CN104504173A - Method for predicting connectivity of titanium alloy pressure connecting interface coupling grain size - Google Patents

Abstract

The invention discloses a method for predicting connectivity of a titanium alloy pressure connecting interface coupling grain size, and aims to solve the technical problem that an existing method cannot predict connectivity of different grain sizes of titanium alloy interfaces. The method includes steps of firstly, simplifying a geometric shape of a cavity, determining materials, connecting process parameters, cavity shape parameters, grain size and a time interval, calculating interface connection length increment within the time interval respectively under the action of plastic deformation and creep dynamics mechanism by adopting a layering partition method on the basis of plastic deformation and steady creep constitutive relation coupled with the grain size according to the creep mechanism dynamic conditions; respectively calculating the interface connecting length increment within the time interval under the mechanism action of an interface source and a surface source according to the surface source mechanism dynamics; calculating total interface connecting length increment by a four-step Runge-Kutta iteration method; calculating the interface connectivity. By the method, prediction of the interface connectivity of different grain sizes of titanium alloy under different connecting process parameters is realized.

Description

The titanium alloy press-in connection interface bonding ratio Forecasting Methodology of coupling crystallite dimension

Technical field

The present invention relates to a kind of titanium alloy press-in connection interface bonding ratio Forecasting Methodology, particularly relate to a kind of titanium alloy press-in connection interface bonding ratio Forecasting Methodology of the crystallite dimension that is coupled.

Background technology

Press-in connection is one of Perfected process being suitable for titanium alloy connection, press joining process can obtain and the microstructure of titanium alloy substrate and the close even consistent jointing of mechanical property, be applicable to the connection of heavy in section and complex internal structure simultaneously, be widely used in Aero-Space lightweight components such as manufacturing hollow blade.

Carry out on the titanium alloy connecting surface microcosmic after Precision Machining uneven, in press-in connection process, on linkage interface, form microscopic holes.Realize microscopic holes to close, thus obtain the prerequisite and key that high interface bonding ratio is formation high-quality metallurgical joint.In actual production, titanium alloy press-in connection interfacial voids closes and usually depends on test and experience with the regulation and control of interface bonding ratio, consumes a large amount of manpower and materials, the production cycle is extended, and cost increases.

In recent years, based on the heightened awareness to void closing process, many scholars propose multiple interface bonding ratio Forecasting Methodology, improve production efficiency, have saved cost.

Document 1 " B.Derby; E.R.Wallach; Theoretical model for diffusion bonding; MetalScience; 1982; 16:49-56 " discloses a kind of interface bonding ratio Forecasting Methodology based on Plastic Deformation Mechanism, creep mechanism, machine-processed and source, the interface mechanism of surface source, but it only considered the impact of different technical parameters on interface bonding ratio, considers that material grains size is on the impact of interface bonding ratio.

Document 2 " Ma Ruifang; Li Miaoquan; Li Hong, Yu Weixin, based on the void closing model of metal diffusion connection mechanism dynamic conditions; Chinese science: technological sciences; 2012,42 (9): 1081-1091 " discloses a kind of interface bonding ratio Forecasting Methodology based on each mechanism of action dynamic conditions, and the method accurately reflects the reach of various connection mechanism, improve the precision of prediction of interface bonding ratio, but unpredictable material Initial Grain Size is on the impact of interface bonding ratio.And crystallite dimension on the impact of interface bonding ratio significantly.

It is that the interface bonding ratio of TC21 alloy under given Joining Technology parameter of 2 μm, 4 μm and 7 μm is respectively 99.5%, 91.8% and 88.7% that document 3 " Yang Yong; Zhou Wenlong; Chen Guoqing; horse Red Army, Han Xiuquan, Li Zhiqiang; the research that thin brilliant TC21 Alloy during Superplastic diffusion connects; Study on manufacturing technology, 2009,3:8-13 " discloses crystallite dimension.The interface bonding ratio of the unpredictable various grain sizes titanium alloy of existing Forecasting Methodology, causes its range of application to have certain limitation.

Summary of the invention

The deficiency of various grain sizes titanium alloy interface bonding ratio prediction cannot be realized to overcome existing method, the invention provides a kind of titanium alloy press-in connection interface bonding ratio Forecasting Methodology of the crystallite dimension that is coupled.First the method simplifies empty geometric configuration; Determine material and Joining Technology parameter, empty form parameter, crystallite dimension and the time interval; According to plastic yield and creep machine braking mechanics condition, based on the coupling plastic yield of crystallite dimension and steady state creep constitutive relation, adopt lamella split plot design, the interface connecting length increment respectively in the computing time of interval under plastic yield and creep machining function; According to source, interface and surface source brake force condition, respectively computing time interval inner boundary source and surface source machining function under interface connecting length increment; Total interface connecting length increment is calculated by quadravalence Runge-Kutta alternative manner; Calculate interface bonding ratio.Compared with prior art, the present invention can realize the Accurate Prediction of the interface bonding ratio of various grain sizes titanium alloy under different Joining Technology parameter.

The technical solution adopted for the present invention to solve the technical problems is: a kind of titanium alloy press-in connection interface bonding ratio Forecasting Methodology of the crystallite dimension that is coupled, and is characterized in adopting following steps:

Step one, be oval by initial empty shape simplification;

Step 2, choose the TC4 alloy with various grain sizes and carry out press-in connection, crystallite dimension is respectively 8.2 μm, 9.8 μm, 12.5 μm and 16.4 μm, and Joining Technology parameter is: connect temperature 850 DEG C, Bonding pressure 30MPa, tie-time 10min.After employing 1000# sand papering, connecting surface roughness is R _a=0.28 μm, R _{λ q}=5.40 μm, initial empty form parameter h ₀=2R _a=0.76 μm, c ₀=R _{λ q}/ 2=2.70 μm.T '=0s, time interval δ t=1s.

