z-axis component；qV(x, y, z) is hard brittle material
Thermal source distribution function.
Assuming thatThen the mathematical description of the heat conduction problem of anisotropy laser-cut material is specific
The equation group (2) seen below, namely equation 2a to 2g：
H in formulax=hx/kx, Hy=hy/ky, Hz=hz/kz；hx, hy, hzIt is each to be distributed as anisotropy laser-cut material
The convection transfer rate of individual side.
In the present embodiment, Laser Focusing light beam is circle symmetrical Gaussian type, i.e. light intensity is in radially Gaussian Profile, vertical along propagating
To exponentially decaying, then thermal source distribution function is：
Q=ξ in formulafPabsThe total amount of heat absorbed for the laser-cut material unit interval, ξfFor thermal conversion factor, laser is represented
Output light energy is converted to the ratio of heat, PabsThe laser output power absorbed by laser-cut material, ωfFor Gauss light
The waist radius of beam, the light intensity at the radius are the 1/e of peak light intensity2, α is the suction of laser-cut material at the wavelength of the laser
Receive coefficient.Q can be provedV(x, y, z) meets ∫ ∫ ∫ qvDV=Q.
Assuming that 3 components of the thermal coefficient of expansion of orthotropy hard brittle material are respectively αx, αy, αz, should in plane
Become and (be applied to when the longitudinal length of the medium more situations bigger than cross sectional dimensions) under assuming, the Thermoelastic Problems in crystal
Basic relational expression is as follows：
Equilibrium equation when ignoring body force (such as gravity) is：
The components of strain meet the equation of comptability：
Under plane strain condition, have：
E in formula is Young's modulus, and ν is Poisson's ratio, so as to have：
σzz=ν (σxx+σyy)-αzEθ (7)
Then the relational expression between thermal strain and thermal stress (generalized Hooke law) is specifically shown in following formula (8), including (8a), (8b)
(8c)：
Airy thermal stress function phis are defined as below,
Then function phi will always meet equilibrium equation (4).If formula (5) is substituted into formula (8), and formula (9) is substituted into it
In the components of stress in, then obtain be on the differential equation of function phi：
In the case where being acted on without external force, the mechanical boundary conditions represented with function phi can be written as：
In formula,
Equation group (2) is solved with integral-transform method below, to Excess temperature θ (x, y, z) positive and negative transform definition such as
Under：
Direct transform to variable x is
Its contravariant is changed to
X (β in formulam, x) and=cos (βmX), m=1,2,3 ... is characterized function, N (βm) be characterized the mould of function, i.e.,
Wherein, βm, m=1,2,3 ... it is characterized value, i.e., the positive root of following transcendental equation：
βm tan(βmA)=Hx (14)
Direct transform for variable y is
Y (γ in formulan, y) and=cos (γnY), n=1,2,3 ... it is characterized function, N (γn) be characterized the mould of function, i.e.,
γn, n=1,2,3 ... it is characterized value, i.e., the positive root of following transcendental equation：
γntan(γnB)=Hy (17)
Integral transformation is implemented to equation group (2) according to defined above, obtains following ODE and boundary condition,
In formula,
Laser Focusing light beam in the present embodiment is Gaussian, carries out integral transformation twice to thermal source distribution function, is changed into：
Formula (19) is substituted into formula (18a), obtained：
In formula,
Its solution under boundary condition (18a) and (18b) is：
In formula
Inverse transformation twice is carried out to formula (21), can obtain the solution of (2) of former problem, namely obtain corresponding three-dimensional temperature field
Expression formula：
Here Airy thermal stress function phis are decomposed into three sub- function phis1、φ2And φ3, and them is met respectively：
φ=φ1+φ2+φ3 (23a)
Wherein φ1And φ2For the general solution of biharmonic equation (23b) and (23c), and φ3It is a particular solution of equation (23d),
Thus φ=φ1+φ2+φ3Equation (10a) will always be met.
According to the characteristics of Temperature Distribution and boundary condition.It can be assumed that φ1Form be
φ1=p (y) cos (β1x) (24)
Formula (24) is substituted into equation (23b), obtained
Thus p (y) must is fulfilled for
This ODE is solved, the general solution for obtaining p (y) is
P (y)=k1cosh(β1y)+k2sinh(β1y)+k3ycosh(β1y)+k4ysinh(β1y) (27)
K in formula1、k2、k3、k4It is unknown constant, will be determined by boundary condition.
