CN104504173B - Couple the titanium alloy press-in connection interface bonding ratio Forecasting Methodology of crystallite dimension - Google Patents
Couple the titanium alloy press-in connection interface bonding ratio Forecasting Methodology of crystallite dimension Download PDFInfo
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Abstract
The invention discloses a kind of titanium alloy press-in connection interface bonding ratio Forecasting Methodology for coupling crystallite dimension, the technical problem of various grain sizes titanium alloy interface bonding ratio prediction can not be realized for solving existing method.Technical scheme is empty geometry simplified first；Determine material and Joining Technology parameter, empty form parameter, crystallite dimension and time interval；According to creep machine braking mechanics condition, based on the plastic deformation and steady state creep constitutive relation of coupling crystallite dimension, using lamella split plot design, the interface connecting length increment in time interval under plastic deformation and creep machining function is calculated respectively；According to surface source brake force condition, the interface connecting length increment under time interval inner boundary source and surface source machining function is calculated respectively；Total interface connecting length increment is calculated by quadravalence Runge Kutta alternative manners；Calculate interface bonding ratio.Realize the prediction of interface bonding ratio of the various grain sizes titanium alloy under different Joining Technology parameters.
Description
Technical Field
The invention relates to a prediction method of titanium alloy pressure connection interface connection rate, in particular to a prediction method of titanium alloy pressure connection interface connection rate coupled with grain size.
Background
The pressure connection is one of ideal methods suitable for titanium alloy connection, a connection joint which is close to or even consistent with the microstructure and the mechanical property of a titanium alloy matrix can be obtained by the pressure connection method, and the pressure connection method is suitable for connection of large sections and complex internal structures and is widely used for manufacturing light-weight components for aerospace, such as hollow blades and the like.
The connecting surface of the titanium alloy after precision processing is microscopically uneven, and a microscopic cavity is formed on the connecting interface in the pressure connecting process. The premise and key to forming a high quality metallurgical joint is to achieve closure of microscopic cavities and thus obtain high interface connectivity. In actual production, the cavity closure of the titanium alloy pressure connection interface and the regulation and control of the interface connection rate generally depend on tests and experiences, a large amount of manpower and material resources are consumed, the production period is prolonged, and the cost is increased.
In recent years, many scholars propose various interface connection rate prediction methods based on deep knowledge of the cavity closing process, so that the production efficiency is improved, and the cost is saved.
Document 1 "b.derby, e.r.wallach, the Theoretical model for differentiation binding, Metal Science, 1982, 16: 49-56 discloses an interface connection rate prediction method based on a plastic deformation mechanism, a creep mechanism, a surface source mechanism and an interface source mechanism, but only the influence of different process parameters on the interface connection rate is considered, and the influence of the grain size of a material on the interface connection rate is not considered.
Document 2 "maruifang, li\28156;, spring, li hong, weixin, void closure model based on metal diffusion bonder braking mechanical conditions, chinese science: technical science, 2012, 42 (9): 1081-. While grain size has a significant effect on the interfacial connectivity.
Document 3 "research on poplar courage, periplon, chengqing, marhongjun, korean modesty, lisseng, superplastic diffusion bonding of fine-grained TC21 alloy, research on manufacturing technology, 2009, 3: 8-13 "discloses that TC21 alloys with grain sizes of 2 μm, 4 μm, and 7 μm have interfacial connectivity of 99.5%, 91.8%, and 88.7%, respectively, for given joining process parameters. The existing prediction method cannot predict the interface connection rate of titanium alloys with different grain sizes, so that the application range of the titanium alloy has certain limitation.
Disclosure of Invention
In order to overcome the defect that the prediction of the interface connection rate of titanium alloys with different grain sizes cannot be realized by the conventional method, the invention provides a prediction method of the interface connection rate of the pressure connection of the titanium alloy with coupled grain sizes. The method firstly simplifies the geometrical shape of the cavity; determining material and connection process parameters, cavity shape parameters, grain size and time interval; according to the dynamic conditions of the plastic deformation and the creep mechanism, based on the plastic deformation and the steady-state creep constitutive relation of the coupled grain size, calculating the increment of the length of the interface connection under the action of the plastic deformation and the creep mechanism in a time interval by adopting a lamella segmentation method; respectively calculating interface connection length increment under the action of an interface source mechanism and a surface source mechanism in a time interval according to the interface source and surface source mechanism dynamic conditions; calculating the total interface connection length increment by a fourth-order Runge-Kutta iteration method; and calculating the interface connection rate. Compared with the prior art, the method can realize the accurate prediction of the interface connection rate of the titanium alloy with different grain sizes under different connection process parameters.
The technical scheme adopted by the invention for solving the technical problems is as follows: a prediction method of the connection rate of a titanium alloy pressure connection interface coupled with the grain size is characterized by comprising the following steps:
step one, simplifying the shape of an initial cavity into an ellipse;
step two, selecting TC4 alloys with different grain sizes for pressure connection, wherein the grain sizes are respectively 8.2 mu m, 9.8 mu m and 12.5 mu mm and 16.4 μm, and the connecting process parameters are as follows: the connection temperature is 850 ℃, the connection pressure is 30MPa, and the connection time is 10 min. The roughness of the connecting surface is R after the 1000# sand paper is adopted for polishinga＝0.28μm，RλqInitial void shape parameter h of 5.40 μm0＝2Ra＝0.76μm，c0＝Rλq2.70 μm. t' is 0s and the time interval t is 1 s.
