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Fracture toughness of layered structures: Embrittlement due to confinement of plasticity
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Engineering Fracture Mechanics 75 (2008) 3743–3754 www.elsevier.com/locate/engfracmech
Fracture toughness of layered structures: Embrittlement due to confinement of plasticity Nils C. Broedling a, Alexander Hartmaier b,*, Huajian Gao c a
b
Max Planck Institute for Metals Research, Heisenbergstrasse 3, 71735 Stuttgart, Germany Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg, Department of Materials Science and Engineering, Martensstrasse 5, 91058 Erlangen, Germany c Division of Engineering, Brown University, Providence, RI 02912, USA Received 4 August 2007; received in revised form 23 October 2007; accepted 25 October 2007 Available online 4 November 2007
Abstract The fracture toughness of a layered composite material is analyzed employing a combined two dimensional dislocation dynamics (DD)–cohesive zone (CZ) model. The fracture mechanism of an elastic–plastic (ductile) material sandwiched within purely elastic layers approaches ideally brittle behaviour with decreasing layer thickness. We investigate the influence of different constitutive parameters concerning dislocation plasticity as well as the effect of cohesive strength of the ductile material on the scaling of fracture toughness with layer thickness. For a constant layer thickness, the results of the numerical model are consistent with the expectation that fracture toughness decreases with increasing yield strength, but increases with the cohesive strength of the material. The scaling behaviour of the fracture toughness with layer thickness depends on these material parameters, but also on the dislocation microstructure in the vicinity of the crack tip. Strain localization due to easy dislocation generation right at the crack tip improves toughness in thin layers and leads to a jumplike increase of fracture toughness with layer thickness. However, the fracture toughness for films that are thick enough to exhibit bulk behaviour proves to be higher when the distribution of dislocations is more homogeneous, because in this case the crack grows in a stable fashion over some distance. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Fracture; Dislocation dynamics; Simulation; Layered structures
1. Introduction Plasticity in geometrically confined layered materials is studied on the basis of a combined dislocation dynamics (DD) and cohesive zone (CZ) method in two dimensions, the formulation of which has been published previously [1]. The model is employed to investigate the interaction of a ductile/elastic interface with a crack and to derive the thickness controlled fracture toughness of a composite layer material within the
*
Corresponding author. Tel.: +49 9131 85 27507; fax: +49 9131 85 27504. E-mail address: [email protected] (A. Hartmaier).
0013-7944/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2007.10.014
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limitations of a two dimensional dislocation representation. The work aims at answering the question in which thickness regime a pre-cracked semi-brittle layer material responds in an ideal brittle manner and how the transition to bulk behaviour occurs. DD simulations describe plastic deformation based on the nucleation, motion and annihilation of discrete dislocations embedded in an elastic continuum. By following the motion of individual dislocations, the dynamic evolution of the microstructure is tracked on time and length scales of fractions of seconds and micrometers. Despite their inherent limitations, two dimensional models have been shown to capture the most important features of plastic deformation in situations of geometrical confinement of plastic zones [1–10]. Therefore they have been used to investigate many features of plasticity in micro- and nano-structured materials where classical continuum plasticity is not applicable. From the modeling point of view, it is rather straightforward to incorporate preferable slip planes, obstacles and to model regions of large plastic deformation or even strain localization. Therefore, DD models are very suitable tools to analyze plasticity in nanostructured materials and to relate macroscopic quantities such as fracture toughness to underlying material lengths scales and microstructures. In plane strain two dimensional (2D) DD models, a dislocation is represented by a point singularity in a linear elastic body. The problem of a finite body with dislocations is formulated by superimposing the singular solutions for the elastic fields, known for a single dislocation in the infinite space, with an auxiliary solution that corrects for the boundary conditions. For the auxiliary solution, a boundary value problem has to be evaluated numerically. A number of problems concerning micro scale plastic deformation has been investigated based on the 2D-DD method developed Van der Giessen et al. [2]. In this work, a technique is introduced where the boundary value problem (BVP) of discrete dislocations is transformed into a weak formulation and then solved within a finite element (FEM) approach. In later work [3,4] a cohesive surface formulation was implemented in the DD method to study fracture accompanied by plastic deformation. A similar approach was used by Chng et al. [5,6] to study interfacial fracture and delamination of thin films. Other workers followed alternative approaches based on Green’s functions to incorporate the evolution of a dislocation microstructure under different boundary conditions into DD models. For example, a procedure was developed to evaluate contact stresses emerging in the wake region of growing fatigue cracks [7], similar methods were used to study nanoindentation [8], DD simulations were performed to study constrained diffusional creep in polycrystalline thin films [9,10]. In the present paper, a combined DD/CZ model is used to investigate the transition behaviour