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Numerical modelling of fatigue crack initiation of martensitic steel
Advances in Engineering Software 41 (2010) 823–829
Contents lists available at ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Numerical modelling of fatigue crack initiation of martensitic steel S. Glodezˇ a,*, N. Jezernik b, J. Kramberger b, T. Lassen c a
University of Maribor, Faculty of Natural Science and Mathematics, Koroška 160, 2000 Maribor, Slovenia University of Maribor, Faculty of Mechanical Engineering, Smetanova 17, 2000 Maribor, Slovenia c University of Agder, Faculty of Engineering and Science, Grooseveien 36, N-4876 Grimstad, Norway b
a r t i c l e
i n f o
Article history: Received 4 September 2009 Accepted 22 December 2009 Available online 2 February 2010 Keywords: Fatigue Crack initiation Numerical modelling
a b s t r a c t Numerical simulation of micro-crack initiation that is based on Tanaka–Mura micro-crack nucleation model is presented. Three improvements were added to this model. Firstly, multiple slip bands where micro-cracks may occur are used in each grain. Second improvement deals with micro-crack coalescence by extending existing micro-cracks along grain boundaries and connecting them into a macro-crack. The third improvement handles segmented micro-crack generation, where a micro-crack is not nucleated in one step like in Tanaka–Mura model, but is instead generated in multiple steps. Numerical simulation of crack-initiation was performed with ABAQUS, using a plug-in that was written specially for handling micro-crack nucleation and coalescence. Since numerical model was directed at simulating fatigue properties of thermally cut steel with martensitic structure, edge properties of specimen were additionally inspected in terms of micro-structural properties, surface roughness and residual stresses. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Current industrial design is heavily concerned with the highcycle fatigue resistance of mechanical components and structures. However, the complete process of fatigue failure of mechanical elements may be divided into the following stages [1,2]: (1) micro-crack nucleation; (2) short crack growth; (3) long crack growth; and (4) occurrence of final failure. In engineering applications the first two stages are usually termed as ‘‘crack initiation period”, while long crack growth is termed as ‘‘crack propagation period”. An exact definition of the transition from initiation to propagation period is usually not possible. However, the crack initiation period generally account for most of the service life, especially in high-cycle fatigue, Fig. 1. The total number of stress cycles N can than be determined from the number of stress cycles Ni required for the fatigue crack initiation and the number of stress cycles Np required for a crack to propagate from the initial to the critical crack length, when the final failure can be expected to occur (N = Ni + Np). This approach is mainly restricted to the crack initiation period, i.e. micro-crack nucleation and short crack growth. The behaviour of short cracks under fatigue loading differs substantially from the behaviour of long-cracks [3]. Existing research in micro-crack nucleation has shown for many materials that micro-cracks are ini-
* Corresponding author. Address: University of Maribor, Faculty of Natural Science and Mathematics, Koroška c. 160, SI-2000 Maribor, Slovenia. Tel.: +386 2 229 3752; fax: +386 2 251 8180. E-mail address: [email protected] (S. Glodezˇ). 0965-9978/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2010.01.002
tiated on slip bands of grains and stretch across whole grain [4]. The often proposed method to solve this problem is by using Tanaka–Mura model for micro-crack initiation [5]. Using this model, the number of stress cycles Nc required for the micro-crack nucleation can be determined as follows:
Nc ¼
8 G Ws
p ð1 mÞ d ðDs 2kÞ2
ð1Þ
Eq. (1) presumes that micro-cracks form along the slip band of grains, depending on slip band length d and average shear stress . Other material constants (shear modurange on the slip band Ds lus G, specific fracture energy per unit area Ws, Poisson’s ratio m, and frictional stress of dislocation on the slip plane k can be for examined material found in specialised literature (for martensite see Ref. [6]). Tanaka–Mura model still has two deficiencies that limit its use. Model deals only with micro-crack nucleation in separate grains and doesn’t deal with macro-crack coalescence from existing micro-cracks. Another problem is that this model uses average shear stress to determinate micro-crack nucleation. In our previous attempts to simulate micro-crack initiation [7], it often happened that an existing micro-crack significantly raised stresses of neighbouring grain, yet the average shear stress in this grain was still below the threshold needed for micro-crack nucleation, therefore no nucleation occurred. This problem becomes increasingly pronounced when using lower loads, that is in high-cycle fatigue regime. The main purpose of this study is to present a few improvements to the Tanaka–Mura model aimed at solving these two problems.
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(with no slip bands indicated), serve as a buffer zone to minimize the numerical mismatch at boundaries (top, right and bottom edge on the right image of Fig. 2) and do not partake in micro-crack evaluation. 2.1. Multiple slip bands
Fig. 1. Schematic representation of the service life of mechanical elements.
