第八章 弹塑性接触_PartII

2) Stresses outside :
求解流程： 1 共轭梯度法求解互补问题得到压力分布 2 基于半解析法求解压力引起的应力 3 基于半解析法求解残余应力 ( σ＝ σ p+ σ r ) 4 径向返回算法求解塑性应变 5 根据体内塑性应变计算表面残余位移。 6 将残余变形叠加到表面形貌
残余应力求解方案
本章研究目标：基于SAM并使用FFT对接触问题进行研究。
Elasto-plastic model
p
Stress Elastic stress
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Elasto-plastic model
p
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p r
i IFFT ( Ai * ~ p)
p
r
~
Residual stress h Plastic strain The plastic strains can be treated as eigenstrains
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Elasto-plastic model
求解线性互补问题：
弹性变形 残余变形
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互补算法推广到弹塑性接触的关键 －表面残余变形的计算 基于SAM实现了残余应力和残余变形 的求解
Cuboidal inclusion in half-space
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H = H0+C·P+D·εp Hi,j≥0, pi,j≥0 Hi,j· pi,j=0
i j ( x ) Fk , k m m i j Fk , k i j (1 )( Fi ,k k j F j , k k i )
3) Stresses inside :
i j ( x ) Fk ,k m m i j Fk ,k i j (1 )( Fi ,k k j F j ,k k i )
Surface gap：h=h0+ue+up h0 - initial body separation between two surfaces ue - the surface displacement due to elastic effect up - the permanent deformation due to plastic effect
) R ( x3 x3
User-defined variables
3) S. Liu et al., recent development (IJP 2012)
Extension of the theory by Liu & Wang (2005) Explicitly shows convolution correlation in the depth direction Four 3D-FFT
Residual stress i C i dx

I 2 i e j [ ij 0,ij x3 1,ij x3 2,ij ]
Residual stress h Plastic strain
h = h0+k·p+D·εp hi,j≥0, pi,j≥0 hi,j· pi,j=0 ΣPi,j=W
统计模型不能得到局部接触信息，粗糙峰之间的相互作用没有考虑。 FEM求解粗糙表面接触需要划分大量网格，效率低。 SAM仅需要在关心的区域划分网格，计算速度快。
Elasto-plastic model
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Elasto-plastic model
p
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半解析法(SAM)：利用解析的方法得到影响系数，使用FFT计算 叠加效应，得到数值解。 本研究通过半解析法得到以下关系： 弹性接触中的应力－位移关系 弹塑性接触中的残余应力和表面残余位移 层状材料弹性接触中的应力－位移关系
Explicit closed form solution In terms of elementary functions Shows convolution/correlation in half space solutions
1/ R
I 1/ RI
)] ln[ R ( x3 x3
Induced by the mirror domain
Half space model When solving multiple cubic eigenstrain problem, one can conduct combined 3D DCFFT for the first term, 3D DCR-FFT for the second term, take x3 out and do 3D DCRFFT for the third term and take x32 out and do 3D DCR-FFT for the forth term.
确定性模型：有限元法(FEM) Wriggers (1995) Liu等 (2001) Pei等 (2005)
半解析法(SAM) 利用解析方法得到影响系数，使用快速傅里 叶变换(FFT)计算叠加效应，得到数值解
Kalker等 (1972) Nogi等 (1997) Polonsky等 (1999) Hu等 (1999)
Kuhn-Tucker complementarity conditions: • Contact regions: p≥0, h=0 • Non-contact regions: p=0, h>0
Residual stress h Plastic strain The plastic strains can be treated as eigenstrains Contact model
Liu, S. B., Jin, X. Q., Wang, Z. J., Keer, L. M., and Wang, Q., 2012, International journal of plasticity Liu, S. B., Jin, X. Q., Wang, Z. J., Keer, L. M., and Wang, Q., 2012, International journal of plasticity
Bi-harmonic potentials
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Stress and displacement or residual stress and displacement due to a cuboidal eigenstrain in a half space Complicated formulation with recursive relations involving Legendre polynomials Did not explicitly demonstrate the convolution/correlation properties in the depth direction.
RI
2 x2 x1 x1
x 2 x3 x 3
2
2
)2 ( x2 x2 )2 ( x3 x3 )2 R ( x1 x1
Harmonic potentials
2) S. Liu & Q. Wang’s solution (JAM 2005)
The elastic field due to eigenstrains can be expressed in terms of Galerkin vectors, F (Yu and Sanday 1991): 1) Displacements:
2 ui (x ) 2(1 ) Fi , j j Fk , k i
2 ei j ek k i j
Displacement and stress induced by eigenstrains
Yu, H. Y., and Sanday, S. C., 1991, Proc. R. Soc. London, Ser. A, 434(1892), pp.521-530.
共轭梯度法求互补问题 FFT算法加快弹性变形(ue)、塑性变形(up)和应力的计算速度
Cuboidal inclusion in half-space
1) Y. P. Chiu’s solution (JAM 1978)
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Cuboidal inclusion in half-space
2016/12/20
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Elasto-plastic model
接触模型介绍：
解析模型： 弹性点、线接触的解析解－Hertz理论
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接触力学
西南交通大学 王战江
统计模型：
Greenwood和Williamson (1966) GW模型 Persson B N J (2001) 分形模型
A cuboidal inclusion in an elastic half-space
R R,11 ; R, 22 ; R,33 ; R, 23 ; R,13 ; R,12
Half space model
H 1
D 1 2
Cuboidal inclusion in half-space
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Cuboidal inclusion in half-space
Out ij
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Surface displacements due to Eigenstrain 王战江 授课
uis 1 U is [e]dx 2


U i [e]dx

I 2 Ui ij 0,ij x3 1,ij x3 2,ij
I 2 Θij ij 0,ij x3 1,ij x3 2,ij
A cuboidal inclusion in an elastic half-space
Induced by the infinite space
Induced by elastic deformation K = the pressure-displacement influence coefficients
Induced by plastic strain D = the plastic-displacement influence coefficients
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Cuboidal inclusion in half-space
ui 1 8H
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In the half space, the Galerkin vectors include two parts, one is induced by the infinite space, and the other by the mirror domain. Finally the displacements and stresses can be expressed as (Liu et all., 2012), 1 Out Θij [e]dx ui Ui [e]dx ij 8H 4H where i and j represent the six components (11, 22, 33, 12, 13, 23) and,
Surface displacement
u = k·p+D·εp
3D FFT for the four terms
1. Conjugate Gradient Method (CGM) is applied to minimize the quadratic function with inequality constrain 2. Fast Fourier Transform (FFT) is used to calculate ue、up and subsurface stresses and residual stress.