CN104330300A - Method for indirectly measuring thermal-damage coupling strength of ultrahigh-temperature ceramic material - Google Patents

Abstract

The invention discloses a method for indirectly measuring the thermal-damage coupling strength of an ultrahigh-temperature ceramic material. The method comprises the steps of establishing a mathematical expression model of the material thermal-damage coupling strength and the elasticity modulus at different temperature according to the measured experimental data of the elasticity modulus of the ultrahigh-temperature ceramic material changing along with temperature and the elasticity modulus at reference temperature, and calculating the material thermal-damage coupling strength at the temperature corresponding to the elasticity modulus of the ultrahigh-temperature ceramic material. The method for indirectly measuring the thermal-damage coupling strength of the ultrahigh-temperature ceramic material has the technical effect that reliable prediction on the material thermal-damage coupling strength at various temperature is realized.

Description

Superhigh temperature ceramic material heat-damage stiffness of coupling indirect measurement method

Technical field

The present invention relates to a kind of intensity indirect measurement method of superhigh temperature ceramic material.

Background technology

Superhigh temperature ceramics (Ultra High Temperature Ceramics, be called for short UHTCs) refer in the most heat-resisting advanced ceramic that still can as usual use under having the conditions such as oxygen more than more than 2000 DEG C, this superhigh temperature ceramic material still has good inoxidizability when temperature reaches 1600 DEG C, and it is mainly used in the thermal protection system of the aircraft such as hypersonic missile, space shuttle as the hot junction of nose of wing, end cap and engine.

Elevated temperature strength described in present patent application refers to the strength of materials of temperature within the scope of 1200 DEG C-3200 DEG C.As the superhigh temperature ceramic material of hypersonic aircraft thermally protective materials, its elevated temperature strength weighs the important parameter of reliability of material.The elevated temperature strength improving superhigh temperature ceramic material is exactly the target that industry is pursued all the time, and this nature is associated with the measurement of elevated temperature strength.At present, about the research of elevated temperature strength lays particular emphasis on experimental study, the temperature of experiment test is also lower, does not reach the test condition of elevated temperature strength far away, and low-temperature test cannot obtain the data of elevated temperature strength.

At present, the difficulty of elevated temperature strength test is very large, and experimental expenses is expensive, does not have unified method of testing and standard, and test value is dispersed large.Intensity experiment belongs to a series of destructive test, needs a series of test specimen, comparatively large owing to there is initial imperfection and process variations to stupalith, by the strength of materials (σ corresponding under causing temperature T _th(T) experimental result) is inaccurate; In contrast, elastic modulus (E (T)) corresponding under temperature T only need use same test specimen to carry out testing, once experiment just can obtain the data of a series of elastic modulus E (T), avoids the inaccurate of the experimental result brought because of test specimen difference.

Ceramic three-point bending test is at high temperature used for elevated temperature strength (σ _th(T) prediction), because superhigh temperature ceramic material its tension and compression anisotropy of rising along with temperature strengthens gradually, need to eliminate the anisotropic impact of its tension and compression to 3 curved strength calculation formula corrections, but, how a reasonably correction inherently still unsolved difficult problem is made to 3 curved strength calculation formula at different temperatures.

The high temperature fracture intensity of the superhigh temperature ceramic material with micro-crack is called heat-damage stiffness of coupling, because the elevated temperature strength prediction of superhigh temperature ceramic material belongs at present still unsolved technical barrier, thus superhigh temperature ceramic material hot-prediction that damages stiffness of coupling is technical barrier in this area equally.

Summary of the invention

Technical matters to be solved by this invention is just to provide a kind of superhigh temperature ceramic material heat-damage stiffness of coupling indirect measurement method, and it can predict material heat-damage stiffness of coupling at each temperature.

Technical matters to be solved by this invention is realized by such technical scheme, according to elastic modulus under the temperature variant experimental data of superhigh temperature ceramic material elastic modulus recorded and reference temperature, set up the mathematical expression model of material heat-damage stiffness of coupling and elastic modulus under different temperatures, calculate material at the temperature corresponding with superhigh temperature ceramic material elastic modulus hot-damage stiffness of coupling.