2. Physical model The problem of micro-crack nucleation and short crack growth is studied on a multi-scale model as shown in Fig. 2. Global model represents a full scale specimen subjected to chosen load level. The proposed micro-crack nucleation model simulates fatigue crack initiation in particular grains, therefore a complete microstructure needs to be modeled. Since it would be impossible to model a detailed microstructure on a global model (due to memory and computational constraints), we decided to use a sub-model in the place where the highest stresses are present (at the inner hole) and formation of cracks is expected. The size of sub-model was chosen to be 0.5 0.5 mm in size, as this would be enough to asses crack initiation period. Loading of a sub-model is implemented by applying stress distribution of global model to the boundary edges of submodel (top, right, and bottom edge on the right image of Fig. 2). Numerical simulation of crack-initiation was performed with ABAQUS [8,9], using a plug-in that was written specially for handling micro-crack nucleation and coalescence. Individual grains are simulated with randomly generated Voronoi tessellation and were given randomly oriented [10,11] orthotropic material properties of martensite [6]: C11 = C22 = C33 = 233 GPa, C12 = C13 = C23 = 135 GPa, C44 = C55 = C66 = 118 GPa. Considering the average grain size of 20 lm the whole model consists of approximately 400 individual grains. The right image in Fig. 2 shows such a model, with slip bands drawn through centres of grains. The surrounding grains
When simulating crack nucleation in particular grain, multiple slip bands were evaluated. For each grain there was one slip band created through the centre of the grain and additional slip bands were created with a 2 lm offset from the first (Fig. 3). Fig. 4 shows a case where the first micro-crack was nucleated in the left grain and influenced its neighbouring grain on the right, so that a micro-crack in it nucleated far from the centre (black dot), close to the first micro-crack. The proximity of nucleated micro-cracks also enables easier micro-crack coalescence than in a model where micro-cracks are allowed to go through grain centres only [6]. 2.2. Crack coalescence A conservative approach was taken to enable crack coalescence. Whenever a new micro-crack was introduced in the model all possible combination of micro-cracks were analysed to see if the average stress on a straight line between two micro-cracks surpasses yield stress of the material. If this was the case, a crack was extended through that line, effectively transforming two microcracks into a single larger crack (Fig. 5). Note significant stress relaxation in the area between the cracks with respect to Fig. 4. When calculating cycles required for crack initiation, no cycles were attributed to crack coalescences (it is simulated as being instantaneous), so the total cycles of crack initiation equal the sum of cycles needed for each micro-crack to nucleate. 2.3. Segmented micro-cracks Fig. 6a shows shear stress range Ds (black solid line) along a slip band of a grain, whose neighbouring grain already has a microcrack. This causes a significant stress concentration on the side of the slip band that is adjacent to the existing micro-crack. Using Tanaka–Mura model would not cause a micro-crack to nucleate in (gray dotted line) is lowthis case, as average shear stress range Ds
Fig. 2. Global model and sub-model of the specimen.
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825
up by keeping track of the damage that was accumulated over previous stages. In the beginning, total deformation energy needed for micro-crack nucleation is calculated for all segments of slip bands of every grain, using following equation:
U th s ¼
8 Ws ds
ð2Þ
where U th s represents a threshold amount of total deformation energy that is required for micro-crack nucleation on a particular segment (index s), ds is the length of that segment and Ws is the specific fracture energy per unit area. Then in each stage the amount of deformation energy acquired per cycle for each segment is calculated: Fig. 3. A crystal grain with multiple slip bands.
DU s ¼
pð1 mÞðDss 2 kÞ2 G
ð3Þ
s and DU s represent average shear stress range and where Ds amount of deformation energy that is accumulated in one loading cycle on a segment s. The number of cycles N is , required for micro-crack nucleation on each segment, is calculated with the following equation:
Nis ¼
i U th s Us DU s
ð4Þ
where U is represents already accumulated damage in each segment of stage i. Then a micro-crack is created on the segment with the smallest Nis . For other segments the accumulated damage for next is increased. stage U iþ1 s
U iþ1 ¼ U is þ DU s Nis;min s Fig. 4. Example of nucleated cracks along slip lines (black dots represent centres of grains).
ð5Þ
Model with a new micro-crack is then recalculated and used for next iteration where the whole process is repeated. In the beginning micro-cracks tend to occur scattered around the model and form preferably in larger grains that are favourably oriented and have higher shear stresses. But after a while existing micro-cracks start coalescing, causing local stress concentrations and amplifying the likelihood of new micro-cracks forming near already coalesced crack. When this crack grows to sufficient length, crack initiation is considered to be finished. The resulting macrocrack is then used as an initial crack for crack propagation. The sum of N is;min for all stages represents total crack initiation life Ni:
Ni ¼
n X
Nis;min
ð6Þ
i¼1
3. Numerical modelling 3.1. Modelling concepts Fig. 5. Example of two micro-cracks coalescing.
er than the required threshold 2k (gray solid line). This problem was solved using segmented micro-cracks. Fig. 6b–d shows the evolution of micro-crack nucleation using four segments along the slip band. Fig. 6a shows that the stress level on the leftmost s (black dotted line) surpasses the threshold stress segment Ds range and a micro-crack can occur there. Fig. 6b shows next iteration where a micro-crack was created on the first segment causing stress relaxation there and a stress increase on the second segment so that the stress range surpasses the threshold. Fig. 6c–d shows the next two iterations where a micro-crack progresses through third and fourth segment and finally forms along whole slip band. Since proposed algorithm for crack nucleation goes through multiple iterations, it is necessary to account for dislocation pile-
We decided to use commercially available software ABAQUS [8], as it enabled us to easily extend it to meet our demands by writing additional routines needed for model generation, adaptation and evaluation. The developed code that can be invoked through GUI, is written in Python language and uses ABAQUS Scripting Interface (ASI) [12,13]. ASI is an object-oriented extension library based on Python, for advanced pre- and post-processing tasks in ABAQUS. Since, in terms of fatigue, we only need to analyse a small portion of a structural element at the place where crack initiation is expected, we decided to use a process of sub-modelling in order to transfer the load of global-model to the boundaries of sub-model. Creating a sub-model is a two-step process and is often used to study in detail an area of interest, usually a region of high stress. First a global model needs to be created and analysed. Then a sub-model is created in place of interest. The boundary nodes of
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Fig. 6. Shear stress distribution along slip band.
