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Fracture Toughness of Si3N4/S45C Joint with an Interface Crack Liedong Fu, Yukio Miyasita and Yoshiharu Mutoh Copyright AD-TECH.; licensee AZoM.com Pty Ltd.
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Abstract
Fracture toughness tests were carried out for Si3N4/S specimens with interface cracks of different lengths. It was found that the specimen with a crack of has higher apparent fracture toughness than those with cracks of and due to the reduction of the residual stress. Fracture propagated into Si3N4 from the crack tip in the direction of 40o for cracks of and while it propagated along the interface for crack of . Elasto-plastic analysis was carried out considering S as the linear hardening material and Si3N4 as the elastic material. It was found that the stress around the crack tip is dominated by an elasto-plastic singular stress field, which is substantially the same as the elastic singular stress field of an interface crack. Evaluation of the fracture path and toughness was carried out based on the stress intensity factors of the elasto-plastic singular stress field. Keywords
Interface Crack, Fracture Toughness, Si3N4/S Joint, Thermal Residual Stress, Elasto-plastic Analysis Introduction
The ceramic/metal joints have been increasingly applied in a wide range of engineering fields because the ceramic has stable mechanical properties at high temperature and good resistance to wear, erosion and oxidation. However, the difference of material properties between metal and ceramic induces stress singularities at the interface edge. Moreover, high thermal residual stress will be induced during the cooling process due to the mismatch of the thermal expansion
coefficients. The stress singularity together with the thermal residual stress degrades the strength of ceramic/metal joint and makes the evaluation of the strength difficult. Many works have been done about the residual stress and the strength evaluation of ceramic/metal joints. For example,
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Kobayashi et al. [1, 2] have investigated the bending strength and residual stress of Si3N4/S joint and the effect of the size of the specimen on the bending strength. Qiu et al. [3] have investigated the influence of residual stress and cyclic load on the strength of Si3N4/S joint. However, due to the complexity of the problem, a generalized evaluation method for the ceramic/metal joint has not yet been proposed.
The elastic solution of the singular stress field of the interface crack has been studied since 1959 [4-9]. Rice [10] has summarized the work in this field and set up the elastic fracture mechanics concepts for interfacial cracks. Yuuki et al. [11, 12] have proposed the maximum normal stress criteria for predicting fracture path and strength of ceramic/metal joint based on the elastic
theory.The plastic deformation of metal will inevitably appear near the crack tip due to the stress singularity. For most of the ceramic/metal joints, the plastic deformation of metal has a significant influence on the strength of the ceramic/metal joint. Due to the analytical complexity, the evaluation of the fracture path and strength of ceramic/metal joint based on the elasto-plastic theory has not yet been made.
In this study, four point bending tests of Si3N4/S joint specimens with an interface crack were carried out. Evaluation of the fracture path and fracture toughness was attempted based on the elasto-plastic analysis. Experimental Specimen Preparation
Figure 1 shows the geometry and dimensions of Si3N4/S joint specimen. The silver based brazing alloy (wt% is: Ag, 71%, Cu, 27%, Ti, 2%) with 60 μm thickness was used for the bonding between Si3N4 ceramics and S steel. Brazing was carried in a vacuum furnace
(2.5x10-5 Torr). The temperature of the furnace was increased at a rate of 20oC/min up to the brazing temperature of 850oC and kept for 10 min, then decreased at a rate of 10oC/min. The joining surfaces were polished with diamond powder of 0.25 μm diameter. During the brazing, a contact pressure of 0.002 MPa was applied.
After brazing, an interface crack was introduced by the electric discharge method with the cutting wire of diameter. Four specimens with different crack lengths were prepared. Two of the specimens had crack lengths of and the other two specimens had crack lengths of and .
Figure 1. Fracture toughness specimen. Experimental Results
Four point bending tests were carried out on the fracture toughness specimens at a crosshead speed of /min. Table 1 shows the results of the fracture toughness. The apparent fracture toughness is defined as:
(1)
with
(2) (3)
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Where Pf is the fracture load, a is the crack length, w the specimen width, t the specimen highness, L2 the outer span and L1 the inner span.
Table 1. Result of the fracture toughness tests.