of the fracture toughness of a semi-brittle layered material. The problem formulation and the method for its solution have already been described in [1]. The model has been developed to study geometrical confinement effects on fracture in ductile layers. An analytical analysis of the problem based on certain simplifying assumptions is given by Hsia et al. [11]. In this work, the effect of pile-up formation at dislocation barriers on fracture in a metal layer is investigated. The analysis is based on the elastic energy balance and shows that the fracture toughness increases proportional to the square-root of layer thickness. After briefly revisiting the numerical method used in the present work, the results of a detailed parametric study on the factors influencing plasticity and fracture toughness of the sandwich structure are presented. From these results, finally, the dominating factors defining the fracture toughness are derived. 2. Dislocation dynamics–cohesive zone model Fig. 1 shows a plane strain mode I crack propagating through a thin elastic–plastic (ductile) layer embedded in an elastic matrix. Shown is only one near-tip-region of a slit-like crack that is chosen so long that its other crack tip does not interfere with that under consideration here. Ahead of the crack tip a cohesive zone is situated in the projection of the crack line. The cohesive zone is characterized by a traction–separation law with theoretical strength rth and the cohesive energy C. The plastic layer and the matrix can have different elastic properties. Dislocations nucleate at randomly placed Frank–Read like sources within the plastic layer. The intrinsic fracture toughness of the material (i.e. its fracture toughness in the absence of plasticity) is specified by the theoretical strength rth and the cohesive energy C. The plastic deformation around the crack tip develops from a distribution of dislocation sources with density qsrc and strength rsrc. The strength of the source is the critical resolved shear stress at which the sources operate and thus corresponds to the yield
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KI
Blunting
h ρ src , σ src , B
σ th , Γ
Fracture 2D Frank-Read Source
Shielding
Fig. 1. Schematics of the model problem. An initial crack is located within a ductile layer. The layer thickness h is the characteristic length scale which controls the fracture behaviour. See text for notations.
strength of the material. The 2D analogon of a dislocation loop is a dislocation dipole. Hence, to simulate a Frank–Read source we place two test dislocations with anti-parallel Burgers vector at the source position and calculate the total force on the dislocations. If this force is positive, i.e. it drives the dislocation dipole apart, the dislocation loop emanating from a Frank–Read source would also expand and thus both dislocations are nucleated. If the dipole collapses under the acting shear stress or by the mutual attraction of the dislocations, a Frank–Read source at this position could not have generated a dislocation loop, and dislocation nucleation is disregarded. The separation of the dislocations in the dipole is calculated such that dislocation start to nucleate if the local resolved shear stress exceeds rsrc. To take into account that a Frank–Read source needs a finite time to generate a dislocation loop, we require that the force on both test dislocations be positive over a given initiation time interval. During the simulation a crack opening tensile stress is applied giving rise to a mode I stress intensity factor K1 I , which is calculated by a J-integral (see [1]). The development of a plastic zone shields the crack tip from the applied load. Dislocation sources subject to the high stresses near the crack tip generate a large number of dislocations and as long as the plastic strain rate at the crack tip is large enough to keep the load on the crack tip below the critical value, the crack is stable. At some stage during the loading the crack starts to propagate. In particular at the beginning of this phase crack propagation is not always critical, but the crack may be stopped again due to groups of dislocations that locally provide a strong elastic shielding. Finally the crack propagates throughout the domain and the applied load at this stage defines the fracture toughness. A cohesive zone segment of 2000 nm ahead of the initial crack tip is discretized with a high resolution of about 8 nm node spacing. This is the predefined maximum distance of allowed crack advance which was found to be sufficient to resolve the stable crack growth prior to ultimate failure. In all simulations with a crack tip source, the stable crack advance occurred within a region of less than 1200 nm. The energy dissipation due to the breaking of atomic bonds defines the lower bound for fracture toughness in absence of any dislocation activity, the intrinsic fracture toughness kIc that corresponds to the Griffith energy. If dislocations are nucleated and provide elastic shielding as well as dissipation of mechanical energy, the fracture toughness increases. For the class of semi-brittle materials considered here the fracture toughness of the bulk material is determined by the competition between the loading rate and the expansion rate of the plastic zone, and sets an upper bound for the fracture toughness. In the case of a thin layer investigated here, the geometrical confinement limits the size of plastic zone. In this case dislocations pile up at the interface to the elastic matrix resulting in a back stress that eventually prevents further dislocation nucleation in the crack tip-region. The critical layer thickness ^ h at which the crack cannot be shielded sufficiently and the material becomes brittle is investigated by the DD simulation. 3. Boundary value problem and constitutive rules In the fully deterministic discrete dislocation model that is considered here, the driving force for dislocation motion is assumed to be completely described within the framework of elasticity. Within this mesoscopic
