- Email: [email protected]
Analysis of void growth in a ductile material in front of a crack tip
Computational Materials Science 26 (2003) 28–35 www.elsevier.com/locate/commatsci
Analysis of void growth in a ductile material in front of a crack tip H. Baaser *, D. Gross Institute of Mechanics, Darmstadt University of Technology, D-64289 Darmstadt, Germany
Abstract The growth of microvoids in a ductile material is analysed in front of a crack tip loaded by a remote KI field. For this purpose, a circular region with radius R around the crack tip containing statistically distributed initially circular microvoids of radii r is discretised. The material behaviour is described by classical J2 plasticity with power hardening law in the scope of finite deformations. The process region is subdivided into a patch of randomly generated polygons simulating a real crystallic microstructure with a characteristic length scale. Within these polygons the material behaviour is represented by the identical, above-mentioned assumptions and constitutive model but with a statistical deviation from their average values. The second part focuses on the representation of geometrically necessary dislocations in the context of the applied finite strain description within the FE formulation. The aim of this investigation is a deeper understanding of correlations between material and geometrical properties of mode I dominated ductile damage and failure processes. Results of the parameter studies show strain distributions around typical microvoids and along the ligament. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 62.20.F; 61.50.C; 61.72; 61.72.Q Keywords: Ductile damage; Void growth; Geometrically necessary dislocations; Voronoi tessellation
1. Introduction The aim of this study is a detailed computational analysis of the growth of microvoids in a ductile material in front of a crack tip loaded by a remote KI field. Damage and failure analysis in ductile metallic materials due to microvoid growth has been subjected to intensive research activities in the past decades. While the models of the con-
*
Corresponding author. E-mail address: [email protected] (H. Baaser).
tinuum damage approach are extended in a steady development, see [12], and many contributions treat with multi-level computations applying different homogenization techniques for micro– macro transitions, see [7] or [10], an increasing number of investigations deal with microstructure computations exclusively on the level of the materials crystallic or grain structure. On this microscale of investigation, many known phenomena of macroscopic material behaviour can reasonably be studied and explained. Objects of interest are e.g. grain boundary sliding, see [16], or dislocation behaviour, see [11], which have an essential influence on damage and failure due to several mechanisms
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 3 8 9 - 0
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
of void growth. Such substructure investigations introduce an additional length scale by defining the geometry (shape, diameter etc.) of the particles building up the microstructure. The first investigations of discrete voids in front of a crack tip assume a homogenized material behaviour in the domain of interest, see [2], while some newer contributions study microstructure behaviour in samples with periodic arrangements of hexagonal cells, see [10] or [16]. Further studies deal with randomly distributed voids and analyse differences of crack propagation in there, [1]. A statistical continuum theory applied to the microstructure of polycrystalline material is given for the case of large plastic deformation by Garmestani et al. [6]. A post-processing procedure for the investigation of the evolution of geometrically necessary dislocations (GND) is described. GND evolution occurs during plastic deformation processes due to the incompatible character of the plastic part of the deformation gradient Fp ¼ Fe1 F, which is not necessary the gradient of a vector field. This topic is the object of many contributions since the works of Kondo [8] and Bilby et al. [3] in the early 1950s and e.g. [9] dealing with the representation of dislocations in the context of differential geometry. A precise review of these developments is given by De Wit [5]. The success in computational mechanics during the last 10–15 years in the treatment of a generalized and unified description of material behaviour based on a multiplicative decomposition of the deformation gradient (see e.g. [14] or [17]) focuses––in quite a natural way––on suitable representations of the internal constraints due to the incompatibility of Fe and Fp . Recent contributions deal with this correlations, see [4,15]. For the purpose of this study, a circular region with radius R around the crack tip containing statistically distributed initially circular microvoids of radii r is discretised by six-noded triangular finite elements under plane strain conditions. The global load is realized by prescribed displacements, representing a KI -dominated far field. The material behaviour is described by classical J2 plasticity with a power hardening law introducing the initial yield stress r0 and the hardening exponent N in the
29