Step 3, based on the coupling plastic yield constitutive relation of crystallite dimension and Plastic Deformation Mechanism action kinetics condition, adopt lamella split plot design, the interface connecting length increment e in computing time interval δ t under Plastic Deformation Mechanism effect ₁, circular is as follows:

Suppose that linkage interface place is deformed into plane strain, obtain equivalent strain rate equivalent stress the plastic yield constitutive relation of following coupling crystallite dimension is adopted to characterize the relation of various grain sizes titanium alloy equivalent stress and equivalent strain rate in press-in connection process:

$\left\{\begin{array}{c}\mathrm{\τ}=n_1{f}_{\mathrm{\α}}+n_2{f}_{\mathrm{\β}}{\mathrm{\τ}}_{\mathrm{\β}}\\ {\mathrm{\τ}}_{\mathrm{\α}}={{\mathrm{\τ}}_{\mathrm{\α}}}^0{[1-{\left(\frac{\mathrm{RT}}{\mathrm{\Δ}{G}_{\mathrm{\α}}}\ln \frac{{\stackrel{\·}{\mathrm{\γ}}}_{\mathrm{\α}0}}{\stackrel{\·}{\mathrm{\γ}}}\right)}^{1/q_{\mathrm{\α}}}]}^{1/p_{\mathrm{\α}}}+\mathrm{a\μb}\sqrt{\mathrm{\ρ}}+{\mathrm{kd}}^{-1/2}\\ {\mathrm{\τ}}_{\mathrm{\β}}={{\mathrm{\τ}}_{\mathrm{\β}}}^0{[1-{\left(\frac{\mathrm{RT}}{\mathrm{\Δ}{G}_{\mathrm{\β}}}\ln \frac{{\stackrel{\·}{\mathrm{\γ}}}_{\mathrm{\β}0}}{\stackrel{\·}{\mathrm{\γ}}}\right)}^{1/q_{\mathrm{\β}}}]}^{1/p_{\mathrm{\β}}}\\ \mathrm{\μ}={\mathrm{\μ}}_0-\frac{s}{\exp (T_r/T)-1}\\ \stackrel{\·}{\mathrm{\ρ}}={\mathrm{\α}}_1\sqrt{\mathrm{\ρ}}\left|{\stackrel{\·}{\mathrm{\ϵ}}}_e\right|-{\mathrm{\α}}_2{e}^{-\frac{Q_{\mathrm{dm}}}{\mathrm{RT}}}\mathrm{\ρ}\\ \stackrel{\·}{d}={\mathrm{\β}}_0{d}^{-{\mathrm{\γ}}_0}{T}^{-1}{e}^{-\frac{Q_{\mathrm{dm}}}{\mathrm{RT}}}+{\mathrm{\β}}_1\left|{\stackrel{\·}{\mathrm{\ϵ}}}_e\right|d^{-{\mathrm{\γ}}_1}-{\mathrm{\β}}_2{\stackrel{\·}{\mathrm{\ρ}}}^{{\mathrm{\γ}}_2}d\\ {\mathrm{\σ}}_e=\mathrm{M\τ}\\ \stackrel{\·}{\mathrm{\γ}}=M{\stackrel{\·}{\mathrm{\ϵ}}}_e\end{array}\right.$

In formula: τ is the shear stress of titanium alloy, units MPa; τ _αand τ _βbe respectively the shear stress of α and β phase, units MPa; f _α, f _βbe respectively the volume fraction of α and β phase, f _α+ f _β=1; n ₁, n ₂for correction factor; with be respectively the limit stress of α and β phase, units MPa; T is for connecting temperature, unit K; R is gas constant, unit 8.3145Jmol ^-1k ^-1; △ G _αwith Δ G _βfor the apparent deformation activation energy of α and β phase, unit kJmol ^-1; for shear strain rate, unit s ^-1; with for the shear strain rate of α during 0K and β phase, unit s ^-1; μ is the modulus of shearing depending on temperature, unit GPa; ρ is dislocation desity, unit cm ^-2; for dislocation desity rate of change, unit cm ^-2s ^-1; B is Bai Shi vector, unit m; D is crystallite dimension, unit μm; for crystallite dimension rate of change, unit μm s ^-1; for equivalent strain rate, unit s ^-1; σ _efor equivalent stress, units MPa; Q _dmfor dislocation motion activation energy, unit 20kJmol ^-1; Q _pdfor grain boundary decision activation energy, unit 677.37kJmol ^-1; M is the Taylor factor; q _α, p _α, q _β, p _β, a, k, s, μ ₀, T _r, α ₁, α ₂, β ₀, β ₁, β ₂, γ ₀, γ ₁, γ ₂for material parameter.

The flow stress of TC4 alloy under different Joining Technology parameter and crystallite dimension is obtained according to above plastic yield constitutive relation.Get flow stress under 0.2% residual deformation as the yield strength σ of TC4 alloy under Joining Technology parameter and crystallite dimension _yield.According to slip line field theory and Mises yield criteria, linkage interface along the stress distribution of empty neck is:

${\mathrm{\σ}}_{\left(x\right)}=2\frac{{\mathrm{\σ}}_{\mathrm{yield}}}{\sqrt{3}}[1+\ln (\frac{x-c}{r_A}+1)](c\≤x\≤c+e)$

In formula, r _afor empty neck radius-of-curvature, unit μm.

Linkage interface contact mean stress under yield situation is:

$\stackrel{\‾}{\mathrm{\σ}}=\frac{{\∫}_c^{c+e}{\mathrm{\σ}}_{\left(x\right)}\mathrm{dx}}{e}=2\frac{{\mathrm{\σ}}_{\mathrm{yield}}}{\sqrt{3}}[1+\frac{r_A}{e}]\ln (1+\frac{e}{r_A})$

Under Bonding pressure, linkage interface contact stress is:

${\mathrm{\σ}}^{\′}=\frac{p{c}_0-\mathrm{\γ}}{e}$

In formula, p is Bonding pressure, units MPa; γ is surface energy, unit Jm ^-2.When time, Plastic Deformation Mechanism continuous action; And work as time, stress state does not meet yield condition, and Plastic Deformation Mechanism fails.