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approach, the microstructure of a solid and the internal elastic fields are associated with a number of discrete defects such as point defects, dislocations and cracks. In response to the thermodynamic force acting on each individual defect the entire microstructure evolves in time and space. To evaluate the interaction between dislocations and crack tip the corresponding mechanical boundary value problem needs to be formulated and solved numerically. The evolution of the microstructure is computed from a chosen initial configuration based on an explicit time integration scheme by successively updating the dislocation position and the displacements in the cohesive zone. The original dislocation-boundary value problem for a semi-infinite body with embedded dislocations and its decomposition is shown in Fig. 2. The decomposition leads to an unbounded problem with homogenous fields, an unbounded problem with singular fields, and a corrective problem with the image fields. In order to reduce the computational costs the model problem is assumed to be mirror-symmetric applying the corresponding boundary conditions at the crack line. The crack is modeled by distributing virtual dislocations along the crack line. Normal and tangential tractions are related to the virtual dislocation distribution in terms of boundary integrals. The system of boundary integral equations is evaluated numerically using the boundary collocation method [1]. The constitutive rules that account for material separation, dislocation motion and nucleation are described next. Material separation near the crack tip is modeled with a cohesive zone that specifies the cohesive strength and the cohesive energy of the ductile material corresponding to the intrinsic fracture toughness or Griffith energy, respectively. The traction separation law, relating the cohesive traction N to the crack opening D, is of trapezoidal shape [12], see Fig. 3. The fracture (Griffith) energy is defined as C ¼ 0:5rth Dc ð1 d1 þ d2 Þ;
ð1Þ
where rth is the cohesive strength, Dc is the crack opening separation above which the cohesive interaction vanishes and d1, d2 are the shape parameters that define the corners of the trapezoid. The corresponding intrinsic plane strain fracture toughness for a purely elastic material is obtained as rffiffiffiffiffiffiffiffiffiffiffiffiffi EC k Ic ¼ ; ð2Þ 1 m2 where C is the cohesive energy, E is the Young’s modulus and m is the Poisson’s ratio. In the presence of plastic deformation in the vicinity of the loaded crack tip, higher fracture toughness values are expected. The Peach–Koehler force (per unit length) which drives the dislocation in the slip direction is expressed as fi ¼ eij3 rjk bk ;
ð3Þ
where eijk is the permutation tensor, bk is the Burgers vector with norm b = 0.25 nm in the examples presented later. The velocity of dislocation (k) is assumed to be in the overdamped viscous glide regime and follows the form
Fig. 2. The boundary value problem of a semi-infinite body B with boundary oB and dislocations. Part of oB with normal vector n is traction-free and the other part is subjected to cohesive tractions N(u). The decomposition leads to an unbounded problem with homogenous fields ()1, an unbounded problem with singular fields ð~Þ, and a corrective problem with the image fields ð^Þ.
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N (Δ)
σ th 1
δ1
δ2
1
Δ Δc
Fig. 3. Traction separation law (N-D-law) with characteristic parameters including the theoretical strength rth and the critical crack tip opening displacement Dc. d1 and d2 are shape parameters giving the fracture energy C.
ðkÞ
vgd ¼
1 ðkÞ f ; B
ð4Þ
where the drag coefficient B = 104 GPa ns expresses the lattice resistance to dislocation glide. Two slip planes at angles of a1 = 45° and a2 = 45° with respect to the crack line are considered. Initially a randomly distributed source configuration is generated keeping a minimum distance of 10b between neighboring slip planes that are associated with the sources. At each time step, a nucleation criterion is evaluated by considering the stability of a test dislocation dipole with the average nucleation distance, as described above. For the near-field dislocation–dislocation interaction, a critical distance of 8b is used as an onset radius below which formations of obstacles or dislocation annihilation is triggered [13]. Obstacle formation occurs when two dislocations approach on different intersecting slip planes. When their distance falls below the critical distance for dislocation–dislocation interaction, their motion is stopped. Obstacle formation leads to a macroscopic hardening effect in the material. The obstacles act as barriers to the glide of further dislocations such that dislocation pile-ups form behind the two crossing dislocations, exerting an increasing driving force on them. An obstacle strength is assumed to be about 50% of the materials cohesive strength. This quantity controls the highest possible magnitude of pile-up stresses within the bulk material due to the formation of locks. Two locking dislocations are disengaged from each other when the driving force acting on them exceeds the obstacle strength. If two dislocations of opposite Burgers vectors are found to be approaching each other on parallel slip planes within the critical distance, they annihilate and are subsequently removed from the simulation. When a single dislocation travels across the crack line it is mirrored, which means that the dislocation is transferred to a slip plane of the alternative type at the same location. This consequently requires that its Burgers vector be rotated accordingly. Thus its shielding effect with respect to the crack tip changes the sign, which in most situations is accompanied by an update of the system matrix. Oscillations by successive back and forth mirroring of a dislocation must be avoided since that could tremendously slow down the solution algorithm. Hence, a driving force that pushes the dislocation across the crack line is required to be present over the initiation time period before mirroring is actually performed. Furthermore, for the dislocationboundary interaction, a onset distance of 40b is considered. The process region has a dimension of 2000 nm in the direction of the crack line. In this region, the moving dislocations are expected to interact strongly with the crack. They may be absorbed if they run into the traction-free crack surface or mirrored where the material is still coherent. 