scope of finite deformations. The process region is subdivided into a patch of randomly generated polygons simulating a real crystallic microstructure with the characteristic length scale L, so that r L R. Within these polygons, the material behaviour again is represented by the identical, above-mentioned assumptions and constitutive model but with a statistical deviation from their average values. In order to represent a crystallographic texture, this seems to be an easy and effective method to describe microscopic inhomogeneity resulting in a macroscopicaly anisotropic behaviour. But it should be clearly stated, that these first assumptions are not able to capture the effects of plastic anisotropy. Further investigations considering mechanisms of crystal plasticity have to follow in this context. Having in mind the simulation of a microstructure as realistic as possible, where the size of grains in typical steel or aluminium applications reaches from about 200 lm down to about 3–5 lm, we investigate, in this contribution, the more or less fine-grained spectrum of metals of interest by choosing two typical average grain sizes of L ’ 8 and 19 lm. Obviously, the length scales of this investigation are directly coupled with those of real metallic material and emphasize the fact, that the length scales of metallic damage and failure processes, which are often assumed to take place in a so-called process zone, are of the same order. Due to these statements a representation of metallic damage and failure occurrence by constitutive models of continuum damage mechanics becomes very questionable, because the material behaviour e.g. near crack tip singularities is still not understood enough and the basic assumptions of continuum theory as continuity of displacement fields and homogeneity in the surrounding of a material point x are violated in these regions. The aim of this investigation is a deeper understanding of basic correlations between material and geometrical properties in the situation of mode I dominated ductile damage and failure processes. Results of the parameter studies and the detailed FE calculations show strain and stress distributions around typical microvoids and along the ligament. In addition, the void growth is shown by illustrative plots of the deformed microstructure
30
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
and void shapes. These representations suggest certain failure modes and seem to be very revealing with respect to a dependence of the intrinsic length scales.
2. Computational treatment 2.1. Generation of heterogene, crystalline microstructure This two-dimensional computational investigation is carried out in a circular KI -dominated remote field with radius R ¼ 2 mm and a crack tip positioned in the origin under plane strain conditions. In the surrounding of the crack tip, a window of approximately 50 50 lm edge length is modeled with a randomly generated crystallinelike substructure, as can be seen in Fig. 1. During the computation the validity of small-scale-yielding is checked by the size of the plastic zone. The evolution of plastic strain is still observed to concentrate in the modeled process zone, so that the small-scale-yielding conditions can be assumed as not violated. For this contribution, we use two different substructures. The first with about 100 (here L ’ 19 lm) and the second with about 300 (L ’ 8 lm) subdomains, each of them represents a single crystallite. These subdomains within the window of interest around the crack tip are generated by a random determination of the node points and a following Delaunay-triangulation in-
cluding its dual Voronoi-diagram. See [13] for the partly used generation code Triangle, which is extended by a special post-processor for the determination of coherent regions of a single crystallite. Assuming ideal conditions for crystal growth, the generated structure can be seen as a synthetic texture representation by grains grown from randomly distributed nucleation, start growing at the same time with constant and equal growing velocity. Within this region containing a grain-like substructure, exemplary eleven microvoids are positioned randomly with different initial radius r near the crack tip, see Fig. 2. The positions and the radii of the assumed microvoids are listed in Table 1. That geometrical arrangement of the voids is used for the both modeled substructures as can be seen in Fig. 2. The resulting regions in this way are discretized by the classical six-noded triangular finite elements, whose constitutive behaviour is assigned to a set of material parameters deviating from average values. In contrast, the constitutive behaviour of the encompassing circular region is modeled by classical J2 plasticity theory for large strains but homogene parameter distribution of averaged values r0 and N . 2.2. FEM formulation and constitutive behaviour In elastic–plastic solids under sufficiently high load finite deformations occur, where the plastic part of the strains usually is large compared to the elastic part. We use the framework of multiplicative elastoplasticity. Its kinematic key assumption is the multiplicative split of the deformation gradient F ¼ Fe Fp
Fig. 1. KI loaded domain. Zoomed out crack tip region.
ð1Þ
into an elastic and a plastic part, providing the basis of a geometrically exact theory and avoiding linearization of any measure of deformation. As a further advantage, fast and numerically stable iterative algorithms, proposed and described in [17], can be used. If the condition U 6 0 (see Eq. (2)) is fulfilled by the current stress state s, this state is possible and is the solution. If, on the other hand, U 6 0 is violated by the trial state, the trial stresses must be projected back on the yield surface U ¼ 0 in an additional step. This ‘‘return mapping’’
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
31
Fig. 2. Crack tip region with same initial void distribution, but different microstructure. Discretization for coarse mesh.
