反平面塑性V形切口尖端应力和位移渐近解

李聪; 胡斌; 牛忠荣

doi:10.21656/1000-0887.420045

反平面塑性V形切口尖端应力和位移渐近解

doi: 10.21656/1000-0887.420045

1.
安徽建筑大学土木工程学院, 合肥 230601
2.
合肥工业大学土木与水利工程学院工程力学系, 合肥 230009

基金项目: 国家自然科学基金 (11272111)

详细信息

作者简介:
李聪(1989—), 女, 讲师, 博士(通讯作者. E-mail：Licong@ahjzu.edu.cn)

中图分类号: O344.3
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- 被引次数: 0
出版历程
- 收稿日期: 2021-02-20
- 录用日期: 2021-02-20
- 修回日期: 2021-05-27
- 网络出版日期: 2021-11-23
- 刊出日期: 2021-12-01

Asymptotic Solutions of Plastic Stress and Displacement at V-Notch Tips Under Anti-Plane Shear

1.
College of Civil Engineering, Anhui Jianzhu University, Hefei 230601, P.R.China
2.
Department of Engineering Mechanics, College of Civil Engineering, Hefei University of Technology, Hefei 230009, P.R.China

摘要

摘要:
提出了一种确定幂硬化材料反平面V形切口尖端应力和位移渐近解的主导项和高阶项的有效方法。首先通过在弹塑性理论基本方程中引入V形切口尖端应力场和位移场的渐近级数展开，建立以应力和位移为特征函数的非线性和线性常微分方程组。然后采用插值矩阵法求解常微分方程组，可得到多阶应力特征指数和其相对应的特征函数。该方法具有通用性强、精度高等优点，可处理任意开口角度和应变硬化指数的V形切口。典型算例验证了该方法的准确性和有效性。
- 弹塑性 /
- 反平面 /
- V形切口 /
- 渐近解 /
- 奇异性
Abstract:
An efficient method was developed to determine the first- and high-order terms of asymptotic solutions of plastic stress and displacement near V-notch tips under anti-plane shear in power-law hardening materials. Through introduction of the asymptotic series expansions of stress and displacement fields around the V-notch tip into the fundamental equations of the elastoplastic theory, the governing ordinary differential equations (ODEs) with the stress and displacement eigen-functions were established. Then the interpolating matrix method was employed to solve the resulting nonlinear and linear ODEs. Consequently, the high-order stress exponents and the associated eigen-solutions were obtained. The presented method, being capable of dealing with the V-notches with arbitrary opening angles and strain hardening indexes under anti-plane shear, has the advantages of great versatility and high accuracy. Typical examples were given to demonstrate the accuracy and effectiveness of this method.
- elastoplasticity /
- anti-plane /
- V-notch /
- asymptotic solution /
- singularity

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图 1 反V形切口尖端区域

Figure 1. An anti-plane V-notch tip

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图 2 插值矩阵法迭代计算Ⅲ型裂纹第1阶应力指数$ {s_1} $收敛情况

Figure 2. First-order stress exponent s₁ in the iterative calculation of the mode Ⅲ crack with the interpolating matrix method

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图 3 幂硬化反平面Ⅲ型裂纹第1阶应力和位移特征角函数

Figure 3. The 1st-order stress and displacement eigen-functions for the mode Ⅲcrack

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图 4 幂硬化反平面Ⅲ型裂纹高阶应力和位移特征角函数，m=7

Figure 4. The high-order stress and displacement eigen-functions for the mode Ⅲ crack with m=7

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图 5 反平面切口第1阶应力特征角函数(m=7)

Figure 5. The 1st-order stress eigen-functions for the mode Ⅲ V-notch with m=7

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图 6 反平面切口第2阶应力特征角函数(m=7)

Figure 6. The 2nd-order stress eigen-functions for the mode Ⅲ V-notch with m=7

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图 7 反平面裂纹第1阶位移和应力特征角函数

Figure 7. The 1st-order stress and displacement eigen-functions for the mode Ⅲ V-notch

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图 8 反平面V形切口第1阶应力特征角函数(m=5)

Figure 8. The 1st-order stress and displacement eigen-functions for the mode Ⅲ V-notch (m=5)

