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Micromorphic continuum. Part III
ARTICLE IN PRESS International Journal of Non-Linear Mechanics 45 (2010) 140–148
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Micromorphic continuum. Part III Small deformation plasticity coupled with damage P. Grammenoudis a,, Ch. Tsakmakis a,, D. Hofer b a b
Darmstadt University of Technology, Institute of Continuum Mechanics, Hochschulstraße 1, D-64289 Darmstadt, Germany Westinghouse Electric Germany GmbH, Abt. PEM, Dudenstr. 44, D-68167 Mannheim, Germany
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 July 2008 Received in revised form 12 October 2009 Accepted 13 October 2009
Properties of the micromorphic theory proposed in Part II are discussed for the case of small deformations. Model responses for beam specimens under bending loading and plates with circular holes under tension loading are calculated by employing the finite element method. The results reported are concerned with the capabilities of the theory to predict size effects. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Micromorphic plasticity Finite deformation Isotropic and kinematic hardening rules Isotropic continuum damage mechanics Finite element implementation
ðdÞ
1. Introduction
T ijk nk ¼ t ij
A finite deformation micromorphic plasticity theory, exhibiting isotropic and kinematic hardening, and incorporating damage effects, has been proposed in Part II. The theory is consistent with the second law of thermodynamics and deals with a plastic micromorphic curvature tensor, which is not required to fulfill some compatibility conditions, i.e., it is not related to some gradient terms. Furthermore, a measure of smallness e has been introduced in Part II, and the theory has been defined to be of small deformations, if terms only up to order OðeÞ are retained. It is a straightforward task to verify that the small deformation version of the micromorphic model proposed, reads as follows (we confine attention to static balance equations and omit the body and double body forces): Equilibrium equations:
ui ¼ u i
@Tij ¼0 @Xj
in RR ;
@T ijk þ Tij Sij ¼ 0 @Xk
ð1Þ
t ðdÞ
h
on @RRij ¼ @RR \@RRij ;
ð4Þ
on @RuRi ;
hij ¼ h ij
ð5Þ
h
on @RRij :
ð6Þ
Kinematics: Hij ¼
@ui ; @Xj
bij ¼
1 ðh þhji Þ; 2 ij
ð7Þ
eij ¼ Hij hij ;
bij ¼ ðbe Þij þ ðbp Þij ;
Kijk ¼
eij ¼ ðee Þij þ ðep Þij ;
@hij ; @Xk
Kijk ¼ ðK e Þijk þðK p Þijk :
ð8Þ ð9Þ
Specific free energy:
C ¼ Ce þ Cis þ Ck :
ð10Þ
Elasticity laws: in RR :
ð2Þ
RCe ¼ ð1 DÞf12 ðAe Þijpq ðee Þij ðee Þpq þ 12ðBe Þijpq ðbe Þij ðbe Þpq þ ðDe Þijpq ðee Þij ðbe Þpq þ 12ðC e Þijpqr ðK e Þijk ðK e Þpqr g;
ð11Þ
Boundary conditions: Tij nj ¼ t i
on @RtRi ¼ @RR \@RuRi ;
ð3Þ
Corresponding authors.
E-mail addresses: [email protected] (P. Grammenoudis), [email protected] (Ch. Tsakmakis). 0020-7462/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2009.10.003
@Ce ¼ ð1 DÞfðBe Þijpq ðbe Þpq þ ðDe Þijpq ðee Þpq g; @ðbe Þij
ð12Þ
@Ce ¼ ð1 DÞfðAe Þijpq ðee Þpq þ ðDe Þijpq ðbe Þpq g; @ðee Þij
ð13Þ
Sij ¼ R Tij ¼ R
ARTICLE IN PRESS P. Grammenoudis et al. / International Journal of Non-Linear Mechanics 45 (2010) 140–148
T ijk ¼ R
@Ce ¼ ð1 DÞðC e Þijkpqr ðK e Þpqr : @ðK e Þijk
ð14Þ
1 k D Þ ððT Tijk ÞD ðAy Þijpq ðTpq Tpq 1 D ij k D k D þ ðSij Sij Þ ðBy Þijpq ðSpq Spq Þ þ ðT ijk T kijk ÞD ðC y Þijkpqr ðT pqr T kpqr ÞD Þ1=2
R þ k0; 1D Flow rule: s_ @f
ðe_ p Þij ¼
z :¼
z @Tij
ðb_ p Þij ¼
;
R k0; 1D
k :¼ R0 þk 0 :
k :¼
s_ @f
z @Sij
;
ð15Þ
ð16Þ
ðK_ p Þijk ¼
s_ @f
z @T ijk
;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @f @f @f @f @f @f þ þ : @Tij @Tij @Sij @Sij @T ijk @T ijk
ð17Þ
ð18Þ
Plasticity: LðtÞ :¼ ½f_ ðtÞs ¼ const: ( s_
40
for f ¼ 0 & L 40;
¼0
otherwise;
Evolution law for damage: _ ¼ a1 s_ R @C : D @D
Yield function: f¼
141
ð19Þ
ð34Þ
The aim of the present paper is to demonstrate the capabilities of this model in describing size effects present in bending of beam specimens and in plates with a hole under tension loading. It should be remarked that Parts I and II made it clear, that micromorphic constitutive theories are very complex and include a large number of material parameters. Therefore, we decided to make transparent capabilities of such theories only for small deformations, excluding from considerations geometrical nonlinearities. Also, several material parameters will be assumed to vanish, in order to reduce the effort of the analysis. Of course, this implies that important capabilities of the model may be not activated. However, the present investigation is not entitled to be complete and will be of qualitative character only. This also concerns the isotropic hardening rule. In fact, isotropic hardening effects due to strains and micromorphic curvature tensors are captured in a unified manner. There are, however, possibilities to account for isotropic hardening effects due to strain and micromorphic curvature effects separately. Such isotropic hardening rules have been elaborated by Grammenoudis and Tsakmakis [1] in micropolar plasticity and are not pursuit here.
ð20Þ 2. Examples
s_ : to be determined from consistency condition f_ ¼ 0:
ð21Þ
Viscoplasticity: s_ :¼
/f Sm ð22Þ
Z0;
Z
/f S : overstress:
ð23Þ
Isotropic hardening:
g RCis ¼ ð1 DÞ ðr2 þ 2r0 rÞ;
ð24Þ
2
R¼R
@Cis ¼ ð1 DÞgðr þ r0 Þ ¼ ð1 DÞðgr þR0 Þ; @r
s_ r_ ¼ ð1 brÞ :
z
Examples illustrating the capabilities of the theory to capture size effects are given in this section and are taken from the doctoral thesis of Hofer [2], where also more details about the implementation are given. Further examples and interesting results on this topic may be found in Dillard et al. [3], Kirchner and Steinmann [4], Neff and Forest [5], Lazar and Maugin [6] as well as Hirschberger and Steinmann [7]. In the ensuing analysis, the chosen values of the material parameters do not reflect some responses of realistic material behavior, i.e., they are only of academic interest and serve to discuss basic features of the model. We set
ð25Þ
Ae1 l ¼ 1:21 105 N=mm2 ;
ð26Þ
m ¼ 8:08 104 N=mm2 ;
Ae2 ¼ m þ a;
RCk ¼ ð1
ðAk Þijpq ðek Þij ðe
uy 1 k Þpq þ 2ðBk Þijpq ðbk Þij ðbk Þpq
þðDk Þijpq ðek Þij ðbk Þpq þ 12ðC k Þijpqr ðK k Þijk ðK k Þpqr g; @Ck ðRk Þij ¼ R ¼ ð1 DÞfðBk Þijpq ðbk Þpq þðDk Þijpq ðek Þpq g; @ðbk Þij ðTk Þij ¼ R
@Ck ¼ ð1 DÞfðAk Þijpq ðek Þpq þðDk Þijpq ðbk Þpq g; @ðek Þij
ð27Þ
y ð28Þ
A
b
a ð30Þ
e_ k ¼ e_ p
s_ fM k ðtrTk Þ1 þ M2k Tk þM3k ðTk ÞT g; 1D 1
ð31Þ
s_ ^ P ½T: K_ k ¼ K_ p 1D k
x
ð29Þ
@Ck ¼ ð1 DÞðC k Þijkpqr ðK k Þpqr ; @ðK k Þijk
s_ fN k ðtrRk Þ1þ 2N2k Rk g; 1D 1
r A
ðT k Þijk ¼ R
b_ k ¼ b_ p
ð35Þ ð36Þ
Kinematic hardening: DÞf12
Ae3 ¼ m a;
ð32Þ
b ð33Þ
Fig. 1. Plane strain problem. The quadratic section with a circular hole is stretched in y direction.