In view of the symmetry of former problem, can remove the odd function item in expression formula (27), obtain
φ1=[k1cosh(β1y)+k4ysinh(β1y)]cos(β1x) (28)
It can equally obtain
φ2=[l1cosh(γ1x)+l4ysinh(γ1x)]cos(γ1y) (29)
L in formula1、l4It is unknown constant, will be determined by boundary condition.
According to the characteristics of Temperature Distribution, it may be assumed that particular solution φ3Form be
And the unity of form of temperature field expression formula is written as
For Laser Focusing Gaussian beam, Kmn(z) can be expressed from the next：
Formula (30) and formula (31a) are substituted into formula (23d), obtained
It can thus obtain
In summary, thermal stress function phi can be written as
Above formula (34) is substituted into boundary condition (10b), with operatorAct on formula (10b) both sides, and profit
With the orthogonality of characteristic function, two linear algebraic equations can be obtained, i.e.,
In formula,
In the same way, above formula (34) is substituted into boundary condition (10c), with operatorAct on formula
The both sides of (10c), and using the orthogonality of characteristic function, two linear algebraic equations can be obtained, i.e.,：
Solve the linear algebraic equation systems being made up of formula (35a), (35b), (36a), (36b), it may be determined that 4 unknown ginsengs
Number k1、k4、l1、l4, the analytic solutions for finally obtaining each component of thermal stress are：
In the present embodiment, by the way that formula (37) is updated in formula (8), also above formula (37a), (37b) and (37c) is substituted into
To formula (8), the analytic solutions of each component of thermal strain can be derived.
It should be pointed out that the formula above in the case where focal beam spot is with Gaussian form on orthotropic media is similarly suitable
For isotropic medium, parameter k therein only need to be madex=k, αx=αy.
Below by taking optical fiber laser cutting sapphire hard brittle material as an example, it is cut sapphire and is absorbing Gauss focusing light
Its temperature field, stress field and strain field are by the analytical expression that derives above after spot thermal source, with Matlab advanced procedures
Language is simulated to it.
Sapphire for numerical analysis herein is orthotropic material, the thermal conductivity factor point in its three directions
Amount is respectively kx=25.2W/ (mk), ky=23.1W/ (mk), kz=23.1W/ (mk), thermal coefficient of expansion component are respectively αx=
6.66×10-6K-1、αy=5 × 10-6K-1、αz=5 × 10-6K-1, the absorption coefficient at 1.064 μm is α=90ppm/cm, poplar
Family name's modulus E=3.45 × 1011, Poisson's ratio ν=0.27.It is assumed that sapphire size is 2a x 2b x h=3mm x 2mm x
6mm, focused spot size ωa=40 μm, Q=20W, hx=hy=3.5 × 105W/(m2K)。
Temperature when Fig. 4 is the sapphire plate material of Gaussian beam cutting anisotropic on laser entrance face (z=0 planes)
Degree distribution, maximum temperaturerise θmaxThe central spot (x=y=0) in the laser entrance face of sapphire flat board, θ occursmaxFor
102.78K, this shows, for given total heat duties Q, Gaussian beam has highest peak light intensity, thus sapphire flat-sheet material
Material has highest local heat source production rate q at the center of laser entrance face (x=y=0) placeV(x,y,z)(W/m3), and herein
Place causes highest θmax。
The thermal stress analytical expression (37) derived more than, when can draw Gaussian beam cutting sapphire sheet material
The distribution situation of each thermal stress component, as shown in Fig. 5~7, respectively illustrate the thermal stress σ of sapphire flat boardxx、σyyAnd σxy
The thermal stress σ of distribution map and sapphire flat boardxx、σyyAnd σxy(negative number representation compression, positive number are represented to draw and answered distribution of contours figure
Power).Maximum pressure thermal stress occurs at point x=y=z=0, respectively σxx=-101.79MPa, σyy=-92.68MPa, σxy
=-26.44MPa.Maximum draws thermal stress to occur on sapphire plate surface, σxx=32.35MPa (y=0), σyy=
31.46MPa (x=0), σxy=8.43MPa (x=0 or y=0), thermal stress σxx、σyyAnd σxyDifference be mainly sapphire sheet material
Caused by the anisotropy of material.