sub-model are driven by macro-model solution. The benefits of sub-modelling are twofold. Firstly, it allows for shorter calculation times as only a part of whole model needs to be analysed. Secondly, a sub-model can be structured in a different way then global model. In presented example a sub-model is modelled with disordered microstructure of randomly oriented orthotropic crystal grains where global model is continuous and isotropic. Due to disparity of model properties at the boundary between global model and sub-model a sufficient buffer zone is needed, a many numerical errors of FEM analysis are present close to the boundaries. In order to asses the micro-crack nucleation within and microcrack coalescence between grains, we used an alternative model description as to what is usually used in ABAQUS. This approach utilises a concept of two-dimensional multi-material system [14] with two sets of components: a set of continua and a set of continuum interfaces (Fig. 7). In our example, a continuum represents a 2D sub-domain of sub-model, that has its own material orientation. A continuum interfaces define interaction between two continua and can exist between two grains of within one grain, if it has a nucleated micro-crack. In order to enable segmented micro-crack nucleation, continuum interface within a grain consists of multiple interface segments with different interaction properties. There are two kinds of interaction properties of continuum interfaces. Initially, the nodes of two continua are aligned at the
interface (similar to TIE command in ABAQUS) and have identical deformation and stress values. Upon crack nucleation within a grain or crack coalescence between grains the appropriate continuum interfaces are considered to fail, causing no interaction between continua through that interface. In our example, where micro-model is loaded with tension only stress field, this is sufficient. But if compression stresses occurred in sub-model, failed interfaces would require some form of contact interaction that would prevent protrusion of one continua into another. 3.2. Implementation The whole process of crack initiation was developed in Python using ABAQUS Scripting Language. In this respect the implementation has two sides. A Python side that controls the entire process and implements novelties described in chapter 2 and ABAQUS side that is used for pre-processing, post-processing and as a solver (Fig. 8). At the beginning a sub-model is generated according to input parameters (grain size, perturbation). In this respect a multitude of crystal grains is created using Voronoi tessellation. Each grain represents one continuum with unique material orientation, while the properties of boundary interface are drawn from macro-model displacement field. Also, a separate file with properties of all po-
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Fig. 7. Sub-model description by continuum decomposition.
recorded. In this manner created model is then used as an input for next iteration. The iteration loop is exited when one of two conditions is met. Eider a predetermined number of loading cycles has passed or a crack of sufficient size has formed so that we can consider crack initiation process finished. Steels are believed to have a fatigue limit at 2 106 cycles. Just to be in the safe side we chose 5 106 to be the maximum amount of cycles a simulation can run through. In cases where simulation exited after 5 106 cycles and no significant crack formation occured it was considered that the load is below fatigue limit. 4. Practical example
Fig. 8. Schematics of implementation in the ABAQUS context.
tential slip bands and slip segments for all grains is created, that holds information about existing micro-crack geometry and accumulated deformation energy for particular slip band segment. This model serves as an initial state for the following crack initiation procedure. Using ABAQUS/CAE side, the model is first automatically meshed and then calculated using ABAQUS Solver, which produces an output database of stress and displacement field. Shear stress field is used by a custom developed routine (Python side) to determinate where in sub-model next micro-crack will occur and how many loading cycles it takes. If a micro-crack occurs on an already existing slip band segment, the appropriate continuum interface is marked as failed. Otherwise, in case a micro-crack occurs in a grain that yet has no slip bands, that grain is split in two parts that have identical material orientation. The interface between these two parts represents a newly created slip band, with a predetermined number of segments where the appropriate segment is marked as failed. Also, all adjacent grains with existing micro-cracks are inspected to determinate if micro-crack coalescence can occur. If it does, the appropriate interface between those grains is also marked as failed. All relevant data regarding new model geometry and deformation energy accumulation for all slip band segments is
The presented model has been used for determination of crack initiation in the specimen made of high strength steel S960. Specimen dimensions were 200 mm 110 mm 5 mm (Fig. 2), with a circular hole of 40 mm. According to the static tensile test, the yield stress of material is Rp0,2 = 1026 MPa and the ultimate tensile stress is Rm = 1064 MPa, with 15% elongation at break. Specimen was manufactured with laser cutting which results in martensitic structure of thermally cut edge. Material parameters of martensitic layer, required for Tanaka–Mura model, were taken from literature [6]: shear modulus G = 118 GPa, specific fracture energy per unit area Ws = 2.0 kJ/m2, Poisson’s ratio m = 0.3, and frictional stress of dislocations on the slip plane k = 108 MPa. Micro-structural analysis was performed to determinate the type of material in heat affected zone (HAZ) and its thickness. With electron microscope imaging it was determined that the average grain size in HAZ is 20 lm and the typical distance between martensitic laths is 2 lm. Fig. 9 shows electron microscope image of investigated material, with some distinct grains shaded for better visualization. These properties were then incorporated in the numerical simulation. Since numerical model was directed at simulating fatigue properties of thermally cut steel, edge properties of test specimen were additionally inspected in terms of surface roughness and residual stresses. Surface roughness of cut edge was determined using a perthometer. Average arithmetic roughness height was found to be Ra = 5.0 lm and maximal roughness height was Rz = 37.6 lm. In simulation, the maximal roughness was modelled, as it can be expected that micro-cracks will likely form where the highest stress concentration exists. No actual measurements of residual stresses were performed, so we obtained appropriate data from other researchers [15]. Through analogy we assumed that maximal compressive stress amounts to 300 MPa at the cut edge and almost linearly drop to zero 0.3 mm from the edge. They achieve maximal tensile value 0.6 mm from cut edge and fade out asymptotically so
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Fig. 11. Micro-crack evaluation for load level 485 MPa.
