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Investigation of strain localization in elastoplastic materials with transversely isotropic elasticity
Available online at www.sciencedirect.com
International Journal of Mechanical Sciences 45 (2003) 217 – 233
Investigation of strain localization in elastoplastic materials with transversely isotropic elasticity Yong-Qiang Zhanga;∗ , Yong Lua , Mao-Hong Yub a
School of Civil and Environmental Engineering, Protective Technology Research Center, Nanyang Technological University, Singapore 639798, Singapore b School of Civil Engineering and Mechanics, Xi’an Jiaotong University, Xi’an 710049, Xi’an, China Received 20 July 2002; received in revised form 28 February 2003; accepted 6 March 2003
Abstract Transversely isotropic materials are of primary interest in many engineering applications. In this paper, general description of the properties of strain localization is deduced for the elastoplastic materials with transversely isotropic elasticity which follow the non-associated plasticity and are subjected to tri-axial stress states. Then the explicit expressions for the direction of localized band and the corresponding hardening modulus at the onset of strain localization are obtained for plane strain condition. Furthermore, the e5ect of deviation from isotropic elasticity in the formulation of strain localization is discussed based on those yield criteria such as Tresca criterion and twin-shear criterion. In the end, the in6uences of yield criteria on the properties of strain localization are elucidated. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Strain localization; Transversely isotropy; Non-associated plasticity; Elastoplastic material
1. Introduction Strain localization of plastic 6ow into shear bands is a ubiquitous feature of elastoplastic materials undergoing non-homogeneous deformation. It can be interpreted physically as the transformation of propagating waves into stationary waves. The onset of localization is indicated by a vanishing wave speed. Within the classical rate-independent continuum theory, strain localization denotes the onset of a discontinuous bifurcation in the form of a jump in the velocity gradient
Corresponding author. Tel.: +65-67906199; fax: +65-67910676. E-mail address: [email protected] (Y.-Q. Zhang).
0020-7403/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0020-7403(03)00055-9
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recent years [1–5]. The basic principles were discussed by Thomas [6], Hill [7], and Mandel [8], in connection with the theory of bifurcation and localization corresponding to stationary acceleration waves. It was shown by Rice [9] that the phenomenon of strain localization may be described as a type of loss of material stability, and simultaneously as the loss of ellipticity of governing di5erential equations, namely formation of discontinuity is due to bifurcation of equilibrium states. Based on Rice’s pioneering work, the conditions for the onset of strain localization have been resolved for a wide variety of constitutive models by Ottosen and Runesson [10], Runesson et al. [11], Bigoni and Hueckel [12], Nielsin and Schreyer [13] and Rizzi et al. [14]. In these works, elasticity is assumed to remain isotropic throughout the loading process. As the strength properties of a wide variety of engineering materials exhibit anisotropic behavior, the extension of such analyses to material models which do include anisotropic elasticity is of interest. Steinmann et al. [15] analyzed the localization condition for an isotropic elastic and orthotropic perfectly plastic material based on a yield criterion of the Hill type under plane stress and plane strain loading and showed that the in-plane bifurcation direction may be considerably modi
2. Transversely isotropic elasticity It is well known that a transversely isotropic elastic material has the rotational symmetry property with reference to a certain axis. The plane perpendicular to this axis is called the basal plane while planes containing the axis of symmetry will be named zonal planes. To better express material properties componentwise, the cartesian axes (e1 ; e2 ; e3 ) are adopted, where e1 and e2 are two arbitrary orthogonal unit vectors of the basal plane and the unit vector e3 denotes the axis of rotational symmetry. As the free energy for the transversely isotropic elastic material inherits the symmetry properties of the material and is a transversely isotropic function of the strain tensor U, the free energy function has the following form [17]: =
c1 2 c2 c4 tr U + tr U2 + c3 tr U tr(M · U) + tr 2 (M · U) + c5 tr(M · U2 ); 2 2 2
(1)
where tensor M = e3 ⊗ e3 with the symbol ‘⊗’ denoting outer product, ci (i ∈ [1; 5]) are
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
219
the trace of tensors. Thus, the stress tensor can be obtained as =
@ = c1 (I : U)I + c2 U + c3 ((M : U)I + (I : U)M) + c4 (M : U)M + c5 (U · M + M · U); @U
(2)
where I designates the second-order identity tensor and the symbol ‘:’ the inner product with double index contraction. Further, the elastic sti5ness takes the form E=
@2 L + c3 (I ⊗ M + M ⊗ I) + c4 M ⊗ M + c5 (I⊗M L + M⊗I); L = c1 I ⊗ I + c2 I⊗I @U ⊗ @U
(3)
L denotes a symmetrized outer product and it has (M⊗I) L ijkl =(Mik Ijl +Mil Ijk )=2. where the symbol ‘⊗’ It is noted from Eq. (3) that the material constants c3 , c4 , c5 might be interpreted as measures of the deviation from isotropic elasticity. In the cartesian axes (e1 , e2 , e3 ), the matrix representation of the constitutive equation = E : U has the following form by using Voigt notation
11
c1 + c2
22 c1 33 c1 + c3 = 23 31
c1
c1 + c 3
c1 + c 2
c1 + c 3
c1 + c 3
c1 + c2 + 2c3 + c4 + 2c5
22 33 : 2 23 2 31
c2 =2 + c5 =2 c2 =2 + c5 =2
12
11
c2 =2
(4)
2 12
From this representation, the relations between the coeNcients ci and the matrix coeNcients Eij can be obtained as E11 = c1 + c2 ;
E12 = c1 ;
E13 = c1 + c3 ;
E33 = c1 + c2 + 2c3 + c4 + 2c5 ;
E44 = c2 =2 + c5 =2:
(5)
The positive de
2 (E11 + E12 )E33 − 2E13 ¿ 0;
E44 ¿ 0:
The material moduli might be related to corresponding parameters of physical meaning. The relations between material moduli and parameters Eij may be obtained by EL =
2 (E11 + E12 )E33 − 2E13 ; E11 + E12
ET =
2 [(E11 + E12 )E33 − 2E13 ](E11 − E12 ) ; 2 E11 E33 − E13
E11 − E12 ; 2
L =
E13 ; E11 + E12
GL = E44 ;
GT =
(6)
where EL and ET , respectively, denote the longitudinal (or axial) elastic modulus and the transverse (or cross-axial) elastic modulus, GL and GT , respectively, denote the longitudinal (or zonal) shear modulus and the transverse (or basal) shear modulus, and L the longitudinal Poisson’s ratio (representing the contraction in the longitudinal direction due to an imposed traction in the basal plane).
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3. Localization condition for non-associated plasticity For simplicity it will be assumed that deformations are small, i.e. con
(7)
where ˙ and U˙ , respectively, designate the stress rate and strain rate tensors, and the tangent sti5ness tensor D is given by 1 D = E − (E : Q) ⊗ (P : E); (8) A where P and Q are the unit outward normals to the yield surface and the plastic potential, respectively, and they can be calculated as follows by using the yield function F and the plastic potential G: @F @F @G @G P= ; Q= ; (9) @ @ @ @ where ‘ · ’ stands for a norm of a tensor. The positive parameter A is de
(10)
where H is the generalized plastic modulus that is positive, zero or negative for hardening, perfect or softening plasticity respectively. When strain localization occurs, the strain rate U˙ across the localized band is discontinuous. The discontinuity of strain rate U˙ must satisfy the Maxwell’s compatibility condition [6] [˙U] = U˙ i − U˙ o = 12 (m ⊗ n + n ⊗ m);
(11)
where the bracket ‘[ · ]’ denotes a jump of the bracketed quantity across the discontinuity surface, U˙ i and U˙ o are, respectively, the strain rates inside and outside the band, vector m represents the mode of discontinuity of the strain rate, and n is the unit normal vector of the band. For this case we have ˙i = Di : U˙ i ; i
˙o = Do : U˙ o
(12)
o
where D and D denote the tangential sti5ness tensors inside and outside the localized band, respectively. It follows from equilibrium considerations that the traction rate across the band must be unique: n · (˙i − ˙o ) = 0:
(13)
Combining Eqs. (11)–(13) yields n · (Di − Do ) : U˙ o + L · m = 0;
(14)
L = n · Di · n
(15)
where denotes the characteristic tangent sti5ness modulus tensor.