The thermal strain distribution of cutting material is to influence a key factor of Quality of Laser Cutting, and Fig. 8~10 are shown respectively
The thermal strain ε of sapphire flat board during Gaussian beam cutting sapphire sheet materialxx、εyyAnd εxyDistribution map and sapphire are put down
The thermal strain ε of platexx、εyyAnd εxyDistribution of contours figure (negative number representation presses thermal strain, and positive number represents to draw thermal strain).Maximum draws heat
Strain occurs at point x=y=z, respectively εxx=-186.21 μm, εyy=-123.65 μm, εxy=-171.83 μm.Maximum is drawn
Thermal strain occurs on sapphire plate surface, εxx=66.34 μm (y=0), εyy=69.37 μm (x=0), εxy=28.58 μm
(x=0 or y=0), thermal strain σxx、σyyAnd σxyDifference be mainly caused by the anisotropy of sapphire sheet material.
Temperature, the thermal stress in three-dimensional orthogonal anisotropy plate material are cut by Gaussian beam focusing disclosed above
With the analytic solutions of thermal strain field, can obtain as drawn a conclusion：First, pressure heat is born in plate material inside by paracentral region should
Power, and bear to draw thermal stress close to the region on surface；Second, under without external force and constraints, thermal coefficient of expansion it is orthogonal each
Anisotropy has a significant impact to the thermal strain component in plate material, and the influence to thermal stress component is then smaller.
The embodiment of the present application also discloses a kind of laser cutting hard brittle material fuel factor analysis model creation method, the wound
Construction method is that the flat board prepared by hard brittle material is carried out actually cutting the method completed before using laser, referring to Figure 11 institutes
Show, the creation method includes：
Step S21：It is determined that the equation of heat conduction corresponding to laser cutting flat board process；Wherein, laser is circle symmetrical Gaussian
The laser of type, hard brittle material are orthotropic material；
Step S22：Using the boundary condition and the equation of heat conduction of flat board, determine flat board when cut inside it
Three-dimensional temperature field expression formula；
Step S23：Based on three-dimensional temperature field expression formula, using Airy thermal stress function methods, determine that flat board is being cut
When corresponding fuel factor analysis model；Wherein, fuel factor analysis model includes thermal stress analytic solutions and thermal strain analytic solutions.
Wherein, above-mentioned thermal stress analytic solutions, it is specially：
In addition, in above-mentioned steps S23, the process of flat board corresponding thermal strain analytic solutions when being cut is determined, including：
Thermal stress analytic solutions are inputted to the relational expression between default thermal strain and thermal stress, obtain thermal strain parsing
Solution；
Wherein, the relational expression between thermal strain and thermal stress, it is specially：
The phase disclosed in previous embodiment is may be referred on each more specifical process of step in above-mentioned creation method
Content is answered, is no longer repeated herein.
Accordingly, the embodiment of the present application also discloses a kind of hard brittle material laser cutting accessory system, the accessory system
The course of work be that the flat board that is prepared by hard brittle material is carried out using laser actually to cut the process completed before；Referring to figure
Shown in 12, above-mentioned accessory system includes：
Equation of heat conduction determining module 11, for determining the equation of heat conduction corresponding to laser cutting flat board process；Wherein,
Laser is the laser of circle symmetrical Gaussian type, and hard brittle material is orthotropic material；
Temperature field expression formula determining module 12, for the boundary condition and the equation of heat conduction using flat board, determine to put down
Three-dimensional temperature field expression formula of the plate when being cut inside it；
Fuel factor analysis model determining module 13, for based on three-dimensional temperature field expression formula, utilizing Airy thermal stress functions
Method, determine flat board corresponding fuel factor analysis model when cut；Wherein, fuel factor analysis model parses including thermal stress
Solution and thermal strain analytic solutions；
Fuel factor distribution map drafting module 14, for drawing fuel factor distribution map corresponding with fuel factor analysis model, and
Fuel factor distribution map is sent to default cutting auxiliary terminal display screen and shown；Wherein, fuel factor distribution map includes heat
Stress envelope and thermal strain distribution map.