Owing to establishing the mathematical expression model of material heat-damage stiffness of coupling and elastic modulus under different temperatures, and without any fitting parameter in this mathematical expression model; In mathematical expression model, the temperature variant experimental data of elasticity modulus of materials easily obtains from experiment, elastic modulus under reference temperature easily obtains by testing, thermal capacitance, Poisson ratio, melting heat etc. can be found from Materials Handbook easily, material heat-damage the stiffness of coupling of computational prediction avoids elevated temperature strength and tests the difficulty brought thus, achieve the prediction carrying out material heat-damage stiffness of coupling under existence conditions with mathematical expression model, mathematical expression model discloses the Physical Mechanism affecting material heat-damage stiffness of coupling.

Technique effect of the present invention is: the reliable prediction achieving material heat-damage stiffness of coupling at each temperature.

Accompanying drawing explanation

Accompanying drawing of the present invention is described as follows:

Fig. 1 is elastic modulus E (T) the temperature variant curve map of TiC stupalith;

Fig. 2 is the elevated temperature strength σ of TiC stupalith prediction _th(T) temperature variant curve map;

Fig. 3 is (the Δ σ under different temperatures T _th/ Δ l) with Δ l variation relation curve;

Fig. 4 is (the Δ σ under different temperatures T _th/ Δ E) with Δ E variation relation curve.

Embodiment

Design of the present invention is: simplify the difficult point that superhigh temperature ceramic material heat-rupture test is measured, from the parameter easily obtained as elasticity modulus of materials, level pressure thermal capacitance, Poisson ratio, melting heat grade, set up superhigh temperature ceramic material heat-damage stiffness of coupling mathematical expression model.

Below in conjunction with drawings and Examples, the invention will be further described:

This method invention is: according to the elastic modulus under the temperature variant experimental data of superhigh temperature ceramic material elastic modulus recorded and reference temperature, set up the mathematical expression model of material heat-damage stiffness of coupling and elastic modulus under different temperatures, calculate material at the temperature corresponding with superhigh temperature ceramic material elastic modulus hot-damage stiffness of coupling.

The step setting up the mathematical expression model of material heat-damage stiffness of coupling and elastic modulus under different temperatures is as follows:

The first step, set up material initial damage state different temperatures under the mathematical expression model of intensity and elastic modulus

The basic imagination that this mathematical expression model is set up is: 1., to a kind of certain material, think that it exists an energy storage limit, namely corresponding changeless Energy maximum value when material occurs to destroy, this maximal value can characterize by strain energy, also can characterize with heat energy.2., from the execution to material, think to there is a kind of quantitative equivalent relation between the heat energy of material storage and strain energy.

The energy storage ultimate value obtaining unit volume material by above imagination is

$W_{\mathrm{TOTAL}}=K(W_T\left(T\right)+\mathrm{\Δ}{H}_M)+W_{{\mathrm{\σ}}_{\mathrm{th}}}\left(T\right)---\left(1\right)$

In formula (1), W _tOTALfor the energy storage ultimate value of unit volume material, T is Current Temperatures, W _t(T) be corresponding heat energy, for strain energy corresponding during material damage, K is the energy conversion factor between heat energy and strain energy; Δ H _mfor melting heat.

Suppose that stupalith is when loading, stress and strain meets linear elasticity relation, then when stupalith destroys, corresponding strain energy is

$W_{{\mathrm{\σ}}_{\mathrm{th}}}\left(T\right)=\frac{{\left({\mathrm{\σ}}_{\mathrm{th}}\left(T\right)\right)}^2}{2E\left(T\right)}---\left(2\right)$

In formula (2), σ _th(T) be the temperature dependency intensity of initial damage state, E (T) is elastic modulus corresponding at T temperature.

With 0 DEG C for reference temperature, corresponding heat energy can be expressed as:

$W_T\left(T\right)={\∫}_0^T{C}_p\left(T\right)\mathrm{dT}---\left(3\right)$

In formula (3), C _p(T) be level pressure thermal capacitance corresponding at T temperature.