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
221
At the onset of strain localization, Do and Di can be regarded the same, namely Do = Di = D. Thus from Eq. (14), it has L·m=0
or
det(L) = 0:
(16)
This is the necessary condition for strain localization, and L = n · D · n:
(17)
When tensor L is singular, the critical hardening modulus can be obtained as follows H = a · R e · b − P : E : Q = hn − h E ;
(18)
e
where hn = a · R · b and hE = P : E : Q, and Re = (Le )−1 = (n · E · n)−1 :
(19)
In addition, the vectors a and b are given by a = n · E : Q;
b = P : E · n:
(20)
Substituting Eq. (3) into Eq. (20) gives a = a 1 n + a 2 Q · n + a 3 e3 + a 4 Q · e 3 ;
b = b 1 n + b 2 P · n + b 3 e3 + b 4 P · e 3
(21)
with a1 = c1 tr Q + c3 (e3 · Q · e3 );
a2 = c2 ;
a3 = (c3 tr Q + c4 (e3 · Q · e3 ))(e3 · n) + c5 (n · Q · e3 );
a4 = c5 (e3 · n)
(22)
b4 = c5 (e3 · n):
(23)
and b1 = c1 tr P + c3 (e3 · P · e3 );
b2 = c2 ;
b3 = (c3 tr P + c4 (e3 · P · e3 ))(e3 · n) + c5 (n · P · e3 ); Due to Eq. (3), we can obtain
hE = c1 (tr P)(tr Q) + c2 P : Q + c3 ((e3 · Q · e3 ) tr P + (e3 · P · e3 ) tr Q) + c4 (e3 · P · e3 )(e3 · Q · e3 ) + 2c5 (e3 · (P · Q) · e3 )
(24)
and the elastic acoustic tensor Le = n · E · n = 1 I + 2 n ⊗ n + 3 (e3 ⊗ n + n ⊗ e3 ) + 4 e3 ⊗ e3 with
(25)
c2 c5 c2 + (e3 · n)2 ¿ 0; 2 = c1 + ; 2 2 2
c5 c5 (e3 · n); 4 = 3 = c3 + (26) + c4 (e3 · n)2 : 2 2 The sign of 1 is implied by the positive de
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
be factorized as det Le = 1 , and thus = 12 (c1 + c2 )(c2 + c5 ) + ((c3 + c5 )(c2 − c3 ) + c4 (c1 + c2 ))(e3 · n)2 + 12 (2c3 (c3 + 2c5 ) − c4 (2c1 + c2 ) + c5 (c4 + 2c5 ))(e3 · n)4 ¿ 0
(27)
is strictly positive as well for all directions n. In addition, the inverse of the elastic acoustic tensor can be obtained as Re =
1I
1
=
1 ; 1
3
=
1 (− 1 3 + ( 2 4 − 32 )(e3 · n)); 1
+
2n
⊗n+
3 (e3
⊗ n + n ⊗ e3 ) +
4 e3
⊗ e3
(28)
with 2
=
1 (− 2 ( 1 + 4 ) + 32 ); 1 4
=
1 (− 4 ( 1 + 2 ) + 32 ): 1
(29)
As strains may localize along a surface of normal n when the elastoplastic acoustic tensor becomes
(30)
n;n·n=1
Let " and # be the LamRe coeNcients of an isotropic reference elastic material with positive de
GL = GT = #;
L =
ET =
4#[#(3" + 2#) + c4 (# + ")] ; 4#(" + #) + c4 (" + 2#)
" : 2(" + #)
For the admissible values of the elastic parameter c4 , there exist lower bound corresponding to the loss of positive de
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
c4LPD and it is given by #(3" + 2#) = −E; c4 ¿ c4LPD = − "+# where E is the Young’s modulus corresponding to the assumed LamRe constants. In view of the above simpli
223
(32)
(33)
with A1 = (2c12 3 Pv Qv + c1 c4 3 (Pv Q33 + Qv P33 ))n3 + (c2 c4 (
1
+
4 )(Q33 P3j nj
+ P33 Q3j nj )
+ c1 c2 4 (Qv P3j nj + Pv Q3j nj ))n3 + (c1 c2 3 (Pv + Qv ) + c2 c4 3 (P33 + Q33 ))(ni Qij nj )n3 ; A2 = (c12 4 Pv Qv + c1 c4 (
2
+
4
+
1 )(Pv Q33
+ Qv P33 ) + c42 (
1
+
2 4 )P33 Q33 )n3
+ c2 c4 2 (Q33 ni Pij nj + P33 ni Qij nj )n23 + c2 c4 3 (P33 Q3j nj + Q33 P3j nj )n23 ; A3 = c42 2 Q33 P33 n43 + c22 4 P3j nj Q3k nk + c1 c2 3 (Q3j nj Pv + P3j nj Qv ) + c22 2 ni Qij nj nk Pkl nl + c22 1 ni Pik Qkj nj + c22 3 (P3j nj nk Qkl nl + Q3j nj nk Pkl nl ); A4 = c12 (
1
+
2 )Pv Qv
+ c 1 c2 (
1
+
2 )(Pv ni Qij nj
+ Qv ni Pij nj )
+ (c1 c4 3 (Qv P33 + Pv Q33 ) + 2c42 3 Q33 P33 )n33 and hE = c1 Pv Qv + c2 Pij Qij + c4 P33 Q33 ;
(34)
where the summation convention is used for Latin indices, and Pv = tr P;
Qv = tr Q:
Assuming stress principal directions are consistent with the symmetrical axes of material, for associated plasticity Eqs. (34) and (35) can be reduced to hn = B1 + B2 + B3 + B4
(35)
with B1 = c12 (
1
2 2 )Pv
+
+ 2c1 c2 (
1
+
2 2 )Pv (P1 n1
+ P2 n22 + P3 n23 )
+ c22 1 (P12 n21 + P22 n22 + P32 n23 ) + c22 2 (P12 n21 + P22 n22 + P32 n23 )2 ; B2 = (2c12 3 Pv2 + 2c1 (c4 + c2 ) 3 Pv P3 )n3 + (2c1 c2 3 Pv + 2c2 c4 3 P3 + 2c22 3 P3 )(P1 n21 + P2 n22 + P3 n23 )n3 ; B3 = (2c2 c4 (
1
+ 2c1 c4 (
+ 1
+
2 4 )P3 2
+
+ 2c1 c2 4 Pv P3 + c12 4 Pv2 4 )P3 Pv
+ c42 (
1
+
+ 2c2 c4 2 P3 (P1 n21 + P2 n22 + P3 n23 )n23 ;
2 4 )P3
+ c22 4 P32 )n23
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
B4 = (2c2 c4 3 P32 + 2c1 c4 3 Pv P3 + 2c42 3 P32 )n33 + c42 2 P32 n43 and hE = c1 Pv2 + c2 (P12 + P22 + P32 ) + c4 P32 :
(36)
4. Critical directions of localized bands and hardening modulus at plane strain For the case of plane strain, assuming the components n1 and n3 are located in the plane of interest and combining its special stress-strain relation, Eqs. (35) and (36) can be reduced to hn = C 1 + C 2 + C 3 + C 4 ;
(37)
where C1 = c12 (
1
2 2 )Pv
+
+ 2c1 c2 (
1
+
2 2 )Pv (P1 n1
+ P3 n23 )
+ c22 1 (P12 n21 + P32 n23 ) + c22 2 (P12 n21 + P32 n23 )2 ; C2 = (2c12 3 Pv2 + 2c1 (c4 + c2 ) 3 Pv P3 )n3 + (2c1 c2 3 Pv + 2c2 c4 3 P3 + 2c22 3 P3 )(P1 n21 + P3 n23 )n3 ; C3 = (2c2 c4 ( + c42 (
1
1
+
+
2 4 )P3 2 4 )P3
+ 2c1 c2 4 Pv P3 + c12 4 Pv2 + 2c1 c4 (
1
+
2
+
4 )P3 Pv
+ c22 4 P32 )n23 + 2c2 c4 2 P3 (P1 n21 + P3 n23 )n23 ;
C4 = (2c2 c4 3 P32 + 2c1 c4 3 Pv P3 + 2c42 3 P32 )n33 + c42 2 P32 n43 and hE = c1 Pv2 + c2 (P12 + P22 + P32 ) + c4 P32 :
(38)