On the more specifical course of work of above-mentioned modules, may be referred to corresponding interior disclosed in previous embodiment
Hold, no longer repeated herein.
Accordingly, the embodiment of the present application also discloses a kind of laser cutting hard brittle material fuel factor analysis model and creates system
System, the course of work of the establishment system are that the flat board prepared by hard brittle material is carried out using laser to complete before actual cutting
Process；Shown in Figure 13, above-mentioned establishment system includes：
Equation of heat conduction determining module 21, for determining the equation of heat conduction corresponding to laser cutting flat board process；Wherein,
Laser is the laser of circle symmetrical Gaussian type, and hard brittle material is orthotropic material；
Temperature field expression formula determining module 22, for the boundary condition and the equation of heat conduction using flat board, determine to put down
Three-dimensional temperature field expression formula of the plate when being cut inside it；
Fuel factor analysis model determining module 23, for based on three-dimensional temperature field expression formula, utilizing Airy thermal stress functions
Method, determine flat board corresponding fuel factor analysis model when cut；Wherein, fuel factor analysis model parses including thermal stress
Solution and thermal strain analytic solutions.
On the more specifical course of work of above-mentioned modules, may be referred to corresponding interior disclosed in previous embodiment
Hold, no longer repeated herein.
Finally, it is to be noted that, herein, such as first and second or the like relational terms be used merely to by
One entity or operation make a distinction with another entity or operation, and not necessarily require or imply these entities or operation
Between any this actual relation or order be present.Moreover, term " comprising ", "comprising" or its any other variant meaning
Covering including for nonexcludability, so that process, method, article or equipment including a series of elements not only include that
A little key elements, but also the other element including being not expressly set out, or also include for this process, method, article or
The intrinsic key element of equipment.In the absence of more restrictions, the key element limited by sentence "including a ...", is not arranged
Except other identical element in the process including the key element, method, article or equipment being also present.
Householder method, model creation method and system are cut by laser to a kind of hard brittle material provided by the present invention above
It is described in detail, specific case used herein is set forth to the principle and embodiment of the present invention, and the above is real
The explanation for applying example is only intended to help the method and its core concept for understanding the present invention；Meanwhile for the general technology of this area
Personnel, according to the thought of the present invention, there will be changes in specific embodiments and applications, in summary, this theory
Bright book content should not be construed as limiting the invention.
Claims (10)
1. a kind of hard brittle material is cut by laser householder method, it is characterised in that the householder method is to by hard using laser
Flat board prepared by fragile material carries out actually cutting the method completed before, including：
Determine the equation of heat conduction corresponding to the laser cutting flat board process；Wherein, the laser is circle symmetrical Gaussian
The laser of type, the hard brittle material are orthotropic material；
Boundary condition and the equation of heat conduction using the flat board, determine the flat board when cut inside it
Three-dimensional temperature field expression formula；
Based on the three-dimensional temperature field expression formula, using Airy thermal stress function methods, determine the flat board in cut phase
The fuel factor analysis model answered；Wherein, the fuel factor analysis model includes thermal stress analytic solutions and thermal strain analytic solutions；
Fuel factor distribution map corresponding with the fuel factor analysis model is drawn, and the fuel factor distribution map is sent to default
Cutting auxiliary terminal display screen shown；Wherein, the fuel factor distribution map includes thermal stress distribution figure and thermal strain point
Butut.
2. hard brittle material according to claim 1 is cut by laser householder method, it is characterised in that the heat transfer side
Journey, it is specially：
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In formula, θ=T-TaRepresent plate material temperature T-phase for air themperature TaTemperature rise；kx、kyAnd kzIt is respectively described hard
The thermal conductivity factor of fragile material along x-axis on the flat board, y-axis, three directions of z-axis component；qV(x, y, z) is described hard crisp
The thermal source distribution function of property material.