When Plastic Deformation Mechanism is done, the convex peak between cavity is divided into the N number of lamella being parallel to linkage interface, the length of i-th lamella is strain rate along the x-axis direction on i-th lamella z direction, suffered stress is approximately:

${\mathrm{\σ}}_{\mathrm{iz}}\≈\frac{p{c}_0}{w_i}=\frac{p{c}_0}{c_0-c\sqrt{1-\frac{{z_i}^2}{h^2}}}$

If the length of t' moment i-th lamella is w _i(t'), calculate according to the plastic yield constitutive relation of coupling crystallite dimension the length that a time step δ t is out of shape rear i-th lamella

According to volume conserva-tion principle before and after lamella distortion, obtain the thickness h of i-th lamella t'+ δ t _i(t'+ δ t)=w _i(t') h _i(t')/w _i(t'+ δ t).

Finally, the gross thickness after the distortion of time interval δ t convex peak is obtained by superposition, the interfacial voids height namely under Plastic Deformation Mechanism effect according to meeting volume conservation principle before and after plastic yield, obtain the interface connecting length increment in time interval δ t under Plastic Deformation Mechanism effect

Step 4, based on the coupling steady state creep constitutive relation of crystallite dimension and creep machining function dynamic conditions, adopt lamella split plot design, the interface connecting length increment e in computing time interval δ t under creep machining function ₂, circular is as follows:

When linkage interface contact stress time, Plastic Deformation Mechanism stops, and creep mechanism starts effect.The steady state creep constitutive relation of coupling crystallite dimension is as follows:

$\stackrel{\·}{\mathrm{\ϵ}}=A\frac{\mathrm{D\μb}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p{\left(\frac{\mathrm{\σ}}{\mathrm{\μ}}\right)}^n$

In formula, for true strain speed, unit s ^-1; σ is trus stress, units MPa; D is coefficient of diffusion, unit m ²s ^-1; K is Boltzmann constant, 1.38 × 10 ^-23; B is Bai Shi vector, unit m; μ is modulus of shearing, units MPa; A, p, n are material parameter.

Material near linkage interface is divided into the N number of lamella be parallel to each other, suffered by each lamella, stress is

${\mathrm{\σ}}_{\mathrm{iz}}\≈\frac{p{c}_0}{w_i}=\frac{p{c}_0}{c_0-c\sqrt{1-\frac{{z_i}^2}{h^2}}}$

The distortion in lamellar spacing direction is adopted to characterize its strain:

According to creep constitutive relation and equivalent strain rate with equivalent stress σ _eexpression formula, obtain the distortion expression formula for i-th lamella: ${\mathrm{dh}}_i=-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\μb}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p\frac{h}{N}{\left(\frac{\sqrt{3}{\mathrm{\σ}}_{\mathrm{iz}}}{2\mathrm{\μ}}\right)}^n\mathrm{dt},$ :

$\begin{array}{c}\mathrm{dh}=\underset{N\→\∞}{\lim }\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{dh}}_i=-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\μb}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p\underset{N\→\∞}{\lim }\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left(\frac{h}{N}\right){\left(\frac{\sqrt{3}{\mathrm{\σ}}_{\mathrm{iz}}}{2\mathrm{\μ}}\right)}^n\mathrm{dt}\\ =-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\μb}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p{\left(\frac{\sqrt{3}p}{2\mathrm{\μ}}\right)}^n\underset{N\→\∞}{\lim }\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left(\frac{h}{N}\right){\left(\frac{c_0}{w_i}\right)}^n\mathrm{dt}\end{array}$

$\frac{\mathrm{dh}}{\mathrm{dt}}=-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\μb}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p{\left(\frac{\sqrt{3}p}{2\mathrm{\μ}}\right)}^n\underset{0}{\overset{h}{\∫}}{\left(\frac{c_0}{w_i}\right)}^n\mathrm{dz}$

Then under creep machining function, empty altitude rate is:

${\stackrel{\·}{h}}_2=-\frac{\sqrt{3}}{2}{A}_c{\left(\frac{\sqrt{3}p}{2\mathrm{\μ}}\right)}^n\underset{0}{\overset{h}{\∫}}{\left(\frac{c_0}{c_0-c\sqrt{1-\frac{z^2}{h^2}}}\right)}^n\mathrm{dz}$

According to material volume principle of invariance, under creep machining function, the rate of change of linkage interface connecting length is:

${\stackrel{\·}{e}}_2=-\frac{{\stackrel{\·}{h}}_2}{h}[c_0(\frac{4}{\mathrm{\π}}-1)+e]$

Interface connecting length increment then in time interval δ t under creep machining function

Step 5, based on source, interface machining function dynamic conditions, the interface connecting length increment e under the machining function of interval δ t inner boundary computing time source ₃;

According to dynamic conditions and the computing method of source, interface machining function, obtain the interface connecting length rate of change under the machining function of source, interface interface connecting length increment then under the machining function of time interval δ t inner boundary source

Step 6, based on surface source machining function dynamic conditions, the interface connecting length increment e under the machining function of interval δ t inside surface computing time source ₄;

According to dynamic conditions and the computing method of surface source machining function, obtain the interface connecting length rate of change under surface source machining function interface connecting length increment then under the machining function of time interval δ t inside surface source

Step 7, by quadravalence Runge-Kutta alternative manner, the interface connecting length increment under each connection mechanism effect to be superposed, obtain the empty form parameter at the end of t '+δ t and the total interface connecting length increment e under each machining function;

If step 8 t '+δ is t<t, then with the empty form parameter at the end of t '+δ t for state parameter, make t '=t '+δ t, repeat step 3 ~ step 7, obtain the interface connecting length increment E under setting Joining Technology parameter and crystallite dimension; Otherwise the total interface connecting length increment at the end of t '+δ t is the interface connecting length increment E under setting Joining Technology parameter and crystallite dimension, interface bonding ratio A _f=E/c ₀.