Comparing Eq. (35) with Eq. (37), it can be found that the plane strain expression (37) is simply a special case of expression (35) for three-dimensional stress and strain states and is obtained when n2 = 0 is assumed. However, the explicit analysis of localization conditions becomes signi
(39)
n21 = 1 − n23 :
(40)
it has
Combining Eqs. (26), (27), (29), (37) and (40) leads to *=
1 hn = (p4 n43 + p2 n23 + p0 ); # #
(41)
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
225
where p4 = −P32 (" + #)c42 + (4#P1 P3 (" + #) − 4#P32 (" + #) − 4#2 P12 − 4"#P1 Pv − "2 Pv2 )c4 − 4#2 (P3 − P1 )2 (" + #); p2 = (" + 2#)P32 c42 + (4#P32 (" + 2#) − 4#P1 P3 (" + #) + 4u2 P12 + "2 Pv2 + 4"#Pv P1 + 2"#P3 Pv )c4 + 4"#2 Pv (P3 − P1 ) + 4#2 (P3 − P1 )((P3 + P1 )(" + 2#) − 2P1 (" + #)); p0 = 4#3 P12 + "2 #Pv2 + 4"#2 P1 Pv
(42)
= #(" + 2#) + c4 (" + 2#)n23 − c4 (" + #)n43 = q0 + q2 n23 + q4 n43 ;
(43)
q0 = #(" + 2#);
(44)
and where q2 = c4 (" + 2#);
q4 = −c4 (" + #):
As can been seen from Eq. (30), the maximum value of * must be derived in order to obtain the critical hardening modulus corresponding to the initiation of strain localization. Using the condition for extreme values of *, namely d*=d(n3 ) = 0, it has −R2 ± R22 − R4 R0 2 2 ; (45) (n3 )I = 0; (n3 )II; III = R4 where R4 = p4 q2 − q4 p2 ; R 2 = p 4 q0 − q 4 p 0 ; R 0 = p 2 q0 − q 2 p 0 :
(46)
Let Y=
d2 * dn23
(47)
and substitute Eq. (45) into Eq. (47), it can be observed that when R0 = p2 q0 − q2 p0 ¡ 0;
(48)
the condition Y ((n23 )I ) ¡ 0 is available. As a result, the maximum value of * exists when n23 = (n23 )I . If 1 denotes the angle in the e1 –e3 plane from the e1 -axis to the normal vector (n1 ; n3 ), then 1 is obtained by tan2 1 =
(n23 )I = 0; 1 − (n23 )I
(49)
namely 1 = 0. The corresponding maximum value of the hardening modulus Hcr can be obtained by substituting (n23 )I into Eq. (30).
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
When R0 ¿ 0, we have Y ((n23 )II ) ¿ 0;
Y ((n23 )III ) ¡ 0:
Thus, the maximum value of * is available when n23 = (n23 )III , and the angle 1 at the onset of strain localization is given as −R2 − R22 − R4 R0 (n23 )III 2 tan 1 = = : (50) 1 − (n23 )III R4 + R2 + R22 − R4 R0 Then substitution of (n23 )III into Eq. (30) gives the corresponding maximum value of the hardening modulus Hcr . 5. Examples of strain localization conditions at plane strain For the case of plastic plane strain, the out-of-plane principal stress whose direction is perpendicular to the plane of interest is given by 2 = k( 1 + 3 )
(51)
and the ratio k changes during deformation and is bounded by 6 k 6 0:5, where is the Poisson’s ratio [18]. For simplicity and without loss of generality, only the stress state 1 ¿ 2 ¿ 3 is considered in the following. 5.1. Tresca criterion When the principal stresses have the relation 1 ¿ 2 ¿ 3 , Tresca criterion is de
(52)
where C is a material constant. We then obtain Pv = tr P = 0;
P1 =
√
2 ; 2
P2 = 0;
P3 = −
√
2 : 2
(53)
Combining Eqs. (42), (44), (46) and (53) yields R0 = 12 #(" + 2#)2 c42 + 2#2 (" + 2#)(2" + 3#)c4 + 8#3 (" + 2#)(" + #); R2 = − 12 #(" + 2#)(" + #)c42 − 2(2#2 (" + 2#)(" + #) + #4 )c4 − 8#3 (" + 2#)(" + #); R4 = −2#2 (" + 2#)c42 − 8#3 (" + #)c4 :
(54)
For engineering materials, normally we have 0 ¡ ¡ 0:5. As " = 2#=(1 − 2) and # ¿ 0, it has " ¿ 0. Thus, we can
(55) (56)
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
227
The value of the corresponding hardening modulus is obtained as follows Hcr = 0:
(57)
As can be seen, based on Tresca criterion, the direction of the localized band on the onset of strain localization and the corresponding hardening modulus are independent of the parameter c4 for the case of plane strain. Consequently, it can be concluded that under Tresca criterion the deviation from isotropic elasticity does not a5ect the properties of strain localization, and the properties of strain localization are just the same as those in the case of elastic isotropy [19]. 5.2. Twin-shear criterion When the principal stresses satisfy 1 ¿ 2 ¿ 3 , Twin-shear criterion (TS criterion) is de
when 2 6 12 ( 1 + 3 );
(58a)
F = 14 ( 1 + 2 ) − 12 3 − C = 0
when 2 ¿ 12 ( 1 + 3 ):
(58b)
From Eq. (51) we know that the yield criterion denoted by Eq. (58a) is valid assuming 2 ¿ 0. We then can show that √ √ 6 6 Pv = 0; P1 = ; P2 = P3 = − (59) 3 6 and R0 = 16 #(" + 2#)2 c42 + 23 #2 (" + 2#)(3" + 4#)c4 + 2#3 (" + 2#)(3" + 2#); R2 = − 16 #(" + 2#)(" + #)c42 − 23 #2 (3"2 + 9#" + 10#2 )c4 − 6#3 (" + 2#)(" + #); R4 = − 43 #2 (" + 3#)c42 − 8#3 (" + #)c4 : As " ¿ 0 and
c4 ¿ c4LPD ,
(60)
we
B = 2# + 16 c4 ;
p0 = 83 #3 ;
p2 = 16 (" + 2#)c42 + 23 #(3" + 8#)c4 + 2#2 (3" + 2#); p4 = − 16 (" + #)c42 − 23 #(3" + 7#)c4 − 6#2 (" + #):
228
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233 44 43 λ / µ =1
o
θ ( )
42 41 40 39 38
-2
-1
0
1
2
3
4
c 4/ µ
Fig. 1. Relationship between angle 1 and parameter c4 for the case "=# = 1. -0.36 -0.37 -0.38
λ / µ =1
Hcr / µ
-0.39 -0.40 -0.41 -0.42 -0.43 -0.44
-2
-1
0
1
2
3
4
c4/ µ
Fig. 2. Relationship between hardening modulus Hcr and parameter c4 for the case "=# = 1.