From formula (3):

W _T(0)＝0 (4)

T _mfor the fusing point of material, due to σ in formula (2) _th(T _m)=0, so

$W_{{\mathrm{\σ}}_{\mathrm{th}}}\left(T_m\right)=0---\left(5\right)$

As T=0, Δ H _m=0, following relational expression can be obtained by formula (4), (5) and (1):

$W_{\mathrm{TOTAL}}=W_{{\mathrm{\σ}}_{\mathrm{th}}}{|}_{T=0}=K(W_T\left(T\right)+{\mathrm{\ΔH}}_M){|}_{T=T_m}---\left(6\right)$

Can be obtained by formula (6):

$K=\frac{W_{{\mathrm{\σ}}_{\mathrm{th}}}{|}_{T=0}}{(W_T\left(T\right)+\mathrm{\Δ}{H}_M){|}_{T=T_M}}=\frac{{\left({\mathrm{\σ}}_{\mathrm{th}}^0\right)}^2}{{2E}_0}/({\∫}_0^{T_m}{C}_p\left(T\right)\mathrm{dT}+\mathrm{\Δ}{H}_M)---\left(7\right)$

In formula (7), e ₀be respectively the strength of materials at reference temperature 0 DEG C and elastic modulus.

To stupalith, due to its tensile property, comparatively compression performance is poor, studies its pulling strengrth more important.Consider uniaxial tension situation:

$\begin{array}{c}{W}_{\mathrm{TOTAL}}=K(W_T\left(T\right)+{\mathrm{\ΔH}}_M)+W_{{\mathrm{\σ}}_{\mathrm{th}}}\left(T\right)\\ =K({\∫}_0^T{C}_p\left(T\right)\mathrm{dT}+\mathrm{\Δ}{H}_M)+\frac{{\left({\mathrm{\σ}}_{\mathrm{th}}\left(T\right)\right)}^2}{2E\left(T\right)}\end{array}---\left(8\right)$

If known K, C _p(T), E (T), Δ H _mσ can be asked by formula (8) _th(T):

${\mathrm{\σ}}_{\mathrm{th}}\left(T\right)={\left[2E\left(T\right)\right[W_{\mathrm{TOTAL}}-K{\∫}_0^T{C}_p\left(T\right)\mathrm{dT}-K{\mathrm{\ΔH}}_M\left]\right]}^{1/2}---\left(9\right)$

Work as T<T _mtime: Δ H _m=0, so

${\mathrm{\&sigma;}}_{\mathrm{th}}\left(T\right)={\left[2E\left(T\right)\right[W_{\mathrm{TOTAL}}-K{\&Integral;}_0^T{C}_p\left(T\right)\mathrm{dT}\left]\right]}^{1/2}---\left(10\right)$

The leading portion equation of wushu (6) and (7) substitute into formula (10) and can obtain:

${\mathrm{\&sigma;}}_{\mathrm{th}}\left(T\right)={\left[\frac{{\left({\mathrm{\&sigma;}}_{\mathrm{th}}^0\right)}^2}{E_0}E\left(T\right)\right[1-\frac{1}{{\&Integral;}_0^{T_m}{C}_p\left(T\right)\mathrm{dT}+{\mathrm{\&Delta;H}}_M}{\&Integral;}_0^T{C}_p\left(T\right)\mathrm{dT}\left]\right]}^{1/2}---\left(11\right)$

In formula (11), σ _th(T) be the strength of materials corresponding under temperature T, E (T) is elastic modulus corresponding under temperature T, for the strength of materials at reference temperature 0 DEG C, E ₀for the elasticity modulus of materials at reference temperature 0 DEG C, C _p(T) be level pressure thermal capacitance corresponding under temperature T, Δ H _mfor melting heat, T _mfor the fusing point of material.

Second step, set up reference temperature under the mathematical expression model of the intensity relevant to material damage

According to the record of teaching material " fracture theory basis " (Fan Tianyou, Science Press, Beijing, 2003), for flat crackle, Critical fracture intensity following relation is there is with l:

${\mathrm{\&sigma;}}_0^f={\left[\frac{E_{l0}{G}_0^f}{2(1-v^2)l}\right]}^{1/2}---\left(12\right)$

In formula (12), E _l0, ν, be respectively the elastic modulus of material original state under reference temperature, Poisson ratio and energy to failure, l is the half long of crackle.

When material damage build-up effect is obvious, consider the impact of material damage accumulation on elastic modulus, if the crackle in material is uniformly distributed, then microcrack parameter l and elasticity modulus of materials E _l0relational expression be:

$E_{l0}=E_0{[1+\frac{16(1-v^2){\mathrm{Nl}}^3}{3}]}^{-1}---\left(13\right)$

In formula (13), E ₀for elastic modulus during material flawless, N is the crackle number in unit volume, i.e. crack density.The Poisson ratio ν of micro-crack on material does not have large impact.