1 2 3 4 1.0 2.0 4.0 4.0 285.4 237.8 1649.0 1744.2 17.128 14.27 98.95 104.65 1.0436 1.0530 1.2561 1.2561 0.9807 1.1607 12.4317 13.1478 As can be seen in Table 1, the specimens with a crack length of indicate a higher fracture load than those with shorter crack lengths of 1.0 and . As the residual stress will redistribute after cutting [2], the relaxation of thermal residual stress for longer crack length may be a possible reason.
Figure 2 shows the macroscopic observation of the fractured specimen. For the specimens with a crack length of 1.0 and , crack propagated into Si3N4 directly from the initial crack tip in the direction of about 40o. For the specimens with a crack length of , the crack propagated along the interface for about and then kinked into Si3N4 in a direction of about 10o to the interface.
(a) a = (b) a = (c) a =
(d) a =
Figure 2. Fractured specimens.
Oscillatory Singular Stress Field of The Interface Crack and The Maximum Normal Stress Criteria The elastic solution of the stress field of an interface crack has been accomplished by the Willims [4], Erdogan [5, 6], England [7] and Sih et al. [8, 9]. It has been found that the stress field near the interface crack tip has the oscillatory singularity. Under the polar coordinate with the origin located at the crack tip, the stress field can be expressed as
(4)
Here is the bi-material constant that can be expressed as
(5)
(6)
where µj and vj are the shear modulus and the Poisson’s ratio of the materials, respectively.
The stress intensity factors of the oscillatory singular stress field are defined as
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(7)
where, l is the reference length to eliminate the dimension of the oscillatory term. Usually l takes the value of the whole crack length, i.e. l=2a.
When the stress along the interface has been known, the stress intensity factors can be can be extrapolated as:
(8)
(9)
Yuuki et al. [11, 12] have proposed up the maximum normal stress criteria for the fracture of interface crack. Considering that the value of is very small, the normal stress can be approximately expressed as
(10)
where
(11)
W1= e-ε(π-θ), W2= eε(π+θ) (12)
(13)
The direction of the maximum normal stress can be determined from:
∂B(θ,ε,y)/∂ θ = 0 (14)
Let θ0 represent the direction of the maximum normal stress, the corresponding stress intensity factor can be expressed as:
(15)
Fracture will occur along the direction of θ0 when Kθmax reaches the KIC value of the base material. It should be noted that fracture may occur along the interface when θ0 becomes smaller than certain value, since the strength of interface is usually lower than that of the base material.
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Elasto-Plastic Singular Stress Field at The Interface Crack Tip
The elasto-plastic singular stress field for a linear hardening material [13] has been found to be substantially the same as that of elastic material whose elastic constants are defined as:
(16)
where E is the Young’s modulus and H’ the hardening coefficient.
Therefore, the elasto-plastic singular stress field at the interface crack tip is substantially the same as the elastic singular stress field of the interface crack tip. The governing region of the elasto-plastic singular stress field will be confined in a small region around the crack tip inside the yield zone. For ceramic/metal joint, considering that the value of hardening coefficient is much less than the value of Young’s modulus, it can be found from Eq. (16) and Eq. (5) that
(17)
FEM Analysis and Evaluation of Fracture Path and Toughness Based on the Elasto-Plastic Stress Intensity Factors
FEM analysis was carried out under plane stress condition using the program of ABAQUS. Si3N4 is assumed as an elastic material whose material constants are independent of temperature and E=289 GPa, v=0.25 and CTE=4.2x10-6. S45C steel is assumed as a linear hardening material with the material constants listed in Table 2 [14]. The stress free temperature is considered to be 550oC for the analysis of the thermal residual stress.
Table 2. Material constants of S45C
E (GPa) v σY- (MPa) H’ (MPa) CTE (10-6) 206 0.3 375 1381 11.71 206 0.3 348 2056 12.17 201 0.3 333 2680 12.63 197 0.3 309 2325 13.09 192 0.3 280 1685 13.55 187 0.3 241 1026 14.01 183 0.3 193 687 14.47 For comparison, the elastic analysis was also carried out. Calculated from the elastic constants of 25oC, the bi-material constantfor elastic case is
0.01588. Table 3 lists the stress intensity factors as well as the direction of the maximum normal stress obtained by the elastic analysis. It can be found that the
value due to residual stress is much higher than
and the values of θ0 due
to the residual stress are almost the same, which are about 70o. The specimen with a crack length of 2.0 mm has the maximum value of Kθmax due to the residual
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stress. The values of Kθmax due to superposition of residual stress and applied stress during fracture toughness test are close to those due to the residual stress. Table 3. Stress intensity factors and the direction of the maximum normal stress
according to the elastic analysis.