Fig. 9. Electron microscope image of investigated material (a few grains are shaded).
that they can no longer be detected 2 mm from the cut edge. Residual stresses were applied to appropriate region in global model. Numerical simulation was performed on four stress levels: 600, 550, 500, and 485 MPa. Fig. 10 shows a few stages of micro-crack evolution for load level 600 MPa. It is evident that in the beginning micro-cracks form near the stress concentration due to surface roughness and where grain orientation is 45° with respect to load so that maximal shear stresses occur. After a while new microcracks nucleate near the existing micro-cracks and start coalescing into a larger macro-crack. When macro-crack reached sufficient length (0.3 mm) simulation was stopped. Fig. 11 shows shear
stress distribution and nucleated micro-cracks for load level 485 MPa after 5 106 cycles. Since no significant crack coalescence occurred it was concluded that crack initiation cannot be completed and that the load level is below fatigue limit. Fig. 12 shows the comparison between numerical and experimental results (detailed experimental procedure is described in [16]). It should be note that numerical results represent the number of stress cycles required for crack initiation (crack length about 0.3 mm), while experimental results represent the number of stress cycles required for final breakage of the specimen. As it is explained in chapter 1, there are not significant differences between them, because crack initiation period generally account for most of the service life, especially in high-cycle fatigue. Table 1 shows the effects of roughness and residual stresses on a number of initiation cycles for 600 MPa load. It is evident that two inspected features oppose each other. While
Fig. 10. Micro-crack evaluation for load level 600 MPa.
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cence is solved very conservatively and some method should be applied to evaluate the number of cycles needed for a crack to extend along grain boundary. Some preliminary testing showed that the number of segments on a slip band influences the rate at which micro-cracks nucleate. Perhaps all segments should not be treated as equally susceptible to micro-crack nucleation, as the proximity of grain boundaries may inhibit the rate at which micro-crack occurs. Fig. 12. Numerical and experimental results with appropriate S–N curve.
References Table 1 Comparing the effects of roughness and residual stresses on a number of initiation cycles for 600 MPa load.
No residual stresses With residual stresses
Smooth
Rough
238 000 296 000
194 000 216 000
compressive residual stresses retard crack initiation, edge roughness accelerates it. 5. Conclusion Crack initiation becomes increasingly important in HCF, as it can amount to more than 90% of components life-cycles. Initiation assessment is also very problematic as it is highly dependent on the minute features, like the microstructure and surface roughness. Using Tanaka–Mura approach to solving crack initiation period still leaves open the problem of micro-crack coalescence. Also, it doesn’t handle the problem of significant stress gradients caused by existing micro-cracks as it uses average stress along slip band. The paper presents a possible solution to this problem, by introducing segmented slip bands, where micro-cracks nucleate in multiple stages. Crack coalescence is solved by connecting two microcracks, if stresses between them surpass yield stress of material. A plug-in for ABAQUS package was created to handle these features. The proposed method shows a reasonably good correlation with experimental testing, but still has some deficiencies. Crack coales-
[1] Stephens RI, Fatemi A, Stephens RR, Fuchs HO. Metal fatigue in engineering. New York: John Wiley & Sons Inc.; 2001. [2] Podrug S, Jelaska D, Glodezˇ S. Influence of different load models on gear crack path shapes and fatigue lives. Fatigue Fract Eng Mater Struct 2008;31:327–39. [3] Miller KJ. The behaviour of short fatigue cracks and their initiation, parts I + II. Fatigue Fract Eng Mater Struct 1987;10:75–113. [4] Tryon RG, Cruse TA. A reliability-based model to predict scatter in fatigue crack nucleation life. Fatigue Fract Eng Mater Struct 1998;21:257–67. [5] Tanaka K, Mura T. A dislocation model for fatigue crack initiation. J Appl Mech 1981;48:97–103. [6] Brückner-Foit A, Huang X. Numerical simulation of micro-crack initiation of martensitic steel under fatigue loading. Int J Fatigue 2006;28:963–71. [7] Jezernik N, Glodezˇ S, Kramberger J. Evaluating microcrack initiation with a finite element model. In: CADAM 2008 proceedings; 2008. p. 25–6. [8] Abaqus 6.7. Theory manual.