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Be1 l;
Be2 m þ b2 ;
De1 l;
De2 m;
Cie ¼ 0
for ia 7;
b2 ¼ 10 m;
ð37Þ
ð38Þ
C7e ¼ c7 Z 0:
ð39Þ
Although for the case (39), important aspects of the constitutive model may be retain inactive, we shall confine ourself on this special case in order to limit the discussion. For what follows, of particular interest is the internal length lc :¼
rffiffiffiffiffi c7
m
;
ð40Þ
suggested by the elasticity laws. Firstly, we shall discuss micromorphic elasticity without damage (pure micromorphic elasticity).
2.1. Pure micromorphic elasticity 2.1.1. Rectangular specimens with circular hole under tension loading Consider the plane strain problem in Fig. 1 where the quadratic section (length b) with a circular hole (radius r) located in the center of the section, is stretched in y direction. With respect to the Cartesian coordinate system x, y, the boundaries x ¼ 7 b=2 are assumed to be traction-free. At the boundary y ¼ b=2 the displacement uy and the traction tx are assumed to vanish, while at the boundary y ¼ b=2, given displacement uy and traction tx ¼ 0 are imposed. The whole circular hole is assumed to be traction free, while the whole boundary is subjected to vanishing double traction tðdÞ . For small circular hole, a nearly uniform stress component s0 in y direction, at y ¼ b=2, will be required to realize the given boundary conditions. In classical elasticity, attention is focussed
3.2 3.5 classical α/μ = 10.0
3
3 T*yy/0 [ ]
α/μ = 100.0
Τyy/0 [ ]
2.5
2.8
2.6
2
2.4
1.5
2.2 10−6
1
spec. 1 spec. 4 spec. 20 spec. 200 10−4
10−2
100
102
104
106
c7/ [mm2]
0.5 0
0.2
0.4
0.6
0.8
1
a [mm] Fig. 2. Distribution of Tyy =s0 , Tyy ¼ Tyy ðy ¼ 0; a Z 0Þ, for c7 =m ¼ 0:1 mm2 and varying values a=m.
Fig. 4. Distributions of Tyy =s0 against c7 =m, for a ¼ m and different specimens. Geometry and boundary conditions of the specimens differ by a factor n ¼ 1, 4, 20, 200, the corresponding specimens being referred to as specimen 1; . . . ; specimen 200, respectively. %
3.2 3.2
3
Tyy /0 [ ]
Τ*yy/0 [ ]
3 2.8
2.8
*
2.6
2.6 α/μ = 0.01
2.4
α/μ = 0.1
2.4
α/μ = 1.0 2.2
α/μ = 10.0
2 10−4
10−3
10−2
10−1
100
101
102
2.2 10-6
10-4
10-2
2
104
106
Fig. 5. Distributions of Tyy =s0 against c7 =m n2 . The results for all specimens (n ¼ 1, 4, 20, 200) are identical. %
%
102 2
c7/(·n ) [mm ]
c7/ [mm2] Fig. 3. Effect of a, c7 on the stress concentration factor Tyy =s0.
100
ARTICLE IN PRESS
*
3.2
3.2
3
3 Tyy /0 [ ]
Tyy /0 [ ]
P. Grammenoudis et al. / International Journal of Non-Linear Mechanics 45 (2010) 140–148
2.8 2.6
*
2.4
143
2.8 2.6 2.4
2.2 0
20
40 60 lm/lc [ ]
80
100
2.2 10-1
100
101 lm/lc[ ]
102
103
Fig. 6. Stress concentration factor Tyy =s0 as a function of the ratio lm =lc at a ¼ m and c7 =m ¼ 0:1 mm2 ; left: linear plot, right: semilogarithmic plot. %
l
1.4
uy
1.2
B A b
1
x uy Fig. 7. Displacement controlled loading of a cantilever rectangular beam, l ¼ 3:4375 mm, b ¼ 1:25 mm, u y ¼ 0:01 mm.
Txx/Txx(class) [ ]
y 0.8 α/µ = 0.1 α/µ = 0.3
0.6
α/µ = 0.4 α/µ = 0.45
0.4
α/µ = 0.5
450 0.2
α/μ = 1.0
400 0
350
10−4
10−2
Txx [N/mm2]
300
100 c7/ [mm2]
102
104
Fig. 9. Effect of material parameters a, c7 on the response of stress component Txx for point A (in the vicinity of the singular point x ¼ l, y ¼ b).
250 200
1.4
150 1.2
100 1
0 0
0.5
1
1.5
2
2.5
3
x [mm] Fig. 8. Distribution of Txx as a function of x, at the upper boundary y ¼ b ¼ 1:25 mm, suggesting a singularity at x ¼ l ða ¼ 108 m; c7 =m ¼ 108 mm2 Þ.
0.8 0.6
s0
Tyy :¼ Tyy ðx ¼ r; y ¼ 0Þ; %
α/μ = 0.3 α/μ = 0.4
0.4 0.2
%
;
α/μ = 0.1
α/μ = 0.45 α/μ = 0.5
on the so-called stress concentration factor Tyy
Txx/Txx(class) [ ]
50
α/μ = 1.0
ð41Þ
which turns out to be equal to 3 (see, e.g., [8, p. 124]), whenever the section is of infinite extension b. In the present context, we refer to as classical, the case where a 0, c7 0, which are approximated numerically for given values of b, r. Particularly, we set b ¼ 2:5 mm and r ¼ 0:25 mm, which imply the value Tyy =s0 ¼ 3:14. Typical properties of micromorphic elasticity may be elucidated by regarding the distribution of Tyy =s0 along the line y ¼ 0 and x Zr, or equivalently a :¼ x r Z 0. For c7 =m ¼ 0:1 mm2 this distribution, parameterized by a=m, is shown in Fig. 2. It can be recognized that increasing values of a=m cause decreasing
0 0.0001
0.01
1 c7/ [mm2]
100
10000
Fig. 10. Effect of material parameters a, c7 on the response of stress component Txx for point B (indicating a larger distance than point A from the singular point x ¼ l, ðclassÞ . y ¼ b). The values of Txx are always smaller than Txx
%
values of Tyy =s0 in the neighborhood of a ¼ 0, and consequently decreasing values of stress concentration factors Tyy =s0 for the micromorphic material. Note that all distributions intersect at a ¼ 0:13 mm. %
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1.25
1
·l []
Txy Tyx
1.2
0.6 0.4 0.2
0.75
Txx
(class)
0.8
1
0.5
Mc/
Txy /Txx(class), Tyx /Txx(class)
1.4
0.25 0
0 10−8
10−4
100 c7/
104
108
10−4
10−2
[mm2]
100 c7/µ
102
104
[mm2]
Fig. 11. Responses of Txy and Tyx (left), as well as Mc (right), at point A ða ¼ 1; 0 mÞ.