a=1mm a=2mm a=4mm K1=1.50 K2=21.05 K1=0.5 K2=25.4 K1=-0.01 K2=24.8 25.79 29.52 28.54 69o 70o 71o K1=2.5 K2=21.7 K1=1.63 K2=25.42 K1=14.0 K2=25.1 26.45 30.21 37.69 68o 69o 61o However, the results of elastic analysis apparently contradict with that the value of Kθmax is much higher than KIC value of Si3N4, which is about 6.0 MPa√m [15]. Also, the elastic analysis cannot explain why the specimen with a=4.0 mm indicates higher fracture load than the specimen with a=1.0 mm since Kθmax due to the residual stress for a=4.0 mm is larger than that for a=1.0 mm.
Figures 3 and 4 show the stress distribution the interface obtained by the
elasto-plastic analysis. A line with the slop of –0.5 is also plotted in the figures for reference. We can see that the curves are almost parallel to the reference line in the region r<10-6m, which indicates that the stress near the crack tip is dominated by the elasto-plastic singular stress field.
Figure 3. Normal stress distribution along the interface.
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Figure 4. Shear stress distribution along the interface.
Figures 5 and 6 show the uncoupled components defined by Eq. (8) and Eq. (9). Different from the elastic case, here the reference length l takes thevalue of 1.0-6 m, which is close to the size of governing region of the elasto-plastic singular stress field. Figure 5 shows stress distribution due to residual stress and Figure 6 shows the stress distribution due to residual stress and applied load. It can be found that the curves are almost parallel to reference line in the region r<1.0-5 m.
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Figure 5. Distribution of the decoupled components along the interface for the residual stress.
Figure 6. Distribution of the decoupled components along the interface at the fracture of specimen.
Table 4 lists the stress intensity factors and the directions of maximum normal stress obtained by the elasto-plastic analysis. It can be found that Kθmax due to residual stress decreases in the sequence of a=2.0 mm, a=1.0 mm and a=4.0 mm. This result can explain why the specimen with a crack length of 4.0 mm indicates higher fracture load compared to the other specimens. The applied load tends to decrease the value of K2. The decrease of K2 for a=4.0 mm is especially obvious and the value of θ0 for a=4.0 mm is 33o, which is much smaller than those for a=1.0 mm and a=2.0 mm. This agrees with the experimental result, where the specimens with a=1.0 mm and a=2.0 mm fractured with an angle of about 40o from the interface, while the specimens of a=4.0 mm fractured along the interface. The values of Kθmax due to residual stress and applied stress, that is when fracture occurred, are almost the same regardless of the crack length. They are also close to the KIC value of Si3N4, although less than it. Kobayashi et al [1] have found in the bending tests of Si3N4/S45C joint that the results can be divided into two groups, in which one shows a relatively high strength while the other shows a very low value. One reason for a low strength is considered to be the existence of a crack in the ceramic, since the cracks are easily initiated from the inherent defect during cutting after joining. This can be also considered to be one of the reasons why the Kθmax values when fracture occurred are less than the KIC value of Si3N4. Table 4. Stress intensity factors and the direction of the maximum normal stress
according to the elasto-plastic analysis.
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a=1mm a=2mm a=4mm K1=-0.15 K2=2.51 K1=-0.22 K2=2.80 K1=-0.30 K2=2.45 2.68 2.95 2.51 71o 71o 72o K1=0.43 K2=2.50 K1=0.78 K2=2.76 K1=3.82 K2=0.76 3.09 3.65 4.28 67o 65o 33o Conclusions
Fracture toughness tests were carried out on Si3N4/S45C joint specimens with interface cracks of different lengths. Evaluation of fracture path and fracture toughness was carried out based on elasto-plastic analysis in which S45C steel was assumed as a linear hardening material. The conclusions obtained can be summarized as:
• The thermal residual stress has a significant effect on the fracture toughness of
the joint. Due to the effect of residual stress, the specimen with a crack length of 4.0 mm has higher fracture toughness than those with crack lengths of 1.0 mm and 2.0 mm. A crack propagated into Si3N4directly from the initial crack tip in the direction of 40o for crack lengths of 1.0 mm or 2.0 mm, while it propagated along the interface for the crack length of 4.0 mm.