Table 1 Position and radius of randomly distributed microvoids (length dimension (lm)) #void
1
2
3
4
5
6
7
8
9
10
11
xpos ypos Radius
42.55 0.1 1.0
18.25 0.0 0.6
7.3 )4.7 1.0
23.4 5.85 0.8
44.3 0.0 0.9
)0.25 8.75 0.8
29.85 )8.55 1.0
14.65 17.3 1.0
22.1 )13.5 1.5
7.2 )12.45 0.6
32.85 2.55 0.85
procedure is used as the integration algorithm for the constitutive equations described below. It should be mentioned that the algorithmic treatment in terms of principal axes has some advantages concerning computational aspects like time and memory saving. The constitutive model used in this study is a classical J2 flow theory. Here, the yield function is written as " #1=N epl eqv U ¼ q r0 Eþ1 ¼ 0; ð2Þ r0 qffiffiffiffiffiffiffiffiffiffi where q ¼ 32 tij tij is the equivalent Kirchhoff stress, p ¼ ð1=3Þsij dij defines the hydrostatic pressure, and tij ¼ sij þ pdij are the components of the Kirchhoff stress deviator. Furthermore, YoungÕs modulus is defined by E, the initial yield stress by r0 and the material hardening by the
exponent N . The set of constitutive equations is complemented by the evolution equations for the plastic strain. The macroscopic plastic strain rate _ pl is determined by the classical associated flow rule _ pl ¼ koU=os, whereas the parameter k is determined through the consistency condition kU_ ¼ 0 during any plastic flow process. 2.3. Applied loading The boundary nodes of the discretized structure positioned at radius R are loaded by the displacement field
ux uy
KI ¼ 2G
rffiffiffiffiffiffi R cos u=2 ð3 4m cos uÞ ; sin u=2 2p ð3Þ
32
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
Table 2 Parameters set for materials (stress dimension (MPa); last line: averaged values) Mix1
Mix2
Mix3
Mix4
Mix5
r0
N
r0
N
r0
N
r0
N
r0
N
460 450 470 490 400 460
7 5 9 11 3 7
480 450 480 480 440 460
8 6 9 9 5 7
464 455 468 458 463 460
7.2 6.7 6.5 7.4 6.8 7.0
480 450 480 480 440 460
7 5 9 11 3 7
460 450 470 490 400 460
8 6 9 9 5 7
455
7
465
7.33
461
6.93
465
7
455
7.33
as given in polar coordinates R, u due to the KI far field for plane strain conditions, see Fig. 1. So the stress intensity factor KI determines the amplitude of the crack tip field, G represents the shear modulus and m the Poisson number. For p this ffiffiffiffi contribution KI is initially set to KIref ¼ 10 MPa m and is incremented stepwise linearly with slope m ¼ KIref /load step. 2.4. Variation of material parameters As described in Section 2.1, each of the modeled grains is performed with a set of material parameters deviating from an averaged value. In this contribution, we assume the different grain structures assembled of six different sets of parameters varying in the initial yield stress rI0 and the hardening exponent N I , I ¼ 1; 2; . . . ; 6, see Eq. (2), while the values for YoungÕs modulus E and the Poisson number m are fixed. Investigating different combinations of the varying sets of parameters, we
define five different mixtures, in the following called mix1 to mix5, where the initial yield stress exists in the range of rI0 ¼ ½430; . . . ; 490 MPa and the hardening exponent N ¼ ½4; . . . ; 11 . Because of nearly the same portion of each material set relative to the crack tip region, one can compute the average values r0 and N by the arithmetic mean of the respective values. The chosen values for the five different mixtures are listed in Table 2 to give an overview on the referenced combinations.
3. Results and comparative studies for different microstructures 3.1. Void growth Several results are described in order to get an impression of the influence of the varying material parameters and the differences in the microstructure
Fig. 3. Evolution of plastic strain and void growth.