4. Results The objective of this work is to carry out parametric studies over a wide parameter range and to examine the characteristics of this generic fracture problem. The results obtained are presented and discussed in two parts. First, the influence of the source distribution on the toughness–thickness dependency is shown. Second, the correlation of the toughness–thickness dependency with cohesive strength and source strength indicated by the results will be pointed out. The source strengths, equivalent to the yield stresses of the model materials, are within the range of 0.05 GPa–1.5 GPa. The cohesive strengths are varied within the range of 0.8 GPa–
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2.4 GPa. The setup of the geometry follows Fig. 1. The simulations start at zero applied stress with an initially dislocation free material and the load is increased monotonically with a constant loading rate of pffiffiffiffiffiffiffi 3 K_ 1 ¼ 10 GPa nm =ns. The material is taken to be elastically homogeneous with shear modulus I G = 26 GPa and Poisson ratio m = 0.33. 4.1. Influence of the microstructural surrounding In this section the influence of the local source distribution as well as that of the source density within the bulk material are investigated. For fracture of thin layers the availability of sources close to the crack tip rather than the overall source density becomes important. In Fig. 4 simulation results for four 1 lm thick layer materials A, B, C and D with different source distributions are plotted. The other material parameters are chosen as rsrc = 0.05 GPa and rth = 0.8 GPa. Fig. 4a and b show clearly the effect of dislocation emission right at the crack tip. The source density of the bulk material in both simulations has been chosen to be 80 lm2 with an overall identical distribution, but in one case (shown in Fig. 4b) a source with a slip plane angle of 45° has been placed at position (x, y) = (40 nm, 50 nm). This crack tip dislocation source is a bulk source that is placed close to the crack tip. The anti-shielding dislocation of a nucleated dipole is absorbed by the free crack surface right after its nucleation. Note, that the serrated shape of the local stress intensity factor in Fig. 4b is a result of the strong dislocation–interface interaction and reflects single nucleation events of the near crack tip source. The consequence of the existence of a crack tip source is that the number of nucleated dislocations has been doubled. The formation of a compact dislocation pile-up of 20 dislocations exerts a strong shielding effect and causes an increase in fracture toughness of about 20%. The results for a higher bulk source density of 120 lm2 are shown in Fig. 4c and for only one crack tip source in Fig. 4d. Another observation in our simulations is that due to the formation of a dislocation array at the crack tip the fracture propagation mechanism depends on the film thickness. For thicker films, in most cases the crack temporally arrests due to the elastic interaction of the growing crack with an existing group of dislocations ahead of the crack. In very thin layers, however, the dislocation microstructure degenerates to a single dislocation array on a single slip plane at the crack tip and, correspondingly, no crack arrest is observed. The resulting fracture toughnesses for the different source distributions are shown in Fig. 5 as a function of film thickness. It is seen that for thicknesses below 1000 nm the crack tip source is dominant, because the curves for the materials with the crack tip source (B, C and D) do not deviate from each other in this regime. Above this thickness the fracture toughness of the material D with only on source close to the crack tip remains approximately constant. The curves of materials B and C start to deviate from each other at a thickness of 2000 nm. Interestingly the material with the higher source density gives the lower fracture toughness, which is explained by the higher dislocation density in this case, hindering the free expansion of the shielding dislocations emanating from the near crack source. Thus the material with a higher source density exhibits a more pronounced work hardening. The comparison of material A and B shows that the bulk fracture toughness of a material with a bulk source density of 80 lm2 increases by about 15% when additionally a crack tip source is considered. The influence of the bulk sources on the fracture toughness of a layered material is noticeable only above a thickness of 1000 nm. The bulk toughness of a material with a crack tip source increases by about 20% (10%) when in addition to the crack tip source a distribution of bulk sources with a density of 80 lm2 (120 lm2) is considered. 4.2. Influence of source strength and cohesive strength We have conducted extensive parameter studies to investigate the effect of cohesive strength, intrinsic fracture toughness as well as source strength on the resulting fracture toughness. Fig. 6 shows toughness–thickness curves for different sets of parameters for a layer material with qsrc = 80 lm2 with crack tip source. Fig. 6a–d reveal the effect of cohesive strength and intrinsic fracture toughness. Both parameters enhance the macroscopic fracture toughness significantly. Fig. 6c shows simulation results for two different intrinsic fracture toughnesses for two different source strengths rsrc. A 40% increase of the intrinsic fracture toughness enhances the bulk toughness by about 10%.