• Stress near the crack tip is dominated by the elasto-plastic singular stress
field. Maximum σθ criterion based on the elasto-plastic singular stress field could be successfully applied for evaluating the fracture path and fracture toughness value. 3. Kθmax value due to the residual stress decreases in the sequence of a=2.0 mm, a=1.0 mm and a=4.0 mm. This is the same sequence of fracture load of the specimens with a=2.0 mm, a=1.0 mm and a=4.0 mm. The applied stress resulted in a decrease of the value of K2. The decrease of K2 for a=4.0 mm was significant and the value of θ0 for a=4.0 mm was much smaller than those for a=1.0 mm and a=2.0 mm. This agrees with the experimental result that the specimens with a=1.0 mm and a=2.0 mm fractured with an angle of about 40o to the interface, while the specimens of a=4.0 mm fractured along the interface. The Kθmax values at the fracture of specimens were almost same for all the specimen and were close to the KIC value of Si3N4. References
1. H. Kobayashi, Y. Arai, H. Nakamura and T. Sato, “Strength Evaluation of Ceramic-Metal Joints”,
Materials Science and Engineering, A143 (1991) 91-102.
2. H. Kobayashi, H., Nakamura, A. Todoroki, W. Park, T. Koide and H. Taniai, Effect of specimen cut off
and size on bending strength of ceramic/metal joints, Trans. of JSME, A60-569 (1994) 65-70. 3. J.H. Qiu, S. Nakamura, M. Kawagoe and M. Morita, “Influence of Joining Strength of Si3N4/S45C on
Residual Stress”, Journal of Inorganic Materials, 13-4 (1998) 167-172.
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4. M.L. Williams, “The Stress Around a Fault or Crack in Dissimilar Media”, Bulletin of the Seismological
Society of America, 49-2 (1959) 199-204.
5. F. Erdogan, “Stress Distribution in Bonded Dissimilar Materials with Cracks”, J. Appl. Mech., 32
(1965) 403-411.
6. F. Erdogan, “Stress Distribution in Bonded Dissimilar Materials Containing Circular Ring-shaped
Cavities”, , 32 (1965) 829-836.
7. A. H. England, “A Crack between Dissimilar Medias”, J. Appl. Mech., 32 (1965) 400-407. 8. G..C. Sih and J. R. Rice, “The Bending of Plates of Dissimilar Materials with Cracks”, J. Appl. Mech.,
31 (1964) 477-483.
9. J. R. Rice and G.C. Sih, “Plane Problems of Cracks in Dissimilar Media”, J. Appl. Mech., 32 (1965)
418-423.
10. J. R. Rice, “Elastic Fracture Mechanics Concepts for Interfacial Cracks”, J. Appl. Mech., 55 (1988)
98-103.
11. R. Yuuki and J.Q. Xu, Eng. Fract. Mech., “Stress Based Criterion for an Interface crack Kinking out of
the Interface in Dissimilar Materials”, 41-5 (1992) 635-644.
12. R. Yuuki, J.Q. Xu and Y. Mutoh, “Evaluation of Fracture and Strength of Metal/Ceramic bonded Joints
Based on Interfacial Fracture Mechanics”, Trans. of JSME, A60-569 (1994) 37-45.
13. J.Q. Xu and L. Fu, “Stress Field near an Interface Edge of Linear Hardening Materials”,Journal of
Zhejiang University: Science V No.3-1 (2002) 13-18.
14. N. Okabe, M. Takahashi, X. Zhu, K. Kagawa and M. Maruyama, “Residual Stress and Fatigue
Strength Properties of Ceramic/Metal joints”, J. Soc. Mat. Sci., Japan, 48-12 (1999) 1416-1422. 15. Y. Mutoh and I. Yumoto, “Fracture Toughness Evaluation for Ceramics/Metal Joints”, Trans. of the Symposium of Material Mechanics of JSME, No.900-50 (1990) 185-190.