2
2
1
1 y [mm]
y [mm]
c7 = 10−8 · mm2
0
0
−1
−1
−2
−2 0
1
2 x [mm]
c7 = 0.1 · mm2
3
0
2
2
1
1
0
−1
−2
−2 1
2 x [mm]
3
0
c7 = 10 · mm2
2
1
2 x [mm]
3
c7 = 108 · mm2 2
1
1 y [mm]
y [mm]
0
−1
0
3
c7 = 5.0 · mm2
y [mm]
y [mm]
c7 = mm2
1 2 x [mm]
0 −1
0 −1
−2
−2 0
1
2 x [mm]
3
0
1
2 x [mm]
3
Fig. 12. Initial and deformed meshes of the rectangular beam for fixed a ¼ m and varying material parameter c7. Displacements uy are presented enlarged, by factor 100. The classical case is approached for c7 -0.
ARTICLE IN PRESS P. Grammenoudis et al. / International Journal of Non-Linear Mechanics 45 (2010) 140–148
The effect of a, c7 on the stress concentration factor is illustrated in Fig. 3. For very large values a=m and values c7 =m Z103 mm2 , the stress concentration factor Tyy =s0 becomes decreasing, whereas for small a=m the value of Tyy =s0 is nearly equal to the classical one. It seems that, at fixed a=m, Tyy =s0 converges for c7 =m against 1 or 0, respectively to limits, the limit for c7 =m-0 being the classical one. To obtain an insight into the size effects due to different, but otherwise similar boundary value problems, we ask for the stress concentration factor Tyy =s0 for the cases where a m and geometry and boundary conditions of the specimens vary from each other according to a factor n ¼ 1, 4, 20, 200. Corresponding results are displayed in Fig. 4, from which we deduce that all distributions are similar. In fact, if the Tyy =s0 values corresponding to the specimen according to factor n are plotted as a function of c7 =ðm n2 Þ, then all plots will coincide (see Fig. 5). In other words, for linear micromorphic elasticity, size effects may be visualized by varying the parameter c7, the other parameters being held fixed. Further size effects may be elucidated by introducing a typical geometry length, as, e.g., lm :¼ 4r ¼ 0:4b. Again we concentrate ourself on specimen geometries and related boundary conditions, differing according to a factor n, with n being now n ¼ 0:0001; 0:01; . . . ; 400; 10 000. On choosing c7 =m ¼ 0:1 mm2 , the internal length lc becomes lc ¼ 0:31623 mm. It can be seen
145
in Fig. 6, that the stress concentration factor Tyy =s0 is a function of the ratio lm =lc (cf. also [9]). %
%
%
%
2.1.2. Displacement controlled loading of cantilever rectangular beam Further features of micromorphic elasticity may be illustrated with the aid of the cantilever rectangular beam shown in Fig. 7. We use Cartesian coordinates x, y and assume plane strain state to apply, with following boundary conditions,
%
x ¼ 0 : uy ¼ u y ;
tðdÞ ¼ 0;
tx ¼ 0;
ð42Þ
x ¼ l : u ¼ 0;
h ¼ 0;
ð43Þ
y ¼ 0 : t ¼ 0;
tðdÞ ¼ 0;
ð44Þ
%
y ¼ l : t ¼ 0;
tðdÞ ¼ 0;
ð45Þ
with the given displacement u y being uniformly distributed along the boundary x ¼ 0. Again we focus attention on the effect of the material parameters a and c7 . Thereby, it is convenient to consider
uy
uy
d
(a) l
b y y x x Fig. 15. Displacement controlled tension loading of rectangular sections with circular hole. All stress responses in the following figures are referred to point ðaÞ on the hole ðx ¼ b þ d=2; y ¼ l=2Þ.
Fig. 13. Displacement controlled uniaxial loading (1 element, plain strain).
600
1
300
α1 = 1.0
0.8
α1 = 0.25
400
d[]
[N/mm2]
500
α1 = 0.5
0.6 0.4
200
α1 = 0.25 α1 = 1.0
100
0.2
α1 = 0.5
0
0 0
0.05
0.1
0.15
= Δl/l0 [ ]
0.2
0.25
0
0.05
0.1
0.15
0.2
0.25
= Δl/l0 [ ]
Fig. 14. Effect of material parameter a1 on the responses of the uniaxial stress s (left) and the damage variable D (right).
ARTICLE IN PRESS P. Grammenoudis et al. / International Journal of Non-Linear Mechanics 45 (2010) 140–148
points, which indicate large amounts of stress gradients. Clearly, the edge point x ¼ l, y ¼ b could be selected for this goal. However, such points will exhibit stress distributions with some singularities. Therefore, we shall confine the discussion on points A, B located at a distance of about 0:02 l and 0:17 l, from the boundary x ¼ l, respectively. Note that the length l and the height b of the beam are chosen to be l ¼ 3:4375 mm and b ¼ 1:25 mm, while the displacement component prescribed on the boundary x ¼ 0 amounts u y ¼ 0:01 mm. Also, A, B are Gauss points with distances from the upper boundary y ¼ b, of about 0:033 b, respectively. However, we shall refer to such points as being located at the upper boundary y ¼ b ¼ 1:25 mm. Accordingly, Fig. 8 displays the stress component Txx at the boundary y ¼ b, as a function of x. It may be seen, that in the neighborhood of x ¼ l, the stress component Txx takes vary large values, which designates the singularity in the distribution of Txx . Once more, we denote by ‘‘classical’’, solutions obtained numerically for very small values a=m and c7 =m. Fig. 9 makes clear, that in the neighborhood of the singularity (point A), stress ðclassÞ , dependent on the component Txx may become larger than Txx material parameters a, c7 . However, with increasing distance from the singularity point, as, e.g., at point B, Txx remains smaller than ðclassÞ , independent of material parameters a, c7 (see Fig. 10). Txx Significant differences between the shear stress components Txy and Tyx may be present, as can be seen in Fig. 11, for point A. Both components approach for very large values of c7 =m, different limits, the one for Txy being vanishing. Of particular interest is also the response of the couple stress Mc :¼ T xyx T yxx , which is shown in Fig. 11 too. It can be recognized that Mc is vanishing for small values c7 =m, while Mc approaches a constant value for very large values c7 =m.
Finally, Fig. 12 illustrates, for fixed a ¼ m, the effect of material parameter c7 on the deformed geometry of the beam. It may be recognized that for small values of c7 the bending mode is dominated, while for very large values of c7 the deformation resamples simple shear mode. 2.2. Micromorphic plasticity coupled with damage In the following, we set Ay1
Ay2 ¼ 1; 5;
0;
By1 0;
By2 ¼ 0;
Ciy 0
for ia 7;
Ay3 ¼ 0;
ð46Þ ð47Þ
C7y ¼ r7 a0;
ð48Þ
k0 ¼ 350 N=mm2
ð49Þ
in the yield function, and
g ¼ 4100 N=mm2
b ¼ 17;
ð50Þ
600
500
400 Tyy [N/mm2]
146
300
200 Table 1 Specimen geometries.
100 Length l (mm)
Width b (mm)
Diameter d (mm)
Spec. Spec. Spec. Spec.