Thirdly, calculating the interface connection length increment e under the action of the plastic deformation mechanism in the time interval t by adopting a lamella segmentation method based on the plastic deformation constitutive relation of the coupled crystal grain size and the action kinetic condition of the plastic deformation mechanism1The specific calculation method is as follows:
assuming that the deformation at the connecting interface is plane strain, obtaining the equivalent strain rateEquivalent stressThe following plastic deformation constitutive relation coupling the grain sizes is adopted to represent the relation between equivalent stress and equivalent strain rate of titanium alloys with different grain sizes in the pressure connection process:
in the formula: tau is the shear stress of the titanium alloy, unit MPa; tau isαAnd τβShear stress in MPa of α and β phases, respectivelyα、fβVolume fractions, f, of α and β phases, respectivelyα+fβ＝1；n1、n2Is a correction factor;andthe critical stresses of the α and β phases respectively,the unit MPa; t is the junction temperature in K; r is a gas constant with the unit of 8.3145 J.mol-1·K-1；△GαAnd Δ GβThe apparent deformation activation energy of α and β phases is expressed in kJ & mol-1；For shear strain rate, in units of s-1；Andshear strain rate in s for the α and β phases at 0K-1(ii) a μ is the temperature-dependent shear modulus in GPa; ρ is the dislocation density in cm-2；Is the rate of change of dislocation density in cm-2·s-1(ii) a b is the Berth vector in m; d is the grain size in μm;is the rate of change of grain size in μm · s-1；Is the equivalent strain rate, in units of s-1；σeEquivalent stress in MPa; qdmThe unit is 20 kJ.mol for dislocation motion activation energy-1；QpdThe unit for activation energy of grain boundary diffusion is 677.37kJ & mol-1(ii) a M is a Taylor factor; q. q.sα、pα、qβ、pβ、a、k、s、μ0、Tr、α1、α2、β0、β1、β2、γ0、γ1、γ2Is a material parameter.
Obtaining TC4 alloy at different connection process parameters and crystals according to the plastic deformation constitutive relationFlow stress at grain size. The flow stress at 0.2% residual deformation was taken as the yield strength σ of the TC4 alloy at the joining process parameters and grain sizeyield. According to the slip line field theory and the Mises yield criterion, the stress distribution of the connecting interface along the neck of the cavity is as follows:
in the formula, rAThe radius of curvature of the neck of the cavity is given in μm.
The average contact stress of the connection interface in a yield state is as follows:
the contact stress of the connection interface under the connection pressure is as follows:
wherein p is the connection pressure in MPa; gamma is surface energy, unit J.m-2. When in useWhen the plastic deformation mechanism is continuously acted; when inWhen the stress state does not satisfy the yield condition, the plastic deformation mechanism stops acting.
When the plastic deformation machine is manufactured, the convex peak between the cavities is divided into N sheet layers parallel to the connecting interface, and the length of the ith sheet layer isStrain rate in the x-axis directionThe z-direction stress of the ith sheet layer is approximately as follows:
let the length of the ith slice at time t' be wi(t'), calculating the length of the ith lamella after one time step t deformation according to the plastic deformation constitutive relation of the coupled crystal grain size
Obtaining the thickness h of the ith lamella at the moment t' + t according to the principle of volume conservation before and after lamella deformationi(t'+t)＝wi(t')·hi(t')/wi(t'+t)。
Finally, the total thickness of the deformed convex peaks in the time interval t is obtained through superposition, namely the height of the interface cavity under the action of a plastic deformation mechanismObtaining the interface connection length increment under the action of the plastic deformation mechanism in the time interval t according to the principle that the volume conservation is met before and after the plastic deformation
Step four, calculating the interface connection length increment e under the action of the creep mechanism in the time interval t by adopting a lamella segmentation method based on the steady-state creep constitutive relation of the coupled grain size and the action dynamic condition of the creep mechanism2The specific calculation method is as follows:
when contact stress of the connection interfaceWhen the plastic deformation mechanism is stopped, the creep mechanism is openedThe effect is started. The steady state creep constitutive relation for the coupled grain sizes is as follows:
in the formula,is the true strain rate, in units of s-1(ii) a Sigma is the true stress, unit MPa; d is the diffusion coefficient in m2·s-1K is Boltzmann's constant, 1.38 × 10-23(ii) a b is the Berth vector in m; μ is shear modulus, in MPa; A. p and n are material parameters.
