Abstract In this contribution we study the microstructurally short crack propagation phase in a grain structure of a duplex stainless steel (DSS). The grains in the DSS are either austenitic or ferritic. In order to take the material microstructure into account in the simulations we generate models of the grain structure using the Voronoi polygonization algorithm. The material behavior of the grains is modelled by crystal plasticity, hence, the crystallographic directions of the grains are modelled explicitly and evolution equations for the plastic slip along each slip-system are defined. Furthermore, the crystal plasticity model is enhanced, in the spirit of the disclocation model proposed by Navarro and De Los Rios [1], to include a crack propagation model taking into account non-local effects of the dislocation density (via the accumulated plastic slip) along a slip direction within a grain. Finally, numerical results are presented which show the influence of material hardening and slip directions on the propagation of a crack in an austenitic grain

Erland Johnson

Kenneth Runesson

Magnus Ekh

Robert Lillbacka

CiteSeerX

ECF15 
MODELLING GROWTH OF SMALL CRACKS IN A 
POLYCRYSTAL 
Robert Lillbacka1,2, Erland Johnson1, Magnus Ekh2 and Kenneth Runesson2 
1, SP Swedish National Testing and Research Institute 
Box 857, SE-501 15 Borås, Sweden 
robert.lillbacka@sp.se 
2, Chalmers University of Technology 
Dept. of Applied Mechanics, SE-412 96 Göteborg, Sweden 
 
 
 
Abstract 
In this contribution we study the microstructurally short crack propagation phase in a grain 
structure of a duplex stainless steel (DSS). The grains in the DSS are either austenitic or 
ferritic. In order to take the material microstructure into account in the simulations we 
generate models of the grain structure using the Voronoi polygonization algorithm. The 
material behavior of the grains is modelled by crystal plasticity, hence, the crystallographic 
directions of the grains are modelled explicitly and evolution equations for the plastic slip 
along each slip-system are defined. Furthermore, the crystal plasticity model is enhanced, in 
the spirit of the disclocation model proposed by Navarro and De Los Rios [1], to include a 
crack propagation model taking into account non-local effects of the dislocation density (via 
the accumulated plastic slip) along a slip direction within a grain. Finally, numerical results 
are presented which show the influence of material hardening and slip directions on the 
propagation of a crack in an austenitic grain. 
 
Introduction  
In many industrial applications, the life of a structural component is controlled by fatigue. 
This life could, from the macroscopic point of view, be divided into the so-called crack 
initiation and propagation phases, see e.g. Suresh [2]. For the crack propagation phase, 
fracture mechanics at the macroscale provides a tool to handle life predictions. However, for 
the initiation phase no such evident tool exists. 
The macroscopic crack initiation phase consists of, firstly, the small crack initiation (at free 
surfaces or defects) due to accumulation of plastic deformation along the slip directions (i.e. 
persistent slip bands) and, secondly, the subsequent propagation in one or several grains. The 
propagation is affected by grain boundaries and defects that act as stress raisers and as 
barriers, which introduce sequence and threshold effects on the macroscale e.g. Miller [3]. 
Therefore, the important characteristica of the material microstructure must necessarily be 
modelled in order to predict the duration of the initiation phase and the early propagation 
phase. Models for the propagation are usually based on the assumption that the cracks 
propagate along slip directions with the expression for the propagation velocity based on 
dislocation mechanics, see Pippan [4] and Navarro and de los Rios [1, 5, 6]. In the model by 
Navarro and de los Rios (the ND-model), the propagation velocity is determined by the 
number of dislocations along a slip line between the crack tip and the grain boundary. 
Initially, slip is assumed to have developed along a primary slip plane in one grain. During 
load cycling, the slip length increases in steps of a grain length when the shear stress has 
 1
ECF15 
reached a critical value in the next unslipped grain. It is assumed that the crack propagates 
with a speed that is proportional to the current plastic displacement φ  along the slip band in 
front of the crack tip, [1], 
da f
dN
φ= ⋅   (1) 
where N denotes load cycle. The constant f could be interpreted as the fraction of dislocations 
in the slip band, which participate in the process of crack extension. All grains are assumed to 
have identical sizes and slip propagation directions.  
The model has been modified to include hardening behavior [7] and, more recently, been 
used for the study of spectrum loading effects [8, 9]. The model can also be used to predict 
the Hall-Petch effect, and the continuum limit after propagation for a couple of grains.  
In the present paper, we will show how it is possible to combine crystal plasticity and a 
crack propagation law in the spirit of the (dislocation mechanics based) ND-model to study 
short crack propagation in a grain structure. The key assumption we make, in order to couple 
the models, is that the accumulated plastic slip in the crystal plasticity model is a measure of 
the dislocation density that is used as the driving force for the crack propagation. The 
combined crystal plasticity and crack propagation law is implememented in a model of a 
microstructure of duplex stainless steel (DSS). In order to illustrate the model capabilities, we 
conclude the paper by showing results for the short crack propagation in grains with different 
hardening characteristica and slip directions. 
 