For the case "=# = 1, we know from Eq. (32) that c4 ¿ − 2:5#. For this case, the in6uence of the parameter c4 on the angle 1 is shown in Fig. 1. As can be seen, the angle 1 decreases with the increase of the parameter c4 . The relation between the hardening modulus Hsb and the parameter c4 is shown in Fig. 2. It can be observed that the hardening modulus Hsb also decreases with the increase of the parameter c4 . 6. In*uence of yield criteria on properties of strain localization It can be found from above analyses that the obtained properties of strain localization are di5erent based on di5erent yield criteria. Consequently, in this section, the in6uence of yield criteria on the values of the direction angle and the corresponding hardening modulus is investigated with reference to a uni
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
229
σ1 TS b = 1
b=3 4
Mises
b =1 2 b =1 4
Tresca b = 0
o
σ2
σ3
Fig. 3. Yield loci on the 2-plane for the UYC.
Based on orthogonal octahedron of the TS element model [20], the uni
1 1 1 − (b 2 + 3 ) − C = 0 2 2(1 + b)
when 2 6
1 + 3 ; 2
(64a)
F=
1 1 ( 1 + b 2 ) − 3 − C = 0 2(1 + b) 2
when 2 ¿
1 + 3 ; 2
(64b)
where b is a parameter that plays an important role in the UYC and it ranges from 0 to 1. It builds a bridge among di5erent yield criteria. When b varies from 0 to 1, a family of convex yield criteria that are suitable for di5erent kinds of materials are deduced. Hence, the UYC is not a single yield criterion but a theoretical system including a series of regular yield criteria with Tresca criterion (b = 0) and TS criterion (b = 1) being its lower and upper bounds, respectively. The parameter b can be expressed as b=
240 − t ; t − 4 0
(65)
where t is uniaxial tensile strength and 40 the shear strength. In practice, when basic material parameters are obtained by experiments, the value of b can be determined through Eq. (65). Whenever parameter b is obtained, the yield criterion for this sort of material is determined and the application is possible. Consequently, b can be regarded as a parameter by which the suitable yield criterion for material of interest can be determined. The yield loci on the 2-plane for the UYC are shown in Fig. 3. It can be observed from Fig. 3 that the UYC is reduced to a criterion which is the linear approximation of von Mises criterion when the value of parameter b is equal to 0.5.
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
From Eq. (51) we know that the yield criterion denoted by Eq. (64a) is valid assuming 2 ¿ 0. Hence we have m1 m2 m3 P1 = ; P2 = ; P3 = ; Pv = 0 (66) w w w with 1 b 1 ; m3 = − ; w = m21 + m22 + m23 m1 = ; m2 = − 2 2(1 + b) 2(1 + b) and R0 = #(" + 2#)2 P32 c42 + 4#2 (" + 2#)P3 ((" + 2#)P3 − (" + #)P1 )c4 + 4#3 (" + 2#)(P3 − P1 )((" + 2#)(P3 + P1 ) − 2(" + #)P1 ); R2 = −P32 #(" + 2#)(" + #)c42 + 4#2 ((" + 2#)(" + #)P1 P3 − (" + 2#)(" + #)P32 − #2 P12 )c4 − 4#3 (" + 2#)(" + #)(P3 − P1 )2 ; R4 = 4#P1 (P3 (" + #)(" + 2#) − #2 P1 − P3 (" + #)2 )c42 + 8P1 #3 (P3 − P1 )(" + #)c4 :
(67)
For " ¿ 0 and c4 ¿ c4LPD , we have R0 ¿ 0. Thus, substitution of Eq. (60) into Eq. (45) gives P3 (" + 2#)c4 + 2#((2# + ")P3 − "P1 ) : (n23 )III = 2(P3 (" + #) − #P1 )c4 + 4#(P3 − P1 )(" + #) Then the direction angle of the localized band on the onset of strain localization can be obtained as P3 (" + 2#)c4 + 2#((2# + ")P3 − "P1 ) tan2 1 = (68) ("P3 − 2#P1 )c4 + 2#("P3 − (" + 2#)P1 ) and the corresponding hardening modulus Hcr =
p 4 A2 + p 2 A + p 0 −B q4 A 2 + q 2 A + q 0
(69)
with A = (n23 )III ;
B = 2# + P32 c4 ;
p0 = 4#3 P12 ;
p2 = (" + 2#)P32 c42 + 4#(P32 (" + 2#) − P1 P3 (" + #) + #P12 )c4 + 4#2 (P3 − P1 )((P3 + P1 )(" + 2#) − 2P1 (" + #)); p4 = −P32 (" + #)c42 + 4#((P1 P3 − P32 )(" + #) − #P12 )c4 − 4#2 (P3 − P1 )2 (" + #): It can be observed that the direction angle and the corresponding hardening modulus, respectively, given by Eqs. (68) and (69) can reduce to those based on Tresca criterion (b = 0) and TS criterion (b = 1). The e5ect of yield criteria on properties of strain localization can be re6ected through the parameter b in accordance with its physical meaning. Assuming "=# = 1, from Eq. (68) the relations
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
231
46 45
c4 = - µ c4 = 0 c4 = 4 µ
44
o
θ ()
43 42 41 40 39 38
0.0
0.2
0. 4
0.6
0. 8
1.0
b
Fig. 4. Relationship between angle 1 and parameter b.
0.0 c4= - µ c4= 0 c4= 4 µ
-0.1
Hcr / µ
-0.2 -0.3 -0.4 -0.5
0.0
0.2
0.4
0.6
0.8
1.0
b
Fig. 5. Relationship between hardening modulus Hcr and parameter b.
between the angle 1 and the parameter b for the cases of three di5erent values of c4 are shown in Fig. 4. It can be observed that the direction angle 1 at the onset of strain localization decreases with the increase of the parameter b. In addition, at the same b the angle 1 also decreases with the increase of the parameter c4 when 0 ¡ b 6 1. From Eq. (69), the relations between the hardening modulus Hcr and the parameter b for the cases of three di5erent values of c4 are shown in Fig. 5. It shows that at the same c4 the hardening modulus Hcr gets smaller with the increase of the parameter b, and at the same b it also becomes smaller with the increase of the parameter c4 when 0 ¡ b 6 1. It should be noted that both the angle 1 and the hardening modulus Hcr are independent of the parameter c4 at b = 0, which is consistent with the conclusion drawn in Section 5. Consequently, it can be concluded from the above analyses that the in6uence of yield criteria on the properties of strain localization is obvious, and a suitable yield criterion for the material of interest is of great importance in the analysis of strain localization. In practice, when basic material parameters are
232
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
obtained by experiments, the value of b can be determined from Eq. (65). With the parameter b, the yield criterion suitable for the material of interest can be determined. 7. Conclusions Under non-associated plasticity and tri-axial stress states, the conditions for localization of deformation into a band in the incremental response of the elastoplastic materials with transversely isotropic elasticity are deduced. The explicit expressions for the direction of localized band and the corresponding hardening modulus at the onset of strain localization are obtained for plane strain condition for associated plasticity. With reference to Tresca criterion, TS criterion and UFC, the e5ect of deviation from isotropic elasticity in the formulation of strain localization is discussed. It is shown that under Tresca criterion (b = 0) the deviation from isotropic elasticity has nothing to do with the properties of strain localization, and the properties of strain localization are just the same as those in the case of elastic isotropy. For other cases, however, the results indicate that the properties of strain localization are dependent on deviation from isotropic elasticity. The in6uence of yield criteria on the properties of strain localization is
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
233
[14] Rizzi E, Maier G, Willam K. On failure indicators in multi-dissipative materials. International Journal of Solids and Structures 1996;33:3187–214. [15] Steinmann P, Miehe CH, Stein E. On the localization analysis of orthotropic Hill-type elastoplastic solids. Journal of the Mechanics and Physics and Solids 1994;42:1969–94. [16] Lee H, Im S, Atluri SN. Strain localization in an orthotropic material with plastic spin. International Journal of Plasticity 1995;11:423–50. [17] Rizzi E, Loret B. Qualitative analysis of strain localization. Part I: Transversely isotropic elasticity and isotropic plasticity. International Journal of Plasticity 1997;13(5):461–99. [18] Lee YK, Ghosh J. Signi
International Journal of Mechanical Sciences 45 (2003) 217 – 233
Investigation of strain localization in elastoplastic materials with transversely isotropic elasticity Yong-Qiang Zhanga;∗ , Yong Lua , Mao-Hong Yub a
School of Civil and Environmental Engineering, Protective Technology Research Center, Nanyang Technological University, Singapore 639798, Singapore b School of Civil Engineering and Mechanics, Xi’an Jiaotong University, Xi’an 710049, Xi’an, China Received 20 July 2002; received in revised form 28 February 2003; accepted 6 March 2003
Abstract Transversely isotropic materials are of primary interest in many engineering applications. In this paper, general description of the properties of strain localization is deduced for the elastoplastic materials with transversely isotropic elasticity which follow the non-associated plasticity and are subjected to tri-axial stress states. Then the explicit expressions for the direction of localized band and the corresponding hardening modulus at the onset of strain localization are obtained for plane strain condition. Furthermore, the e5ect of deviation from isotropic elasticity in the formulation of strain localization is discussed based on those yield criteria such as Tresca criterion and twin-shear criterion. In the end, the in6uences of yield criteria on the properties of strain localization are elucidated. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Strain localization; Transversely isotropy; Non-associated plasticity; Elastoplastic material
1. Introduction Strain localization of plastic 6ow into shear bands is a ubiquitous feature of elastoplastic materials undergoing non-homogeneous deformation. It can be interpreted physically as the transformation of propagating waves into stationary waves. The onset of localization is indicated by a vanishing wave speed. Within the classical rate-independent continuum theory, strain localization denotes the onset of a discontinuous bifurcation in the form of a jump in the velocity gradient
Corresponding author. Tel.: +65-67906199; fax: +65-67910676. E-mail address: [email protected] (Y.-Q. Zhang).