Dividing the material near the connecting interface into N parallel sheets, each sheet being stressed by
And (3) adopting deformation of the sheet layer in the thickness direction to represent the strain:
according to creep constitutive relation and equivalent strain rateAnd equivalent stress sigmaeThe deformation expression for the ith slice is obtained:obtaining:
the rate of change of the cavity height under the action of the creep mechanism is:
according to the principle that the volume of the material is unchanged, the change rate of the connection length of the connection interface under the action of a creep mechanism is as follows:
the length of the interface connection increases under the action of the creep mechanism within the time interval t
Step five, calculating the interface connection length increment e under the action of the interface source mechanism in the time interval t based on the action kinetic condition of the interface source mechanism3；
Obtaining the change rate of the interface connection length under the action of the interface source mechanism according to the dynamic condition and the calculation method of the interface source mechanism actionThe length increment of the interface connection under the action of the interface source mechanism within the time interval t
Step six, calculating the interface connection length increment e under the action of the surface source mechanism in the time interval t based on the surface source mechanism action dynamics condition4；
Kinetic conditions and computational methods based on surface source mechanism actionObtaining the interface connection length change rate under the action of the surface source mechanismThe interface connection length under the action of the surface source mechanism is increased in the time interval t
Step seven, overlapping the interface connection length increment under the action of each connection mechanism by a fourth-order Runge-Kutta iteration method to obtain a cavity shape parameter at the end of the t' + t moment and a total interface connection length increment e under the action of each mechanism;
step eight, if t' + t<t, taking the cavity shape parameter at the end of the time t '+ t as a state parameter, changing t' + t, and repeating the third step to the seventh step to obtain an interface connection length increment E under the set connection process parameter and the set grain size; otherwise, the total interface connection length increment at the end of the t' + t moment is the interface connection length increment E under the set connection process parameters and the grain size, and the interface connection rate Af＝E/c0。
The invention has the beneficial effects that: the method firstly simplifies the geometrical shape of the cavity; determining material and connection process parameters, cavity shape parameters, grain size and time interval; according to the dynamic conditions of the plastic deformation and the creep mechanism, based on the plastic deformation and the steady-state creep constitutive relation of the coupled grain size, calculating the increment of the length of the interface connection under the action of the plastic deformation and the creep mechanism in a time interval by adopting a lamella segmentation method; respectively calculating interface connection length increment under the action of an interface source mechanism and a surface source mechanism in a time interval according to the interface source and surface source mechanism dynamic conditions; calculating the total interface connection length increment by a fourth-order Runge-Kutta iteration method; and calculating the interface connection rate. Compared with the prior art, the method realizes the accurate prediction of the interface connection rate of the titanium alloy with different grain sizes under different connection process parameters.
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
Drawings
FIG. 1 is a flow chart of the method for predicting the connectivity of the pressure connection interface of the titanium alloy coupled with the grain size according to the invention.
FIG. 2 shows the initial void shape of the connection interface of the method of the present invention, wherein h0Half the height of the cavity, c0Half the width of the cavity, RaFor surface profile arithmetic mean square error, RλqIs the surface profile root mean square wavelength.
FIG. 3 is a schematic diagram of the calculation of plastic deformation and creep mechanism by the sheet segmentation method according to the method of the present invention. Wherein z isiIs the distance of the ith sheet from the bonding interface, wiIs the width of the ith slice, hiIs the thickness (h) of the ith sheeti＝h/N)。
Detailed Description
Reference is made to fig. 1-3. The prediction method of the titanium alloy pressure connection interface connection rate coupled with the grain size comprises the following specific steps:
step one, simplifying the shape of an initial cavity into an ellipse;
and step two, selecting TC4 alloys with different grain sizes for pressure connection, wherein the grain sizes are respectively 8.2 μm, 9.8 μm, 12.5 μm and 16.4 μm, and the material parameters required by calculation are shown in Table 1. The selected connection process parameters are as follows: the connection temperature is 850 ℃, the connection pressure is 30MPa, and the connection time is 10 min. The roughness of the connecting surface is R after the 1000# sand paper is adopted for polishinga＝0.28μm，RλqInitial void shape parameter h of 5.40 μm0＝2Ra＝0.76μm，c0＝Rλq2.70 μm. t' is 0s and the time interval t is 1 s.