Crystal plasticity framework  
The applied model is based on crystal plasticity and is thoroughly discussed in, e.g. Asaro 
[10], Miehe et al. [11] and Ekh et al. [12]. Here we shall only summarize the main ideas. The 
model is based on the classical multiplicative split of the deformation gradient pe FFF ⋅= . 
The kinematic assumption is that the plastic deformation gradient  describes the plastic 
slip along the slip-systems, whereas the elastic deformation gradient F
pF
e describes the crystal 
lattice distortion. The plastic slip on a slip-system ( αs , αm ) is driven by the the resolved 
shear (or Schmid) stress ατ , which is calculated as 
def
α α ατ = ⋅ ⋅s τ m  (2) 
where  is the Kirchhoff stress tensor, and (τ αs , αm ) define the slip-direction and slip-plane 
normal, respectively, in the slip-system.  Hence, the yield function on each slip-system can be 
defined as 
( )Yα α α αϕ τ κ= − +   (3) 
where ακ  is the drag-stress corresponding to latent hardening, and Yα is the initial yield 
stress. 
Further, from the principle of maximum dissipation, the plastic velocity gradient (describing 
the evolution of the plastic deformation gradient) becomes  
,  f = [F ]  (4) p p e T e T[ ] (( [ ] ) ([ ] ))α αα
α
µ
• •
−⋅ = ⋅ ⊗ ⋅∑F f s F F m p p 1−
Regarding the evolution for the hardening ακ , we adopt the model proposed by Chang and 
Asaro [13]. This model takes cross-hardening effects into account. 
 2
ECF15 
 
Short crack propagation model 
A new crack propagation model is developed in the spirit of the ND-model. A grain structure 
with both ferritic and austenitic grains is generated through Voronoi polygonization, 
Lillbacka et al. [14] (for a description of the algorithm c.f. Cannmo [15]). It is discretized 
using CST-triangles as illustrated in Figure 1. 
 
Figure 1. From left to right: initial grain structure with 16 grains (14 ferritic grains (light 
grey) and 2 austenitic grains (dark grey)) obtained from Voronoi polygonization, grain 
structure with predefined crack path and a FE- discretized grain structure. 
 
In the numerical examples we will predefine a crack propagation direction (that coincides 
with a slip plane) inside an austenitic grain of the grain structure, see Figure 1. We will also 
assume that the crack will initiate at the left grain boundary and then propagate through the 
centre of that grain in the predefined direction. In order to accomplish crack growth in the FE 
simulations, double nodes are inserted along the crack path, and a penalty formulation is used 
to control the crack opening and closure. 
In the crystal plasticity model the amount of accumulated plastic slip ( αµ ) is calculated in 
the chosen slip directions. It is possible to obtain a measure of the relative plastic 
displacement (between two slip planes) in the slip direction as  
tan( )b bα α αφ µ µ= ≈   (5) 
where b is the magnitude of a Burgers vector. The mean value of the plastic displacement 
along the specific slip direction between the crack tip at x = a and the tip of the slip line at x = 
c is obtained through integration  
( )
c
a
b x dx
c aα α
φ µ=
− ∫  (6) 
This measure is then used together with Eq. (1) to define the crack propagation law. 
 
Numerical examples 
In the numerical examples plane strain is assumed, and the twenty-four slip systems in the 
ferrite and austenite grains are reduced to planar double and triple slip systems, respectively, 
c.f. McDowell et al [16]. In the case of hardening, the actual material parameter values from 
calibration of a duplex stainless steel were used. Moreover, one of the slip systems is aligned 
with the crack direction, thus allowing for the calculation of the plastic displacement along 
the crack path from Eq. (6). The loading on the microstructure is a prescribed cyclic uniaxial 
 3
ECF15 
strain ε  in the range 0 0.25%ε≤ ≤  along the vertical axis, leading to an almost elastic 
(macroscopic) response.  
A parameter study was conducted with respect to the choice of the crack angle and the 
influence of hardening. These results are shown in Figures 2 and 3. The crack growth is 
comparatively high when the crack starts to propagate from a grain boundary while it is 
decelerated as it approaches the next grain boundary, c.f. right Figure 2. The crack angle 45 
degrees results in higher shear stress (due to the applied loading) and, thus, higher crack 
growth rate than does 40 degrees. Comparing the crack growth rate vs. crack length with the 
same type of curves for the ND-model e.g. Navarro and de Los Rios [1, 5, 6], we note that the 
growth rate is not as smooth in our model as in the ND-model. This is an effect of the fact 
that we really calculate a fluctuating plastic displacement, whereas in the ND case this is 
assumed to be constant across the grain. As to the influence of hardening, the obvious result 
that reducing hardening promotes crack growth was confirmed. 
 