Due to the accumulative process that high temperature is the microstructure damage that extensively distributes, and it is insensitive to single crack size, therefore for considering that damage accumulation is on the impact of intensity, formula (12) is utilized, fracture strength expression formula that (13) can be revised:

${\mathrm{\&sigma;}}_0^f={\left[\frac{E_0{G}_0^f}{2(1-v^2)l}{[1+\frac{16(1-v^2){\mathrm{Nl}}^3}{3}]}^{-1}\right]}^{1/2}---\left(14\right)$

In formula (14), for Critical fracture intensity, for energy to failure, E ₀for elastic modulus during material flawless, ν is Poisson ratio, and l is half length of crackle, and N is the crackle number in unit volume.

3rd step, set up superhigh temperature ceramic material heat-damage stiffness of coupling mathematical expression model

Along with the rising of temperature, the high temperature of hard brittle material is the accumulative process of the microstructure damage that extensively distributes, and insensitive to single crack size, high-temperature damage crackle is tending towards elliposoidal.

Work as T<T _mtime high temperature fracture intensity expression formula be:

$\begin{array}{c}{\mathrm{\&sigma;}}_{\mathrm{th}}(T,l)={\mathrm{\&sigma;}}_{\mathrm{th}}\left(T\right){\mathrm{\&sigma;}}_0^f/{\mathrm{\&sigma;}}_{\mathrm{th}}^0\\ ={\left[\frac{G_0^{f}E\left(T\right)}{2(1-v^2)l}\right[1-\frac{1}{{\&Integral;}_0^{T_m}{C}_p\left(T\right)\mathrm{dT}+{\mathrm{\&Delta;H}}_M}{\&Integral;}_0^T{C}_p\left(T\right)\mathrm{dT}\left]{[1+\frac{16(1-v^2){\mathrm{Nl}}^3}{3}]}^{-1}\right]}^{1/2}\end{array}---\left(15\right)$

(15) formula is simplified further:

The elastic modulus recorded by experiment is obviously temperature and the coefficient result of material damage, therefore, can be designated as E (T, N, l) by testing the elastic modulus recorded:

${E(T,N,l)=E\left(T\right)[1+\frac{16(1-v^2){\mathrm{Nl}}^3}{3}]}^{-1}---\left(16\right)$

Be out of shape by (16) formula:

$E\left(T\right)=E(T,N,l)[1+\frac{16(1-v^2){\mathrm{Nl}}^3}{3}]---\left(17\right)$

Wushu (17) substitutes into formula (15) and can obtain, T<T _mtime high temperature fracture intensity expression formula:

${\mathrm{\&sigma;}}_{\mathrm{th}}(T,l)={\left[\frac{G_0^{f}E(T,N,l)}{2(1-v^2)l}\right[1-\frac{1}{{\&Integral;}_0^{T_m}{C}_p\left(T\right)\mathrm{dT}+{\mathrm{\&Delta;H}}_M}{\&Integral;}_0^T{C}_p\left(T\right)\mathrm{dT}\left]\right]}^{1/2}---\left(18\right)$

For plane stress situation:

$G_0^f=\frac{K_{\mathrm{IC}}^2}{E_0}---\left(19\right)$

In formula (19), K _iCit is material plane strain fracture toughness.Research shows, along with the elevated insult crackle of temperature is tending towards elliposoidal, l is tending towards the radius of ball, so

${\mathrm{\&sigma;}}_{\mathrm{th}}(T,l)={\left[\frac{K_{\mathrm{IC}}^{2}E(T,N,l)}{2E_0(1-v^2){\left(\frac{3V}{4\mathrm{\&pi;\&alpha;}}\right)}^{1/3}}\right[1-\frac{1}{{\&Integral;}_0^{T_m}{C}_p\left(T\right)\mathrm{dT}+{\mathrm{\&Delta;H}}_M}{\&Integral;}_0^T{C}_p\left(T\right)\mathrm{dT}\left]\right]}^{1/2}---\left(20\right)$

In formula (20), σ _th(T, l) is heat-damage stiffness of coupling, K _iCfor material plane strain fracture toughness, E (T, N, l) is corresponding elastic modulus when there is crackle under temperature T, E ₀for the elastic modulus under reference temperature during material flawless, ν is Poisson ratio, C _p(T) be level pressure thermal capacitance corresponding under temperature T, Δ H _mfor melting heat, T _mfor the fusing point of material, V is single lesion volume, single lesion volume refers to the volume of a damage in test specimen, α is lesion shape coefficient, in order to describe the change of crack shape, observe half long l and the lesion shape factor alpha that can obtain crackle by experiment, major axis and the minor axis length of elliposoidal crackle can be determined by l and α, thus calculate single elliposoidal lesion volume V.