TABLE 1 Material parameters for TC4 alloys
Thirdly, calculating the interface connection length increment e under the action of the plastic deformation mechanism in the time interval t by adopting a lamella segmentation method based on the plastic deformation constitutive relation of the coupled crystal grain size and the action kinetic condition of the plastic deformation mechanism1The specific calculation method is as follows:
assuming that the deformation at the connecting interface is plane strain, obtaining the equivalent strain rateEquivalent stressThe following plastic deformation constitutive relation coupling the grain sizes is adopted to represent the relation between equivalent stress and equivalent strain rate of titanium alloys with different grain sizes in the pressure connection process:
in the formula: τ is the shear stress (MPa) of the titanium alloy; tau isαAnd τβShear stress (MPa) of α and β phases, respectivelyα、fβVolume fractions (f) of α and β phases, respectivelyα+fβ＝1)；n1、n2Is a correction factor;andcritical stress (MPa) of α and β phases respectively, T is connection temperature (K), R is gas constant (8.3145J. mol)-1·K-1)；△GαAnd Δ GβIs apparent change of α and β phasesShape activation energy (kJ. mol)-1)；Is the shear strain rate(s)-1)；Andshear strain rate(s) of α and β phases at 0K-1) (ii) a μ is a temperature-dependent shear modulus (GPa); ρ is the dislocation density (cm)-2)；Is dislocation density change rate (cm)-2·s-1) (ii) a b is a Berth vector (m); d is the grain size (μm);is the rate of change of crystal grain size (. mu.m.s)-1)；Is equivalent strain rate(s)-1)；σeEquivalent stress (MPa); qdmActivation energy for dislocation motion (20 kJ. mol)-1)；QpdActivation energy for grain boundary diffusion (677.37 kJ. mol)-1) (ii) a M is a Taylor factor; q. q.sα、pα、qβ、pβ、a、k、s、μ0、Tr、α1、α2、β0、β1、β2、γ0、γ1、γ2Is a material parameter. The TC4 alloy plastic constitutive relation material parameters determined by the method of solving the minimum value of the objective function by the optimization technology in the embodiment are shown in table 2.
TABLE 2 Material parameters of TC4 alloy plasticity constitutive relation
And obtaining the flow stress of the TC4 alloy under different connection process parameters and grain sizes according to the plastic deformation constitutive relation. Taking the flow stress under 0.2 percent of residual deformation as the yield strength sigma of the TC4 alloy under certain connection process parameters and grain sizesyield. According to the slip line field theory and the Mises yield criterion, the stress distribution of the connecting interface along the neck of the cavity is as follows:
in the formula, rAThe radius of curvature (μm) of the neck of the cavity.
The average contact stress of the connection interface in a yield state is as follows:
the contact stress of the connection interface under the connection pressure is as follows:
wherein p is a joining pressure (MPa); gamma is surface energy (J.m)-2). When in useWhen the plastic deformation mechanism is continuously acted; when inWhen the stress state does not satisfy the yield condition, the plastic deformation mechanism stops acting.
When the plastic deformation machine is manufactured, the convex peak between the cavities is divided into N sheet layers parallel to the connecting interface, and the length of the ith sheet layer isStrain rate in the x-axis directionThe z-direction stress of the ith sheet layer is approximately as follows:
let the length of the ith slice at time t' be wi(t'), calculating the length of the ith lamella after one time step t deformation according to the plastic deformation constitutive relation of the coupled crystal grain size
Obtaining the thickness h of the ith lamella at the moment t' + t according to the principle of volume conservation before and after lamella deformationi(t'+t)＝wi(t')·hi(t')/wi(t'+t)。
Finally, the total thickness of the deformed convex peaks in the time interval t is obtained through superposition, namely the height of the interface cavity under the action of a plastic deformation mechanismObtaining the interface connection length increment under the action of the plastic deformation mechanism in the time interval t according to the principle that the volume conservation is met before and after the plastic deformation
Step four, calculating the creep mechanism action in the time interval t by adopting a lamella segmentation method based on the steady-state creep constitutive relation of the coupled grain size and the creep mechanism action dynamics conditionsLower interface connection length increment e2The specific calculation method is as follows:
when contact stress of the connection interfaceAt this time, the plastic deformation mechanism stops and the creep mechanism starts to act. The steady state creep constitutive relation for the coupled grain sizes is as follows:
in the formula,is the true strain rate(s)-1) (ii) a σ is the true stress (MPa); d is the diffusion coefficient (m)2·s-1) K is Boltzmann's constant (1.38 × 10)-23) (ii) a b is a Berth vector (m); μ is shear modulus (MPa); A. p and n are material parameters. The material parameters of the creep constitutive relation of the TC4 alloy determined in the present embodiment are shown in table 3.
TABLE 3 Material parameters of TC4 alloy creep constitutive equation
| Symbol | Unit of | Numerical value |
| A | — | 6.80 |
| D | m2·s-1 | 1.477×10-11 |
| p | — | 2 |
| n | — | 2 |
Dividing the material near the connecting interface into N parallel sheets, each sheet being stressed by
And (3) adopting deformation of the sheet layer in the thickness direction to represent the strain:
according to creep constitutive relation and equivalent strain rateAnd equivalent stress sigmaeThe deformation expression for the ith slice is obtained:obtaining:
the rate of change of the cavity height under the action of the creep mechanism is:
according to the principle that the volume of the material is unchanged, the change rate of the connection length of the connection interface under the action of a creep mechanism is as follows:
the length of the interface connection increases under the action of the creep mechanism within the time interval t
Step five, calculating the interface connection length increment e under the action of the interface source mechanism in the time interval t based on the action kinetic condition of the interface source mechanism3；
According to the document 2 "maruifang, li\28156;, spring, li hong, wen wei xin, cavity closure model based on the mechanical condition of metal diffusion bonder braking, chinese science: technical science, 2012, 42 (9): the dynamic condition and the calculation method of the interface source mechanism action proposed in 1081-The length increment of the interface connection under the action of the interface source mechanism within the time interval t
Step six, calculating the interface connection length increment e under the action of the surface source mechanism in the time interval t based on the surface source mechanism action dynamics condition4；
According to the document 2 "maruifang, li\28156;, spring, li hong, wen wei xin, cavity closure model based on the mechanical condition of metal diffusion bonder braking, chinese science: technical science, 2012, 42 (9): the dynamic condition and the calculation method of the surface source mechanism action proposed in 1081-The interface connection length under the action of the surface source mechanism is increased in the time interval t
Step seven, overlapping the interface connection length increment under the action of each connection mechanism by a fourth-order Runge-Kutta iteration method to obtain a cavity shape parameter at the end of the t' + t moment and a total interface connection length increment e under the action of each mechanism;
step eight, if t' + t<t (the set connection time), taking the cavity shape parameter at the end of the time t '+ t as a state parameter, making t' + t, and repeating the third step to the seventh step to obtain an interface connection length increment E under the set connection process parameter and the grain size; otherwise, the total interface connection length increment at the end of the t' + t moment is the interface connection length increment E under the set connection process parameters and the grain size, and the interface connection rate Af＝E/c0。
According to the above steps, the predicted value of the interface connection rate and the comparison result with the test value when the alloy with different grain sizes TC4 is connected under the conditions of 850 ℃ of connection temperature, 30MPa of connection pressure and 10min of connection time are shown in Table 4.