N.C. Broedling et al. / Engineering Fracture Mechanics 75 (2008) 3743–3754 Shielding
a
b kI / kIc
kI / kIc
2
1
0
0
0.5
1
1.5
Shielding 2
1
0
2
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0
0.5
KI / kIc
1
1.5
2
KI / kIc
Opening Stress Component
Opening Stress Component
1.0 0.9 0.8 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.1 0.0
total 24 absorbed 2
mirrored 3 dissociated obstacles 138
total 58 absorbed 20
Shielding
c
d
kI / kIc
2
kI / kIc
1.0 0.9 0.8 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.1 0.0
1
0 0
0.5
1
1.5
Shielding 2
1
0
2
mirrored 1 dissociated obstacles 224
0
0.5
1
1.5
Opening Stress Component
Opening Stress Component
1.0 0.9 0.8 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.1 0.0
total 53 absorbed 23
mirrored 4 dissociated obstacles 5
2
KI / kIc
KI / kIc
1.0 0.9 0.8 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.1 0.0
total 21 absorbed 19
mirrored 0 dissociated obstacles 0
Fig. 4. Comparison of four different source distributions for a 1 lm thick layer. Red and green dots denote dislocation sources associated with the two different slip systems. Shown is shielding and final microstructure for each distribution together with the corresponding variation of the opening stress component: (a) qsrc = 80 lm2 without crack tip source, (b) qsrc = 80 lm2 with crack tip source, (c) qsrc = 120 lm2 with crack tip source, (d) only a single crack tip source.
We define the critical layer thickness to be that thickness where the fracture toughness of the layer material reaches 95% of the materials bulk toughness, see Fig. 7a. The bulk toughness as well as the critical layer
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1.8
K Ic /k Ic
1.6
1.4
ρsrc = 80, no CT source ρsrc = 80, with CT source ρsrc = 120, with CT source
1.2
only CT source
1
0
5000
10000
h [nm]
Fig. 5. Normalized fracture toughness as a function of layer thickness, cf. plots of four different source distributions for a 1 lm thick layer in Fig. 4. The lines connecting the data points are mere visual guides.
b 2.4 2.2 2 1.8 1.6 1.4 σ th = 0.8, kIc= 0.37 σ th = 1.6, kIc= 0.52 σ th = 2.4, kIc= 0.64
1.2 1
0
5000
Norm. fracture toughness KIc/kIc
Norm. fracture toughness KIc/kIc
a
2.4 2.2 2 1.8 1.6 1.4
1
10000
σsrc= 0.05, k Ic=0.52 σsrc= 0.10, k Ic=0.52 σsrc= 0.15, k Ic=0.52
1.2 0
Layer thickness [nm]
d
2.5 σ th = 0.8, σ th = 0.8, σ th = 1.6, σ th = 1.6, kIc= 0.37
2.25 2
σ src = 0.10 σ src = 0.15 σ src = 0.10 σ src = 0.15
1.75 1.5 1.25 1
0
5000
10000
Layer thickness [nm]
Norm. fracture toughness KIc/kIc
Norm. fracture toughness KIc/kIc
c
5000
10000
Layer thickness [nm]
2.5 σ th = 1.6, σ th = 1.6, σ th = 1.6, σ th = 0.8,
2.25
σ src = 0.10, kIc =0.37 σ src = 0.15, kIc =0.37 σ src = 0.10, kIc =0.52 σ src = 0.15, kIc =0.52
2 1.75 1.5 1.25 1
0
5000
10000
Layer thickness [nm]
Fig. 6. Toughness–thickness dependencies for a layered material with qsrc = 120 lm2. In the legends, the stresses rth and rsrc are given in units of 1 GPa, fracture energies kIc have the unit 1 MPa m1/2. (a) Three different sets of cohesive parameters and constant source strength of 0.1 GPa. (b) Influence of source strength for a constant cohesive strength of 1.6 GPa. (c) Influence of cohesive strength for two different source strengths of 0.1 GPa and 0.15 GPa. (d) Influence of cohesive energy for two sets of source and cohesive strengths.
N.C. Broedling et al. / Engineering Fracture Mechanics 75 (2008) 3743–3754 4500
Critical layer thickness [nm]
Norm. bulk toughness KbulkIc /k Ic
2.5 2.25 2 1.75 1.5
σsrc
ρsrc
1.25
0.05 0.05 0.1 0.1 0.15 0.15
80 120 80 120 80 120
1
0
0.5
1
1.5
2
2.5
4000 3500 3000
σsrc
ρsrc
0.05 0.05 0.1 0.1 0.15 0.15
80 120 80 120 80 120
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2500 2000 1500 1000 500 0
3
Cohesive strength [GPa]
0
0.5
1
1.5
2
2.5
3
Cohesive strength [GPa]
Fig. 7. Bulk fracture toughness (a) and critical layer thickness (b) as a function of cohesive strength for different source strengths. Curves correspond to two source densities qsrc = 80 lm2 (filled symbols), qsrc = 120 lm2 (open symbols) and three different source strengths rsrc 0.05 GPa, 0.1 GPa and 0.15 GPa, respectively.