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表 1 反平面Ⅲ型裂纹前4阶应力特征指数随硬化指数m的变化

Table 1. Stress exponent $ {s_k} $ of the mode Ⅲ crack with various m values

$ {s_k} $	method	m=3	m=5	m=7	m=10	m=13
$ {s_{\text{1}}} $	present (Q=20)	−0.249 999	−0.166 666	−0.124 999	−0.090 907	−0.071 426
	present (Q =40)	−0.249 999	−0.166 667	−0.125 000	−0.090 909	−0.071 429
	present (Q =80)	−0.249 999	−0.166 667	−0.125 000	−0.090 909	−0.071 429
	ref. [14]	−0.250 000	−0.166 667	−0.125 000	−0.090 909	−0.071 429
$ {s_{\text{2}}} $	present (Q =20)	0.249999^#	0.499998^#	0.437490	0.365759	0.314434
	present (Q =40)	0.249 999^#	0.500 001^#	0.435 761	0.363 758	0.312 255
	present (Q =80)	0.249 999^#	0.500 001^#	0.435 666	0.363 646	0.312 133
	ref. [14]	0.250 000^#	0.500 000^#	0.435 660	0.363 636	0.312 094
$ {s_{\text{3}}} $	present (Q =20)	0.574 150	0.501 559	0.624 995^#	0.727 256^#	0.700 294^×
	present (Q =40)	0.572 948	0.500 086	0.625 000^#	0.727 272^#	0.695 939^×
	present (Q =80)	0.572 879	0.500 004	0.625 000^#	0.727 272^#	0.695 695^×
	ref. [14]	0.572 876	0.500 000	0.625 000^#	0.727 273^#	0.695 617^×
$ {s_{\text{4}}} $	present (Q =20)	0.749 997^#	1.166 662^#	0.999 979^×	0.822 425^×	0.785 686^#
	present (Q =40)	0.749 997^#	1.166 669^#	0.996 522^×	0.818 425^×	0.785 719^#
	present (Q =80)	0.749 997^#	1.166 669^#	0.996 332^×	0.818 201^×	0.785 719^#
	ref. [14]	0.750 000^#	1.166 667^#	0.996 320^×	0.818 182^×	0.785 714^#

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表 2 反平面V形切口($ {\text{2}}\alpha {\text{ = 3}}{{\text{0}}^ \circ } $)前4阶应力特征指数随硬化指数m的变化

Table 2. Stress exponent $ {s_k} $ of the mode Ⅲ V-notch with $ {\text{2}}\alpha {\text{ = 3}}{{\text{0}}^ \circ } $ for various m values

$ {s_k} $	method	m=3	m=5	m=7	m=10	m=13
$ {s_{\text{1}}} $	present(Q=20)	−0.232204	−0.157527	−0.119464	−0.087783	−0.069413
	present(Q=40)	−0.232206	−0.157527	−0.119465	−0.087784	−0.069415
	present(Q=80)	−0.232206	−0.157527	−0.119465	−0.087784	−0.069415
	ref. [17]	−0.232206	−0.157564	−0.119502	−0.087824	−0.069457
$ {s_{\text{2}}} $	present(Q=20)	0.232204^#	0.472581^#	0.554001	0.472991	0.413208
	present(Q=40)	0.232206^#	0.472581^#	0.551878	0.470411	0.410316
	present(Q=80)	0.232206^#	0.472581^#	0.551876	0.470409	0.410314
	ref. [17]	−	−	0.551851	0.470288	0.409963
$ {s_{\text{3}}} $	present(Q=20)	0.696612^×	0.623794	0.597320^#	0.702264^#	0.763543^#
	present(Q=40)	0.696618^×	0.621968	0.597325^#	0.702272^#	0.763565^#
	present(Q=80)	0.696618^×	0.621972	0.597325^#	0.702272^#	0.763565^#
	ref. [17]	−	0.622051	−	−	−
$ {s_{\text{4}}} $	present(Q=20)	0.700023	1.102689^#	1.227466^×	1.033765^×	0.895829^×
	present(Q=40)	0.698686	1.102689^#	1.223221^×	1.028606^×	0.890047^×
	present(Q=80)	0.698682	1.102689^#	1.223217^×	1.028602^×	0.890043^×
	ref. [17]	0.698604	−	−	−	−