Contents lists available at ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Numerical modelling of fatigue crack initiation of martensitic steel S. Glodezˇ a,*, N. Jezernik b, J. Kramberger b, T. Lassen c a
University of Maribor, Faculty of Natural Science and Mathematics, Koroška 160, 2000 Maribor, Slovenia University of Maribor, Faculty of Mechanical Engineering, Smetanova 17, 2000 Maribor, Slovenia c University of Agder, Faculty of Engineering and Science, Grooseveien 36, N-4876 Grimstad, Norway b
a r t i c l e
i n f o
Article history: Received 4 September 2009 Accepted 22 December 2009 Available online 2 February 2010 Keywords: Fatigue Crack initiation Numerical modelling
a b s t r a c t Numerical simulation of micro-crack initiation that is based on Tanaka–Mura micro-crack nucleation model is presented. Three improvements were added to this model. Firstly, multiple slip bands where micro-cracks may occur are used in each grain. Second improvement deals with micro-crack coalescence by extending existing micro-cracks along grain boundaries and connecting them into a macro-crack. The third improvement handles segmented micro-crack generation, where a micro-crack is not nucleated in one step like in Tanaka–Mura model, but is instead generated in multiple steps. Numerical simulation of crack-initiation was performed with ABAQUS, using a plug-in that was written specially for handling micro-crack nucleation and coalescence. Since numerical model was directed at simulating fatigue properties of thermally cut steel with martensitic structure, edge properties of specimen were additionally inspected in terms of micro-structural properties, surface roughness and residual stresses. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Current industrial design is heavily concerned with the highcycle fatigue resistance of mechanical components and structures. However, the complete process of fatigue failure of mechanical elements may be divided into the following stages [1,2]: (1) micro-crack nucleation; (2) short crack growth; (3) long crack growth; and (4) occurrence of final failure. In engineering applications the first two stages are usually termed as ‘‘crack initiation period”, while long crack growth is termed as ‘‘crack propagation period”. An exact definition of the transition from initiation to propagation period is usually not possible. However, the crack initiation period generally account for most of the service life, especially in high-cycle fatigue, Fig. 1. The total number of stress cycles N can than be determined from the number of stress cycles Ni required for the fatigue crack initiation and the number of stress cycles Np required for a crack to propagate from the initial to the critical crack length, when the final failure can be expected to occur (N = Ni + Np). This approach is mainly restricted to the crack initiation period, i.e. micro-crack nucleation and short crack growth. The behaviour of short cracks under fatigue loading differs substantially from the behaviour of long-cracks [3]. Existing research in micro-crack nucleation has shown for many materials that micro-cracks are ini-
* Corresponding author. Address: University of Maribor, Faculty of Natural Science and Mathematics, Koroška c. 160, SI-2000 Maribor, Slovenia. Tel.: +386 2 229 3752; fax: +386 2 251 8180. E-mail address: [email protected] (S. Glodezˇ). 0965-9978/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2010.01.002
tiated on slip bands of grains and stretch across whole grain [4]. The often proposed method to solve this problem is by using Tanaka–Mura model for micro-crack initiation [5]. Using this model, the number of stress cycles Nc required for the micro-crack nucleation can be determined as follows:
Nc ¼
8 G Ws
p ð1 mÞ d ðDs 2kÞ2
ð1Þ
Eq. (1) presumes that micro-cracks form along the slip band of grains, depending on slip band length d and average shear stress . Other material constants (shear modurange on the slip band Ds lus G, specific fracture energy per unit area Ws, Poisson’s ratio m, and frictional stress of dislocation on the slip plane k can be for examined material found in specialised literature (for martensite see Ref. [6]). Tanaka–Mura model still has two deficiencies that limit its use. Model deals only with micro-crack nucleation in separate grains and doesn’t deal with macro-crack coalescence from existing micro-cracks. Another problem is that this model uses average shear stress to determinate micro-crack nucleation. In our previous attempts to simulate micro-crack initiation [7], it often happened that an existing micro-crack significantly raised stresses of neighbouring grain, yet the average shear stress in this grain was still below the threshold needed for micro-crack nucleation, therefore no nucleation occurred. This problem becomes increasingly pronounced when using lower loads, that is in high-cycle fatigue regime. The main purpose of this study is to present a few improvements to the Tanaka–Mura model aimed at solving these two problems.
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(with no slip bands indicated), serve as a buffer zone to minimize the numerical mismatch at boundaries (top, right and bottom edge on the right image of Fig. 2) and do not partake in micro-crack evaluation. 2.1. Multiple slip bands
Fig. 1. Schematic representation of the service life of mechanical elements.