0020-7403/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0020-7403(03)00055-9
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
recent years [1–5]. The basic principles were discussed by Thomas [6], Hill [7], and Mandel [8], in connection with the theory of bifurcation and localization corresponding to stationary acceleration waves. It was shown by Rice [9] that the phenomenon of strain localization may be described as a type of loss of material stability, and simultaneously as the loss of ellipticity of governing di5erential equations, namely formation of discontinuity is due to bifurcation of equilibrium states. Based on Rice’s pioneering work, the conditions for the onset of strain localization have been resolved for a wide variety of constitutive models by Ottosen and Runesson [10], Runesson et al. [11], Bigoni and Hueckel [12], Nielsin and Schreyer [13] and Rizzi et al. [14]. In these works, elasticity is assumed to remain isotropic throughout the loading process. As the strength properties of a wide variety of engineering materials exhibit anisotropic behavior, the extension of such analyses to material models which do include anisotropic elasticity is of interest. Steinmann et al. [15] analyzed the localization condition for an isotropic elastic and orthotropic perfectly plastic material based on a yield criterion of the Hill type under plane stress and plane strain loading and showed that the in-plane bifurcation direction may be considerably modi
2. Transversely isotropic elasticity It is well known that a transversely isotropic elastic material has the rotational symmetry property with reference to a certain axis. The plane perpendicular to this axis is called the basal plane while planes containing the axis of symmetry will be named zonal planes. To better express material properties componentwise, the cartesian axes (e1 ; e2 ; e3 ) are adopted, where e1 and e2 are two arbitrary orthogonal unit vectors of the basal plane and the unit vector e3 denotes the axis of rotational symmetry. As the free energy for the transversely isotropic elastic material inherits the symmetry properties of the material and is a transversely isotropic function of the strain tensor U, the free energy function has the following form [17]: =
c1 2 c2 c4 tr U + tr U2 + c3 tr U tr(M · U) + tr 2 (M · U) + c5 tr(M · U2 ); 2 2 2
(1)
where tensor M = e3 ⊗ e3 with the symbol ‘⊗’ denoting outer product, ci (i ∈ [1; 5]) are
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
219
the trace of tensors. Thus, the stress tensor can be obtained as =
@ = c1 (I : U)I + c2 U + c3 ((M : U)I + (I : U)M) + c4 (M : U)M + c5 (U · M + M · U); @U
(2)
where I designates the second-order identity tensor and the symbol ‘:’ the inner product with double index contraction. Further, the elastic sti5ness takes the form E=
@2 L + c3 (I ⊗ M + M ⊗ I) + c4 M ⊗ M + c5 (I⊗M L + M⊗I); L = c1 I ⊗ I + c2 I⊗I @U ⊗ @U
(3)
L denotes a symmetrized outer product and it has (M⊗I) L ijkl =(Mik Ijl +Mil Ijk )=2. where the symbol ‘⊗’ It is noted from Eq. (3) that the material constants c3 , c4 , c5 might be interpreted as measures of the deviation from isotropic elasticity. In the cartesian axes (e1 , e2 , e3 ), the matrix representation of the constitutive equation = E : U has the following form by using Voigt notation
11
c1 + c2
22 c1 33 c1 + c3 = 23 31
c1
c1 + c 3
c1 + c 2
c1 + c 3
c1 + c 3
c1 + c2 + 2c3 + c4 + 2c5
22 33 : 2 23 2 31
c2 =2 + c5 =2 c2 =2 + c5 =2
12
11
c2 =2
(4)
2 12
From this representation, the relations between the coeNcients ci and the matrix coeNcients Eij can be obtained as E11 = c1 + c2 ;
E12 = c1 ;
E13 = c1 + c3 ;
E33 = c1 + c2 + 2c3 + c4 + 2c5 ;
E44 = c2 =2 + c5 =2:
(5)
The positive de
2 (E11 + E12 )E33 − 2E13 ¿ 0;
E44 ¿ 0:
The material moduli might be related to corresponding parameters of physical meaning. The relations between material moduli and parameters Eij may be obtained by EL =
2 (E11 + E12 )E33 − 2E13 ; E11 + E12
ET =
2 [(E11 + E12 )E33 − 2E13 ](E11 − E12 ) ; 2 E11 E33 − E13
E11 − E12 ; 2
L =
E13 ; E11 + E12
GL = E44 ;
GT =
(6)
where EL and ET , respectively, denote the longitudinal (or axial) elastic modulus and the transverse (or cross-axial) elastic modulus, GL and GT , respectively, denote the longitudinal (or zonal) shear modulus and the transverse (or basal) shear modulus, and L the longitudinal Poisson’s ratio (representing the contraction in the longitudinal direction due to an imposed traction in the basal plane).
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
3. Localization condition for non-associated plasticity For simplicity it will be assumed that deformations are small, i.e. con
(7)
where ˙ and U˙ , respectively, designate the stress rate and strain rate tensors, and the tangent sti5ness tensor D is given by 1 D = E − (E : Q) ⊗ (P : E); (8) A where P and Q are the unit outward normals to the yield surface and the plastic potential, respectively, and they can be calculated as follows by using the yield function F and the plastic potential G: @F @F @G @G P= ; Q= ; (9) @ @ @ @ where ‘ · ’ stands for a norm of a tensor. The positive parameter A is de
(10)
where H is the generalized plastic modulus that is positive, zero or negative for hardening, perfect or softening plasticity respectively. When strain localization occurs, the strain rate U˙ across the localized band is discontinuous. The discontinuity of strain rate U˙ must satisfy the Maxwell’s compatibility condition [6] [˙U] = U˙ i − U˙ o = 12 (m ⊗ n + n ⊗ m);
(11)
where the bracket ‘[ · ]’ denotes a jump of the bracketed quantity across the discontinuity surface, U˙ i and U˙ o are, respectively, the strain rates inside and outside the band, vector m represents the mode of discontinuity of the strain rate, and n is the unit normal vector of the band. For this case we have ˙i = Di : U˙ i ; i
˙o = Do : U˙ o
(12)
o
where D and D denote the tangential sti5ness tensors inside and outside the localized band, respectively. It follows from equilibrium considerations that the traction rate across the band must be unique: n · (˙i − ˙o ) = 0:
(13)
Combining Eqs. (11)–(13) yields n · (Di − Do ) : U˙ o + L · m = 0;
(14)
L = n · Di · n
(15)
where denotes the characteristic tangent sti5ness modulus tensor.