3. hard brittle material according to claim 2 is cut by laser householder method, it is characterised in that the three-dimensional temperature field
Expression formula, it is specially：
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4. the hard brittle material laser cutting householder method according to any one of claims 1 to 3, it is characterised in that described
Thermal stress analytic solutions, it is specially：
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<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msub>
<mi>&beta;</mi>
<mi>m</mi>
</msub>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<msub>
<mi>R</mi>
<mrow>
<mi>m</mi>
<mi>n</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mi>m</mi>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>.</mo>
</mrow>
5. hard brittle material according to claim 4 is cut by laser householder method, it is characterised in that determines the flat board
The process of corresponding thermal strain analytic solutions when being cut, including：
The thermal stress analytic solutions are inputted to the relational expression between default thermal strain and thermal stress, obtain the thermal strain solution
Analysis solution；
Wherein, the relational expression between the thermal strain and thermal stress, it is specially：
<mrow>
<msub>
<mi>&epsiv;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>-</mo>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>E</mi>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>-</mo>
<mfrac>
<mi>v</mi>
<mrow>
<mn>1</mn>
<mo>-</mo>
<mi>v</mi>
</mrow>
</mfrac>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>v</mi>
<mo>)</mo>
</mrow>
<msub>
<mi>&alpha;</mi>
<mi>x</mi>
</msub>
<mi>&theta;</mi>
<mo>;</mo>
</mrow>
<mrow>
<msub>
<mi>&epsiv;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>-</mo>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>E</mi>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>-</mo>
<mfrac>
<mi>v</mi>
<mrow>
<mn>1</mn>
<mo>-</mo>
<mi>v</mi>
</mrow>
</mfrac>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>v</mi>
<mo>)</mo>
</mrow>
<msub>
<mi>&alpha;</mi>
<mi>y</mi>
</msub>
<mi>&theta;</mi>
<mo>;</mo>
</mrow>
<mrow>
<msub>
<mi>&epsiv;</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>v</mi>
</mrow>
<mi>E</mi>
</mfrac>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>.</mo>
</mrow>
6. one kind laser cutting hard brittle material fuel factor analysis model creation method, it is characterised in that the creation method is
The flat board prepared by hard brittle material is carried out actually cutting the method completed before using laser, including：
Determine the equation of heat conduction corresponding to the laser cutting flat board process；Wherein, the laser is circle symmetrical Gaussian
The laser of type, the hard brittle material are orthotropic material；
Boundary condition and the equation of heat conduction using the flat board, determine the flat board when cut inside it
Three-dimensional temperature field expression formula；
Based on the three-dimensional temperature field expression formula, using Airy thermal stress function methods, determine the flat board in cut phase
The fuel factor analysis model answered；Wherein, the fuel factor analysis model includes thermal stress analytic solutions and thermal strain analytic solutions.
7. laser cutting hard brittle material fuel factor analysis model creation method according to claim 6, it is characterised in that
The thermal stress analytic solutions, it is specially：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mi>&phi;</mi>
</mrow>
<mrow>
<mo>&part;</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mo>&lsqb;</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<mi>cosh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>k</mi>
<mn>4</mn>
</msub>
<mi>y</mi>
<mi> </mi>
<mi>sinh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<msup>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mn>2</mn>
</msup>
<mi>cos</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>+</mo>
<mo>{</mo>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
<msubsup>
<mi>&gamma;</mi>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mi>cosh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>l</mi>
<mn>4</mn>
</msub>
<mo>&lsqb;</mo>
<mn>2</mn>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>cosh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msubsup>
<mi>&gamma;</mi>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mi>x</mi>
<mi> </mi>
<mi>sinh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>}</mo>
<mi>cos</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>&beta;</mi>
<mi>m</mi>
<mn>2</mn>
</msubsup>
<msub>
<mi>R</mi>
<mrow>
<mi>m</mi>
<mi>n</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mi>m</mi>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>;</mo>
</mrow>
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mi>&phi;</mi>
</mrow>
<mrow>
<mo>&part;</mo>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>=</mo>
<mo>{</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<msubsup>
<mi>&beta;</mi>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mi>cosh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>k</mi>
<mn>4</mn>
</msub>
<mo>&lsqb;</mo>
<mn>2</mn>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>cosh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msubsup>
<mi>&beta;</mi>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mi>y</mi>
<mi> </mi>
<mi>sinh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>}</mo>
<mi>cos</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mo>&lsqb;</mo>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
<mi>cosh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>l</mi>
<mn>4</mn>
</msub>
<mi>x</mi>