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
due to the coarse and the finer grain size in front of the crack tip. Fig. 2 shows the discretization and the coarse grain structure in the zoomed-out crack tip region. The completely discretized circular KI -crack tip problem for the coarse substructure consists of about 33 000 nodes and over 16 000 elements, while for the discretization of the finer grains within the KI field over 70 000 nodes and nearly 35 000 elements are needed. Again, for the coarse grain substructure, the evolution of the plastic strain in front of the crack tip and especially around the microvoids near the ligament is plotted in Fig. 3 in the deformed configuration for the load steps 120, 160 and 200. Obviously, the high increased values are directly on the elongation of the existing crack tip. In Fig. 4, the void volume (fraction) for the different materials sets and the different grain sizes are plotted against the load level. As it can be seen, no significant differences can be found. This is explained by the displacement controlled loading, which effects just the elastic part of the volume change due to the assumption of plastic incompressibility. Fig. 5 shows the stress
33
component ryy along the ligament for both discretizations at the load level 60 and 120. Additionally the near tip field solution for this stress component ryy is plotted, where not only the asymptotic behaviour but also the oscillations near the void locations of the actual solution can be seen. In evidence are the strong oscillations in front of the crack tip, where the microvoids are located. In contrast, for a larger distance from the crack tip, the stress values converge to a smooth decreasing trend due to the absence of perturbation caused by microvoids. 3.2. Geometrically necessary dislocations As last result demonstrated in this article, Fig. 6 shows pffiffiffiffiffiffiffiffiffiffiffiffithe zones of concentrations of the norm Gij Gij of the dislocation tensor G ¼ J e Fe1 curl Fe1
ð4Þ
as proposed by Cermelli and Gurtin [4] at the load level 120. Essential is the fact that the grain boundaries acts as concentrators of the quantity,
Fig. 4. Void volume fraction for different material sets.
34
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
Fig. 5. Asymptotic behaviour of stress component ryy along ligament for both discretizations and different mixtures at load level 60 and 120 to elastic solutions. Additionally two void positions near the ligament.
Fig. 6. Geometrically necessary dislocations at load 120 KI =KIref .
which is obvious due to the different appearances for the coarse and the fine crystalline substructure.
4. Summary In this contribution we show a study of void growth near a crack tip in the scope of a KI far field. Additionally, the near crack tip region is
built up by a synthetic crystalline microstructure with two different length scales representing different mean values of crystal diameter. Therefore, the void position and radius and as well the material properties of the modeled grains in the surrounding of the crack tip vary randomly. The results confirm the suggestion that in this loading regime just the voids in the direct elongation of the crack tip grow dramatically, while the others
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
beside the ligament are affected non-essentially. The end of the investigation deals with the representation of so-called GND, which give some insight into the internal constraints within the crystals due to plastic flow. This type of microdefects accumulates in the surrounding of the crack tip and especially at the grain boundaries. These two results may help to understand and to model material behaviour near crack tip singularities in a more accurate and responsible way. Further investigations have to respect in more detail constitutive characteristics of material behaviour on the microscale in zones of high stress singularities. References [1] A. Al-Ostaz, I. Jasiuk, Crack initiation and propagation in materials with randomly distributed holes, Engineering Fracture Mechanics 58 (5–6) (1997) 395–420. [2] N. Aravas, R.M. McMeeking, Microvoid growth and failure in the ligament between a hole and a blunt crack tip, International Journal of Fracture 29 (1985) 21–38. [3] B.A. Bilby, R. Bullough, E. Smith, Continuous distribution of dislocations: A new application of the methods of nonRiemannian geometry, Proceedings of the Royal Society of London Series A 231 (1955) 263–273. [4] P. Cermelli, M.E. Gurtin, On the characterization of geometrically necessary dislocations in finite plasticity, Journal of the Mechanics and Physics of Solids 49 (2001) 1539–1568. [5] R. De Wit, A view of the relation between the continuum theory of lattice defects and non-Euklidian geometry in the linear approximation, International Journal of Engineering Science 19 (12) (1981) 1475–1506. [6] H. Garmestani, S. Lin, B.L. Adams, S. Ahzi, Statistical continuum theory for large plastic deformation of poly-
[7]
[8]
[9] [10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
35
crystalline materials, Journal of the Mechanics and Physics of Solids 49 (2001) 589–607. S. Ghosh, K. Lee, P. Raghavan, A multi-level computational model for multi-scale damage analysis in composite and porous materials, International Journal of Solids and Structures 38 (2001) 2335–2385. K. Kondo, On the geometrical and physical foundation of the theory of yielding, in: Proc. 2. Japan Nat. Congress Appl. Mech, Science Council of Japan, 1952, pp. 41–47. E. Kr€ oner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer, 1958. C. Miehe, J. Schr€ oder, J. Schotte, Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials, Computer Methods in Applied Mechanics and Engineering 171 (1999) 387– 418. M. Ortiz, E.A. Repetto, L. Stainier, A theory of subgrain dislocation structures, Journal of the Mechanics and Physics of Solids 48 (2000) 2077–2114. T. Pardoen, J.W. Hutchinson, An extended model for void growth and coalescence, Journal of the Mechanics and Physics of Solids 48 (2000) 2467–2512. J.R. Shewchuk, Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator, First Workshop on Applied Computational Geometry, Philadelphia, Pennsylvania, 1996, pp. 124–133; see also http://www.cs.cmu.edu/
quake/triangle.research.html. J.C. Simo, Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory, Computer Methods in Applied Mechanics and Engineering 99 (1992) 61–112. P. Steinmann, Views on multiplicative elastoplasticity and the continuum theory of dislocations, International Journal of Engineering Science 34 (15) (1996) 1717–1735. E. van der Giessen, V. Tvergaard, Interaction of cavitating grain boundary facets in creeping polycrystals, Mechanics of Materials 17 (1994) 47–69. G. Weber, L. Anand, Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic–viscoplastic solids, Computer Methods in Applied Mechanics and Engineering 79 (1990) 173–202.