2. Physical model The problem of micro-crack nucleation and short crack growth is studied on a multi-scale model as shown in Fig. 2. Global model represents a full scale specimen subjected to chosen load level. The proposed micro-crack nucleation model simulates fatigue crack initiation in particular grains, therefore a complete microstructure needs to be modeled. Since it would be impossible to model a detailed microstructure on a global model (due to memory and computational constraints), we decided to use a sub-model in the place where the highest stresses are present (at the inner hole) and formation of cracks is expected. The size of sub-model was chosen to be 0.5 0.5 mm in size, as this would be enough to asses crack initiation period. Loading of a sub-model is implemented by applying stress distribution of global model to the boundary edges of submodel (top, right, and bottom edge on the right image of Fig. 2). Numerical simulation of crack-initiation was performed with ABAQUS [8,9], using a plug-in that was written specially for handling micro-crack nucleation and coalescence. Individual grains are simulated with randomly generated Voronoi tessellation and were given randomly oriented [10,11] orthotropic material properties of martensite [6]: C11 = C22 = C33 = 233 GPa, C12 = C13 = C23 = 135 GPa, C44 = C55 = C66 = 118 GPa. Considering the average grain size of 20 lm the whole model consists of approximately 400 individual grains. The right image in Fig. 2 shows such a model, with slip bands drawn through centres of grains. The surrounding grains
When simulating crack nucleation in particular grain, multiple slip bands were evaluated. For each grain there was one slip band created through the centre of the grain and additional slip bands were created with a 2 lm offset from the first (Fig. 3). Fig. 4 shows a case where the first micro-crack was nucleated in the left grain and influenced its neighbouring grain on the right, so that a micro-crack in it nucleated far from the centre (black dot), close to the first micro-crack. The proximity of nucleated micro-cracks also enables easier micro-crack coalescence than in a model where micro-cracks are allowed to go through grain centres only [6]. 2.2. Crack coalescence A conservative approach was taken to enable crack coalescence. Whenever a new micro-crack was introduced in the model all possible combination of micro-cracks were analysed to see if the average stress on a straight line between two micro-cracks surpasses yield stress of the material. If this was the case, a crack was extended through that line, effectively transforming two microcracks into a single larger crack (Fig. 5). Note significant stress relaxation in the area between the cracks with respect to Fig. 4. When calculating cycles required for crack initiation, no cycles were attributed to crack coalescences (it is simulated as being instantaneous), so the total cycles of crack initiation equal the sum of cycles needed for each micro-crack to nucleate. 2.3. Segmented micro-cracks Fig. 6a shows shear stress range Ds (black solid line) along a slip band of a grain, whose neighbouring grain already has a microcrack. This causes a significant stress concentration on the side of the slip band that is adjacent to the existing micro-crack. Using Tanaka–Mura model would not cause a micro-crack to nucleate in (gray dotted line) is lowthis case, as average shear stress range Ds
Fig. 2. Global model and sub-model of the specimen.
S. Glodezˇ et al. / Advances in Engineering Software 41 (2010) 823–829
825
up by keeping track of the damage that was accumulated over previous stages. In the beginning, total deformation energy needed for micro-crack nucleation is calculated for all segments of slip bands of every grain, using following equation:
U th s ¼
8 Ws ds
ð2Þ
where U th s represents a threshold amount of total deformation energy that is required for micro-crack nucleation on a particular segment (index s), ds is the length of that segment and Ws is the specific fracture energy per unit area. Then in each stage the amount of deformation energy acquired per cycle for each segment is calculated: Fig. 3. A crystal grain with multiple slip bands.
DU s ¼
pð1 mÞðDss 2 kÞ2 G
ð3Þ
s and DU s represent average shear stress range and where Ds amount of deformation energy that is accumulated in one loading cycle on a segment s. The number of cycles N is , required for micro-crack nucleation on each segment, is calculated with the following equation:
Nis ¼
i U th s Us DU s
ð4Þ
where U is represents already accumulated damage in each segment of stage i. Then a micro-crack is created on the segment with the smallest Nis . For other segments the accumulated damage for next is increased. stage U iþ1 s
U iþ1 ¼ U is þ DU s Nis;min s Fig. 4. Example of nucleated cracks along slip lines (black dots represent centres of grains).
ð5Þ
Model with a new micro-crack is then recalculated and used for next iteration where the whole process is repeated. In the beginning micro-cracks tend to occur scattered around the model and form preferably in larger grains that are favourably oriented and have higher shear stresses. But after a while existing micro-cracks start coalescing, causing local stress concentrations and amplifying the likelihood of new micro-cracks forming near already coalesced crack. When this crack grows to sufficient length, crack initiation is considered to be finished. The resulting macrocrack is then used as an initial crack for crack propagation. The sum of N is;min for all stages represents total crack initiation life Ni:
Ni ¼
n X
Nis;min
ð6Þ
i¼1
3. Numerical modelling 3.1. Modelling concepts Fig. 5. Example of two micro-cracks coalescing.
er than the required threshold 2k (gray solid line). This problem was solved using segmented micro-cracks. Fig. 6b–d shows the evolution of micro-crack nucleation using four segments along the slip band. Fig. 6a shows that the stress level on the leftmost s (black dotted line) surpasses the threshold stress segment Ds range and a micro-crack can occur there. Fig. 6b shows next iteration where a micro-crack was created on the first segment causing stress relaxation there and a stress increase on the second segment so that the stress range surpasses the threshold. Fig. 6c–d shows the next two iterations where a micro-crack progresses through third and fourth segment and finally forms along whole slip band. Since proposed algorithm for crack nucleation goes through multiple iterations, it is necessary to account for dislocation pile-
We decided to use commercially available software ABAQUS [8], as it enabled us to easily extend it to meet our demands by writing additional routines needed for model generation, adaptation and evaluation. The developed code that can be invoked through GUI, is written in Python language and uses ABAQUS Scripting Interface (ASI) [12,13]. ASI is an object-oriented extension library based on Python, for advanced pre- and post-processing tasks in ABAQUS. Since, in terms of fatigue, we only need to analyse a small portion of a structural element at the place where crack initiation is expected, we decided to use a process of sub-modelling in order to transfer the load of global-model to the boundaries of sub-model. Creating a sub-model is a two-step process and is often used to study in detail an area of interest, usually a region of high stress. First a global model needs to be created and analysed. Then a sub-model is created in place of interest. The boundary nodes of
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Fig. 6. Shear stress distribution along slip band.