TABLE 4 comparison of predicted and tested values of compressive bonding interfacial connectivity of TC4 alloy at different grain sizes
By comparing the predicted value and the test value of the interface connection rate, the average relative error is found to be 5.48 percent, and the maximum relative error is found to be 10.03 percent. The method has higher accuracy and reliability, and realizes the prediction of the connection rate of the pressure connection interface of the titanium alloy with different grain sizes.
Claims (1)
1. A prediction method of the connection rate of a titanium alloy pressure connection interface coupled with grain size is characterized by comprising the following steps:
step one, simplifying the shape of an initial cavity into an ellipse;
step two, selecting TC4 alloys with different grain sizes for pressure connection, wherein the grain sizes are respectively 8.2 μm, 9.8 μm, 12.5 μm and 16.4 μm, and the connection process parameters are as follows: the connection temperature is 850 ℃, the connection pressure is 30MPa, and the connection time is 10 min; the roughness of the connecting surface is R after the 1000# sand paper is adopted for polishinga＝0.28μm，RλqInitial void shape parameter h of 5.40 μm0＝2Ra＝0.76μm，c0＝Rλq2.70 μm; t' is 0s, and the time interval t is 1 s;
thirdly, calculating the interface connection length increment e under the action of the plastic deformation mechanism in the time interval t by adopting a lamella segmentation method based on the plastic deformation constitutive relation of the coupled crystal grain size and the action kinetic condition of the plastic deformation mechanism1The specific calculation method is as follows:
assuming that the deformation at the connecting interface is plane strain, obtaining the equivalent strain rateEquivalent stressThe following plastic deformation constitutive relation coupling the grain sizes is adopted to represent the relation between equivalent stress and equivalent strain rate of titanium alloys with different grain sizes in the pressure connection process:
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>&tau;</mi> <mo>=</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <msub> <mi>f</mi> <mi>&alpha;</mi> </msub> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <mo>+</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <msub> <mi>f</mi> <mi>&beta;</mi> </msub> <msub> <mi>&tau;</mi> <mi>&beta;</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <mo>=</mo> <msup> <msub> <mi>&tau;</mi> <mi>&alpha;</mi> </msub> <mn>0</mn> </msup> <msup> <mrow> <mo>&lsqb;</mo> <mn>1</mn> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> </mrow> <mrow> <msub> <mi>&Delta;G</mi> <mi>&alpha;</mi> </msub> </mrow> </mfrac> <mi>l</mi> <mi>n</mi> <mfrac> <msub> <mover> <mi>&gamma;</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>&alpha;</mi> <mn>0</mn> </mrow> </msub> <mover> <mi>&gamma;</mi> <mo>&CenterDot;</mo> </mover> </mfrac> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <msub> <mi>q</mi> <mi>&alpha;</mi> </msub> </mrow> </msup> <mo>&rsqb;</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <msub> <mi>p</mi> <mi>&alpha;</mi> </msub> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>&mu;</mi> <mi>b</mi> <msqrt> <mi>&rho;</mi> </msqrt> <mo>+</mo> <msup> <mi>kd</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&tau;</mi> <mi>&beta;</mi> </msub> <mo>=</mo> <msup> <msub> <mi>&tau;</mi> <mi>&beta;</mi> </msub> <mn>0</mn> </msup> <msup> <mrow> <mo>&lsqb;</mo> <mn>1</mn> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> </mrow> <mrow> <msub> <mi>&Delta;G</mi> <mi>&beta;</mi> </msub> </mrow> </mfrac> <mi>l</mi> <mi>n</mi> <mfrac> <msub> <mover> <mi>&gamma;</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>&beta;</mi> <mn>0</mn> </mrow> </msub> <mover> <mi>&gamma;</mi> <mo>&CenterDot;</mo> </mover> </mfrac> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <msub> <mi>q</mi> <mi>&beta;</mi> </msub> </mrow> </msup> <mo>&rsqb;</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <msub> <mi>p</mi> <mi>&beta;</mi> </msub> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&mu;</mi> <mo>=</mo> <msub> <mi>&mu;</mi> <mn>0</mn> </msub> <mo>-</mo> <mfrac> <mi>s</mi> <mrow> <mi>exp</mi> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mi>r</mi> </msub> <mo>/</mo> <mi>T</mi> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&rho;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>&alpha;</mi> <mn>1</mn> </msub> <msqrt> <mi>&rho;</mi> </msqrt> <mo>|</mo> <msub> <mover> <mi>&epsiv;</mi> <mo>&CenterDot;</mo> </mover> <mi>e</mi> </msub> <mo>|</mo> <mo>-</mo> <msub> <mi>&alpha;</mi> <mn>2</mn> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>Q</mi> <mrow> <mi>d</mi> <mi>m</mi> </mrow> </msub> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </msup> <mi>&rho;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>d</