Analysis of void growth in a ductile material in front of a crack tip H. Baaser *, D. Gross Institute of Mechanics, Darmstadt University of Technology, D-64289 Darmstadt, Germany
Abstract The growth of microvoids in a ductile material is analysed in front of a crack tip loaded by a remote KI field. For this purpose, a circular region with radius R around the crack tip containing statistically distributed initially circular microvoids of radii r is discretised. The material behaviour is described by classical J2 plasticity with power hardening law in the scope of finite deformations. The process region is subdivided into a patch of randomly generated polygons simulating a real crystallic microstructure with a characteristic length scale. Within these polygons the material behaviour is represented by the identical, above-mentioned assumptions and constitutive model but with a statistical deviation from their average values. The second part focuses on the representation of geometrically necessary dislocations in the context of the applied finite strain description within the FE formulation. The aim of this investigation is a deeper understanding of correlations between material and geometrical properties of mode I dominated ductile damage and failure processes. Results of the parameter studies show strain distributions around typical microvoids and along the ligament. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 62.20.F; 61.50.C; 61.72; 61.72.Q Keywords: Ductile damage; Void growth; Geometrically necessary dislocations; Voronoi tessellation
1. Introduction The aim of this study is a detailed computational analysis of the growth of microvoids in a ductile material in front of a crack tip loaded by a remote KI field. Damage and failure analysis in ductile metallic materials due to microvoid growth has been subjected to intensive research activities in the past decades. While the models of the con-
*
Corresponding author. E-mail address: [email protected] (H. Baaser).
tinuum damage approach are extended in a steady development, see [12], and many contributions treat with multi-level computations applying different homogenization techniques for micro– macro transitions, see [7] or [10], an increasing number of investigations deal with microstructure computations exclusively on the level of the materials crystallic or grain structure. On this microscale of investigation, many known phenomena of macroscopic material behaviour can reasonably be studied and explained. Objects of interest are e.g. grain boundary sliding, see [16], or dislocation behaviour, see [11], which have an essential influence on damage and failure due to several mechanisms
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 3 8 9 - 0
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
of void growth. Such substructure investigations introduce an additional length scale by defining the geometry (shape, diameter etc.) of the particles building up the microstructure. The first investigations of discrete voids in front of a crack tip assume a homogenized material behaviour in the domain of interest, see [2], while some newer contributions study microstructure behaviour in samples with periodic arrangements of hexagonal cells, see [10] or [16]. Further studies deal with randomly distributed voids and analyse differences of crack propagation in there, [1]. A statistical continuum theory applied to the microstructure of polycrystalline material is given for the case of large plastic deformation by Garmestani et al. [6]. A post-processing procedure for the investigation of the evolution of geometrically necessary dislocations (GND) is described. GND evolution occurs during plastic deformation processes due to the incompatible character of the plastic part of the deformation gradient Fp ¼ Fe1 F, which is not necessary the gradient of a vector field. This topic is the object of many contributions since the works of Kondo [8] and Bilby et al. [3] in the early 1950s and e.g. [9] dealing with the representation of dislocations in the context of differential geometry. A precise review of these developments is given by De Wit [5]. The success in computational mechanics during the last 10–15 years in the treatment of a generalized and unified description of material behaviour based on a multiplicative decomposition of the deformation gradient (see e.g. [14] or [17]) focuses––in quite a natural way––on suitable representations of the internal constraints due to the incompatibility of Fe and Fp . Recent contributions deal with this correlations, see [4,15]. For the purpose of this study, a circular region with radius R around the crack tip containing statistically distributed initially circular microvoids of radii r is discretised by six-noded triangular finite elements under plane strain conditions. The global load is realized by prescribed displacements, representing a KI -dominated far field. The material behaviour is described by classical J2 plasticity with a power hardening law introducing the initial yield stress r0 and the hardening exponent N in the
29