10 40 200 2000
2.5 10 50 500
1 4 20 200
1 4 20 200
0 0
0.4
0.6
0.8
1
Fig. 17. Response of the stress component Tyy on the hole at point ðaÞ as a function of the global strain Dl=l ða1 ¼ 1:0; r7 ¼ 10 mm2 Þ.
700
700
600
600
500
500
400 300
400 300 r7 = 1.0 1/mm2
200
spec. 1 spec. 4 spec. 20 spec. 200
100
0.2
Δl/l0 [%]
Tyy [N/mm2]
Tyy [N/mm2]
Size factor n (number of specimen)
200
spec. 1 spec. 4 spec. 20 spec. 200
r7 = 10.0 1/mm2 100
0
r7 = 100.0 1/mm2
0 0
1
2
3
4
5
Δl/l0 [%] Fig. 16. Response of the stress component Tyy on the hole at point ðaÞ as a function of the global strain Dl=l ða1 ¼ 0:1; r7 ¼ 10 mm2 Þ.
0
0.5
1
1.5
2
2.5 3 Δl/l0 [%]
3.5
4
4.5
Fig. 18. Effect of the material parameter r7 on the response of stress component Tyy at ðaÞ (specimen 4, a1 ¼ 0; 1).
ARTICLE IN PRESS P. Grammenoudis et al. / International Journal of Non-Linear Mechanics 45 (2010) 140–148
in the rule for isotropic hardening. Moreover, we fix the values of a and c7 in the elasticity laws by
a ¼ 0:1 m;
c7
m
¼ 0:1 mm2 :
ð51Þ
2.2.1. Uniaxial loading First, we present calculations for homogeneous uniaxial tension loading of a rectangular specimen (plane strain), according to Fig. 13. At the bottom of the specimen it is given uy ¼ 0, tx ¼ 0, tðdÞ ¼ 0, while at the top it is uy ¼ u y , tx ¼ 0, tðdÞ ¼ 0. The remaining boundaries are subject to the conditions t ¼ 0 and tðdÞ ¼ 0. The aim is to demonstrate the capabilities of the damage model. To this end, it suffices to concentrate on isotropic hardening only. Further, as the deformations are homogeneous, no material parameters of terms related to micromorphic curvature tensors are involved. Fig. 14 shows the effect of the damage parameter a1 (cf. Eq. (34)) on the responses of the uniaxial stress s and the damage variable D. Further discussion
600
¨ about the damage law for the classical case is provided in Lammer and Tsakmakis [10]. 2.2.2. Rectangular specimens with circular hole under tension loading We consider again the boundary value problem of Section 2.1.1, but now with respect to the specimen geometry displayed in Fig. 15 (length l differs from width b). In order to elucidate the capabilities of the micromorphic theory in predicting size effects, four specimen geometries are considered, referred to as specimens 1, 4, 20 and 200 (see Table 1). First, only isotropic hardening is addressed, with material parameters as given in Section 2.2, and r7 ¼ 10 mm2 . The discussion is referred to the stress component Tyy at point ðaÞ (see Fig. 15). It can be recognized from Fig. 16 that softening for large specimens begins earlier than for small ones. Comparison of Fig. 16 ða1 ¼ 0:1Þ with Fig. 17 ða1 ¼ 1:0Þ suggests that the form of the responses is strong dependent on the damage parameter a1. Moreover, Fig. 18 illustrates that maximal values of stresses and maximal global strains depend on the material parameter r7, present in the yield function. Next, we assume the micromorphic model material to exhibit kinematic hardening only, governed by the material parameters r7 ¼ 10 mm2 ;
500
P k7 ¼ 500=N;
Tyy [N/mm2]
400
M2k ¼ 50 mm2 =N; C7k ¼ 200 N=mm;
Ak2 ¼ Ak3 ¼ 200 N=mm2 ;
ð52Þ ð53Þ
the remaining material parameters related to kinematical hardening being vanishing. Similar to the case of pure isotropic hardening, Fig. 19 suggests that softening for large specimens begins earlier than for small ones. Fig. 20 confirms that this holds also for the case of combined isotropic and kinematic hardening.
300
200 spec. 1 spec. 4 spec. 20 spec. 200
100
3. Concluding remarks
0 0
Fig. 19. Responses of ða1 ¼ 1; k0 ¼ 500 N=mm2 Þ.
0.2
Tyy
0.4 Δl/l0 [%] at
ðaÞ
for
0.6
pure
0.8
kinematic
hardening
700 600 500 Tyy [N/mm2]
147
400 300 200 100
spec. 20 spec. 200
0 0
0.5
1
1.5
2
Δl/l0 [%] Fig. 20. Responses of Tyy at ðaÞ for combined isotropic and kinematic hardening ða1 ¼ 1; k0 ¼ 350 N=mm2 Þ.
A general framework for micromorphic plasticity has been formulated in Parts I, II, incorporating isotropic and kinematic hardening. The hardening laws are of the Armstrong–Frederick type and the yield function is a generalization of the classical v. Mises yield function. Some properties of the resulting theory, concerning prediction of size effects for small deformations, are reported in Part III. However, no comparison with experimental data is available, so that it is not possible to evaluate the appropriateness of the chosen constitutive functions. This concerns over all the yield function and the isotropic hardening, the latter being unifiedly postulated. Further studies, with reference to experimental results will help to clarify such issues, but this is beyond the scope of the present paper. All the discussions in the three articles make clear that phenomenological micromorphic theories (at least plasticity theories) are very complicated and involve a large number of material parameters. Therefore, it will be useful to clarify in future works, if it is possible to approximate the essential material responses predicted by micromorphic theories by some simpler gradient models, which deal with classical stresses only, and involve a smaller number of material parameters. Also it is of interest to answer the following question. Is the micromorphic model appropriate enough to describe all known size effects to the necessary degree? References [1] P. Grammenoudis, C. Tsakmakis, Isotropic hardening in micropolar plasticity, Archive of Applied Mechanics 79 (2009) 323–334. ¨ [2] D. Hofer, Simulation von großeneffekten mit mikromorphen theorien, Doctoral Thesis, TU Darmstadt, 2003.
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[3] T. Dillard, S. Forest, P. Ienny, Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams, European Journal of Mechanics A—Solids 25 (2006) 526–549. [4] N. Kirchner, P. Steinmann, A unifying treatise on variational principles for gradient and micromorphic continua, Philosophical Magazine 85 (2005) 3875–3895. [5] P. Neff, S. Forest, A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results, Journal of Elasticity 87 (2007) 239–276.
[6] M. Lazar, G. Maugin, On microcontinuum field theories: the eshelby stress tensor and incompatibility conditions, Philosophical Magazine 87 (2007) 3853–3870. [7] C.B. Hirschberger, K.E., Steinmann, P., On deformational and configurational mechanics of micromorphic hyperelasticity—theory and computation. Computer Methods in Applied Mechanics and Engineering 196 (2007) 4027–4044. [8] P. Gould, Introduction to Linear Elasticity, Springer, New York, Berlin, 1983. [9] R. Mindlin, Influence of couple-stresses on stress concentrations, Experimental Mechanics 3 (1963) 1–7. ¨ [10] H. Lammer, C. Tsakmakis, Discussion of coupled elastoplasticity and damage constitutive equations for small and finite deformations, International Journal of Plasticity 16 (2000) 495–523.