The invention has the beneficial effects as follows: first the method simplifies empty geometric configuration; Determine material and Joining Technology parameter, empty form parameter, crystallite dimension and the time interval; According to plastic yield and creep machine braking mechanics condition, based on the coupling plastic yield of crystallite dimension and steady state creep constitutive relation, adopt lamella split plot design, the interface connecting length increment respectively in the computing time of interval under plastic yield and creep machining function; According to source, interface and surface source brake force condition, respectively computing time interval inner boundary source and surface source machining function under interface connecting length increment; Total interface connecting length increment is calculated by quadravalence Runge-Kutta alternative manner; Calculate interface bonding ratio.Compared with prior art, present invention achieves the Accurate Prediction of the interface bonding ratio of various grain sizes titanium alloy under different Joining Technology parameter.

Below in conjunction with the drawings and specific embodiments, the present invention is elaborated.

Accompanying drawing explanation

Fig. 1 is that the present invention is coupled the process flow diagram of titanium alloy press-in connection interface bonding ratio Forecasting Methodology of crystallite dimension.

Fig. 2 is the initial empty shape of the inventive method linkage interface, wherein, and h ₀for the half of cavity height, c ₀for the half of empty width, R _afor surface profile arithmetic mean variance, R _{λ q}for surface profile root mean square wavelength.

Fig. 3 is the distortion of the inventive method lamella split plot design computational plasticity and creep machining function schematic diagram.Wherein, z _ibe the distance of i-th lamella apart from linkage interface, w _ibe the width of i-th lamella, h _ibe the thickness (h of i-th lamella _i=h/N).

Embodiment

With reference to Fig. 1-3.The be coupled titanium alloy press-in connection interface bonding ratio Forecasting Methodology concrete steps of crystallite dimension of the present invention are as follows:

Step one, be oval by initial empty shape simplification;

Step 2, choose the TC4 alloy with various grain sizes and carry out press-in connection, crystallite dimension is respectively 8.2 μm, 9.8 μm, 12.5 μm and 16.4 μm, and the material parameter needed for calculating is as shown in table 1.The Joining Technology parameter chosen is: connect temperature 850 DEG C, Bonding pressure 30MPa, tie-time 10min.After employing 1000# sand papering, connecting surface roughness is R _a=0.28 μm, R _{λ q}=5.40 μm, initial empty form parameter h ₀=2R _a=0.76 μm, c ₀=R _{λ q}/ 2=2.70 μm.T '=0s, time interval δ t=1s.

The material parameter of table 1 TC4 alloy

Step 3, based on the coupling plastic yield constitutive relation of crystallite dimension and Plastic Deformation Mechanism action kinetics condition, adopt lamella split plot design, the interface connecting length increment e in computing time interval δ t under Plastic Deformation Mechanism effect ₁, circular is as follows:

Suppose that linkage interface place is deformed into plane strain, obtain equivalent strain rate equivalent stress the plastic yield constitutive relation of following coupling crystallite dimension is adopted to characterize the relation of various grain sizes titanium alloy equivalent stress and equivalent strain rate in press-in connection process:

$\left\{\begin{array}{c}\mathrm{\&tau;}=n_1{f}_{\mathrm{\&alpha;}}+n_2{f}_{\mathrm{\&beta;}}{\mathrm{\&tau;}}_{\mathrm{\&beta;}}\\ {\mathrm{\&tau;}}_{\mathrm{\&alpha;}}={{\mathrm{\&tau;}}_{\mathrm{\&alpha;}}}^0{[1-{\left(\frac{\mathrm{RT}}{\mathrm{\&Delta;}{G}_{\mathrm{\&alpha;}}}\ln \frac{{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}_{\mathrm{\&alpha;}0}}{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}\right)}^{1/q_{\mathrm{\&alpha;}}}]}^{1/p_{\mathrm{\&alpha;}}}+\mathrm{a\&mu;b}\sqrt{\mathrm{\&rho;}}+{\mathrm{kd}}^{-1/2}\\ {\mathrm{\&tau;}}_{\mathrm{\&beta;}}={{\mathrm{\&tau;}}_{\mathrm{\&beta;}}}^0{[1-{\left(\frac{\mathrm{RT}}{\mathrm{\&Delta;}{G}_{\mathrm{\&beta;}}}\ln \frac{{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}_{\mathrm{\&beta;}0}}{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}\right)}^{1/q_{\mathrm{\&beta;}}}]}^{1/p_{\mathrm{\&beta;}}}\\ \mathrm{\&mu;}={\mathrm{\&mu;}}_0-\frac{s}{\exp (T_r/T)-1}\\ \stackrel{\&CenterDot;}{\mathrm{\&rho;}}={\mathrm{\&alpha;}}_1\sqrt{\mathrm{\&rho;}}\left|{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_e\right|-{\mathrm{\&alpha;}}_2{e}^{-\frac{Q_{\mathrm{dm}}}{\mathrm{RT}}}\mathrm{\&rho;}\\ \stackrel{\&CenterDot;}{d}={\mathrm{\&beta;}}_0{d}^{-{\mathrm{\&gamma;}}_0}{T}^{-1}{e}^{-\frac{Q_{\mathrm{dm}}}{\mathrm{RT}}}+{\mathrm{\&beta;}}_1\left|{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_e\right|d^{-{\mathrm{\&gamma;}}_1}-{\mathrm{\&beta;}}_2{\stackrel{\&CenterDot;}{\mathrm{\&rho;}}}^{{\mathrm{\&gamma;}}_2}d\\ {\mathrm{\&sigma;}}_e=\mathrm{M\&tau;}\\ \stackrel{\&CenterDot;}{\mathrm{\&gamma;}}=M{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_e\end{array}\right.$