Embodiment

Utilize the superhigh temperature ceramic material heat-damage stiffness of coupling mathematical expression model of formula (20) below, for titanium carbide (TiC) pottery, implement this material heat-damage stiffness of coupling to predict, and the Sensitivity Analysis to intensity correlation parameter.

1, the elastic modulus under the temperature variant experimental data of elastic modulus of superhigh temperature ceramic material and reference temperature is tested

The temperature variant experiment test of elastic modulus adopts Chinese patent literature CN 102944466 A disclosed for the Mechanics Performance Testing apparatus and method under superhigh temperature well-oxygenated environment, TiC ceramic test piece is installed on proving installation, close the fire door of high temperature furnace, be heated to different predetermined temperatures, insulation, load, the experimental data of record load and test specimen elongation, by data analysis obtain TiC stupalith elastic modulus E (T) temperature variant data, depict its change curve as shown in Figure 1, in FIG, due to the shortage of elastic modulus E (T) high temperature experimental data, according to existing experimental data, elastic modulus is extrapolated, so that next step utilizes mathematical expression model prediction TiC stupalith heat-damage stiffness of coupling.

Titanium carbide (TiC) stupalith and material heat-damage the relevant parameter of stiffness of coupling in table 1.

The material parameter of table 1 TiC

Material parameter	Parameter value
		E 0(GPa)	444
ν	0.195
		K IC(MPa·m 1/2)	3.82
T m(℃)	3016.85
		ΔH M(cal/mol)	17000
C p(T)(cal/mol)	11.94+0.23×10 -3T-3.53×10 5T -2+0.45×10 -6T 2

In upper table, E ₀obtained by experiment, ν, K _iC, T _m, Δ H _m, C _p(T) by consulting handbook, document obtains.

2, by the mathematical expression model of formula (20), calculate at the temperature corresponding with superhigh temperature ceramic material elastic modulus material heat-damage stiffness of coupling

By elastic modulus E (T) the temperature variant data of Fig. 1, according to the mathematical expression model of formula (20), calculate material heat-damage stiffness of coupling, obtain the material heat-damage stiffness of coupling σ predicted _th(T, l) temperature variant curve map, as shown in Figure 2, uses σ in Fig. 2, Fig. 3 and Fig. 4 _threpresent material heat-damage stiffness of coupling σ _th(T, l).As can be seen from Figure 2: predicted value and experiment value coincide better.So the prediction of this method to superhigh temperature ceramic material heat-damage stiffness of coupling is reliable.

Heat-damage stiffness of coupling to the susceptibility of crack size as shown in Figure 3, Δ σ in figure _thfor intensity increment, Δ l is the long increment of crackle half.In Fig. 3, curve a, b, c, d, e, f be respectively T=24 DEG C, T=903 DEG C, T=1502 DEG C, T=2102 DEG C, T=2902 DEG C, T=3017 DEG C Δ σ _ththe relation that/Δ l changes with Δ l.

As can be seen from Figure 3, along with the rising of temperature, material heat-damage stiffness of coupling σ _thmore and more lower to the susceptibility of crack size; Can find out that sensitivity increases and decreases the variation tendency of amplitude, ao l with l by the variation tendency of curve corresponding to lower temperature, l increasing degree is more little more remarkable; Reduction amplitude is more large more remarkable.And temperature higher time, material heat-damage stiffness of coupling σ _thlower and change is little to the susceptibility of crackle size increase Δ l.

Heat-damage stiffness of coupling to the susceptibility of elastic modulus as shown in Figure 4, Δ σ in figure _thfor intensity increment, Δ E is elastic modulus increment.In Fig. 4, curve g, h, i, j, k, l be respectively T=24 DEG C, T=903 DEG C, T=1502 DEG C, T=2102 DEG C, T=2902 DEG C, T=3017 DEG C Δ σ _ththe relation that/Δ E changes with Δ E.