scope of finite deformations. The process region is subdivided into a patch of randomly generated polygons simulating a real crystallic microstructure with the characteristic length scale L, so that r L R. Within these polygons, the material behaviour again is represented by the identical, above-mentioned assumptions and constitutive model but with a statistical deviation from their average values. In order to represent a crystallographic texture, this seems to be an easy and effective method to describe microscopic inhomogeneity resulting in a macroscopicaly anisotropic behaviour. But it should be clearly stated, that these first assumptions are not able to capture the effects of plastic anisotropy. Further investigations considering mechanisms of crystal plasticity have to follow in this context. Having in mind the simulation of a microstructure as realistic as possible, where the size of grains in typical steel or aluminium applications reaches from about 200 lm down to about 3–5 lm, we investigate, in this contribution, the more or less fine-grained spectrum of metals of interest by choosing two typical average grain sizes of L ’ 8 and 19 lm. Obviously, the length scales of this investigation are directly coupled with those of real metallic material and emphasize the fact, that the length scales of metallic damage and failure processes, which are often assumed to take place in a so-called process zone, are of the same order. Due to these statements a representation of metallic damage and failure occurrence by constitutive models of continuum damage mechanics becomes very questionable, because the material behaviour e.g. near crack tip singularities is still not understood enough and the basic assumptions of continuum theory as continuity of displacement fields and homogeneity in the surrounding of a material point x are violated in these regions. The aim of this investigation is a deeper understanding of basic correlations between material and geometrical properties in the situation of mode I dominated ductile damage and failure processes. Results of the parameter studies and the detailed FE calculations show strain and stress distributions around typical microvoids and along the ligament. In addition, the void growth is shown by illustrative plots of the deformed microstructure
30
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
and void shapes. These representations suggest certain failure modes and seem to be very revealing with respect to a dependence of the intrinsic length scales.
2. Computational treatment 2.1. Generation of heterogene, crystalline microstructure This two-dimensional computational investigation is carried out in a circular KI -dominated remote field with radius R ¼ 2 mm and a crack tip positioned in the origin under plane strain conditions. In the surrounding of the crack tip, a window of approximately 50 50 lm edge length is modeled with a randomly generated crystallinelike substructure, as can be seen in Fig. 1. During the computation the validity of small-scale-yielding is checked by the size of the plastic zone. The evolution of plastic strain is still observed to concentrate in the modeled process zone, so that the small-scale-yielding conditions can be assumed as not violated. For this contribution, we use two different substructures. The first with about 100 (here L ’ 19 lm) and the second with about 300 (L ’ 8 lm) subdomains, each of them represents a single crystallite. These subdomains within the window of interest around the crack tip are generated by a random determination of the node points and a following Delaunay-triangulation in-
cluding its dual Voronoi-diagram. See [13] for the partly used generation code Triangle, which is extended by a special post-processor for the determination of coherent regions of a single crystallite. Assuming ideal conditions for crystal growth, the generated structure can be seen as a synthetic texture representation by grains grown from randomly distributed nucleation, start growing at the same time with constant and equal growing velocity. Within this region containing a grain-like substructure, exemplary eleven microvoids are positioned randomly with different initial radius r near the crack tip, see Fig. 2. The positions and the radii of the assumed microvoids are listed in Table 1. That geometrical arrangement of the voids is used for the both modeled substructures as can be seen in Fig. 2. The resulting regions in this way are discretized by the classical six-noded triangular finite elements, whose constitutive behaviour is assigned to a set of material parameters deviating from average values. In contrast, the constitutive behaviour of the encompassing circular region is modeled by classical J2 plasticity theory for large strains but homogene parameter distribution of averaged values r0 and N . 2.2. FEM formulation and constitutive behaviour In elastic–plastic solids under sufficiently high load finite deformations occur, where the plastic part of the strains usually is large compared to the elastic part. We use the framework of multiplicative elastoplasticity. Its kinematic key assumption is the multiplicative split of the deformation gradient F ¼ Fe Fp
Fig. 1. KI loaded domain. Zoomed out crack tip region.