sub-model are driven by macro-model solution. The benefits of sub-modelling are twofold. Firstly, it allows for shorter calculation times as only a part of whole model needs to be analysed. Secondly, a sub-model can be structured in a different way then global model. In presented example a sub-model is modelled with disordered microstructure of randomly oriented orthotropic crystal grains where global model is continuous and isotropic. Due to disparity of model properties at the boundary between global model and sub-model a sufficient buffer zone is needed, a many numerical errors of FEM analysis are present close to the boundaries. In order to asses the micro-crack nucleation within and microcrack coalescence between grains, we used an alternative model description as to what is usually used in ABAQUS. This approach utilises a concept of two-dimensional multi-material system [14] with two sets of components: a set of continua and a set of continuum interfaces (Fig. 7). In our example, a continuum represents a 2D sub-domain of sub-model, that has its own material orientation. A continuum interfaces define interaction between two continua and can exist between two grains of within one grain, if it has a nucleated micro-crack. In order to enable segmented micro-crack nucleation, continuum interface within a grain consists of multiple interface segments with different interaction properties. There are two kinds of interaction properties of continuum interfaces. Initially, the nodes of two continua are aligned at the
interface (similar to TIE command in ABAQUS) and have identical deformation and stress values. Upon crack nucleation within a grain or crack coalescence between grains the appropriate continuum interfaces are considered to fail, causing no interaction between continua through that interface. In our example, where micro-model is loaded with tension only stress field, this is sufficient. But if compression stresses occurred in sub-model, failed interfaces would require some form of contact interaction that would prevent protrusion of one continua into another. 3.2. Implementation The whole process of crack initiation was developed in Python using ABAQUS Scripting Language. In this respect the implementation has two sides. A Python side that controls the entire process and implements novelties described in chapter 2 and ABAQUS side that is used for pre-processing, post-processing and as a solver (Fig. 8). At the beginning a sub-model is generated according to input parameters (grain size, perturbation). In this respect a multitude of crystal grains is created using Voronoi tessellation. Each grain represents one continuum with unique material orientation, while the properties of boundary interface are drawn from macro-model displacement field. Also, a separate file with properties of all po-
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Fig. 7. Sub-model description by continuum decomposition.
recorded. In this manner created model is then used as an input for next iteration. The iteration loop is exited when one of two conditions is met. Eider a predetermined number of loading cycles has passed or a crack of sufficient size has formed so that we can consider crack initiation process finished. Steels are believed to have a fatigue limit at 2 106 cycles. Just to be in the safe side we chose 5 106 to be the maximum amount of cycles a simulation can run through. In cases where simulation exited after 5 106 cycles and no significant crack formation occured it was considered that the load is below fatigue limit. 4. Practical example
Fig. 8. Schematics of implementation in the ABAQUS context.
tential slip bands and slip segments for all grains is created, that holds information about existing micro-crack geometry and accumulated deformation energy for particular slip band segment. This model serves as an initial state for the following crack initiation procedure. Using ABAQUS/CAE side, the model is first automatically meshed and then calculated using ABAQUS Solver, which produces an output database of stress and displacement field. Shear stress field is used by a custom developed routine (Python side) to determinate where in sub-model next micro-crack will occur and how many loading cycles it takes. If a micro-crack occurs on an already existing slip band segment, the appropriate continuum interface is marked as failed. Otherwise, in case a micro-crack occurs in a grain that yet has no slip bands, that grain is split in two parts that have identical material orientation. The interface between these two parts represents a newly created slip band, with a predetermined number of segments where the appropriate segment is marked as failed. Also, all adjacent grains with existing micro-cracks are inspected to determinate if micro-crack coalescence can occur. If it does, the appropriate interface between those grains is also marked as failed. All relevant data regarding new model geometry and deformation energy accumulation for all slip band segments is
The presented model has been used for determination of crack initiation in the specimen made of high strength steel S960. Specimen dimensions were 200 mm 110 mm 5 mm (Fig. 2), with a circular hole of 40 mm. According to the static tensile test, the yield stress of material is Rp0,2 = 1026 MPa and the ultimate tensile stress is Rm = 1064 MPa, with 15% elongation at break. Specimen was manufactured with laser cutting which results in martensitic structure of thermally cut edge. Material parameters of martensitic layer, required for Tanaka–Mura model, were taken from literature [6]: shear modulus G = 118 GPa, specific fracture energy per unit area Ws = 2.0 kJ/m2, Poisson’s ratio m = 0.3, and frictional stress of dislocations on the slip plane k = 108 MPa. Micro-structural analysis was performed to determinate the type of material in heat affected zone (HAZ) and its thickness. With electron microscope imaging it was determined that the average grain size in HAZ is 20 lm and the typical distance between martensitic laths is 2 lm. Fig. 9 shows electron microscope image of investigated material, with some distinct grains shaded for better visualization. These properties were then incorporated in the numerical simulation. Since numerical model was directed at simulating fatigue properties of thermally cut steel, edge properties of test specimen were additionally inspected in terms of surface roughness and residual stresses. Surface roughness of cut edge was determined using a perthometer. Average arithmetic roughness height was found to be Ra = 5.0 lm and maximal roughness height was Rz = 37.6 lm. In simulation, the maximal roughness was modelled, as it can be expected that micro-cracks will likely form where the highest stress concentration exists. No actual measurements of residual stresses were performed, so we obtained appropriate data from other researchers [15]. Through analogy we assumed that maximal compressive stress amounts to 300 MPa at the cut edge and almost linearly drop to zero 0.3 mm from the edge. They achieve maximal tensile value 0.6 mm from cut edge and fade out asymptotically so
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Fig. 11. Micro-crack evaluation for load level 485 MPa.