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表 3 反平面V形切口($ {\text{2}}\alpha {\text{ = 15}}{{\text{0}}^ \circ } $)前4阶应力特征指数随硬化指数m的变化

Table 3. Stress exponent $ {s_k} $ of the mode Ⅲ V-notch with $ {\text{2}}\alpha {\text{ = 15}}{{\text{0}}^ \circ } $ for various m values

$ {s_k} $	method	m=3	m=5	m=7	m=10	m=13
$ {s_{\text{1}}} $	present(Q=20)	−0.080378	−0.060350	−0.049587	−0.039967	−0.033919
	present(Q=40)	−0.080383	−0.060356	−0.049564	−0.039974	−0.033925
	present(Q=80)	−0.080383	−0.060356	−0.049593	−0.039974	−0.033925
	ref. [17]	−0.080384	−0.060369	−0.049614	−0.040000	−0.033956
$ {s_{\text{2}}} $	present(Q=20)	0.080378^#	0.181051^#	0.247935^#	0.319736^#	0.373109^#`
	present(Q=40)	0.080383^#	0.181068^#	0.247970^#	0.319792^#	0.373175^#`
	present(Q=80)	0.080383^#	0.181068^#	0.247965^#	0.319792^#	0.373175^#`
	ref. [17]	−	−	−	−	−
$ {s_{\text{3}}} $	present(Q=20)	0.241134^#	0.422452^#	0.545457^#	0.679439^#	0.780137^#
	present(Q=40)	0.241149^#	0.422492^#	0.545534^#	0.679558^#	0.780275^#
	present(Q=80)	0.241149^#	0.422492^#	0.545523^#	0.679558^#	0.780275^#
	ref. [17]	−	−	−	−	−
$ {s_{\text{4}}} $	present(Q=20)	1.589836	1.556131	1.520007	1.470114	1.426214
	present(Q=40)	1.587821	1.553694	1.516597	1.464221	1.417315
	present(Q=80)	1.587820	1.553693	1.516597	1.464220	1.417314
	ref. [17]	1.587794	1.553562	1.516332	1.463944	1.416870

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表 4 反平面V形切口($ {\text{2}}\alpha {\text{ = 6}}{{\text{0}}^ \circ } $)前4阶应力特征指数随硬化指数m的变化(Q=40)

Table 4. Stress exponent $ {s_k} $ of the mode Ⅲ V-notch with $ {\text{2}}\alpha {\text{ = 6}}{{\text{0}}^ \circ } $ for various m values (Q=40)

$ {s_k} $	method	m=3	m=5	m=7	m=10	m=13
$ {s_{\text{1}}} $	present	−0.209034	−0.144787	−0.111379	−0.083025	−0.066275
$ {s_{\text{1}}} $	ref. [17]	−0.209035	−0.144820	−0.111421	−0.083068	−0.066323
$ {s_{\text{2}}} $	present	0.209034^#	0.434361^#	0.556895^#	0.613636	0.545403
$ {s_{\text{2}}} $	ref. [17]	−	−	−	0.613578	0.545247
$ {s_{\text{3}}} $	present	0.627102^#	0.775407	0.702317	0.664200^#	0.729025^#
$ {s_{\text{3}}} $	ref. [17]	−	0.775369	0.702299	−	−
$ {s_{\text{4}}} $	present	0.851750	1.013509^#	1.225169^#	1.310297^×	1.157081^×
$ {s_{\text{4}}} $	ref. [17]	0.851659	−	−	−	−

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表 5 反平面V形切口($ {\text{2}}\alpha {\text{ = 9}}{{\text{0}}^ \circ } $)前4阶应力特征指数随硬化指数m的变化(Q=40)

Table 5. Stress exponent $ {s_k} $ of the mode Ⅲ V-notch with $ {\text{2}}\alpha {\text{ = 9}}{{\text{0}}^ \circ } $ for various m values (Q=40)