Contents lists available at ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Micromorphic continuum. Part III Small deformation plasticity coupled with damage P. Grammenoudis a,, Ch. Tsakmakis a,, D. Hofer b a b
Darmstadt University of Technology, Institute of Continuum Mechanics, Hochschulstraße 1, D-64289 Darmstadt, Germany Westinghouse Electric Germany GmbH, Abt. PEM, Dudenstr. 44, D-68167 Mannheim, Germany
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 July 2008 Received in revised form 12 October 2009 Accepted 13 October 2009
Properties of the micromorphic theory proposed in Part II are discussed for the case of small deformations. Model responses for beam specimens under bending loading and plates with circular holes under tension loading are calculated by employing the finite element method. The results reported are concerned with the capabilities of the theory to predict size effects. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Micromorphic plasticity Finite deformation Isotropic and kinematic hardening rules Isotropic continuum damage mechanics Finite element implementation
ðdÞ
1. Introduction
T ijk nk ¼ t ij
A finite deformation micromorphic plasticity theory, exhibiting isotropic and kinematic hardening, and incorporating damage effects, has been proposed in Part II. The theory is consistent with the second law of thermodynamics and deals with a plastic micromorphic curvature tensor, which is not required to fulfill some compatibility conditions, i.e., it is not related to some gradient terms. Furthermore, a measure of smallness e has been introduced in Part II, and the theory has been defined to be of small deformations, if terms only up to order OðeÞ are retained. It is a straightforward task to verify that the small deformation version of the micromorphic model proposed, reads as follows (we confine attention to static balance equations and omit the body and double body forces): Equilibrium equations:
ui ¼ u i
@Tij ¼0 @Xj
in RR ;
@T ijk þ Tij Sij ¼ 0 @Xk
ð1Þ
t ðdÞ
h
on @RRij ¼ @RR \@RRij ;
ð4Þ
on @RuRi ;
hij ¼ h ij
ð5Þ
h
on @RRij :
ð6Þ
Kinematics: Hij ¼
@ui ; @Xj
bij ¼
1 ðh þhji Þ; 2 ij
ð7Þ
eij ¼ Hij hij ;
bij ¼ ðbe Þij þ ðbp Þij ;
Kijk ¼
eij ¼ ðee Þij þ ðep Þij ;
@hij ; @Xk
Kijk ¼ ðK e Þijk þðK p Þijk :
ð8Þ ð9Þ
Specific free energy:
C ¼ Ce þ Cis þ Ck :
ð10Þ
Elasticity laws: in RR :
ð2Þ
RCe ¼ ð1 DÞf12 ðAe Þijpq ðee Þij ðee Þpq þ 12ðBe Þijpq ðbe Þij ðbe Þpq þ ðDe Þijpq ðee Þij ðbe Þpq þ 12ðC e Þijpqr ðK e Þijk ðK e Þpqr g;
ð11Þ
Boundary conditions: Tij nj ¼ t i
on @RtRi ¼ @RR \@RuRi ;
ð3Þ
Corresponding authors.
E-mail addresses: [email protected] (P. Grammenoudis), [email protected] (Ch. Tsakmakis). 0020-7462/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2009.10.003
@Ce ¼ ð1 DÞfðBe Þijpq ðbe Þpq þ ðDe Þijpq ðee Þpq g; @ðbe Þij
ð12Þ
@Ce ¼ ð1 DÞfðAe Þijpq ðee Þpq þ ðDe Þijpq ðbe Þpq g; @ðee Þij
ð13Þ
Sij ¼ R Tij ¼ R
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T ijk ¼ R
@Ce ¼ ð1 DÞðC e Þijkpqr ðK e Þpqr : @ðK e Þijk
ð14Þ
1 k D Þ ððT Tijk ÞD ðAy Þijpq ðTpq Tpq 1 D ij k D k D þ ðSij Sij Þ ðBy Þijpq ðSpq Spq Þ þ ðT ijk T kijk ÞD ðC y Þijkpqr ðT pqr T kpqr ÞD Þ1=2
R þ k0; 1D Flow rule: s_ @f
ðe_ p Þij ¼
z :¼
z @Tij
ðb_ p Þij ¼
;
R k0; 1D
k :¼ R0 þk 0 :
k :¼
s_ @f
z @Sij
;
ð15Þ
ð16Þ
ðK_ p Þijk ¼
s_ @f
z @T ijk
;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @f @f @f @f @f @f þ þ : @Tij @Tij @Sij @Sij @T ijk @T ijk
ð17Þ
ð18Þ
Plasticity: LðtÞ :¼ ½f_ ðtÞs ¼ const: ( s_
40
for f ¼ 0 & L 40;
¼0
otherwise;
Evolution law for damage: _ ¼ a1 s_ R @C : D @D
Yield function: f¼
141
ð19Þ
ð34Þ
The aim of the present paper is to demonstrate the capabilities of this model in describing size effects present in bending of beam specimens and in plates with a hole under tension loading. It should be remarked that Parts I and II made it clear, that micromorphic constitutive theories are very complex and include a large number of material parameters. Therefore, we decided to make transparent capabilities of such theories only for small deformations, excluding from considerations geometrical nonlinearities. Also, several material parameters will be assumed to vanish, in order to reduce the effort of the analysis. Of course, this implies that important capabilities of the model may be not activated. However, the present investigation is not entitled to be complete and will be of qualitative character only. This also concerns the isotropic hardening rule. In fact, isotropic hardening effects due to strains and micromorphic curvature tensors are captured in a unified manner. There are, however, possibilities to account for isotropic hardening effects due to strain and micromorphic curvature effects separately. Such isotropic hardening rules have been elaborated by Grammenoudis and Tsakmakis [1] in micropolar plasticity and are not pursuit here.
ð20Þ 2. Examples
s_ : to be determined from consistency condition f_ ¼ 0:
ð21Þ
Viscoplasticity: s_ :¼
/f Sm ð22Þ
Z0;
Z
/f S : overstress:
ð23Þ
Isotropic hardening:
g RCis ¼ ð1 DÞ ðr2 þ 2r0 rÞ;
ð24Þ
2
R¼R
@Cis ¼ ð1 DÞgðr þ r0 Þ ¼ ð1 DÞðgr þR0 Þ; @r
s_ r_ ¼ ð1 brÞ :
z
Examples illustrating the capabilities of the theory to capture size effects are given in this section and are taken from the doctoral thesis of Hofer [2], where also more details about the implementation are given. Further examples and interesting results on this topic may be found in Dillard et al. [3], Kirchner and Steinmann [4], Neff and Forest [5], Lazar and Maugin [6] as well as Hirschberger and Steinmann [7]. In the ensuing analysis, the chosen values of the material parameters do not reflect some responses of realistic material behavior, i.e., they are only of academic interest and serve to discuss basic features of the model. We set
ð25Þ
Ae1 l ¼ 1:21 105 N=mm2 ;
ð26Þ
m ¼ 8:08 104 N=mm2 ;
Ae2 ¼ m þ a;
RCk ¼ ð1
ðAk Þijpq ðek Þij ðe
uy 1 k Þpq þ 2ðBk Þijpq ðbk Þij ðbk Þpq
þðDk Þijpq ðek Þij ðbk Þpq þ 12ðC k Þijpqr ðK k Þijk ðK k Þpqr g; @Ck ðRk Þij ¼ R ¼ ð1 DÞfðBk Þijpq ðbk Þpq þðDk Þijpq ðek Þpq g; @ðbk Þij ðTk Þij ¼ R
@Ck ¼ ð1 DÞfðAk Þijpq ðek Þpq þðDk Þijpq ðbk Þpq g; @ðek Þij
ð27Þ
y ð28Þ
A
b
a ð30Þ
e_ k ¼ e_ p
s_ fM k ðtrTk Þ1 þ M2k Tk þM3k ðTk ÞT g; 1D 1
ð31Þ
s_ ^ P ½T: K_ k ¼ K_ p 1D k
x
ð29Þ
@Ck ¼ ð1 DÞðC k Þijkpqr ðK k Þpqr ; @ðK k Þijk
s_ fN k ðtrRk Þ1þ 2N2k Rk g; 1D 1
r A
ðT k Þijk ¼ R
b_ k ¼ b_ p
ð35Þ ð36Þ
Kinematic hardening: DÞf12
Ae3 ¼ m a;
ð32Þ
b ð33Þ
Fig. 1. Plane strain problem. The quadratic section with a circular hole is stretched in y direction.