Si3N4/S45C的断裂韧性界面裂纹
Liedong富，鸠山由纪夫Miyasita和吉春Mutoh 题目 摘要 关键词 导言 实验 样品制备 实验结果
振荡奇异应力场的界面裂纹和最大正应力准则 弹塑性应力场奇异界面裂纹尖端
有限元分析与评价路径和断裂韧性基于弹塑性应力强度因子 结论 参考文献 联系方式 摘要
断裂韧性试验，进行了Si3N4/S45C标本界面裂纹的长度不同。 结果发现，试样的裂纹的4毫米具有较高的断裂韧性明显高于裂缝的1毫米和2毫米由于减少残余应力。 骨折繁殖到第3硅ñ 4裂纹尖端的方向40 °的裂缝1毫米和2毫米，而它繁殖沿界面裂纹的4毫米。 弹塑性分析进行了审议S45C的线性硬化材料和Si 3 ñ 4弹性材料。 结果发现，
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周围的应力裂纹尖端主要是由弹塑性奇异应力场，这是大致相同的弹性应力场奇异的界面裂纹。 评价断裂韧性和道路进行了基于应力强度因子的弹塑性奇异应力场。 关键词
界面裂纹，断裂韧性，热残余应力，弹塑性分析 导言
陶瓷与金属的联合已越来越多地应用在广泛的工程领域，因为陶瓷具有稳定的力学性能在高温下和良好的抗磨损，侵蚀和氧化。 然而，不同的材料性能之间的金属，陶瓷诱导应力奇异界面优势。 此外，较高的热残余应力会引起在冷却过程由于不匹配的热膨胀系数。应力一道奇异的热残余应力降低强度的陶瓷与金属的联合，使评价兵力困难。许多工程已完成的残余应力和强度评价陶瓷与金属接头。举例来说，小林等人。调查的抗弯强度和残余应力的Si3N4/S45C联合和影响大小的标本的抗弯强度。调查的影响，残余应力和循环荷载强度的Si3N4/S45C联合。但是，由于问题的复杂性，普遍评价方法的陶瓷/金属联合尚未提出。弹性解决奇异应力场的界面裂纹，研究了自1959年以来， 赖斯总结了在这一领域的工作，并建立弹性断裂力学的概念，界面裂缝。 佑辉等人提出了最高标准的正常应力性骨折的预测路径和强度的陶瓷与金属的联合为基础的弹性理论。 塑性变形的金属将不可避免地出现附近的裂纹尖端由于应力奇异。对于大多数陶瓷与金属接头的塑性变形金属有重大影响的力量，陶瓷与金属联合。由于复杂的分析，评价骨折的路径和强度的陶瓷/金属的联合为基础的弹塑性理论尚未作出。在这项研究中，四点弯曲试验的标本的界面裂纹进行。评价断裂的路线和断裂韧性尝试基于弹塑性分析。 实验 标本 制备
图1
图1显示了几何形状和尺寸的标本。 基于银钎焊合金（银， 71 ％ ，铜， 27 ％ ，钛， 2 ％ ）与60微米的厚度用于粘接硅之间的陶瓷和S45C钢。 钎焊是在真空炉
摄氏度摄氏度
（ 2.5x10 -5子） 。 的温度炉提高率为20 /分钟到钎焊温度为850 ，并保持
摄氏度
10分钟，然后在一个下降率为10 /分钟。 加入表面抛光钻石粉直径0.25微米。 在钎焊，一个接触压力0.002兆帕适用。
钎焊后，一个界面裂纹介绍了放电的方法切割线的直径0.1毫米。 四标本不同裂纹长度准备。两个标本了裂纹长度为4.0毫米，而其他两个标本了裂纹长度为1.0毫米和2.0毫米。 实验结果
四点弯曲试验进行了断裂韧性标本在十字头速度为0.5毫米/分钟。 表1显示的结果，断裂韧性。
表观断裂韧性的定义为：
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其中p f是断裂负荷，一个是裂纹长度，瓦特试样宽度，吨试样高度，L2 外跨度和L 1 ，内跨度。
表1 。 结果断裂韧性试验。
一 二 三 四 1.0 2.0 4.0 4.0 285.4 237.8 1649.0 1744.2 17.128 14.27 98.95 104.65 1.0436 1.0530 1.2561 1.2561 0.9807 1.1607 12.4317 13.1478 可以看出，在表1 ，标本与裂纹长度的四点○毫米表明较高的断裂载荷比那些短的裂纹长度的1.0和2.0毫米。 由于残余应力重新分配后，将削减放宽对热残余应力的长期裂纹长度可能是一个可能的原因。图2显示了宏观观测裂缝标本。裂缝延续到标本的裂纹长度的1.0和2.0毫米，硅直接从第四处初始裂纹尖端的方向约40°。对于标本的裂纹长度为4.0毫米，沿裂纹传播的接口约1.0毫米，然后扭折到四个方向约10°接口。