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
221
At the onset of strain localization, Do and Di can be regarded the same, namely Do = Di = D. Thus from Eq. (14), it has L·m=0
or
det(L) = 0:
(16)
This is the necessary condition for strain localization, and L = n · D · n:
(17)
When tensor L is singular, the critical hardening modulus can be obtained as follows H = a · R e · b − P : E : Q = hn − h E ;
(18)
e
where hn = a · R · b and hE = P : E : Q, and Re = (Le )−1 = (n · E · n)−1 :
(19)
In addition, the vectors a and b are given by a = n · E : Q;
b = P : E · n:
(20)
Substituting Eq. (3) into Eq. (20) gives a = a 1 n + a 2 Q · n + a 3 e3 + a 4 Q · e 3 ;
b = b 1 n + b 2 P · n + b 3 e3 + b 4 P · e 3
(21)
with a1 = c1 tr Q + c3 (e3 · Q · e3 );
a2 = c2 ;
a3 = (c3 tr Q + c4 (e3 · Q · e3 ))(e3 · n) + c5 (n · Q · e3 );
a4 = c5 (e3 · n)
(22)
b4 = c5 (e3 · n):
(23)
and b1 = c1 tr P + c3 (e3 · P · e3 );
b2 = c2 ;
b3 = (c3 tr P + c4 (e3 · P · e3 ))(e3 · n) + c5 (n · P · e3 ); Due to Eq. (3), we can obtain
hE = c1 (tr P)(tr Q) + c2 P : Q + c3 ((e3 · Q · e3 ) tr P + (e3 · P · e3 ) tr Q) + c4 (e3 · P · e3 )(e3 · Q · e3 ) + 2c5 (e3 · (P · Q) · e3 )
(24)
and the elastic acoustic tensor Le = n · E · n = 1 I + 2 n ⊗ n + 3 (e3 ⊗ n + n ⊗ e3 ) + 4 e3 ⊗ e3 with
(25)
c2 c5 c2 + (e3 · n)2 ¿ 0; 2 = c1 + ; 2 2 2
c5 c5 (e3 · n); 4 = 3 = c3 + (26) + c4 (e3 · n)2 : 2 2 The sign of 1 is implied by the positive de
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
be factorized as det Le = 1 , and thus = 12 (c1 + c2 )(c2 + c5 ) + ((c3 + c5 )(c2 − c3 ) + c4 (c1 + c2 ))(e3 · n)2 + 12 (2c3 (c3 + 2c5 ) − c4 (2c1 + c2 ) + c5 (c4 + 2c5 ))(e3 · n)4 ¿ 0
(27)
is strictly positive as well for all directions n. In addition, the inverse of the elastic acoustic tensor can be obtained as Re =
1I
1
=
1 ; 1
3
=
1 (− 1 3 + ( 2 4 − 32 )(e3 · n)); 1
+
2n
⊗n+
3 (e3
⊗ n + n ⊗ e3 ) +
4 e3
⊗ e3
(28)
with 2
=
1 (− 2 ( 1 + 4 ) + 32 ); 1 4
=
1 (− 4 ( 1 + 2 ) + 32 ): 1
(29)
As strains may localize along a surface of normal n when the elastoplastic acoustic tensor becomes
(30)
n;n·n=1
Let " and # be the LamRe coeNcients of an isotropic reference elastic material with positive de
GL = GT = #;
L =
ET =
4#[#(3" + 2#) + c4 (# + ")] ; 4#(" + #) + c4 (" + 2#)
" : 2(" + #)
For the admissible values of the elastic parameter c4 , there exist lower bound corresponding to the loss of positive de
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
c4LPD and it is given by #(3" + 2#) = −E; c4 ¿ c4LPD = − "+# where E is the Young’s modulus corresponding to the assumed LamRe constants. In view of the above simpli
223
(32)
(33)
with A1 = (2c12 3 Pv Qv + c1 c4 3 (Pv Q33 + Qv P33 ))n3 + (c2 c4 (
1
+
4 )(Q33 P3j nj
+ P33 Q3j nj )
+ c1 c2 4 (Qv P3j nj + Pv Q3j nj ))n3 + (c1 c2 3 (Pv + Qv ) + c2 c4 3 (P33 + Q33 ))(ni Qij nj )n3 ; A2 = (c12 4 Pv Qv + c1 c4 (
2
+
4
+
1 )(Pv Q33
+ Qv P33 ) + c42 (
1
+
2 4 )P33 Q33 )n3
+ c2 c4 2 (Q33 ni Pij nj + P33 ni Qij nj )n23 + c2 c4 3 (P33 Q3j nj + Q33 P3j nj )n23 ; A3 = c42 2 Q33 P33 n43 + c22 4 P3j nj Q3k nk + c1 c2 3 (Q3j nj Pv + P3j nj Qv ) + c22 2 ni Qij nj nk Pkl nl + c22 1 ni Pik Qkj nj + c22 3 (P3j nj nk Qkl nl + Q3j nj nk Pkl nl ); A4 = c12 (
1
+
2 )Pv Qv
+ c 1 c2 (
1
+
2 )(Pv ni Qij nj
+ Qv ni Pij nj )
+ (c1 c4 3 (Qv P33 + Pv Q33 ) + 2c42 3 Q33 P33 )n33 and hE = c1 Pv Qv + c2 Pij Qij + c4 P33 Q33 ;
(34)
where the summation convention is used for Latin indices, and Pv = tr P;
Qv = tr Q:
Assuming stress principal directions are consistent with the symmetrical axes of material, for associated plasticity Eqs. (34) and (35) can be reduced to hn = B1 + B2 + B3 + B4
(35)
with B1 = c12 (
1
2 2 )Pv
+
+ 2c1 c2 (
1
+
2 2 )Pv (P1 n1
+ P2 n22 + P3 n23 )
+ c22 1 (P12 n21 + P22 n22 + P32 n23 ) + c22 2 (P12 n21 + P22 n22 + P32 n23 )2 ; B2 = (2c12 3 Pv2 + 2c1 (c4 + c2 ) 3 Pv P3 )n3 + (2c1 c2 3 Pv + 2c2 c4 3 P3 + 2c22 3 P3 )(P1 n21 + P2 n22 + P3 n23 )n3 ; B3 = (2c2 c4 (
1
+ 2c1 c4 (
+ 1
+
2 4 )P3 2
+
+ 2c1 c2 4 Pv P3 + c12 4 Pv2 4 )P3 Pv
+ c42 (
1
+
+ 2c2 c4 2 P3 (P1 n21 + P2 n22 + P3 n23 )n23 ;
2 4 )P3
+ c22 4 P32 )n23
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
B4 = (2c2 c4 3 P32 + 2c1 c4 3 Pv P3 + 2c42 3 P32 )n33 + c42 2 P32 n43 and hE = c1 Pv2 + c2 (P12 + P22 + P32 ) + c4 P32 :
(36)
4. Critical directions of localized bands and hardening modulus at plane strain For the case of plane strain, assuming the components n1 and n3 are located in the plane of interest and combining its special stress-strain relation, Eqs. (35) and (36) can be reduced to hn = C 1 + C 2 + C 3 + C 4 ;
(37)
where C1 = c12 (
1
2 2 )Pv
+
+ 2c1 c2 (
1
+
2 2 )Pv (P1 n1
+ P3 n23 )
+ c22 1 (P12 n21 + P32 n23 ) + c22 2 (P12 n21 + P32 n23 )2 ; C2 = (2c12 3 Pv2 + 2c1 (c4 + c2 ) 3 Pv P3 )n3 + (2c1 c2 3 Pv + 2c2 c4 3 P3 + 2c22 3 P3 )(P1 n21 + P3 n23 )n3 ; C3 = (2c2 c4 ( + c42 (
1
1
+
+
2 4 )P3 2 4 )P3
+ 2c1 c2 4 Pv P3 + c12 4 Pv2 + 2c1 c4 (
1
+
2
+
4 )P3 Pv
+ c22 4 P32 )n23 + 2c2 c4 2 P3 (P1 n21 + P3 n23 )n23 ;
C4 = (2c2 c4 3 P32 + 2c1 c4 3 Pv P3 + 2c42 3 P32 )n33 + c42 2 P32 n43 and hE = c1 Pv2 + c2 (P12 + P22 + P32 ) + c4 P32 :
(38)