As can be seen from Figure 4, along with the rising of temperature, material heat-damage stiffness of coupling σ _thmore and more stronger to the susceptibility of elastic modulus, namely improve elastic modulus during high temperature to material heat-damage stiffness of coupling σ _thimprove successful; As can be seen from the variation tendency of curve corresponding to higher temperature, along with the increase of elastic modulus amplification Δ E, material heat-damage stiffness of coupling σ _thits sensitivity is reduced gradually; And temperature lower time, material heat-damage stiffness of coupling σ _thlower and change is little to the susceptibility of elastic modulus amplification Δ E.

Compared with prior art, advantage of the present invention is:

(1) C is passed through _p(T) temperature is separated with defective effect, and C _p(T) can obtain from Materials Handbook easily;

(2) energy to failure G _fvariation with temperature trend is more complicated, and experiment is difficult to obtain, and this mathematical expression model only uses the energy to failure under reference temperature, avoids the dependence to its temperature dependency;

(3) mathematical expression model discloses the Physical Mechanism controlling elevated temperature strength better;

(4) whole temperature history is applicable to;

(5) can transform to various situation easily, be Florence Griffith strength model when not considering temperature impact, when not considering the accumulation that damages and the change of shape.

Claims (2)

1. superhigh temperature ceramic material heat-damage stiffness of coupling indirect measurement method, it is characterized in that: according to the elastic modulus under the temperature variant experimental data of superhigh temperature ceramic material elastic modulus recorded and reference temperature, set up the mathematical expression model of material heat-damage stiffness of coupling and elastic modulus under different temperatures, calculate material at the temperature corresponding with superhigh temperature ceramic material elastic modulus hot-damage stiffness of coupling.

2. superhigh temperature ceramic material according to claim 1 heat-damage stiffness of coupling indirect measurement method, the step that it is characterized in that setting up the mathematical expression model of material heat-damage stiffness of coupling and elastic modulus under different temperatures comprises:

The first step, set up material initial damage state different temperatures under the mathematical expression model of intensity and elastic modulus be

${\mathrm{\&sigma;}}_{\mathrm{th}}\left(T\right)={\left[\frac{{\left({\mathrm{\&sigma;}}_{\mathrm{th}}^2\right)}^2}{E_0}E\left(T\right)\right[1-\frac{1}{{{\&Integral;}_0^{T_m}C}_p\left(T\right)\mathrm{dT}+{\mathrm{\&Delta;H}}_M}{\&Integral;}_0^T{C}_p\left(T\right)\mathrm{dT}\left]\right]}^{1/2}$

In formula, σ _th(T) be the strength of materials corresponding under temperature T, E (T) is elastic modulus corresponding under temperature T, for the strength of materials at reference temperature 0 DEG C, E ₀for the elasticity modulus of materials at reference temperature 0 DEG C, C _p(T) be level pressure thermal capacitance corresponding under temperature T, Δ H _mfor melting heat, T _mfor the fusing point of material;

Second step, set up reference temperature under the mathematical expression model of the intensity relevant to material damage be

${\mathrm{\&sigma;}}_0^f={\left[\frac{E_0{G}_0^f}{2(1-v^2)l}{[1+\frac{16(1-v^2){\mathrm{Nl}}^3}{3}]}^{-1}\right]}^{1/2}$

In formula, for Critical fracture intensity, for energy to failure, E ₀for the elastic modulus under reference temperature during material flawless, ν is Poisson ratio, and l is half length of crackle, and N is the crackle number in unit volume;

3rd step, set up superhigh temperature ceramic material heat-damage stiffness of coupling mathematical expression model and be

${\mathrm{\&sigma;}}_{\mathrm{th}}(T,l)={\left[\frac{K_{\mathrm{IC}}^{2}E(T,N,l)}{{2E}_0(1-v^2){\left(\frac{3V}{4\mathrm{\&pi;\&alpha;}}\right)}^{1/3}}\right[1-\frac{1}{{\&Integral;}_0^{T_m}{C}_p\left(T\right)\mathrm{dT}+{\mathrm{\&Delta;H}}_M}{\&Integral;}_0^T{C}_p\left(T\right)\mathrm{dT}\left]\right]}^{1/2}$

In formula, σ _th(T, l) is heat-damage stiffness of coupling, K _iCfor material plane strain fracture toughness, E (T, N, l) is corresponding elastic modulus when there is crackle under temperature T, E ₀for the elastic modulus under reference temperature during material flawless, ν is Poisson ratio, and V is single lesion volume, and α is lesion shape coefficient, C _p(T) be level pressure thermal capacitance corresponding under temperature T, Δ H _mfor melting heat, T _mfor the fusing point of material.