$ {s_k} $	method	m=3	m=5	m=7	m=10	m=13
$ {s_{\text{1}}} $	present	−0.178394	−0.126571	−0.099142	−0.075360	−0.060999
$ {s_{\text{1}}} $	ref. [17]	−0.178395	−0.126599	−0.099179	−0.075403	−0.061047
$ {s_{\text{2}}} $	present	0.178394^#	0.379713^#	0.495710^#	0.602880^#	0.670989^#
$ {s_{\text{2}}} $	ref. [17]	−	−	−	−	−
$ {s_{\text{3}}} $	present	0.535182^#	0.885997^#	0.899889	0.809746	0.736893
$ {s_{\text{3}}} $	ref. [17]	−	−	0.899826	0.809670	0.736783
$ {s_{\text{4}}} $	present	1.040992	0.970537	1.090562^#	1.281120^#	1.402977^#
$ {s_{\text{4}}} $	ref. [17]	1.040889	0.970465	−	−	−

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表 6 反平面V形切口($ {\text{2}}\alpha {\text{ = 12}}{{\text{0}}^ \circ } $)前4阶应力特征指数随硬化指数m的变化(Q=40)

Table 6. Stress exponents $ {s_k} $ of the mode Ⅲ V-notch with $ {\text{2}}\alpha {\text{ = 12}}{{\text{0}}^ \circ } $ for various m values (Q=40)

$ {s_k} $	method	m=3	m=5	m=7	m=10	m=13
$ {s_{\text{1}}} $	present	−0.137145	−0.099974	−0.080061	−0.060462	−0.051585
$ {s_{\text{1}}} $	ref. [17]	−0.137146	−0.100000	−0.080094	−0.062500	−0.051628
$ {s_{\text{2}}} $	present	0.137145^#	0.299922^#	0.400305^#	0.483696^#	0.567435^#
$ {s_{\text{2}}} $	ref. [17]	−	−	−	−	−
$ {s_{\text{3}}} $	present	0.411435^#	0.699818^#	0.880671^#	1.027854^#	1.013736
$ {s_{\text{3}}} $	ref. [17]	−	−	−	−	1.013494
$ {s_{\text{4}}} $	present	1.279549	1.222471	1.162660	1.082304	1.186455^#
$ {s_{\text{4}}} $	ref. [17]	1.279434	1.222375	1.162546	1.082129	−

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表 7 反平面V形切口前4阶应力特征指数随硬化指数m和张角2α的变化 ( Q =40)

Table 7. Stress exponents $ {s_k} $ of the mode Ⅲ V-notch for various m values and $ {\text{2}}\alpha $ (Q=40)

$ {s_k} $	$ {\text{2}}\alpha $/(°)	m=3	m=5	m=7	m=10	m=13
$ {s_{\text{1}}} $	0	−0.316987	−0.194766	−0.140313	−0.098797	−0.076224
	30	−0.313242	−0.193473	−0.139669	−0.098487	−0.076043
	90	−0.300944	−0.188989	−0.137387	−0.097374	−0.075386
	150	−0.274774	−0.178251	−0.131633	−0.094459	−0.073631
$ {s_{\text{2}}} $	0	−0.049037	−0.026170	−0.017810	−0.012034	−0.009085
	30	−0.000702	0.008722	0.009252	0.008153	0.006996
	90	0.138739	0.116011	0.095654	0.074849	0.061274
	150	0.274774^#	0.321572	0.272393	0.220346	0.184745
$ {s_{\text{3}}} $	0	0.218913^×	0.142426^×	0.104693^×	0.074729^×	0.058054^×
	30	0.311838^×	0.210917^×	0.158173^×	0.114793^×	0.090035^×
	90	0.300944^×	0.421011^×	0.328695^×	0.247072^×	0.197934^×
	150	0.380293	0.534753^#	0.658165^#	0.535151^×	0.443121^×
$ {s_{\text{4}}} $	0	0.285422	0.213317	0.169289	0.129282	0.104644
	30	0.313242^#	0.285697	0.228736	0.176303	0.143628
	90	0.638404	0.50008	0.410023	0.323996	0.268649
	150	0.824322^×	0.821395^×	0.676419^×	0.623227	0.531023

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