ð1Þ
into an elastic and a plastic part, providing the basis of a geometrically exact theory and avoiding linearization of any measure of deformation. As a further advantage, fast and numerically stable iterative algorithms, proposed and described in [17], can be used. If the condition U 6 0 (see Eq. (2)) is fulfilled by the current stress state s, this state is possible and is the solution. If, on the other hand, U 6 0 is violated by the trial state, the trial stresses must be projected back on the yield surface U ¼ 0 in an additional step. This ‘‘return mapping’’
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
31
Fig. 2. Crack tip region with same initial void distribution, but different microstructure. Discretization for coarse mesh.
Table 1 Position and radius of randomly distributed microvoids (length dimension (lm)) #void
1
2
3
4
5
6
7
8
9
10
11
xpos ypos Radius
42.55 0.1 1.0
18.25 0.0 0.6
7.3 )4.7 1.0
23.4 5.85 0.8
44.3 0.0 0.9
)0.25 8.75 0.8
29.85 )8.55 1.0
14.65 17.3 1.0
22.1 )13.5 1.5
7.2 )12.45 0.6
32.85 2.55 0.85
procedure is used as the integration algorithm for the constitutive equations described below. It should be mentioned that the algorithmic treatment in terms of principal axes has some advantages concerning computational aspects like time and memory saving. The constitutive model used in this study is a classical J2 flow theory. Here, the yield function is written as " #1=N epl eqv U ¼ q r0 Eþ1 ¼ 0; ð2Þ r0 qffiffiffiffiffiffiffiffiffiffi where q ¼ 32 tij tij is the equivalent Kirchhoff stress, p ¼ ð1=3Þsij dij defines the hydrostatic pressure, and tij ¼ sij þ pdij are the components of the Kirchhoff stress deviator. Furthermore, YoungÕs modulus is defined by E, the initial yield stress by r0 and the material hardening by the
exponent N . The set of constitutive equations is complemented by the evolution equations for the plastic strain. The macroscopic plastic strain rate _ pl is determined by the classical associated flow rule _ pl ¼ koU=os, whereas the parameter k is determined through the consistency condition kU_ ¼ 0 during any plastic flow process. 2.3. Applied loading The boundary nodes of the discretized structure positioned at radius R are loaded by the displacement field
ux uy
KI ¼ 2G
rffiffiffiffiffiffi R cos u=2 ð3 4m cos uÞ ; sin u=2 2p ð3Þ
32
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
Table 2 Parameters set for materials (stress dimension (MPa); last line: averaged values) Mix1
Mix2
Mix3
Mix4
Mix5
r0
N
r0
N
r0
N
r0
N
r0
N
460 450 470 490 400 460
7 5 9 11 3 7
480 450 480 480 440 460
8 6 9 9 5 7
464 455 468 458 463 460
7.2 6.7 6.5 7.4 6.8 7.0
480 450 480 480 440 460
7 5 9 11 3 7
460 450 470 490 400 460
8 6 9 9 5 7
455
7
465
7.33
461
6.93
465
7
455
7.33
as given in polar coordinates R, u due to the KI far field for plane strain conditions, see Fig. 1. So the stress intensity factor KI determines the amplitude of the crack tip field, G represents the shear modulus and m the Poisson number. For p this ffiffiffiffi contribution KI is initially set to KIref ¼ 10 MPa m and is incremented stepwise linearly with slope m ¼ KIref /load step. 2.4. Variation of material parameters As described in Section 2.1, each of the modeled grains is performed with a set of material parameters deviating from an averaged value. In this contribution, we assume the different grain structures assembled of six different sets of parameters varying in the initial yield stress rI0 and the hardening exponent N I , I ¼ 1; 2; . . . ; 6, see Eq. (2), while the values for YoungÕs modulus E and the Poisson number m are fixed. Investigating different combinations of the varying sets of parameters, we
define five different mixtures, in the following called mix1 to mix5, where the initial yield stress exists in the range of rI0 ¼ ½430; . . . ; 490 MPa and the hardening exponent N ¼ ½4; . . . ; 11 . Because of nearly the same portion of each material set relative to the crack tip region, one can compute the average values r0 and N by the arithmetic mean of the respective values. The chosen values for the five different mixtures are listed in Table 2 to give an overview on the referenced combinations.
