Strain gradient near interface of coaxial cylinders in torsion

Theoretical and Applied Fracture Mechanics 36 (2001) 195±202

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Strain gradient near interface of coaxial cylinders in torsion G.F. Wang, S.W. Yu *, X.Q. Feng Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

Abstract Classical elastoplastic theory predicts that the rotation angle near an interface between two mismatched materials is discontinuous under shear. The strain gradient e€ects, however, can be signi®cant within a narrow region near the interface. This can be shown by application of the strain gradient plasticity. The matching expansion method was used to obtain asymptotic results. Comparison is then made with those found numerically for the interface torsion problem of a two-layered cylindrical tube. The strain gradient plasticity theory solution di€ers from that of the classical elastoplastic theory solution, depending on the properties aside from the interface behavior and the loading mode. A failure criterion is also proposed that accounts for the strain gradients. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Classical mechanics theories possess no length scale and cannot predict scale e€ects. However, experimental observations do display strong size e€ects, especially when the characteristic geometry dimensions are of the order of microns. For example, torsion tests of thin copper wires [1] have shown that normalized torsion hardening increases with decrease in the wire diameter. Indentation tests in [2] revealed that the hardness would increase as the size of the indentor is decreased. Micro-bending tests [3] also showed that the normalized bending hardness increased as the beam thickness decreases. In terms of statistically stored dislocations and geometrically induced dislocations [4], a strain gradient theory of plasticity has been developed [1,5] that includes a length parameter l in the spirit of the couple stress theory [6,7]. When the length scale reduces to zero, this theory reduces to the J2 deformation theory. Some test data associated with size e€ects have been interpreted successfully by this theory. Additional works can be found in [8,9]. For particle-reinforced and multi-layered materials, discontinuities of material properties prevail across interfaces. Geometric in compatibility and applied stresses (shear) may cause dislocations near interfaces such that the density is comparable with the statistically stored dislocation. Hence, the strain gradient e€ects should be taken into account for interface problems. In what follows, the quasi-axis-symmetrical problem is investigated by adopting the strain gradient plasticity theory. The perturbation method is used to obtain the asymptotic characteristics of the defor-

*

Corresponding author. Fax: +86-10-62770349. E-mail address: [email protected] (S.W. Yu).

0167-8442/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 1 ) 0 0 0 7 1 - 4

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G.F. Wang et al. / Theoretical and Applied Fracture Mechanics 36 (2001) 195±202

mation ®eld. The results are then compared with those from numerical calculations. A new failure criterion with strain gradient e€ects will be suggested. 2. Equations and solutions 2.1. Basic equations The strain gradient could be signi®cant for certain problems with interfaces. Consider a plane quasi-axis-symmetrical problem in polar coordinates (r; h) with the displacements uh and ur can be expressed as uh ˆ uh …r†; by

ur ˆ 0:

…1†

From the kinematic equations, the non-vanishing strains eab , rotation angle x and curvature jrr are given 1 erh ˆ ehr ˆ 2



duh dr

 uh ; r

1 xˆ 2



 duh uh ‡ ; dr r

1 jrr ˆ 2



d2 uh duh ‡ dr2 r dr

 uh : r2

…2†

The equilibrium equations of stresses take the forms dtrh trh ‡ thr ‡ ˆ 0; dr r

trh

thr ‡

dmrr mrr ‡ ˆ 0; dr r

…3†

where trh and thr are the unsymmetrical Cauchy stresses and mrr is the couple stress. For an elastic-power law hardening material with strain gradient e€ects the strain energy density has the form [1]:
…n‡1†=2n 1 2 n r0 e2e ‡ l2 j2e wˆ ‡ kem ; …4† 1‡n 2 in which e2e ˆ

 2 eij eij 3

 1 eii ejj ; 3

2 j2e ˆ jij jij : 3

…5†

Note that em is equal to ekk ; n the hardening exponent; r0 the tensile yielding stress; k the bulk modulus and l is an intrinsic material length, about several microns for metals. For plane strain, the constitutive relation can be expressed as  

…1 n†=2n 2 1 2 2 2 2 …1 n†=2n rij ˆ r0 ee ‡ l je em dij ‡ kem dij ; lij ˆ l2 r0 e2e ‡ l2 j2e eij jij ; …6† 3 3 3 where rij is the symmetric part of the unsymmetrical Cauchy stresses tij . It can be shown from Eqs. (2), (3) and (6) that uh for the considered quasi-axis-symmetrical problem satis®es the following equation:       d …1 n†=n duh uh 2 …1 n†=n duh uh ‡ dr r dr r dr r  2   2  1 2 d d 1 duh uh …1 n†=n d uh l ‡ ‡ ˆ 0; …7† 2 dr2 r dr r2 dr2 r dr r2