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>&beta;</mi> <mn>0</mn> </msub> <msup> <mi>d</mi> <mrow> <mo>-</mo> <msub> <mi>&gamma;</mi> <mn>0</mn> </msub> </mrow> </msup> <msup> <mi>T</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <msub> <mi>Q</mi> <mrow> <mi>p</mi> <mi>d</mi> </mrow> </msub> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </msup> <mo>+</mo> <msub> <mi>&beta;</mi> <mn>1</mn> </msub> <mo>|</mo> <msub> <mover> <mi>&epsiv;</mi> <mo>&CenterDot;</mo> </mover> <mi>e</mi> </msub> <mo>|</mo> <msup> <mi>d</mi> <mrow> <mo>-</mo> <msub> <mi>&gamma;</mi> <mn>1</mn> </msub> </mrow> </msup> <mo>-</mo> <msub> <mi>&beta;</mi> <mn>2</mn> </msub> <msup> <mover> <mi>&rho;</mi> <mo>&CenterDot;</mo> </mover> <msub> <mi>&gamma;</mi> <mn>2</mn> </msub> </msup> <mi>d</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&sigma;</mi> <mi>e</mi> </msub> <mo>=</mo> <mi>M</mi> <mi>&tau;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&gamma;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mi>M</mi> <msub> <mover> <mi>&epsiv;</mi> <mo>&CenterDot;</mo> </mover> <mi>e</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
in the formula: tau is the shear stress of the titanium alloy, unit MPa; tau isαAnd τβShear stress in MPa of α and β phases, respectivelyα、fβVolume fractions, f, of α and β phases, respectivelyα+fβ＝1；n1、n2Is a correction factor;andcritical stress of α and β phases in MPa, T in K, and R in 8.3145J. mol-1·K-1；ΔGαAnd Δ GβThe apparent deformation activation energy of α and β phases is expressed in kJ & mol-1；For shear strain rate, in units of s-1；Andshear strain rate in s for the α and β phases at 0K-1(ii) a μ is the temperature-dependent shear modulus in GPa; ρ is the dislocation density in cm-2；Is the rate of change of dislocation density in cm-2·s-1(ii) a b is the Berth vector in m; d is the grain size in μm;is the rate of change of grain size in μm · s-1；Is the equivalent strain rate, in units of s-1；σeEquivalent stress in MPa; qdmThe unit is 20 kJ.mol for dislocation motion activation energy-1；QpdThe unit for activation energy of grain boundary diffusion is 677.37kJ & mol-1(ii) a M is a Taylor factor; q. q.sα、pα、qβ、pβ、a、k、s、μ0、Tr、α1、α2、β0、β1、β2、γ0、γ1、γ2Is a material parameter;
obtaining the flow stress of the TC4 alloy under different connection process parameters and grain sizes according to the plastic deformation constitutive relation; the flow stress at 0.2% residual deformation was taken as the yield strength σ of the TC4 alloy at the joining process parameters and grain sizeyield(ii) a According to the slip line field theory and Mises yield criterion, the connecting interface is along the neck of the hollow holeThe stress distribution of (A) is:
<mrow> <msub> <mi>&sigma;</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mfrac> <msub> <mi>&sigma;</mi> <mrow> <mi>y</mi> <mi>i</mi> <mi>e</mi> <mi>l</mi> <mi>d</mi> </mrow> </msub> <msqrt> <mn>3</mn> </msqrt> </mfrac> <mo>&lsqb;</mo> <mn>1</mn> <mo>+</mo> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>x</mi> <mo>-</mo> <mi>c</mi> </mrow> <msub> <mi>r</mi> <mi>A</mi> </msub> </mfrac> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>,</mo> <mrow> <mo>(</mo> <mi>c</mi> <mo>&le;</mo> <mi>x</mi> <mo>&le;</mo> <mi>c</mi> <mo>+</mo> <mi>e</mi> <mo>)</mo> </mrow> </mrow>
in the formula, rAThe radius of curvature of the neck of the cavity is in units of μm; c representsA void width; e represents the total interfacing length increment;
the average contact stress of the connection interface in a yield state is as follows:
<mrow> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <mfrac> <mrow> <msubsup> <mo>&Integral;</mo> <mi>c</mi> <mrow> <mi>c</mi> <mo>+</mo> <mi>e</mi> </mrow> </msubsup> <msub> <mi>&sigma;</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </msub> <mi>d</mi> <mi>x</mi> </mrow> <mi>e</mi> </mfrac> <mo>=</mo> <mn>2</mn> <mfrac> <msub> <mi>&sigma;</mi> <mrow> <mi>y</mi> <mi>i</mi> <mi>e</mi> <mi>l</mi> <mi>d</mi> </mrow> </msub> <msqrt> <mn>3</mn> </msqrt> </mfrac> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <msub> <mi>r</mi> <mi>A</mi> </msub> <mi>e</mi> </mfrac> </mrow> <mo>&rsqb;</mo> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mi>e</mi> <msub> <mi>r</mi> <mi>A</mi> </msub> </mfrac> <mo>)</mo> </mrow> </mrow>
the contact stress of the connection interface under the connection pressure is as follows:
<mrow> <msup> <mi>&sigma;</mi> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <mrow> <msub> <mi>pc</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>&gamma;</mi> </mrow> <mi>e</mi> </mfrac> </mrow>
wherein p is the connection pressure in MPa; gamma is surface energy, unit J.m-2(ii) a When in useWhen the plastic deformation mechanism is continuously acted; when inWhen the stress state does not meet the yield condition, the plastic deformation mechanism stops acting;