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Be1 l;
Be2 m þ b2 ;
De1 l;
De2 m;
Cie ¼ 0
for ia 7;
b2 ¼ 10 m;
ð37Þ
ð38Þ
C7e ¼ c7 Z 0:
ð39Þ
Although for the case (39), important aspects of the constitutive model may be retain inactive, we shall confine ourself on this special case in order to limit the discussion. For what follows, of particular interest is the internal length lc :¼
rffiffiffiffiffi c7
m
;
ð40Þ
suggested by the elasticity laws. Firstly, we shall discuss micromorphic elasticity without damage (pure micromorphic elasticity).
2.1. Pure micromorphic elasticity 2.1.1. Rectangular specimens with circular hole under tension loading Consider the plane strain problem in Fig. 1 where the quadratic section (length b) with a circular hole (radius r) located in the center of the section, is stretched in y direction. With respect to the Cartesian coordinate system x, y, the boundaries x ¼ 7 b=2 are assumed to be traction-free. At the boundary y ¼ b=2 the displacement uy and the traction tx are assumed to vanish, while at the boundary y ¼ b=2, given displacement uy and traction tx ¼ 0 are imposed. The whole circular hole is assumed to be traction free, while the whole boundary is subjected to vanishing double traction tðdÞ . For small circular hole, a nearly uniform stress component s0 in y direction, at y ¼ b=2, will be required to realize the given boundary conditions. In classical elasticity, attention is focussed
3.2 3.5 classical α/μ = 10.0
3
3 T*yy/0 [ ]
α/μ = 100.0
Τyy/0 [ ]
2.5
2.8
2.6
2
2.4
1.5
2.2 10−6
1
spec. 1 spec. 4 spec. 20 spec. 200 10−4
10−2
100
102
104
106
c7/ [mm2]
0.5 0
0.2
0.4
0.6
0.8
1
a [mm] Fig. 2. Distribution of Tyy =s0 , Tyy ¼ Tyy ðy ¼ 0; a Z 0Þ, for c7 =m ¼ 0:1 mm2 and varying values a=m.
Fig. 4. Distributions of Tyy =s0 against c7 =m, for a ¼ m and different specimens. Geometry and boundary conditions of the specimens differ by a factor n ¼ 1, 4, 20, 200, the corresponding specimens being referred to as specimen 1; . . . ; specimen 200, respectively. %
3.2 3.2
3
Tyy /0 [ ]
Τ*yy/0 [ ]
3 2.8
2.8
*
2.6
2.6 α/μ = 0.01
2.4
α/μ = 0.1
2.4
α/μ = 1.0 2.2
α/μ = 10.0
2 10−4
10−3
10−2
10−1
100
101
102
2.2 10-6
10-4
10-2
2
104
106
Fig. 5. Distributions of Tyy =s0 against c7 =m n2 . The results for all specimens (n ¼ 1, 4, 20, 200) are identical. %
%
102 2
c7/(·n ) [mm ]
c7/ [mm2] Fig. 3. Effect of a, c7 on the stress concentration factor Tyy =s0.
100
ARTICLE IN PRESS
*
3.2
3.2
3
3 Tyy /0 [ ]
Tyy /0 [ ]
P. Grammenoudis et al. / International Journal of Non-Linear Mechanics 45 (2010) 140–148
2.8 2.6
*
2.4
143
2.8 2.6 2.4
2.2 0
20
40 60 lm/lc [ ]
80
100
2.2 10-1
100
101 lm/lc[ ]
102
103
Fig. 6. Stress concentration factor Tyy =s0 as a function of the ratio lm =lc at a ¼ m and c7 =m ¼ 0:1 mm2 ; left: linear plot, right: semilogarithmic plot. %
l
1.4
uy
1.2
B A b
1
x uy Fig. 7. Displacement controlled loading of a cantilever rectangular beam, l ¼ 3:4375 mm, b ¼ 1:25 mm, u y ¼ 0:01 mm.
Txx/Txx(class) [ ]
y 0.8 α/µ = 0.1 α/µ = 0.3
0.6
α/µ = 0.4 α/µ = 0.45
0.4
α/µ = 0.5
450 0.2
α/μ = 1.0
400 0
350
10−4
10−2
Txx [N/mm2]
300
100 c7/ [mm2]
102
104
Fig. 9. Effect of material parameters a, c7 on the response of stress component Txx for point A (in the vicinity of the singular point x ¼ l, y ¼ b).
250 200
1.4
150 1.2
100 1
0 0
0.5
1
1.5
2
2.5
3
x [mm] Fig. 8. Distribution of Txx as a function of x, at the upper boundary y ¼ b ¼ 1:25 mm, suggesting a singularity at x ¼ l ða ¼ 108 m; c7 =m ¼ 108 mm2 Þ.
0.8 0.6
s0
Tyy :¼ Tyy ðx ¼ r; y ¼ 0Þ; %
α/μ = 0.3 α/μ = 0.4
0.4 0.2
%
;
α/μ = 0.1
α/μ = 0.45 α/μ = 0.5
on the so-called stress concentration factor Tyy
Txx/Txx(class) [ ]
50
α/μ = 1.0
ð41Þ
which turns out to be equal to 3 (see, e.g., [8, p. 124]), whenever the section is of infinite extension b. In the present context, we refer to as classical, the case where a 0, c7 0, which are approximated numerically for given values of b, r. Particularly, we set b ¼ 2:5 mm and r ¼ 0:25 mm, which imply the value Tyy =s0 ¼ 3:14. Typical properties of micromorphic elasticity may be elucidated by regarding the distribution of Tyy =s0 along the line y ¼ 0 and x Zr, or equivalently a :¼ x r Z 0. For c7 =m ¼ 0:1 mm2 this distribution, parameterized by a=m, is shown in Fig. 2. It can be recognized that increasing values of a=m cause decreasing
0 0.0001
0.01
1 c7/ [mm2]
100
10000
Fig. 10. Effect of material parameters a, c7 on the response of stress component Txx for point B (indicating a larger distance than point A from the singular point x ¼ l, ðclassÞ . y ¼ b). The values of Txx are always smaller than Txx
%
values of Tyy =s0 in the neighborhood of a ¼ 0, and consequently decreasing values of stress concentration factors Tyy =s0 for the micromorphic material. Note that all distributions intersect at a ¼ 0:13 mm. %
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1.25
1
·l []
Txy Tyx
1.2
0.6 0.4 0.2
0.75
Txx
(class)
0.8
1
0.5
Mc/
Txy /Txx(class), Tyx /Txx(class)
1.4
0.25 0
0 10−8
10−4
100 c7/
104
108
10−4
10−2
[mm2]
100 c7/µ
102
104
[mm2]
Fig. 11. Responses of Txy and Tyx (left), as well as Mc (right), at point A ða ¼ 1; 0 mÞ.