G.F. Wang et al. / Theoretical and Applied Fracture Mechanics 36 (2001) 195±202

in which

"  2 1 duh -2 ˆ 3 2 dr

uh r

2

l2 ‡ 4



d2 uh duh ‡ dr2 r dr

uh r2

197

2 # :

…8†

2.2. Asymptotic solution The parameter l in Eq. (8) is much smaller than the characteristic size of typical engineering structures or components. The perturbation theory [10,11] suggests a boundary-layer solution approach. This involves the assumption of a smaller couple stress on the boundaries. The stress equilibrium conditions become trh ˆ s1 ; mrr ˆ m1 h at r ˆ b;  2 b trh ˆ s1 ; mrr ˆ m2 h at r ˆ a; a

…9†

where b and a are the internal and external radii of the cylinder, respectively, and h ˆ l=a  1. Using the perturbation method, the matching solutions of the zeroth order can be obtained:  n 1 s 1 b2 uh =a ˆ c3 n 3…1‡n†=2 n1 2n ; 2n r0 a2   n 1 n s 1 b2 x ˆ c3 3…1‡n†=2 n2…1 n† 2n r0 a2   n=2  1 1 m2 m1 …n 1†=2 …n‡1†=2 E…nn2 † E…nn1 † ; ‡ …1 ‡ n† 3 2 2n r0 a r0 a mrr ˆ ‰ m1 E…n1 † ‡ m2 E…n2 †Šh;  2  2  1=2 h b b 2 m2 E…n2 † ; thr ˆ s1 ‡ trh ˆ s1 an an n a i

…10†

i m1 E…n1 † : a

1=2

In Eq. (10), E…na † ˆ exp‰… 1† …2=n† na Š denotes the boundary-layer function. The dimensionless coordinate n ˆ r=a while na ˆ …n nCa †=e (a ˆ 1; 2) stands for the stretching variables near the boundary with nC1 ˆ b=a and nC2 ˆ 1. The rigid rotation is c3 . 2.3. Numerical results In the quasi-axis-symmetrical problem, the non-vanishing stresses trh and thr and the couple stress mrr are functions of only the coordinate r. The compatibility condition requires jrr ˆ

derh 2erh ‡ : dr r

And the constitutive relation becomes "   2 #…n 1†=2 2 3 re me sij 1 ‡ eij ˆ ‡ rmm dij ; 2 r0 r0 l r0 9k

…11†

3 jij ˆ 2

"

re r0

2

 ‡

me r0 l

2 #…n

1†=2

mij ; r0

…12†

where r2e ˆ 32 …sij sij † and m2e ˆ 32 mij mij are the von Misses e€ective stress and the e€ective couple stress, and sij are the deviatoric stresses.

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G.F. Wang et al. / Theoretical and Applied Fracture Mechanics 36 (2001) 195±202

In what follows, the normalized quantities n ˆ r=a;

trh ˆ trh =r0 ;

h ˆ l=a;

thr ˆ thr =r0 ;

 rr ˆ mrr =r0 l m

…13†

will be introduced. For conciseness, the overbar ``)'' will be omitted. It can be shown from Eqs. (2), (11) and (12) that  1 ‡ …n dtrh ˆ dn

 3r2rh dthr 2mrr ˆ h R2 dn 2rrh dmrr 1 ; ˆ …trh h n dn 1†

3 …n 2 thr †

1†

2 2 mrr …trh thr † ‡ …n 2 h R

2 2† rrh ; n

mrr : n

…14†

Note also that 3 R2 ˆ 3r2rh ‡ m2rr 2

and

rrh ˆ …thr ‡ trh †=2:

…15†

For example, consider the following boundary conditions: trh ˆ 4; mrr ˆ 1 trh ˆ 1

at n ˆ 0:5;

at n ˆ 1:0:

…16†

Since the parameter h is very small, Eqs. (14) are not only nonlinear but also sti€. An adaptive step fourth-order implicit Runge±Kutta method is thus employed. The shooting method is used to deal with the two-point boundary-value problem. In the calculation, let h ˆ 0:01. In the shooting process, let thr ˆ 3:4655 as the initial predicted value at n ˆ 0:5. This is obtained from the asymptotic solutions. After 10 iterations, the value of thr at n ˆ 0:5 approaches 3.5411 with a relative error at n ˆ 1:0 of less than 10 7 . The asymptotic and the numerical results of the stresses are plotted in Fig. 1. Their maximum relative error between the asymptotic and the numerical results is less than 2.134%. This indicates that the asymptotic solutions provide a good approximation to the real solution of the nonlinear equations.