 
Figure 2. Left figure shows normalized crack length vs cycle number for a crack that is 
inclined 40 and 45 degrees against the horizontal axes.  Right figure shows normalized crack 
growth vs normalized crack length for the same cases.  
 
 
Figure 3. Left figure shows normalized crack length vs cycle number for a crack with 45 
degrees alignment against the horizontal axes for perfectly plastic and hardening material 
behavior. Right figure shows normalized crack growth vs normalized crack length for the 
same cases.  
 
Furthermore, the crack extension and effective plastic slip field, which is the “driving force” 
behind the crack propagation law are shown in Figure 4.   
 4
ECF15 
 
 
Figure 4. Crack growth through the first grain in the grain structure for a 45 degree crack 
with hardening. Crack and effective plastic slip field after the fifth and the final cycle from 
left to right (dark areas indicate high levels of plastic slip).  
 
Conclusions and outlook 
In this paper we have exploited the ND-model in the context of crystal plasticity in order to 
predict the propagation of short cracks in a grain of a polycrystalline material. Preliminary FE 
simulations have shown that (for constant load amplitude) the propagation rate decreases as 
the crack approaches the next grain boundary. 
Future developments would have to include provision for the crack to extend to the 
neighbouring grains with a new direction determined by the appropriate crystallography. 
Moreover, a restriction of the present approach is that Dirichlet boundary conditions are 
applied to the microstructure. Hence, the crack can never extend to the boundary of the 
microstructure. To circumvent this drawback, other types of variational settings may be 
utilized. 
Acknowledgements 
Part of the support for this work was provided by Sandvik Materials Technology which is 
gratefully acknowledged. 
References 
1. Navarro, A. and de los Rios, E.R., Fatigue Fract. Engng Mater. Struct., Vol. 10, 169-186,     
1987. 
2. Suresh, S., Fatigue of materials, Cambridge University Press, Cambridge, 2001.  
3. Miller, K.J., Fatigue Fract. Engng Mater. Struct., Vol. 16, 931-939, 1993. 
4. Pippan, R., Acta metall. mater, Vol. 39, 255-262, 1991. 
5. Navarro, A. and de los Rios, E.R., Fatigue Fract. Engng Mater. Struct., Vol. 11, 383-396,       
1988. 
6. Navarro, A. and de los Rios, E.R., Phil. Mag., A57, 15-36, 1988. 
7. de los Rios, E.R. and Navarro, A., Proc. R. Soc. (London), A272, 304, 1963.  
 5
ECF15 
8. James, M.N. and de los Rios, E.R., Fatigue Fract. Engng Mater. Struct., Vol. 19, 413-426, 
1996. 
9. Wei, L.W., de los Rios, E.R. and James, M.N., Int. J. Fatigue, Vol. 24, 963-975, 2002.  
10. Asaro, R., Adv. Appl. Mech., Vol. 23, 1-115, 1983. 
11. Miehe, C., Schröder, J., and Schotte, J., Comput. Methods Appl. Mech. Engrg., Vol. 171, 
387-418, 1999 
12. Ekh, M., Lillbacka, R., and Runesson, K., Int. J. Plast, Accepted, 2002. 
13. Chang, Y.W., and Asaro, R.J., Acta Metall, Vol. 29,  241-257, 1981 
14. Lillbacka, R.,  Ekh, M., Johnson, E. and Runesson, K., In Proceedings of Fatigue 2003,   
edited by M.R. Bache, P.A. Blackmore, J. Draper, J.H. Edwards, P. Roberts and J.R. 
Yates, Engineering Integrity Society, Cambridge, 2003, 475-484. 
15. Cannmo, P., PhD thesis, Chalmers University of Technology, 1997 
16. Morrissey, R., Goh, C. and McDowell, D.L., Mech. of Mat., Vol. 35, 295-311, 2003    
 6


MODELLING GROWTH OF SMALL CRACKS IN A POLYCRYSTAL

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=EC63FEB6AB5269D61E6BF6AB984FA092?doi=10.1.1.1051.861&rep=rep1&type=pdf

MODELLING GROWTH OF SMALL CRACKS IN A POLYCRYSTAL

Abstract

Similar works

Full text

Available Versions

CiteSeerX