(a) a = 1.0mm
(b) a = 2.0mm
(c) a = 4.0mm
(d) a = 4.0mm
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振荡奇异应力场的位界面裂纹和最大正应力准则
Willims ，埃尔多安，英格兰等，已经完成了弹性的解决方案的应力场的界面裂纹分析。人们发现，应力场附近的界面裂纹尖端的裂纹样子奇特。根据极坐标位于裂纹尖端的应力场可表示为
这里 是材料常数，可表示为
而μ j 和 V j的剪切模量和泊松比的材料，分别。 应力强度因子的应力场被界定为
在那里， l是长度的参考，以消除层面的振荡任期。 1-5 升的价值需要对整个裂纹长度，即升= 2A型。
当应力沿界面已众所周知，应力强度因子可以可以推断为：
佑辉等人提出了最大正应力准则骨折的界面裂纹。考虑到非常小，正常的压力大约可以表示为
W1= e-ε(π-θ), W2= eε(π+θ)
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该方向的最大正应力可确定：
∂B(θ,ε,y)/∂ θ = 0
让θ 0代表方向的最大正应力，相应的应力强度因子可以表示
为：
骨折将发生方向的 θ 0 当k θmax 达到的KIC的基础材料。 应该指出的是，可能会发生断裂沿界面 θ 0 时 变得小于一定的价值，因为强度的界面通常是低于基础材料。 弹塑性应力场奇异界面裂纹尖端
弹塑性奇异应力场的线性硬化材料已被发现基本上相同的弹性材料的弹性常数的定义是：
其中E是杨氏模量和H '硬化系数。
因此，弹塑性应力场奇异的界面裂纹尖端是大致相同的弹性应力场奇异的界面裂纹尖端。 管辖区域的弹塑性奇异应力场将限于在一个小附近地区的内裂纹尖端区的产量。 陶瓷与金属联合，考虑到价值的硬化系数是远远低于价值的杨氏模量，可以发
现
有限元分析法和评价路径和断裂韧性基于弹塑性应力强度因子
有限元分析是进行平面应力条件下使用的程序。 弹性材料，其材料常数是完全独立的温度E = 289千兆，W= 0.25和热膨胀系数= 4.2x10 -6 。 S45C钢承担作为一个线性硬化材料与材料常数列于表2 [ 14 ] 。 免费的温度应力被认为是550° ，用于分析的热残余应力。
表2 。 材料常数S45C
E (GPa) v σY- (MPa) H’ (MPa) CTE (10-6) 206 0.3 375 1381 11.71 206 0.3 348 2056 12.17 201 0.3 333 2680 12.63 197 0.3 309 2325 13.09 摄氏度
192 0.3 280 1685 13.55 187 0.3 241 1026 14.01 183 0.3 193 687 14.47 作为比较，弹性分析还开展。 计算出的弹性常数25
，双向材料常数 弹性是
0.01588案件。 表3列出的应力强度因子，以及方向的最大正应力得到了弹性分析。它
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可发现，价值由于残余应力远远高于 和价值观θ0由于残余应力几乎是相同的，这是约
70o试样的裂纹长度的2.0毫米的最高值的K θmax由于残余应力。K θmax由于叠加的残余应力和应用应力断裂韧性测试已接近那些由于残余应力。
表3 。 应力强度因子和方向的最大正应力按照弹性分析。
a= 1mm a=2mm a=4mm K1=1.50 K2=21.05 K1=0.5 K2=25.4 K1=-0.01 K2=24.8 25.79 29.52 28.54 69 ö 70 ö 71 ö K1=2.5 K2=21.7 K1=1.63 K2=25.42 K1=14.0 K2=25.1 26.45 30.21 37.69 68 ö 69 ö 61 ö 然而，结果显然弹性分析矛盾与价值的K θmax远远高于K 集成电路价值的硅三语4 ，大约是6.0
Mpa/m。 此外，弹性分析不能解释为什么标本与= 4.0毫米表明较高的断裂载荷比试样与a=1.0毫米自K θmax由于残余应力为= 4.0毫米，大于一个= 1.0毫米。