when the plastic deformation machine is manufactured, the convex peak between the cavities is divided into N sheet layers parallel to the connecting interface, and the length of the ith sheet layer isWherein z isiDenotes the distance of the ith sheet from the joining interface, and h denotesHeight of void, strain rate in x-axis directionThe z-direction stress of the ith sheet layer is approximately as follows:
<mrow> <msub> <mi>&sigma;</mi> <mrow> <mi>i</mi> <mi>z</mi> </mrow> </msub> <mo>&ap;</mo> <mfrac> <mrow> <msub> <mi>pc</mi> <mn>0</mn> </msub> </mrow> <msub> <mi>w</mi> <mi>i</mi> </msub> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>pc</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>c</mi> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <mrow> <msup> <msub> <mi>z</mi> <mi>i</mi> </msub> <mn>2</mn> </msup> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mfrac> </mrow>
let the length of the ith slice at time t' be wi(t'), calculating the length of the ith lamella after one time step t deformation according to the plastic deformation constitutive relation of the coupled crystal grain size
Obtaining the thickness h of the ith lamella at the moment t' + t according to the principle of volume conservation before and after lamella deformationi(t'+t)＝wi(t')·hi(t')/wi(t'+t)；
Finally, the time interval t inner convex is obtained through superpositionTotal thickness after peak deformation, i.e. interfacial cavity height under the action of plastic deformation mechanismObtaining the interface connection length increment under the action of the plastic deformation mechanism in the time interval t according to the principle that the volume conservation is met before and after the plastic deformation
Step four, calculating the interface connection length increment e under the action of the creep mechanism in the time interval t by adopting a lamella segmentation method based on the steady-state creep constitutive relation of the coupled grain size and the action dynamic condition of the creep mechanism2The specific calculation method is as follows:
when contact stress of the connection interfaceWhen the plastic deformation mechanism stops, the creep mechanism starts to act; the steady state creep constitutive relation for the coupled grain sizes is as follows:
<mrow> <mover> <mi>&epsiv;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mi>A</mi> <mfrac> <mrow> <mi>D</mi> <mi>&mu;</mi> <mi>b</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mi>b</mi> <mi>d</mi> </mfrac> <mo>)</mo> </mrow> <mi>p</mi> </msup> <msup> <mrow> <mo>(</mo> <mfrac> <mi>&sigma;</mi> <mi>&mu;</mi> </mfrac> <mo>)</mo> </mrow> <mi>n</mi> </msup> </mrow>
in the formula,is the true strain rate, in units of s-1(ii) a Sigma is the true stress, unit MPa; d is the diffusion coefficient of the light-emitting diode,unit m2·s-1K is Boltzmann's constant, 1.38 × 10-23(ii) a b is the Berth vector in m; μ is shear modulus, in MPa; A. p and n are material parameters;
dividing the material near the connecting interface into N parallel sheets, each sheet being stressed by
<mrow> <msub> <mi>&sigma;</mi> <mrow> <mi>i</mi> <mi>z</mi> </mrow> </msub> <mo>&ap;</mo> <mfrac> <mrow> <msub> <mi>pc</mi> <mn>0</mn> </msub> </mrow> <msub> <mi>w</mi> <mi>i</mi> </msub> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>pc</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>c</mi> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <mrow> <msup> <msub> <mi>z</mi> <mi>i</mi> </msub> <mn>2</mn> </msup> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mfrac> </mrow>
And (3) adopting deformation of the sheet layer in the thickness direction to represent the strain:
according to creep constitutive relation and equivalent strain rateAnd equivalent stress sigmaeThe deformation expression for the ith slice is obtained:obtaining:
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>d</mi> <mi>h</mi> <mo>=</mo> <munder> <mi>lim</mi> <mrow> <mi>N</mi> <mo>&RightArrow;</mo> <mi>&infin;</mi> </mrow> </munder> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>dh</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> <mi>A</mi> <mfrac> <mrow> <mi>D</mi> <mi>&mu;</mi> <mi>b</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mi>b</mi> <mi>d</mi> </mfrac> <mo>)</mo> </mrow> <mi>p</mi> </msup> <munder> <mi>lim</mi> <mrow> <mi>N</mi> <mo>&RightArrow;</mo> <mi>&infin;</mi> </mrow> </munder> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <mrow> <mo>(</mo> <mfrac> <mi>h</mi> <mi>N</mi> </mfrac> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> <msub> <mi>&sigma;</mi> <mrow> <mi>i</mi> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>&mu;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mi>d</mi> <mi>t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> <mi>A</mi> <mfrac> <mrow> <mi>D</mi> <mi>&mu;</mi> <mi>b</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mi>b</mi> <mi>d</