thickness is increasing with the cohesive strength, see Fig. 7b. A 100% increase of the cohesive strength enhances the bulk toughness about 10% and the critical layer thickness about 20%. More significant is the influence of the source strength, corresponding to the initial yield strength of the material. When dislocation nucleation is harder (100% higher yield strength), the bulk fracture toughness drops about 20% and the critical layer thickness decreases about 80%. 5. Discussion
1 80 with CT ; σ th = 0.8 80 with CT ; σ th = 1.6 80 with CT ; σ th = 2.4
0.75
0.5
0.25
0
0
2500
5000
7500
Layer thickness [nm]
10000
Norm. max. tensile stress on interface
Norm. max. tensile stress on interface
The semi-brittle bulk materials considered here show a rather low fracture toughness determined by the competition of loading rate and expansion rate of the plastic zone. Due to this rate effect the layer materials reach their bulk fracture toughness at a layer thickness of a few micrometers at the high loading rate applied. In this model the dislocations nucleated close to the crack tip and piling up against the interface to the elastic region do not cause decohesion at the ductile/elastic interface, nor cracking of the brittle layer. Fig. 8 shows that the stresses at the dislocation pile-up stays below the theoretical strength of the cohesive zone model even for the smallest layer thicknesses. The results show that crack initiation at the ductile/elastic interface is only expected if the interface strength is less than 50% of the bulk cohesive strength. For numerical purposes dislocations pile-up at a minimum distance of 40b to the interface, see [1].
1 80 with CT; 80 with CT; 80 with CT;
σ src = 0.05 σ src = 0.1 σ src = 0.15
0.75
0.5
0.25
0
0
2500
5000
7500
10000
Layer thickness [nm]
Fig. 8. Maximum tensile stress at the ductile/elastic interface as a function of layer thickness for a source distribution with qsrc = 80 lm2 and a crack tip source. The stress is normalized with respect to the cohesive strength employed in the cohesive zone model. Curves are plotted for different cohesive strengths (a) and for different source strengths (b).
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In the following we analyze the importance of dislocation nucleation right at the crack tip for the overall fracture toughness of the layer materials. Various simulations have been performed with the present DD model suggesting that for thick layers a propagating crack temporarily arrests when it comes close to a source. This is caused by the sudden increase in the local stress at the source due to the approaching crack tip, which causes the source to emit a number of dislocations within a short period of time. These dislocations suddenly exert a strong shielding effect on the crack tip, thus causing its arrest. This mechanism has also been described by Cleveringa et al. [3], where crack growth in a bulk material was studied. However, we observe that for thin layers the back stress from a dislocation pile-up at the elastic–elastic/plastic interface not only inhibits further dislocation generation at the source, but also increases the local stress intensity factor at the crack tip. Therefore, crack arrest seems to be more difficult in thin layers. The crack tip source implemented in this model in order to enable dislocation activity right at the crack tip nucleates complete dislocation dipoles, representing dislocation loops in three dimensions. However, the antishielding dislocations emitted from this source are absorbed from the free crack surface right after each nucleation event and hence the results are identical to the monopole source model used in other work [14,15]. The strongest effect of enabling dislocation nucleation right at the crack is that the materials fracture toughness increases more rapidly with the layer thickness. At a thickness of about 300 nm the materials fracture toughness can already reach 120% of the intrinsic fracture toughness, whereas a material without crack tip source still exhibits an almost brittle response. The analytical ‘‘super dislocation” model in [11] considers the total dislocation density and hence the origin of the shielding factor to be concentrated at a point where the slip plane that is associated with the crack tip intersects the ductile/elastic interface. The super dislocation model underestimates the shielding effect of a compact dislocation pile-up right at the crack tip that we found to be the reason for a jump-like increase of fracture toughness with the layer thickness. Furthermore, in our multi source model the back stress originating from dislocation pile-ups at the interface triggers source operation close to the interface and thus remote to the crack tip. If the source generates dislocations on a slip plane parallel to that of the piled-up dislocations, the newly created dislocations move back to the crack tip and increase the local stress at the source that generated the piled-up dislocations. Thus the secondary dislocations reduce the back stress of the pile-up and enable further generation of shielding dislocations. The activation of the second slip system furthermore produces dislocation groups that travel in front of the crack tip and exert a strong local shielding effect on the approaching crack tip. Consequently the crack tip is trapped by the group of dislocations until the external load again exceeds a critical value, which causes a jump-like mechanism of crack advance. This unsteady crack motion becomes even more pronounced if a nearby dislocation source can be activated. In any case, dislocation nucleation at the interface enhances the toughness either by reactivating crack tip sources or by immediately shielding the advancing crack. Hence the present numerical model predicts a rapid transition at smaller layer thicknesses