2
2
1
1 y [mm]
y [mm]
c7 = 10−8 · mm2
0
0
−1
−1
−2
−2 0
1
2 x [mm]
c7 = 0.1 · mm2
3
0
2
2
1
1
0
−1
−2
−2 1
2 x [mm]
3
0
c7 = 10 · mm2
2
1
2 x [mm]
3
c7 = 108 · mm2 2
1
1 y [mm]
y [mm]
0
−1
0
3
c7 = 5.0 · mm2
y [mm]
y [mm]
c7 = mm2
1 2 x [mm]
0 −1
0 −1
−2
−2 0
1
2 x [mm]
3
0
1
2 x [mm]
3
Fig. 12. Initial and deformed meshes of the rectangular beam for fixed a ¼ m and varying material parameter c7. Displacements uy are presented enlarged, by factor 100. The classical case is approached for c7 -0.
ARTICLE IN PRESS P. Grammenoudis et al. / International Journal of Non-Linear Mechanics 45 (2010) 140–148
The effect of a, c7 on the stress concentration factor is illustrated in Fig. 3. For very large values a=m and values c7 =m Z103 mm2 , the stress concentration factor Tyy =s0 becomes decreasing, whereas for small a=m the value of Tyy =s0 is nearly equal to the classical one. It seems that, at fixed a=m, Tyy =s0 converges for c7 =m against 1 or 0, respectively to limits, the limit for c7 =m-0 being the classical one. To obtain an insight into the size effects due to different, but otherwise similar boundary value problems, we ask for the stress concentration factor Tyy =s0 for the cases where a m and geometry and boundary conditions of the specimens vary from each other according to a factor n ¼ 1, 4, 20, 200. Corresponding results are displayed in Fig. 4, from which we deduce that all distributions are similar. In fact, if the Tyy =s0 values corresponding to the specimen according to factor n are plotted as a function of c7 =ðm n2 Þ, then all plots will coincide (see Fig. 5). In other words, for linear micromorphic elasticity, size effects may be visualized by varying the parameter c7, the other parameters being held fixed. Further size effects may be elucidated by introducing a typical geometry length, as, e.g., lm :¼ 4r ¼ 0:4b. Again we concentrate ourself on specimen geometries and related boundary conditions, differing according to a factor n, with n being now n ¼ 0:0001; 0:01; . . . ; 400; 10 000. On choosing c7 =m ¼ 0:1 mm2 , the internal length lc becomes lc ¼ 0:31623 mm. It can be seen
145
in Fig. 6, that the stress concentration factor Tyy =s0 is a function of the ratio lm =lc (cf. also [9]). %
%
%
%
2.1.2. Displacement controlled loading of cantilever rectangular beam Further features of micromorphic elasticity may be illustrated with the aid of the cantilever rectangular beam shown in Fig. 7. We use Cartesian coordinates x, y and assume plane strain state to apply, with following boundary conditions,
%
x ¼ 0 : uy ¼ u y ;
tðdÞ ¼ 0;
tx ¼ 0;
ð42Þ
x ¼ l : u ¼ 0;
h ¼ 0;
ð43Þ
y ¼ 0 : t ¼ 0;
tðdÞ ¼ 0;
ð44Þ
%
y ¼ l : t ¼ 0;
tðdÞ ¼ 0;
ð45Þ
with the given displacement u y being uniformly distributed along the boundary x ¼ 0. Again we focus attention on the effect of the material parameters a and c7 . Thereby, it is convenient to consider
uy
uy
d
(a) l
b y y x x Fig. 15. Displacement controlled tension loading of rectangular sections with circular hole. All stress responses in the following figures are referred to point ðaÞ on the hole ðx ¼ b þ d=2; y ¼ l=2Þ.
Fig. 13. Displacement controlled uniaxial loading (1 element, plain strain).
600
1
300
α1 = 1.0
0.8
α1 = 0.25
400
d[]
[N/mm2]
500
α1 = 0.5
0.6 0.4
200
α1 = 0.25 α1 = 1.0
100
0.2
α1 = 0.5
0
0 0
0.05
0.1
0.15
= Δl/l0 [ ]
0.2
0.25
0
0.05
0.1
0.15
0.2
0.25
= Δl/l0 [ ]
Fig. 14. Effect of material parameter a1 on the responses of the uniaxial stress s (left) and the damage variable D (right).
ARTICLE IN PRESS P. Grammenoudis et al. / International Journal of Non-Linear Mechanics 45 (2010) 140–148
points, which indicate large amounts of stress gradients. Clearly, the edge point x ¼ l, y ¼ b could be selected for this goal. However, such points will exhibit stress distributions with some singularities. Therefore, we shall confine the discussion on points A, B located at a distance of about 0:02 l and 0:17 l, from the boundary x ¼ l, respectively. Note that the length l and the height b of the beam are chosen to be l ¼ 3:4375 mm and b ¼ 1:25 mm, while the displacement component prescribed on the boundary x ¼ 0 amounts u y ¼ 0:01 mm. Also, A, B are Gauss points with distances from the upper boundary y ¼ b, of about 0:033 b, respectively. However, we shall refer to such points as being located at the upper boundary y ¼ b ¼ 1:25 mm. Accordingly, Fig. 8 displays the stress component Txx at the boundary y ¼ b, as a function of x. It may be seen, that in the neighborhood of x ¼ l, the stress component Txx takes vary large values, which designates the singularity in the distribution of Txx . Once more, we denote by ‘‘classical’’, solutions obtained numerically for very small values a=m and c7 =m. Fig. 9 makes clear, that in the neighborhood of the singularity (point A), stress ðclassÞ , dependent on the component Txx may become larger than Txx material parameters a, c7 . However, with increasing distance from the singularity point, as, e.g., at point B, Txx remains smaller than ðclassÞ , independent of material parameters a, c7 (see Fig. 10). Txx Significant differences between the shear stress components Txy and Tyx may be present, as can be seen in Fig. 11, for point A. Both components approach for very large values of c7 =m, different limits, the one for Txy being vanishing. Of particular interest is also the response of the couple stress Mc :¼ T xyx T yxx , which is shown in Fig. 11 too. It can be recognized that Mc is vanishing for small values c7 =m, while Mc approaches a constant value for very large values c7 =m.
Finally, Fig. 12 illustrates, for fixed a ¼ m, the effect of material parameter c7 on the deformed geometry of the beam. It may be recognized that for small values of c7 the bending mode is dominated, while for very large values of c7 the deformation resamples simple shear mode. 2.2. Micromorphic plasticity coupled with damage In the following, we set Ay1
Ay2 ¼ 1; 5;
0;
By1 0;
By2 ¼ 0;
Ciy 0
for ia 7;
Ay3 ¼ 0;
ð46Þ ð47Þ
C7y ¼ r7 a0;
ð48Þ
k0 ¼ 350 N=mm2
ð49Þ
in the yield function, and
g ¼ 4100 N=mm2
b ¼ 17;
ð50Þ
600
500
400 Tyy [N/mm2]
146
300
200 Table 1 Specimen geometries.
100 Length l (mm)
Width b (mm)
Diameter d (mm)
Spec. Spec. Spec. Spec.