图3
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图4
图3和图4表明，应力分布的界面获得的弹塑性分析。曲线上的-0.5也是推测的数字，以供参考。我们可以看到，曲线几乎平行的参考线在该地区r < 10-6米，这表明，附近的应力裂纹尖端占主导地位的是弹塑性奇异应力场。
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图
5图6
图5和图6显示解耦组件定义的方程。不同的弹性情况下，这里的参考长度 L需要的价值1.0 -6米，接近大小的规划的弹塑性奇异应力场。 图5显示应力分布由于残余应力和图6显示的应力分布由于残余应力和应用负载。它可以发现，曲线几乎平行的参考线在该地区r = 1.0 -5米。表4列出了应力强度因子和方向的最大正应力得到了弹塑性分析。可以发现K θmax 由于残余应力下降的顺序a= 2.0毫米，a=1.0毫米，a= 4.0毫米。这一结果可以解释为什么试样的裂纹长度的4.0毫米表明较高的断裂载荷相比，其他标本。应用负荷趋于减少的价值的K 2 。减少的K 2 ， a= 4.0 mm是尤为明显的价值θ 0为= 4.0毫米，这是远远小于一个a= 1.0毫米，a= 2.0毫米。 这同意的实验结果，那里的标本与= 1.0毫米，a = 2.0毫米的裂缝角度约40°的接口，而标本的=四点零毫米裂缝沿界面。值的K θmax 由于残余应力，这是发生骨折时，几乎是相同的，不论裂纹的长度。 KIC 等值的Si3N4，虽然低于它。小林等人[ 1 ]发现在弯曲试验的Si3N4/S45C联合的结果可分为两个群体，其中显示了相对较高的强度，而其他显示了非常低的价值。一个原因是低强度被认为是存在裂纹的陶瓷，因为很容易裂缝开始从固有缺陷在切割后加入。这也可以认为是原因之一的Kθmax 价值观发生断裂时，小于的 KIC价值的Si3N4。
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表4 。 应力强度因子和方向的最大正应力根据弹塑性分析。
K1=-0.15 K2=2.51 K1=-0.22 a= 2mm K2=2.80 K1=-0.30 a = 4mm K2=2.45 a = 1mm 结论
断裂韧性试验进行了四Si3N4/S45C联合标本界面裂纹的长度不同。评价断裂的路线和断裂韧性进行了基于弹塑性分析中Si3N4钢承担作为一个线性硬化材料。 所得结论可以概括为： 热残余应力有重大影响的断裂韧性的联合。由于影响的残余应力，试样的裂纹长度的4.0毫米具有较高的断裂韧性比那些裂纹长度为1.0毫米和2.0毫米。裂纹传播Si3N4/S45C直接从最初的裂纹尖端的方向40°的裂纹长度为1.0毫米或2.0毫米，而它繁殖沿界面裂纹长度为4.0毫米。 不久应力裂纹尖端占主导地位的是弹塑性奇异应力
场。 最大 σ θ 标准为基础的弹塑性应力场奇异可以成功地用于评估骨折的道路和断裂韧性值。 Kθmax 价值由于残余应力下降的顺序a= 2.0毫米，a=1.0毫米，a = 4.0毫米。 这是同一序列的断裂载荷的标本与a= 2.0毫米，为1.0毫米，a = 4.0毫米。 应用应力导致减少的价值的K 2 。减少的K 2 = 4.0毫米显着的价值θ0= 4.0毫米远远小于一个a= 1.0毫米，a= 2.0毫米。这同意实验结果的标本与a= 1.0毫米，a = 2.0毫米的裂缝角度约40°的接口，而标本的a= 4.0毫米裂缝沿界面方向。 θmax 的K值在骨折的标本几乎相同的所有标本和接近的KIC。 参考文献
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2.68 2.95 2.51 71° 71° 72° K1=0.43 K2=2.50 K1=0.78 K2=2.76 K1=3.82 K2=0.76 3.09 3.65 4.28 67° 65° 33° 文档来源为:从网络收集整理.word版本可编辑.欢迎下载支持.
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