whereas the analytical super dislocation model predicts a rather gradual toughness–thickness dependency. Moreover, the super dislocation model does not account for rate effects due to the competition between loading rate and expansion rate of the plastic zone and does not indicate saturation of fracture toughness at large layer thicknesses. Fig. 9a shows that in the multiple source model the number of total dislocations that participate in the system increases strongly with increasing thickness. However, not all of the nucleated dislocations contribute equally to the overall shielding factor, in fact some dislocations do not contribute at all, as can be seen by comparing the large differences in the total number of dislocations plotted in Fig. 9a to the relatively small difference in the fracture toughness of the different source distributions. The comparison reveals that the crack tip source provides the most efficient shielding and thus the best ratio of fracture toughness versus number of dislocations. In Fig. 9b it is shown that the number of blunting dislocations gives a much better correspondence to the fracture toughness than the total number of dislocations. Blunting dislocations are defined as those dislocations that are absorbed by the open crack flanks, their number is consequently also related to the magnitude of the net Burgers vector within the plastic zone and hence to the density of geometrically necessary dislocations. In all cases where dislocation generation at the crack tip is considered, the shielding is mainly caused by the dislocations emanating from that source. If dislocation nucleation directly at the crack tip is not enabled, the net Burgers vector at failure is lower, which means that the shielding has to be produced by non-blunting dislocations. Our results show that fracture toughness in the regime below the critical layer thickness is strongly dependent on the availability and activity
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2
Norm. fracture toughness KIc/kIc
Total dislocations at failure
1400 120 with CT source 80 with CT source only CT source 80 without CT source
1200 1000 800 600 400 200 0
0
2500
5000
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10000
1.75
1.5
1.25
1
120 with CT source 80 with CT source only CT source 80 without CT source
0
10
20
30
40
Blunting dislocations
Fig. 9. Comparison of four different source distributions (see text) (a) Number of dislocations present at failure as a function of layer thickness. (b) Normalized fracture toughness as a function of blunting dislocations. The values in the legends are the densities of the bulk source distribution per lm2.
of dislocation sources right at the crack tip. Consequently it can be argued that since a propagating crack tip can be stopped when it comes close to a dislocation source this configuration, which yields the maximum fracture toughness, dominates the fracture process in layered structures. 6. Conclusions A combined two dimensional dislocation dynamics (DD)–cohesive zone (CZ) model is employed to study fracture of a layered composite material. Frank–Read like dislocation sources are distributed homogenously in an elastic–plastic layer sandwiched within elastic regions. It is assumed that two slip systems are active in the elastic–plastic material. A cohesive zone model of trapezoidal shape, which needs fracture energy and maximum tensile strength of the interface as input parameters, describes crack propagation. The competition between nucleation and motion of shielding dislocations and the propagation of the crack determines the amount of plastic energy dissipation and thus the fracture toughness of the layered structure. A general result of the numerical model is that fracture toughness scales with the size of the plastic zone. This leads to an embrittlement of the layered structure due to geometrical confinement. However, even in the absence of geometrical confinement it is found that the size of the plastic zone, now determined by the yield strength and the cohesive strength of the material, decides on its bulk fracture toughness. The results of the numerical model show that fracture of the layered structure undergoes a transition from ideally brittle to bulk behaviour with increasing layer thickness. The critical layer thickness at which the bulk fracture toughness of the elastic–plastic material is reached as well as the bulk fracture toughness itself increase with the cohesive strength of the interface, but become smaller for higher yield strengths. Fracture toughness as a function of layer thickness saturates gradually if dislocation activity is dispersed, dilute and not compact around the crack tip and increases abruptly within the layer thickness range of 300–1000 nm when dislocation activity right at the crack tip is possible and a compact pile-up of shielding dislocations forms near the crack tip. While the slip localization is beneficial for the fracture toughness at small layer thicknesses, the bulk fracture toughness is found to be slightly lower compared to a model with a homogeneous source distribution. Nucleation of dislocations on sources away from the crack tip increases the density of shielding dislocations in front of the crack and thus shields the crack after it starts to propagate. This leads to a jump like crack advance, with periods of crack arrest in regions where groups of dislocations elastically shield the crack from the applied load. While in thicker layers crack growth is thus stable over some distance, crack arrest in very thin layer materials seems to be more difficult. This is attributed to the back stress from the dislocation pile-up at the layer interface that adds to the applied load and to the comparatively small density of shielding dislocations in front of the crack tip in very thin films.