Fig. 9. Electron microscope image of investigated material (a few grains are shaded).
that they can no longer be detected 2 mm from the cut edge. Residual stresses were applied to appropriate region in global model. Numerical simulation was performed on four stress levels: 600, 550, 500, and 485 MPa. Fig. 10 shows a few stages of micro-crack evolution for load level 600 MPa. It is evident that in the beginning micro-cracks form near the stress concentration due to surface roughness and where grain orientation is 45° with respect to load so that maximal shear stresses occur. After a while new microcracks nucleate near the existing micro-cracks and start coalescing into a larger macro-crack. When macro-crack reached sufficient length (0.3 mm) simulation was stopped. Fig. 11 shows shear
stress distribution and nucleated micro-cracks for load level 485 MPa after 5 106 cycles. Since no significant crack coalescence occurred it was concluded that crack initiation cannot be completed and that the load level is below fatigue limit. Fig. 12 shows the comparison between numerical and experimental results (detailed experimental procedure is described in [16]). It should be note that numerical results represent the number of stress cycles required for crack initiation (crack length about 0.3 mm), while experimental results represent the number of stress cycles required for final breakage of the specimen. As it is explained in chapter 1, there are not significant differences between them, because crack initiation period generally account for most of the service life, especially in high-cycle fatigue. Table 1 shows the effects of roughness and residual stresses on a number of initiation cycles for 600 MPa load. It is evident that two inspected features oppose each other. While
Fig. 10. Micro-crack evaluation for load level 600 MPa.
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cence is solved very conservatively and some method should be applied to evaluate the number of cycles needed for a crack to extend along grain boundary. Some preliminary testing showed that the number of segments on a slip band influences the rate at which micro-cracks nucleate. Perhaps all segments should not be treated as equally susceptible to micro-crack nucleation, as the proximity of grain boundaries may inhibit the rate at which micro-crack occurs. Fig. 12. Numerical and experimental results with appropriate S–N curve.
References Table 1 Comparing the effects of roughness and residual stresses on a number of initiation cycles for 600 MPa load.
No residual stresses With residual stresses
Smooth
Rough
238 000 296 000
194 000 216 000
compressive residual stresses retard crack initiation, edge roughness accelerates it. 5. Conclusion Crack initiation becomes increasingly important in HCF, as it can amount to more than 90% of components life-cycles. Initiation assessment is also very problematic as it is highly dependent on the minute features, like the microstructure and surface roughness. Using Tanaka–Mura approach to solving crack initiation period still leaves open the problem of micro-crack coalescence. Also, it doesn’t handle the problem of significant stress gradients caused by existing micro-cracks as it uses average stress along slip band. The paper presents a possible solution to this problem, by introducing segmented slip bands, where micro-cracks nucleate in multiple stages. Crack coalescence is solved by connecting two microcracks, if stresses between them surpass yield stress of material. A plug-in for ABAQUS package was created to handle these features. The proposed method shows a reasonably good correlation with experimental testing, but still has some deficiencies. Crack coales-
[1] Stephens RI, Fatemi A, Stephens RR, Fuchs HO. Metal fatigue in engineering. New York: John Wiley & Sons Inc.; 2001. [2] Podrug S, Jelaska D, Glodezˇ S. Influence of different load models on gear crack path shapes and fatigue lives. Fatigue Fract Eng Mater Struct 2008;31:327–39. [3] Miller KJ. The behaviour of short fatigue cracks and their initiation, parts I + II. Fatigue Fract Eng Mater Struct 1987;10:75–113. [4] Tryon RG, Cruse TA. A reliability-based model to predict scatter in fatigue crack nucleation life. Fatigue Fract Eng Mater Struct 1998;21:257–67. [5] Tanaka K, Mura T. A dislocation model for fatigue crack initiation. J Appl Mech 1981;48:97–103. [6] Brückner-Foit A, Huang X. Numerical simulation of micro-crack initiation of martensitic steel under fatigue loading. Int J Fatigue 2006;28:963–71. [7] Jezernik N, Glodezˇ S, Kramberger J. Evaluating microcrack initiation with a finite element model. In: CADAM 2008 proceedings; 2008. p. 25–6. [8] Abaqus 6.7. Theory manual.