3. Results and comparative studies for different microstructures 3.1. Void growth Several results are described in order to get an impression of the influence of the varying material parameters and the differences in the microstructure
Fig. 3. Evolution of plastic strain and void growth.
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
due to the coarse and the finer grain size in front of the crack tip. Fig. 2 shows the discretization and the coarse grain structure in the zoomed-out crack tip region. The completely discretized circular KI -crack tip problem for the coarse substructure consists of about 33 000 nodes and over 16 000 elements, while for the discretization of the finer grains within the KI field over 70 000 nodes and nearly 35 000 elements are needed. Again, for the coarse grain substructure, the evolution of the plastic strain in front of the crack tip and especially around the microvoids near the ligament is plotted in Fig. 3 in the deformed configuration for the load steps 120, 160 and 200. Obviously, the high increased values are directly on the elongation of the existing crack tip. In Fig. 4, the void volume (fraction) for the different materials sets and the different grain sizes are plotted against the load level. As it can be seen, no significant differences can be found. This is explained by the displacement controlled loading, which effects just the elastic part of the volume change due to the assumption of plastic incompressibility. Fig. 5 shows the stress
33
component ryy along the ligament for both discretizations at the load level 60 and 120. Additionally the near tip field solution for this stress component ryy is plotted, where not only the asymptotic behaviour but also the oscillations near the void locations of the actual solution can be seen. In evidence are the strong oscillations in front of the crack tip, where the microvoids are located. In contrast, for a larger distance from the crack tip, the stress values converge to a smooth decreasing trend due to the absence of perturbation caused by microvoids. 3.2. Geometrically necessary dislocations As last result demonstrated in this article, Fig. 6 shows pffiffiffiffiffiffiffiffiffiffiffiffithe zones of concentrations of the norm Gij Gij of the dislocation tensor G ¼ J e Fe1 curl Fe1
ð4Þ
as proposed by Cermelli and Gurtin [4] at the load level 120. Essential is the fact that the grain boundaries acts as concentrators of the quantity,
Fig. 4. Void volume fraction for different material sets.
34
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
Fig. 5. Asymptotic behaviour of stress component ryy along ligament for both discretizations and different mixtures at load level 60 and 120 to elastic solutions. Additionally two void positions near the ligament.
Fig. 6. Geometrically necessary dislocations at load 120 KI =KIref .
which is obvious due to the different appearances for the coarse and the fine crystalline substructure.
4. Summary In this contribution we show a study of void growth near a crack tip in the scope of a KI far field. Additionally, the near crack tip region is
built up by a synthetic crystalline microstructure with two different length scales representing different mean values of crystal diameter. Therefore, the void position and radius and as well the material properties of the modeled grains in the surrounding of the crack tip vary randomly. The results confirm the suggestion that in this loading regime just the voids in the direct elongation of the crack tip grow dramatically, while the others
H. Baaser, D. Gross / Computational Materials Science 26 (2003) 28–35
beside the ligament are affected non-essentially. The end of the investigation deals with the representation of so-called GND, which give some insight into the internal constraints within the crystals due to plastic flow. This type of microdefects accumulates in the surrounding of the crack tip and especially at the grain boundaries. These two results may help to understand and to model material behaviour near crack tip singularities in a more accurate and responsible way. Further investigations have to respect in more detail constitutive characteristics of material behaviour on the microscale in zones of high stress singularities. References [1] A. Al-Ostaz, I. Jasiuk, Crack initiation and propagation in materials with randomly distributed holes, Engineering Fracture Mechanics 58 (5–6) (1997) 395–420. [2] N. Aravas, R.M. McMeeking, Microvoid growth and failure in the ligament between a hole and a blunt crack tip, International Journal of Fracture 29 (1985) 21–38. [3] B.A. Bilby, R. Bullough, E. Smith, Continuous distribution of dislocations: A new application of the methods of nonRiemannian geometry, Proceedings of the Royal Society of London Series A 231 (1955) 263–273. [4] P. Cermelli, M.E. Gurtin, On the characterization of geometrically necessary dislocations in finite plasticity, Journal of the Mechanics and Physics of Solids 49 (2001) 1539–1568. [5] R. De Wit, A view of the relation between the continuum theory of lattice defects and non-Euklidian geometry in the linear approximation, International Journal of Engineering Science 19 (12) (1981) 1475–1506. [6] H. Garmestani, S. Lin, B.L. Adams, S. Ahzi, Statistical continuum theory for large plastic deformation of poly-
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