Comparing Eq. (35) with Eq. (37), it can be found that the plane strain expression (37) is simply a special case of expression (35) for three-dimensional stress and strain states and is obtained when n2 = 0 is assumed. However, the explicit analysis of localization conditions becomes signi
(39)
n21 = 1 − n23 :
(40)
it has
Combining Eqs. (26), (27), (29), (37) and (40) leads to *=
1 hn = (p4 n43 + p2 n23 + p0 ); # #
(41)
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
225
where p4 = −P32 (" + #)c42 + (4#P1 P3 (" + #) − 4#P32 (" + #) − 4#2 P12 − 4"#P1 Pv − "2 Pv2 )c4 − 4#2 (P3 − P1 )2 (" + #); p2 = (" + 2#)P32 c42 + (4#P32 (" + 2#) − 4#P1 P3 (" + #) + 4u2 P12 + "2 Pv2 + 4"#Pv P1 + 2"#P3 Pv )c4 + 4"#2 Pv (P3 − P1 ) + 4#2 (P3 − P1 )((P3 + P1 )(" + 2#) − 2P1 (" + #)); p0 = 4#3 P12 + "2 #Pv2 + 4"#2 P1 Pv
(42)
= #(" + 2#) + c4 (" + 2#)n23 − c4 (" + #)n43 = q0 + q2 n23 + q4 n43 ;
(43)
q0 = #(" + 2#);
(44)
and where q2 = c4 (" + 2#);
q4 = −c4 (" + #):
As can been seen from Eq. (30), the maximum value of * must be derived in order to obtain the critical hardening modulus corresponding to the initiation of strain localization. Using the condition for extreme values of *, namely d*=d(n3 ) = 0, it has −R2 ± R22 − R4 R0 2 2 ; (45) (n3 )I = 0; (n3 )II; III = R4 where R4 = p4 q2 − q4 p2 ; R 2 = p 4 q0 − q 4 p 0 ; R 0 = p 2 q0 − q 2 p 0 :
(46)
Let Y=
d2 * dn23
(47)
and substitute Eq. (45) into Eq. (47), it can be observed that when R0 = p2 q0 − q2 p0 ¡ 0;
(48)
the condition Y ((n23 )I ) ¡ 0 is available. As a result, the maximum value of * exists when n23 = (n23 )I . If 1 denotes the angle in the e1 –e3 plane from the e1 -axis to the normal vector (n1 ; n3 ), then 1 is obtained by tan2 1 =
(n23 )I = 0; 1 − (n23 )I
(49)
namely 1 = 0. The corresponding maximum value of the hardening modulus Hcr can be obtained by substituting (n23 )I into Eq. (30).
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
When R0 ¿ 0, we have Y ((n23 )II ) ¿ 0;
Y ((n23 )III ) ¡ 0:
Thus, the maximum value of * is available when n23 = (n23 )III , and the angle 1 at the onset of strain localization is given as −R2 − R22 − R4 R0 (n23 )III 2 tan 1 = = : (50) 1 − (n23 )III R4 + R2 + R22 − R4 R0 Then substitution of (n23 )III into Eq. (30) gives the corresponding maximum value of the hardening modulus Hcr . 5. Examples of strain localization conditions at plane strain For the case of plastic plane strain, the out-of-plane principal stress whose direction is perpendicular to the plane of interest is given by 2 = k( 1 + 3 )
(51)
and the ratio k changes during deformation and is bounded by 6 k 6 0:5, where is the Poisson’s ratio [18]. For simplicity and without loss of generality, only the stress state 1 ¿ 2 ¿ 3 is considered in the following. 5.1. Tresca criterion When the principal stresses have the relation 1 ¿ 2 ¿ 3 , Tresca criterion is de
(52)
where C is a material constant. We then obtain Pv = tr P = 0;
P1 =
√
2 ; 2
P2 = 0;
P3 = −
√
2 : 2
(53)
Combining Eqs. (42), (44), (46) and (53) yields R0 = 12 #(" + 2#)2 c42 + 2#2 (" + 2#)(2" + 3#)c4 + 8#3 (" + 2#)(" + #); R2 = − 12 #(" + 2#)(" + #)c42 − 2(2#2 (" + 2#)(" + #) + #4 )c4 − 8#3 (" + 2#)(" + #); R4 = −2#2 (" + 2#)c42 − 8#3 (" + #)c4 :
(54)
For engineering materials, normally we have 0 ¡ ¡ 0:5. As " = 2#=(1 − 2) and # ¿ 0, it has " ¿ 0. Thus, we can
(55) (56)
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227
The value of the corresponding hardening modulus is obtained as follows Hcr = 0:
(57)
As can be seen, based on Tresca criterion, the direction of the localized band on the onset of strain localization and the corresponding hardening modulus are independent of the parameter c4 for the case of plane strain. Consequently, it can be concluded that under Tresca criterion the deviation from isotropic elasticity does not a5ect the properties of strain localization, and the properties of strain localization are just the same as those in the case of elastic isotropy [19]. 5.2. Twin-shear criterion When the principal stresses satisfy 1 ¿ 2 ¿ 3 , Twin-shear criterion (TS criterion) is de
when 2 6 12 ( 1 + 3 );
(58a)
F = 14 ( 1 + 2 ) − 12 3 − C = 0
when 2 ¿ 12 ( 1 + 3 ):
(58b)
From Eq. (51) we know that the yield criterion denoted by Eq. (58a) is valid assuming 2 ¿ 0. We then can show that √ √ 6 6 Pv = 0; P1 = ; P2 = P3 = − (59) 3 6 and R0 = 16 #(" + 2#)2 c42 + 23 #2 (" + 2#)(3" + 4#)c4 + 2#3 (" + 2#)(3" + 2#); R2 = − 16 #(" + 2#)(" + #)c42 − 23 #2 (3"2 + 9#" + 10#2 )c4 − 6#3 (" + 2#)(" + #); R4 = − 43 #2 (" + 3#)c42 − 8#3 (" + #)c4 : As " ¿ 0 and
c4 ¿ c4LPD ,
(60)
we
B = 2# + 16 c4 ;
p0 = 83 #3 ;
p2 = 16 (" + 2#)c42 + 23 #(3" + 8#)c4 + 2#2 (3" + 2#); p4 = − 16 (" + #)c42 − 23 #(3" + 7#)c4 − 6#2 (" + #):
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233 44 43 λ / µ =1
o
θ ( )
42 41 40 39 38
-2
-1
0
1
2
3
4
c 4/ µ
Fig. 1. Relationship between angle 1 and parameter c4 for the case "=# = 1. -0.36 -0.37 -0.38
λ / µ =1
Hcr / µ
-0.39 -0.40 -0.41 -0.42 -0.43 -0.44
-2
-1
0
1
2
3
4
c4/ µ
Fig. 2. Relationship between hardening modulus Hcr and parameter c4 for the case "=# = 1.