<mi> </mi>
<mi>sinh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<msubsup>
<mi>&gamma;</mi>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mi>cos</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>&gamma;</mi>
<mi>n</mi>
<mn>2</mn>
</msubsup>
<msub>
<mi>R</mi>
<mrow>
<mi>m</mi>
<mi>n</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mi>m</mi>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>;</mo>
</mrow>
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mi>&phi;</mi>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
<mo>&part;</mo>
<mi>y</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>{</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>sinh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>k</mi>
<mn>4</mn>
</msub>
<mo>&lsqb;</mo>
<mi>sinh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mi> </mi>
<mi>cosh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>}</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>+</mo>
<mo>{</mo>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>sinh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>l</mi>
<mn>4</mn>
</msub>
<mo>&lsqb;</mo>
<mi>sinh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mi> </mi>
<mi>cosh</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>}</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msub>
<mi>&beta;</mi>
<mi>m</mi>
</msub>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<msub>
<mi>R</mi>
<mrow>
<mi>m</mi>
<mi>n</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mi>m</mi>
</msub>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>.</mo>
</mrow>
8. laser cutting hard brittle material fuel factor analysis model creation method according to claim 7, it is characterised in that
The process of the flat board corresponding thermal strain analytic solutions when being cut is determined, including：
The thermal stress analytic solutions are inputted to the relational expression between default thermal strain and thermal stress, obtain the thermal strain solution
Analysis solution；
Wherein, the relational expression between the thermal strain and thermal stress, it is specially：
<mrow>
<msub>
<mi>&epsiv;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>-</mo>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>E</mi>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>-</mo>
<mfrac>
<mi>v</mi>
<mrow>
<mn>1</mn>
<mo>-</mo>
<mi>v</mi>
</mrow>
</mfrac>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>v</mi>
<mo>)</mo>
</mrow>
<msub>
<mi>&alpha;</mi>
<mi>x</mi>
</msub>
<mi>&theta;</mi>
<mo>;</mo>
</mrow>
<mrow>
<msub>
<mi>&epsiv;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>-</mo>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>E</mi>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>-</mo>
<mfrac>
<mi>v</mi>
<mrow>
<mn>1</mn>
<mo>-</mo>
<mi>v</mi>
</mrow>
</mfrac>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>v</mi>
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9. a kind of hard brittle material is cut by laser accessory system, it is characterised in that the course of work of the accessory system is utilization
Laser carries out actually cutting the process completed before to the flat board prepared by hard brittle material；The accessory system includes：
Equation of heat conduction determining module, for determining the equation of heat conduction corresponding to the laser cutting flat board process；Its
In, the laser is the laser of circle symmetrical Gaussian type, and the hard brittle material is orthotropic material；
Temperature field expression formula determining module, for the boundary condition using the flat board and the equation of heat conduction, determine
Three-dimensional temperature field expression formula of the flat board when being cut inside it；
Fuel factor analysis model determining module, for based on the three-dimensional temperature field expression formula, using Airy thermal stress function methods,
Determine the flat board corresponding fuel factor analysis model when cut；Wherein, the fuel factor analysis model includes hot answer
Power analytic solutions and thermal strain analytic solutions；
Fuel factor distribution map drafting module, for drawing fuel factor distribution map corresponding with the fuel factor analysis model, and will
The fuel factor distribution map sends to default cutting auxiliary terminal display screen and shown；Wherein, the fuel factor distribution map
Including thermal stress distribution figure and thermal strain distribution map.
10. one kind laser cutting hard brittle material fuel factor analysis model creates system, it is characterised in that the establishment system
The course of work is that the flat board prepared by hard brittle material is carried out actually cutting the process completed before using laser；The establishment
System includes：
Equation of heat conduction determining module, for determining the equation of heat conduction corresponding to the laser cutting flat board process；Its
In, the laser is the laser of circle symmetrical Gaussian type, and the hard brittle material is orthotropic material；
Temperature field expression formula determining module, for the boundary condition using the flat board and the equation of heat conduction, determine
Three-dimensional temperature field expression formula of the flat board when being cut inside it；
Fuel factor analysis model determining module, for based on the three-dimensional temperature field expression formula, using Airy thermal stress function methods,
Determine the flat board corresponding fuel factor analysis model when cut；Wherein, the fuel factor analysis model includes hot answer
Power analytic solutions and thermal strain analytic solutions.
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