In formula: τ is the shear stress (MPa) of titanium alloy; τ _αand τ _βbe respectively the shear stress (MPa) of α and β phase; f _α, f _βbe respectively the volume fraction (f of α and β phase _α+ f _β=1); n ₁, n ₂for correction factor; with be respectively the limit stress (MPa) of α and β phase; T is for connecting temperature (K); R is gas constant (8.3145Jmol ^-1k ^-1); △ G _αwith Δ G _βfor the apparent deformation activation energy (kJmol of α and β phase ^-1); for shear strain rate (s ^-1); with for the shear strain rate (s of α during 0K and β phase ^-1); μ is the modulus of shearing (GPa) depending on temperature; ρ is dislocation desity (cm ^-2); for dislocation desity rate of change (cm ^-2s ^-1); B is Bai Shi vector (m); D is crystallite dimension (μm); for crystallite dimension rate of change (μm s ^-1); for equivalent strain rate (s ^-1); σ _efor equivalent stress (MPa); Q _dmfor dislocation motion activation energy (20kJmol ^-1); Q _pdfor grain boundary decision activation energy (677.37kJmol ^-1); M is the Taylor factor; q _α, p _α, q _β, p _β, a, k, s, μ ₀, T _r, α ₁, α ₂, β ₀, β ₁, β ₂, γ ₀, γ ₁, γ ₂for material parameter.It is as shown in table 2 that present embodiment adopts optimisation technique to solve the TC4 alloy plasticity constitutive relation material parameter that objective function minimum value method determines.

The material parameter of table 2 TC4 alloy plasticity constitutive relation

The flow stress of TC4 alloy under different Joining Technology parameter and crystallite dimension is obtained according to above plastic yield constitutive relation.Get flow stress under 0.2% residual deformation as the yield strength σ of TC4 alloy under certain Joining Technology parameter and crystallite dimension _yield.According to slip line field theory and Mises yield criteria, linkage interface along the stress distribution of empty neck is:

${\mathrm{\&sigma;}}_{\left(x\right)}=2\frac{{\mathrm{\&sigma;}}_{\mathrm{yield}}}{\sqrt{3}}[1+\ln (\frac{x-c}{r_A}+1)](c\&le;x\&le;c+e)$

In formula, r _afor empty neck radius-of-curvature (μm).

Linkage interface contact mean stress under yield situation is:

$\stackrel{\&OverBar;}{\mathrm{\&sigma;}}=\frac{{\&Integral;}_c^{c+e}{\mathrm{\&sigma;}}_{\left(x\right)}\mathrm{dx}}{e}=2\frac{{\mathrm{\&sigma;}}_{\mathrm{yield}}}{\sqrt{3}}[1+\frac{r_A}{e}]\ln (1+\frac{e}{r_A})$

Under Bonding pressure, linkage interface contact stress is:

${\mathrm{\&sigma;}}^{\&prime;}=\frac{p{c}_0-\mathrm{\&gamma;}}{e}$

In formula, p is Bonding pressure (MPa); γ is surface energy (Jm ^-2).When time, Plastic Deformation Mechanism continuous action; And work as time, stress state does not meet yield condition, and Plastic Deformation Mechanism fails.

When Plastic Deformation Mechanism is done, the convex peak between cavity is divided into the N number of lamella being parallel to linkage interface, the length of i-th lamella is strain rate along the x-axis direction on i-th lamella z direction, suffered stress is approximately:

${\mathrm{\&sigma;}}_{\mathrm{iz}}\&ap;\frac{p{c}_0}{w_i}=\frac{p{c}_0}{c_0-c\sqrt{1-\frac{{z_i}^2}{h^2}}}$

If the length of t' moment i-th lamella is w _i(t'), calculate according to the plastic yield constitutive relation of coupling crystallite dimension the length that a time step δ t is out of shape rear i-th lamella

According to volume conserva-tion principle before and after lamella distortion, obtain the thickness h of i-th lamella t'+ δ t _i(t'+ δ t)=w _i(t') h _i(t')/w _i(t'+ δ t).

Finally, the gross thickness after the distortion of time interval δ t convex peak is obtained by superposition, the interfacial voids height namely under Plastic Deformation Mechanism effect according to meeting volume conservation principle before and after plastic yield, obtain the interface connecting length increment in time interval δ t under Plastic Deformation Mechanism effect

Step 4, based on the coupling steady state creep constitutive relation of crystallite dimension and creep machining function dynamic conditions, adopt lamella split plot design, the interface connecting length increment e in computing time interval δ t under creep machining function ₂, circular is as follows:

When linkage interface contact stress time, Plastic Deformation Mechanism stops, and creep mechanism starts effect.The steady state creep constitutive relation of coupling crystallite dimension is as follows:

$\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}=A\frac{\mathrm{D\&mu;b}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p{\left(\frac{\mathrm{\&sigma;}}{\mathrm{\&mu;}}\right)}^n$

In formula, for true strain speed (s ^-1); σ is trus stress (MPa); D is coefficient of diffusion (m ²s ^-1); K is Boltzmann constant (1.38 × 10 ^-23); B is Bai Shi vector (m); μ is modulus of shearing (MPa); A, p, n are material parameter.The material parameter of the TC4 alloy creep constitutive relation that present embodiment is determined is as shown in table 3.

The material parameter of table 3 TC4 alloy creep constitutive equation

Symbol	Unit	Numerical value
			A	—	6.80
D	m 2·s -1	1.477×10 -11
			p	—	2
n	—	2

Material near linkage interface is divided into the N number of lamella be parallel to each other, suffered by each lamella, stress is

${\mathrm{\&sigma;}}_{\mathrm{iz}}\&ap;\frac{p{c}_0}{w_i}=\frac{p{c}_0}{c_0-c\sqrt{1-\frac{{z_i}^2}{h^2}}}$

The distortion in lamellar spacing direction is adopted to characterize its strain:

According to creep constitutive relation and equivalent strain rate with equivalent stress σ _eexpression formula, obtain the distortion expression formula for i-th lamella: ${\mathrm{dh}}_i=-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\&mu;b}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p\frac{h}{N}{\left(\frac{\sqrt{3}{\mathrm{\&sigma;}}_{\mathrm{iz}}}{2\mathrm{\&mu;}}\right)}^n\mathrm{dt},$ :