Fig. 1. Comparison between the asymptotic and the numerical results.

G.F. Wang et al. / Theoretical and Applied Fracture Mechanics 36 (2001) 195±202

199

3. Boundary layer near interface The asymptotic solution will be used for the interface problem of a two-layered cylindrical tube, as shown in Fig. 2. The inner and outer surfaces of the tube are subjected to the shear stresses s1 and s2 , respectively. The quantities referred to the two materials are denoted by r01 ; n1 ; l1 and r02 ; n2 ; l2 . From the continuous conditions of uh , x, trh and lrr on the interface, the couple stress and the Cauchy stresses within the cylinders can be determined by "   8  # 1=2 1=2 > n 2 r b 2 > > H sb l2 exp …c 6 r 6 b†; > < 2 n1 l1 " # …17† lrr ˆ  1=2    n 1=2 > > 2 b r 2 > > …b 6 r 6 a†; H sb l2 exp : 2 n2 l2  2 b …c 6 r 6 a†; r "   8 (  2  1=2  #) 1=2 > b n2 l2 2 r b > > ‡H exp > < sb r n1 l1 n1 l1 (  "   #) thr ˆ  2 1=2 > > b 2 b r > > H exp : sb r n2 l2 trh ˆ sb

…c 6 r 6 b†; …18† …b 6 r 6 a†;

where the parameter H is given by   p 1 3=2 Hˆ 2 n2 h
2 i  r n1  s n1 n2 n h
i n 1 b 2  02 b 2 3…n2 n1 †=2 1 ‡ …n2 1† ba 1 ‡ …n 1† 1 a r01 r02 n1 sb 2  ;  n1 =2    n2 =2 r02 …n 1†=2 r02 l2 1 1 …n2 n1 †=2 …1 ‡ n †…n2 1†=2 …1 ‡ n1 † 1 ‡ 3 2 2n1 2n2 r01 l1 2

and sb ˆ s2 …a=b† is the shear stress on the interface without considering the strain gradient e€ects.

Fig. 2. Two-layer cylinder under shear stresses.

…19†

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G.F. Wang et al. / Theoretical and Applied Fracture Mechanics 36 (2001) 195±202

Fig. 3. Distributions of mrr .

Fig. 4. Distributions of thr .

If the strain hardening in the two materials is the same, then n1 ˆ n2 ˆ n. Figs. 3 and 4 give the distributions of mrr and thr with a ˆ 2b ˆ 4c, s2 ˆ 0:2r02 , r01 ˆ 2r02 and l=b ˆ 100. It can be seen that the e€ects of boundary layer are of signi®cance. The variations of H with material properties are plotted in Figs. 5 and 6.

4. Failure criterion with strain gradient e€ects Since the stresses near an interface are unsymmetrical, a new criterion may be necessary for analyzing the interface failure where strain gradient e€ects prevail. It was suggested in [12] that for ®nite deformation jrhj ˆ jrhjcr ;

…20†

where j sin hj ˆ 12 jr  uj and jrhjcr is a material constant. For small deformation of plane problem, the above criterion reduces to

G.F. Wang et al. / Theoretical and Applied Fracture Mechanics 36 (2001) 195±202

201

Fig. 5. Variations of H to n.

Fig. 6. Variations of H to r02 =r01 .

jje j ˆ jjcr j

…21†

In the strain gradient plasticity theory, the generalized e€ective plastic strain is de®ned by -2 ˆ e2e ‡ l2 j2e ;

…22†

which accounts for both statistically stored dislocations and geometrically induced dislocations. From the criterion in [12] and von Misses yielding condition, a more general failure criterion can be deduced. In the form of the generalized e€ective plastic strain, there results - ˆ -cr :

…23†

In the form of the generalized e€ective stress, it is found that R ˆ rcr ; 2

…24†

where R ˆ r2e ‡ l 2 m2e , -cr and rcr are two material constants. They can be determined from the uniaxial tension tests.

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G.F. Wang et al. / Theoretical and Applied Fracture Mechanics 36 (2001) 195±202

5. Conclusions The rotation angle near an interface is an important consideration for analyzing interface failure. The strain gradient e€ects must be considered when shear stresses are applied. According to the strain gradient plasticity theory, a boundary layer plays a role near an interface. The local shear stresses are unsymmetrical. The boundary-layer solution obtained from the strain gradient plasticity theory di€ers signi®cantly from that of the classical theory depending on material properties besides the interface and the loading mode. The size e€ect sensitive strain gradient solution provides further insights into the failure behavior of interfaces.

Acknowledgements The project is supported by the National Natural Science Foundation (under grant no. 19891180) and the Education Ministry of China.

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