10 40 200 2000
2.5 10 50 500
1 4 20 200
1 4 20 200
0 0
0.4
0.6
0.8
1
Fig. 17. Response of the stress component Tyy on the hole at point ðaÞ as a function of the global strain Dl=l ða1 ¼ 1:0; r7 ¼ 10 mm2 Þ.
700
700
600
600
500
500
400 300
400 300 r7 = 1.0 1/mm2
200
spec. 1 spec. 4 spec. 20 spec. 200
100
0.2
Δl/l0 [%]
Tyy [N/mm2]
Tyy [N/mm2]
Size factor n (number of specimen)
200
spec. 1 spec. 4 spec. 20 spec. 200
r7 = 10.0 1/mm2 100
0
r7 = 100.0 1/mm2
0 0
1
2
3
4
5
Δl/l0 [%] Fig. 16. Response of the stress component Tyy on the hole at point ðaÞ as a function of the global strain Dl=l ða1 ¼ 0:1; r7 ¼ 10 mm2 Þ.
0
0.5
1
1.5
2
2.5 3 Δl/l0 [%]
3.5
4
4.5
Fig. 18. Effect of the material parameter r7 on the response of stress component Tyy at ðaÞ (specimen 4, a1 ¼ 0; 1).
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in the rule for isotropic hardening. Moreover, we fix the values of a and c7 in the elasticity laws by
a ¼ 0:1 m;
c7
m
¼ 0:1 mm2 :
ð51Þ
2.2.1. Uniaxial loading First, we present calculations for homogeneous uniaxial tension loading of a rectangular specimen (plane strain), according to Fig. 13. At the bottom of the specimen it is given uy ¼ 0, tx ¼ 0, tðdÞ ¼ 0, while at the top it is uy ¼ u y , tx ¼ 0, tðdÞ ¼ 0. The remaining boundaries are subject to the conditions t ¼ 0 and tðdÞ ¼ 0. The aim is to demonstrate the capabilities of the damage model. To this end, it suffices to concentrate on isotropic hardening only. Further, as the deformations are homogeneous, no material parameters of terms related to micromorphic curvature tensors are involved. Fig. 14 shows the effect of the damage parameter a1 (cf. Eq. (34)) on the responses of the uniaxial stress s and the damage variable D. Further discussion
600
¨ about the damage law for the classical case is provided in Lammer and Tsakmakis [10]. 2.2.2. Rectangular specimens with circular hole under tension loading We consider again the boundary value problem of Section 2.1.1, but now with respect to the specimen geometry displayed in Fig. 15 (length l differs from width b). In order to elucidate the capabilities of the micromorphic theory in predicting size effects, four specimen geometries are considered, referred to as specimens 1, 4, 20 and 200 (see Table 1). First, only isotropic hardening is addressed, with material parameters as given in Section 2.2, and r7 ¼ 10 mm2 . The discussion is referred to the stress component Tyy at point ðaÞ (see Fig. 15). It can be recognized from Fig. 16 that softening for large specimens begins earlier than for small ones. Comparison of Fig. 16 ða1 ¼ 0:1Þ with Fig. 17 ða1 ¼ 1:0Þ suggests that the form of the responses is strong dependent on the damage parameter a1. Moreover, Fig. 18 illustrates that maximal values of stresses and maximal global strains depend on the material parameter r7, present in the yield function. Next, we assume the micromorphic model material to exhibit kinematic hardening only, governed by the material parameters r7 ¼ 10 mm2 ;
500
P k7 ¼ 500=N;
Tyy [N/mm2]
400
M2k ¼ 50 mm2 =N; C7k ¼ 200 N=mm;
Ak2 ¼ Ak3 ¼ 200 N=mm2 ;
ð52Þ ð53Þ
the remaining material parameters related to kinematical hardening being vanishing. Similar to the case of pure isotropic hardening, Fig. 19 suggests that softening for large specimens begins earlier than for small ones. Fig. 20 confirms that this holds also for the case of combined isotropic and kinematic hardening.
300
200 spec. 1 spec. 4 spec. 20 spec. 200
100
3. Concluding remarks
0 0
Fig. 19. Responses of ða1 ¼ 1; k0 ¼ 500 N=mm2 Þ.
0.2
Tyy
0.4 Δl/l0 [%] at
ðaÞ
for
0.6
pure
0.8
kinematic
hardening
700 600 500 Tyy [N/mm2]
147
400 300 200 100
spec. 20 spec. 200
0 0
0.5
1
1.5
2
Δl/l0 [%] Fig. 20. Responses of Tyy at ðaÞ for combined isotropic and kinematic hardening ða1 ¼ 1; k0 ¼ 350 N=mm2 Þ.
A general framework for micromorphic plasticity has been formulated in Parts I, II, incorporating isotropic and kinematic hardening. The hardening laws are of the Armstrong–Frederick type and the yield function is a generalization of the classical v. Mises yield function. Some properties of the resulting theory, concerning prediction of size effects for small deformations, are reported in Part III. However, no comparison with experimental data is available, so that it is not possible to evaluate the appropriateness of the chosen constitutive functions. This concerns over all the yield function and the isotropic hardening, the latter being unifiedly postulated. Further studies, with reference to experimental results will help to clarify such issues, but this is beyond the scope of the present paper. All the discussions in the three articles make clear that phenomenological micromorphic theories (at least plasticity theories) are very complicated and involve a large number of material parameters. Therefore, it will be useful to clarify in future works, if it is possible to approximate the essential material responses predicted by micromorphic theories by some simpler gradient models, which deal with classical stresses only, and involve a smaller number of material parameters. Also it is of interest to answer the following question. Is the micromorphic model appropriate enough to describe all known size effects to the necessary degree? References [1] P. Grammenoudis, C. Tsakmakis, Isotropic hardening in micropolar plasticity, Archive of Applied Mechanics 79 (2009) 323–334. ¨ [2] D. Hofer, Simulation von großeneffekten mit mikromorphen theorien, Doctoral Thesis, TU Darmstadt, 2003.
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[3] T. Dillard, S. Forest, P. Ienny, Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams, European Journal of Mechanics A—Solids 25 (2006) 526–549. [4] N. Kirchner, P. Steinmann, A unifying treatise on variational principles for gradient and micromorphic continua, Philosophical Magazine 85 (2005) 3875–3895. [5] P. Neff, S. Forest, A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results, Journal of Elasticity 87 (2007) 239–276.
[6] M. Lazar, G. Maugin, On microcontinuum field theories: the eshelby stress tensor and incompatibility conditions, Philosophical Magazine 87 (2007) 3853–3870. [7] C.B. Hirschberger, K.E., Steinmann, P., On deformational and configurational mechanics of micromorphic hyperelasticity—theory and computation. Computer Methods in Applied Mechanics and Engineering 196 (2007) 4027–4044. [8] P. Gould, Introduction to Linear Elasticity, Springer, New York, Berlin, 1983. [9] R. Mindlin, Influence of couple-stresses on stress concentrations, Experimental Mechanics 3 (1963) 1–7. ¨ [10] H. Lammer, C. Tsakmakis, Discussion of coupled elastoplasticity and damage constitutive equations for small and finite deformations, International Journal of Plasticity 16 (2000) 495–523.