mi> </mfrac> <mo>)</mo> </mrow> <mi>p</mi> </msup> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> <mi>p</mi> </mrow> <mrow> <mn>2</mn> <mi>&mu;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>n</mi> </msup> <munder> <mi>lim</mi> <mrow> <mi>N</mi> <mo>&RightArrow;</mo> <mi>&infin;</mi> </mrow> </munder> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <mrow> <mo>(</mo> <mfrac> <mi>h</mi> <mi>N</mi> </mfrac> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>c</mi> <mn>0</mn> </msub> <msub> <mi>w</mi> <mi>i</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mi>d</mi> <mi>t</mi> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mrow> <mfrac> <mrow> <mi>d</mi> <mi>h</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> <mi>A</mi> <mfrac> <mrow> <mi>D</mi> <mi>&mu;</mi> <mi>b</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mi>b</mi> <mi>d</mi> </mfrac> <mo>)</mo> </mrow> <mi>p</mi> </msup> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> <mi>p</mi> </mrow> <mrow> <mn>2</mn> <mi>&mu;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>n</mi> </msup> <munderover> <mo>&Integral;</mo> <mn>0</mn> <mi>h</mi> </munderover> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>c</mi> <mn>0</mn> </msub> <msub> <mi>w</mi> <mi>i</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mi>d</mi> <mi>z</mi> </mrow>
the rate of change of the cavity height under the action of the creep mechanism is:
<mrow> <msub> <mover> <mi>h</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> <msub> <mi>A</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> <mi>p</mi> </mrow> <mrow> <mn>2</mn> <mi>&mu;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>n</mi> </msup> <munderover> <mo>&Integral;</mo> <mn>0</mn> <mi>h</mi> </munderover> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>c</mi> <mn>0</mn> </msub> <mrow> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>c</mi> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>z</mi> <mn>2</mn> </msup> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mi>d</mi> <mi>z</mi> </mrow>
wherein,
according to the principle that the volume of the material is unchanged, the change rate of the connection length of the connection interface under the action of a creep mechanism is as follows:
<mrow> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <msub> <mover> <mi>h</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mi>h</mi> </mfrac> <mo>&lsqb;</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mfrac> <mn>4</mn> <mi>&pi;</mi> </mfrac> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mi>e</mi> <mo>&rsqb;</mo> </mrow>
the length of the interface connection increases under the action of the creep mechanism within the time interval t
Step five, calculating the interface connection length increment e under the action of the interface source mechanism in the time interval t based on the action kinetic condition of the interface source mechanism3；
Obtaining the change rate of the interface connection length under the action of the interface source mechanism according to the dynamic condition and the calculation method of the interface source mechanism actionThe length increment of the interface connection under the action of the interface source mechanism within the time interval t
Step six, based on surface source mechanism action dynamicsCondition, calculating the interface connection length increment e under the action of surface source mechanism in time interval t4；
Obtaining the interface connection length change rate under the action of the surface source mechanism according to the dynamic condition and the calculation method of the surface source mechanism actionThe interface connection length under the action of the surface source mechanism is increased in the time interval t
Step seven, overlapping the interface connection length increment under the action of each connection mechanism by a fourth-order Runge-Kutta iteration method to obtain a cavity shape parameter at the end of the t' + t moment and a total interface connection length increment e under the action of each mechanism;
step eight, if t' + t<t, taking the cavity shape parameter at the end of the time t '+ t as a state parameter, changing t' + t, and repeating the third step to the seventh step to obtain an interface connection length increment E under the set connection process parameter and the set grain size; otherwise, the total interface connection length increment at the end of the t' + t moment is the interface connection length increment E under the set connection process parameters and the grain size, and the interface connection rate Af＝E/c0。
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| CN107505444B (en) * | 2017-07-31 | 2019-10-22 | 西北工业大学 | Titanium alloy press-in connection interface bonding ratio prediction technique |
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