$\begin{array}{c}\mathrm{dh}=\underset{N\&RightArrow;\&infin;}{\lim }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{dh}}_i=-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\&mu;b}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p\underset{N\&RightArrow;\&infin;}{\lim }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\left(\frac{h}{N}\right){\left(\frac{\sqrt{3}{\mathrm{\&sigma;}}_{\mathrm{iz}}}{2\mathrm{\&mu;}}\right)}^n\mathrm{dt}\\ =-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\&mu;b}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p{\left(\frac{\sqrt{3}p}{2\mathrm{\&mu;}}\right)}^n\underset{N\&RightArrow;\&infin;}{\lim }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\left(\frac{h}{N}\right){\left(\frac{c_0}{w_i}\right)}^n\mathrm{dt}\end{array}$

$\frac{\mathrm{dh}}{\mathrm{dt}}=-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\&mu;b}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p{\left(\frac{\sqrt{3}p}{2\mathrm{\&mu;}}\right)}^n\underset{0}{\overset{h}{\&Integral;}}{\left(\frac{c_0}{w_i}\right)}^n\mathrm{dz}$

Then under creep machining function, empty altitude rate is:

${\stackrel{\&CenterDot;}{h}}_2=-\frac{\sqrt{3}}{2}{A}_c{\left(\frac{\sqrt{3}p}{2\mathrm{\&mu;}}\right)}^n\underset{0}{\overset{h}{\&Integral;}}{\left(\frac{c_0}{c_0-c\sqrt{1-\frac{z^2}{h^2}}}\right)}^n\mathrm{dz}$

According to material volume principle of invariance, under creep machining function, the rate of change of linkage interface connecting length is:

${\stackrel{\&CenterDot;}{e}}_2=-\frac{{\stackrel{\&CenterDot;}{h}}_2}{h}[c_0(\frac{4}{\mathrm{\&pi;}}-1)+e]$

Interface connecting length increment then in time interval δ t under creep machining function

Step 5, based on source, interface machining function dynamic conditions, the interface connecting length increment e under the machining function of interval δ t inner boundary computing time source ₃;

According to dynamic conditions and the computing method of source, the interface machining function proposed in document 2 " Ma Ruifang; Li Miaoquan; Li Hong; Yu Weixin; based on the void closing model of metal diffusion connection mechanism dynamic conditions, Chinese science: technological sciences, 2012; 42 (9): 1081-1091 ", obtain the interface connecting length rate of change under the machining function of source, interface interface connecting length increment then under the machining function of time interval δ t inner boundary source

Step 6, based on surface source machining function dynamic conditions, the interface connecting length increment e under the machining function of interval δ t inside surface computing time source ₄;

According to dynamic conditions and the computing method of the surface source machining function proposed in document 2 " Ma Ruifang; Li Miaoquan; Li Hong; Yu Weixin; based on the void closing model of metal diffusion connection mechanism dynamic conditions, Chinese science: technological sciences, 2012; 42 (9): 1081-1091 ", obtain the interface connecting length rate of change under surface source machining function interface connecting length increment then under the machining function of time interval δ t inside surface source

Step 7, by quadravalence Runge-Kutta alternative manner, the interface connecting length increment under each connection mechanism effect to be superposed, obtain the empty form parameter at the end of t '+δ t and the total interface connecting length increment e under each machining function;

If step 8 t '+δ is t<t (tie-time of setting), then with the empty form parameter at the end of t '+δ t for state parameter, make t '=t '+δ t, repeat step 3 ~ step 7, obtain the interface connecting length increment E under setting Joining Technology parameter and crystallite dimension; Otherwise the total interface connecting length increment at the end of t '+δ t is the interface connecting length increment E under setting Joining Technology parameter and crystallite dimension, interface bonding ratio A _f=E/c ₀.

According to above-mentioned steps, various grain sizes TC4 alloy is 850 DEG C in connection temperature, and Bonding pressure is 30MPa, and the tie-time is interface bonding ratio predicted value and as shown in table 4 with the comparing result of trial value when connecting under 10min condition.

The lower TC4 alloy press-in connection interface bonding ratio predicted value of table 4 various grain sizes and the comparing result of trial value

Can find that its average relative error is 5.48% by the predicted value and trial value that contrast interface bonding ratio, maximum relative error is 10.03%.Show that the inventive method has higher accuracy and reliability, achieve the prediction of various grain sizes titanium alloy press-in connection interface bonding ratio.

Claims (1)

1. a titanium alloy press-in connection interface bonding ratio Forecasting Methodology for the crystallite dimension that is coupled, is characterized in that comprising the following steps:

Step one, be oval by initial empty shape simplification;

Step 2, choose the TC4 alloy with various grain sizes and carry out press-in connection, crystallite dimension is respectively 8.2 μm, 9.8 μm, 12.5 μm and 16.4 μm, and Joining Technology parameter is: connect temperature 850 DEG C, Bonding pressure 30MPa, tie-time 10min; After employing 1000# sand papering, connecting surface roughness is R _a=0.28 μm, R _{λ q}=5.40 μm, initial empty form parameter h ₀=2R _a=0.76 μm, c ₀=R _{λ q}/ 2=2.70 μm; T '=0s, time interval δ t=1s;

Step 3, based on the coupling plastic yield constitutive relation of crystallite dimension and Plastic Deformation Mechanism action kinetics condition, adopt lamella split plot design, the interface connecting length increment e in computing time interval δ t under Plastic Deformation Mechanism effect ₁, circular is as follows:

Suppose that linkage interface place is deformed into plane strain, obtain equivalent strain rate equivalent stress the plastic yield constitutive relation of following coupling crystallite dimension is adopted to characterize the relation of various grain sizes titanium alloy equivalent stress and equivalent strain rate in press-in connection process:

$\left\{\begin{array}{c}\mathrm{\&tau;}=n_1{f}_{\mathrm{\&alpha;}}{\mathrm{\&tau;}}_{\mathrm{\&alpha;}}+n_2{f}_{\mathrm{\&beta;}}{\mathrm{\&tau;}}_{\mathrm{\&beta;}}\\ {\mathrm{\&tau;}}_{\mathrm{\&alpha;}}={{\mathrm{\&tau;}}_{\mathrm{\&alpha;}}}^0{[1-{\left(\frac{\mathrm{RT}}{\mathrm{\&Delta;}{G}_{\mathrm{\&alpha;}}}\ln \frac{{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}_{\mathrm{\&alpha;}0}}{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}\right)}^{1/q_{\mathrm{\&alpha;}}}]}^{1/p_{\mathrm{\&alpha;}}}+\mathrm{a\&mu;b}\sqrt{\mathrm{\&rho;}}+{\mathrm{kd}}^{-1/2}\\ {\mathrm{\&tau;}}_{\mathrm{\&beta;}}={{\mathrm{\&tau;}}_{\mathrm{\&beta;}}}^0{[1-{\left(\frac{\mathrm{RT}}{\mathrm{\&Delta;}{G}_{\mathrm{\&beta;}}}\ln \frac{{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}_{\mathrm{\&beta;}0}}{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}\right)}^{1/q_{\mathrm{\&beta;}}}]}^{1/p_{\mathrm{\&beta;}}}\\ \mathrm{\&mu;}={\mathrm{\&mu;}}_0-\frac{s}{\exp (T_r/T)-1}\\ \stackrel{\&CenterDot;}{\mathrm{\&rho;}}={\mathrm{\&alpha;}}_1\sqrt{\mathrm{\&rho;}}\left|{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_e\right|-{\mathrm{\&alpha;}}_2{e}^{\frac{Q_{\mathrm{dm}}}{\mathrm{RT}}}\mathrm{\&rho;}\\ \stackrel{\&CenterDot;}{d}={\mathrm{\&beta;}}_0{d}^{-{\mathrm{\&gamma;}}_0}{T}^{-1}{e}^{\frac{Q_{\mathrm{pd}}}{\mathrm{RT}}}+{\mathrm{\&beta;}}_1\left|{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_e\right|d^{-{\mathrm{\&gamma;}}_1}{\mathrm{\&beta;}}_2{\stackrel{\&CenterDot;}{\mathrm{\&rho;}}}^{{\mathrm{\&gamma;}}_2}d\\ {\mathrm{\&sigma;}}_e=\mathrm{M\&tau;}\\ \stackrel{\&CenterDot;}{\mathrm{\&gamma;}}=M{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_e\end{array}\right.$

In formula: τ is the shear stress of titanium alloy, units MPa; τ _αand τ _βbe respectively the shear stress of α and β phase, units MPa; f _α, f _βbe respectively the volume fraction of α and β phase, f _α+ f _β=1; n ₁, n ₂for correction factor; with be respectively the limit stress of α and β phase, units MPa; T is for connecting temperature, unit K; R is gas constant, unit 8.3145Jmol ^-1k ^-1; △ G _αwith Δ G _βfor the apparent deformation activation energy of α and β phase, unit kJmol ^-1; for shear strain rate, unit s ^-1; with for the shear strain rate of α during 0K and β phase, unit s ^-1; μ is the modulus of shearing depending on temperature, unit GPa; ρ is dislocation desity, unit cm ^-2; for dislocation desity rate of change, unit cm ^-2s ^-1; B is Bai Shi vector, unit m; D is crystallite dimension, unit μm; for crystallite dimension rate of change, unit μm s ^-1; for equivalent strain rate, unit s ^-1; σ _efor equivalent stress, units MPa; Q _dmfor dislocation motion activation energy, unit 20kJmol ^-1; Q _pdfor grain boundary decision activation energy, unit 677.37kJmol ^-1; M is the Taylor factor; q _α, p _α, q _β, p _β, a, k, s, μ ₀, T _r, α ₁, α ₂, β ₀, β ₁, β ₂, γ ₀, γ ₁, γ ₂for material parameter;

The flow stress of TC4 alloy under different Joining Technology parameter and crystallite dimension is obtained according to above plastic yield constitutive relation; Get flow stress under 0.2% residual deformation as the yield strength σ of TC4 alloy under Joining Technology parameter and crystallite dimension _yield; According to slip line field theory and Mises yield criteria, linkage interface along the stress distribution of empty neck is:

${\mathrm{\&sigma;}}_{\left(x\right)}=2\frac{{\mathrm{\&sigma;}}_{\mathrm{yield}}}{\sqrt{3}}[1+\ln (\frac{x-c}{r_A}+1)](c\&le;x\&le;c+e)$

In formula, r _afor empty neck radius-of-curvature, unit μm;

Linkage interface contact mean stress under yield situation is:

$\stackrel{\&OverBar;}{\mathrm{\&sigma;}}=\frac{{\&Integral;}_c^{c+e}{\mathrm{\&sigma;}}_{\left(x\right)}\mathrm{dx}}{e}=2\frac{{\mathrm{\&sigma;}}_{\mathrm{yield}}}{\sqrt{3}}[1+\frac{r_A}{e}]\ln (1+\frac{e}{r_A})$

Under Bonding pressure, linkage interface contact stress is:

${\mathrm{\&sigma;}}^{\&prime;}=\frac{{\mathrm{pc}}_0-\mathrm{\&gamma;}}{e}$

In formula, p is Bonding pressure, units MPa; γ is surface energy, unit Jm ^-2; When time, Plastic Deformation Mechanism continuous action; And work as time, stress state does not meet yield condition, and Plastic Deformation Mechanism fails;

When Plastic Deformation Mechanism is done, the convex peak between cavity is divided into the N number of lamella being parallel to linkage interface, the length of i-th lamella is strain rate along the x-axis direction on i-th lamella z direction, suffered stress is approximately:

${\mathrm{\&sigma;}}_{\mathrm{iz}}\&ap;\frac{{\mathrm{pc}}_0}{w_i}=\frac{{\mathrm{pc}}_0}{c_0-c\sqrt{1-\frac{{z_i}^2}{h^2}}}$

If the length of t ' moment i-th lamella is w _i(t '), calculates according to the plastic yield constitutive relation of coupling crystallite dimension the length that a time step δ t is out of shape rear i-th lamella

According to volume conserva-tion principle before and after lamella distortion, obtain the thickness h of i-th lamella t '+δ t _i(t '+δ t)=w _i(t ') h _i(t ')/w _i(t '+δ t);

Finally, the gross thickness after the distortion of time interval δ t convex peak is obtained by superposition, the interfacial voids height namely under Plastic Deformation Mechanism effect according to meeting volume conservation principle before and after plastic yield, obtain the interface connecting length increment in time interval δ t under Plastic Deformation Mechanism effect

Step 4, based on the coupling steady state creep constitutive relation of crystallite dimension and creep machining function dynamic conditions, adopt lamella split plot design, the interface connecting length increment e in computing time interval δ t under creep machining function ₂, circular is as follows:

When linkage interface contact stress time, Plastic Deformation Mechanism stops, and creep mechanism starts effect; The steady state creep constitutive relation of coupling crystallite dimension is as follows:

$\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}=A\frac{\mathrm{D\&mu;b}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p{\left(\frac{\mathrm{\&sigma;}}{\mathrm{\&mu;}}\right)}^n$

In formula, for true strain speed, unit s ^-1; σ is trus stress, units MPa; D is coefficient of diffusion, unit m ²s ^-1; K is Boltzmann constant, 1.38 × 10 ^-23; B is Bai Shi vector, unit m; μ is modulus of shearing, units MPa; A, p, n are material parameter;

Material near linkage interface is divided into the N number of lamella be parallel to each other, suffered by each lamella, stress is

${\mathrm{\&sigma;}}_{\mathrm{iz}}\&ap;\frac{{\mathrm{pc}}_0}{w_i}=\frac{{\mathrm{pc}}_0}{c_0-c\sqrt{1-\frac{{z_i}^2}{h^2}}}$

The distortion in lamellar spacing direction is adopted to characterize its strain:

According to creep constitutive relation and equivalent strain rate with equivalent stress σ _eexpression formula, obtain the distortion expression formula for i-th lamella: ${\mathrm{dh}}_i=-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\&mu;b}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p\frac{h}{N}{\left(\frac{\sqrt{3}{\mathrm{\&sigma;}}_{\mathrm{iz}}}{2\mathrm{\&mu;}}\right)}^n\mathrm{dt},$ :

$\begin{array}{c}\mathrm{dh}=\underset{N\&RightArrow;\&infin;}{\lim }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{dh}}_i=-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\&mu;b}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p\underset{N\&RightArrow;\&infin;}{\lim }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\left(\frac{h}{N}\right){\left(\frac{\sqrt{3}{\mathrm{\&sigma;}}_{\mathrm{iz}}}{2\mathrm{\&mu;}}\right)}^n\mathrm{dt}\\ =-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\&mu;b}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p{\left(\frac{\sqrt{3}p}{2\mathrm{\&mu;}}\right)}^n\underset{N\&RightArrow;\&infin;}{\lim }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\left(\frac{h}{N}\right){\left(\frac{c_0}{w_i}\right)}^n\mathrm{dt}\\ \frac{\mathrm{dh}}{\mathrm{dt}}=-\frac{\sqrt{3}}{2}A\frac{\mathrm{D\&mu;b}}{\mathrm{kT}}{\left(\frac{b}{d}\right)}^p{\left(\frac{\sqrt{3}p}{2\mathrm{\&mu;}}\right)}^n\underset{0}{\overset{h}{\&Integral;}}{\left(\frac{c_0}{w_i}\right)}^n\mathrm{dz}\end{array}$

Then under creep machining function, empty altitude rate is:

${\stackrel{\&CenterDot;}{h}}_2=-\frac{\sqrt{3}}{2}{A}_c{\left(\frac{\sqrt{3}p}{2\mathrm{\&mu;}}\right)}^n\underset{0}{\overset{h}{\&Integral;}}{\left(\frac{c_0}{c_0-c\sqrt{1-\frac{z^2}{h^2}}}\right)}^n\mathrm{dz}$

According to material volume principle of invariance, under creep machining function, the rate of change of linkage interface connecting length is:

${\stackrel{\&CenterDot;}{e}}_2=-\frac{{\stackrel{\&CenterDot;}{h}}_2}{h}[c_0(\frac{4}{\mathrm{\&pi;}}-1)+e]$

Interface connecting length increment then in time interval δ t under creep machining function

Step 5, based on source, interface machining function dynamic conditions, the interface connecting length increment e under the machining function of interval δ t inner boundary computing time source ₃;

According to dynamic conditions and the computing method of source, interface machining function, obtain the interface connecting length rate of change under the machining function of source, interface interface connecting length increment then under the machining function of time interval δ t inner boundary source

Step 6, based on surface source machining function dynamic conditions, the interface connecting length increment e under the machining function of interval δ t inside surface computing time source ₄;

According to dynamic conditions and the computing method of surface source machining function, obtain the interface connecting length rate of change under surface source machining function interface connecting length increment then under the machining function of time interval δ t inside surface source

Step 7, by quadravalence Runge-Kutta alternative manner, the interface connecting length increment under each connection mechanism effect to be superposed, obtain the empty form parameter at the end of t '+δ t and the total interface connecting length increment e under each machining function;

If step 8 t '+δ is t<t, then with the empty form parameter at the end of t '+δ t for state parameter, make t '=t '+δ t, repeat step 3 ~ step 7, obtain the interface connecting length increment E under setting Joining Technology parameter and crystallite dimension; Otherwise the total interface connecting length increment at the end of t '+δ t is the interface connecting length increment E under setting Joining Technology parameter and crystallite dimension, interface bonding ratio A _f=E/c ₀.