For the case "=# = 1, we know from Eq. (32) that c4 ¿ − 2:5#. For this case, the in6uence of the parameter c4 on the angle 1 is shown in Fig. 1. As can be seen, the angle 1 decreases with the increase of the parameter c4 . The relation between the hardening modulus Hsb and the parameter c4 is shown in Fig. 2. It can be observed that the hardening modulus Hsb also decreases with the increase of the parameter c4 . 6. In*uence of yield criteria on properties of strain localization It can be found from above analyses that the obtained properties of strain localization are di5erent based on di5erent yield criteria. Consequently, in this section, the in6uence of yield criteria on the values of the direction angle and the corresponding hardening modulus is investigated with reference to a uni
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
229
σ1 TS b = 1
b=3 4
Mises
b =1 2 b =1 4
Tresca b = 0
o
σ2
σ3
Fig. 3. Yield loci on the 2-plane for the UYC.
Based on orthogonal octahedron of the TS element model [20], the uni
1 1 1 − (b 2 + 3 ) − C = 0 2 2(1 + b)
when 2 6
1 + 3 ; 2
(64a)
F=
1 1 ( 1 + b 2 ) − 3 − C = 0 2(1 + b) 2
when 2 ¿
1 + 3 ; 2
(64b)
where b is a parameter that plays an important role in the UYC and it ranges from 0 to 1. It builds a bridge among di5erent yield criteria. When b varies from 0 to 1, a family of convex yield criteria that are suitable for di5erent kinds of materials are deduced. Hence, the UYC is not a single yield criterion but a theoretical system including a series of regular yield criteria with Tresca criterion (b = 0) and TS criterion (b = 1) being its lower and upper bounds, respectively. The parameter b can be expressed as b=
240 − t ; t − 4 0
(65)
where t is uniaxial tensile strength and 40 the shear strength. In practice, when basic material parameters are obtained by experiments, the value of b can be determined through Eq. (65). Whenever parameter b is obtained, the yield criterion for this sort of material is determined and the application is possible. Consequently, b can be regarded as a parameter by which the suitable yield criterion for material of interest can be determined. The yield loci on the 2-plane for the UYC are shown in Fig. 3. It can be observed from Fig. 3 that the UYC is reduced to a criterion which is the linear approximation of von Mises criterion when the value of parameter b is equal to 0.5.
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
From Eq. (51) we know that the yield criterion denoted by Eq. (64a) is valid assuming 2 ¿ 0. Hence we have m1 m2 m3 P1 = ; P2 = ; P3 = ; Pv = 0 (66) w w w with 1 b 1 ; m3 = − ; w = m21 + m22 + m23 m1 = ; m2 = − 2 2(1 + b) 2(1 + b) and R0 = #(" + 2#)2 P32 c42 + 4#2 (" + 2#)P3 ((" + 2#)P3 − (" + #)P1 )c4 + 4#3 (" + 2#)(P3 − P1 )((" + 2#)(P3 + P1 ) − 2(" + #)P1 ); R2 = −P32 #(" + 2#)(" + #)c42 + 4#2 ((" + 2#)(" + #)P1 P3 − (" + 2#)(" + #)P32 − #2 P12 )c4 − 4#3 (" + 2#)(" + #)(P3 − P1 )2 ; R4 = 4#P1 (P3 (" + #)(" + 2#) − #2 P1 − P3 (" + #)2 )c42 + 8P1 #3 (P3 − P1 )(" + #)c4 :
(67)
For " ¿ 0 and c4 ¿ c4LPD , we have R0 ¿ 0. Thus, substitution of Eq. (60) into Eq. (45) gives P3 (" + 2#)c4 + 2#((2# + ")P3 − "P1 ) : (n23 )III = 2(P3 (" + #) − #P1 )c4 + 4#(P3 − P1 )(" + #) Then the direction angle of the localized band on the onset of strain localization can be obtained as P3 (" + 2#)c4 + 2#((2# + ")P3 − "P1 ) tan2 1 = (68) ("P3 − 2#P1 )c4 + 2#("P3 − (" + 2#)P1 ) and the corresponding hardening modulus Hcr =
p 4 A2 + p 2 A + p 0 −B q4 A 2 + q 2 A + q 0
(69)
with A = (n23 )III ;
B = 2# + P32 c4 ;
p0 = 4#3 P12 ;
p2 = (" + 2#)P32 c42 + 4#(P32 (" + 2#) − P1 P3 (" + #) + #P12 )c4 + 4#2 (P3 − P1 )((P3 + P1 )(" + 2#) − 2P1 (" + #)); p4 = −P32 (" + #)c42 + 4#((P1 P3 − P32 )(" + #) − #P12 )c4 − 4#2 (P3 − P1 )2 (" + #): It can be observed that the direction angle and the corresponding hardening modulus, respectively, given by Eqs. (68) and (69) can reduce to those based on Tresca criterion (b = 0) and TS criterion (b = 1). The e5ect of yield criteria on properties of strain localization can be re6ected through the parameter b in accordance with its physical meaning. Assuming "=# = 1, from Eq. (68) the relations
Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
231
46 45
c4 = - µ c4 = 0 c4 = 4 µ
44
o
θ ()
43 42 41 40 39 38
0.0
0.2
0. 4
0.6
0. 8
1.0
b
Fig. 4. Relationship between angle 1 and parameter b.
0.0 c4= - µ c4= 0 c4= 4 µ
-0.1
Hcr / µ
-0.2 -0.3 -0.4 -0.5
0.0
0.2
0.4
0.6
0.8
1.0
b
Fig. 5. Relationship between hardening modulus Hcr and parameter b.
between the angle 1 and the parameter b for the cases of three di5erent values of c4 are shown in Fig. 4. It can be observed that the direction angle 1 at the onset of strain localization decreases with the increase of the parameter b. In addition, at the same b the angle 1 also decreases with the increase of the parameter c4 when 0 ¡ b 6 1. From Eq. (69), the relations between the hardening modulus Hcr and the parameter b for the cases of three di5erent values of c4 are shown in Fig. 5. It shows that at the same c4 the hardening modulus Hcr gets smaller with the increase of the parameter b, and at the same b it also becomes smaller with the increase of the parameter c4 when 0 ¡ b 6 1. It should be noted that both the angle 1 and the hardening modulus Hcr are independent of the parameter c4 at b = 0, which is consistent with the conclusion drawn in Section 5. Consequently, it can be concluded from the above analyses that the in6uence of yield criteria on the properties of strain localization is obvious, and a suitable yield criterion for the material of interest is of great importance in the analysis of strain localization. In practice, when basic material parameters are
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Y.-Q. Zhang et al. / International Journal of Mechanical Sciences 45 (2003) 217 – 233
obtained by experiments, the value of b can be determined from Eq. (65). With the parameter b, the yield criterion suitable for the material of interest can be determined. 7. Conclusions Under non-associated plasticity and tri-axial stress states, the conditions for localization of deformation into a band in the incremental response of the elastoplastic materials with transversely isotropic elasticity are deduced. The explicit expressions for the direction of localized band and the corresponding hardening modulus at the onset of strain localization are obtained for plane strain condition for associated plasticity. With reference to Tresca criterion, TS criterion and UFC, the e5ect of deviation from isotropic elasticity in the formulation of strain localization is discussed. It is shown that under Tresca criterion (b = 0) the deviation from isotropic elasticity has nothing to do with the properties of strain localization, and the properties of strain localization are just the same as those in the case of elastic isotropy. For other cases, however, the results indicate that the properties of strain localization are dependent on deviation from isotropic elasticity. The in6uence of yield criteria on the properties of strain localization is
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