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Acknowledgement AH and NB gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) under contract number HA 2925/8-1. References [1] Broedling NC, Hartmaier A, Gao H. A combined dislocation–cohesive zone model for constraint plastic deformation at crack tips in layered structures. Int J Fract 2006;140:169–81. [2] Van der Giessen E, Needleman A. Discrete dislocation plasticity: a simple planar model. Model Simul Mater Sci Engng 1995;3:689–735. [3] Cleveringa HHM, Van der Giessen E, Needleman A. A discrete dislocation analysis of mode I crack growth. J Mech Phys Solids 2000;48:1133–57. [4] Deshpande VS, Needleman A, Van der Giessen E. Discrete dislocation plasticity modeling of short cracks in single crystals. Acta Mater 2003;51:1–15. [5] Chng AC, O’Day MP, Curtin WA, Tay AAO, Meng Lim K. Fracture in confined thin films: a discrete dislocation study. Acta Mater 2006;54:1017–27. [6] Chng AC, Curtin WA, Tay AAO, Lim KM. Effect of residual stress on fracture in confined thin films: a discrete dislocation study. Model Simul Mater Sc 2008;16:1. [7] Riemelmoser FO, Pippan R, Kolednik O. Cyclic crack growth in elastic plastic solids: a description in terms of dislocation theory. Comput Mech 1997;20:139–44. [8] Kreuzer HGM, Pippan R. Discrete dislocation simulation of nanoindentation: the effect of moving conditions and indenter shape. Mater Sci Engng A 2004;387–389:254–6. [9] Hartmaier A, Buehler MJ, Gao H. A discrete dislocation plasticity model of creep in polycrystalline thin films. Defect Diffus Forum 2003;224–225:107. [10] Hartmaier A, Buehler MJ, Gao H. Multiscale modeling of deformation in polycrystalline thin metal films on substrates. Adv Engng Mater 2005;7:165–9. [11] Hsia KJ, Suo Z, Yang W. Cleavage due to dislocation confinement in layered materials. J Mech Phys Solids 1994;6:877–96. [12] Tvergaard V, Hutchinson JW. The influence of plasticity on mixed mode interface toughness. J Mech Phys Solids 1993;41:1119–35. [13] Essmann U, Mughrabi H. Annihilation of dislocations during tensile and cyclic deformation and limits of dislocation densities. Philos Mag A 1979;40:731–56. [14] Roberts SG, Booth AS, Hirsch PB. Dislocation activity and brittle–ductile transitions in single crystals. Mater Sci Engng 1994;A176:91–8. [15] Hartmaier A, Gumbsch P. Thermal activation of crack-tip plasticity: the brittle or ductile response of a stationary crack loaded to failure. Phys Rev B 2005;71:024108-1–024108-11.
Engineering Fracture Mechanics 75 (2008) 3743–3754 www.elsevier.com/locate/engfracmech
Fracture toughness of layered structures: Embrittlement due to confinement of plasticity Nils C. Broedling a, Alexander Hartmaier b,*, Huajian Gao c a
b
Max Planck Institute for Metals Research, Heisenbergstrasse 3, 71735 Stuttgart, Germany Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg, Department of Materials Science and Engineering, Martensstrasse 5, 91058 Erlangen, Germany c Division of Engineering, Brown University, Providence, RI 02912, USA Received 4 August 2007; received in revised form 23 October 2007; accepted 25 October 2007 Available online 4 November 2007
Abstract The fracture toughness of a layered composite material is analyzed employing a combined two dimensional dislocation dynamics (DD)–cohesive zone (CZ) model. The fracture mechanism of an elastic–plastic (ductile) material sandwiched within purely elastic layers approaches ideally brittle behaviour with decreasing layer thickness. We investigate the influence of different constitutive parameters concerning dislocation plasticity as well as the effect of cohesive strength of the ductile material on the scaling of fracture toughness with layer thickness. For a constant layer thickness, the results of the numerical model are consistent with the expectation that fracture toughness decreases with increasing yield strength, but increases with the cohesive strength of the material. The scaling behaviour of the fracture toughness with layer thickness depends on these material parameters, but also on the dislocation microstructure in the vicinity of the crack tip. Strain localization due to easy dislocation generation right at the crack tip improves toughness in thin layers and leads to a jumplike increase of fracture toughness with layer thickness. However, the fracture toughness for films that are thick enough to exhibit bulk behaviour proves to be higher when the distribution of dislocations is more homogeneous, because in this case the crack grows in a stable fashion over some distance. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Fracture; Dislocation dynamics; Simulation; Layered structures
1. Introduction Plasticity in geometrically confined layered materials is studied on the basis of a combined dislocation dynamics (DD) and cohesive zone (CZ) method in two dimensions, the formulation of which has been published previously [1]. The model is employed to investigate the interaction of a ductile/elastic interface with a crack and to derive the thickness controlled fracture toughness of a composite layer material within the
*
Corresponding author. Tel.: +49 9131 85 27507; fax: +49 9131 85 27504. E-mail address: [email protected] (A. Hartmaier).
0013-7944/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2007.10.014
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limitations of a two dimensional dislocation representation. The work aims at answering the question in which thickness regime a pre-cracked semi-brittle layer material responds in an ideal brittle manner and how the transition to bulk behaviour occurs. DD simulations describe plastic deformation based on the nucleation, motion and annihilation of discrete dislocations embedded in an elastic continuum. By following the motion of individual dislocations, the dynamic evolution of the microstructure is tracked on time and length scales of fractions of seconds and micrometers. Despite their inherent limitations, two dimensional models have been shown to capture the most important features of plastic deformation in situations of geometrical confinement of plastic zones [1–10]. Therefore they have been used to investigate many features of plasticity in micro- and nano-structured materials where classical continuum plasticity is not applicable. From the modeling point of view, it is rather straightforward to incorporate preferable slip planes, obstacles and to model regions of large plastic deformation or even strain localization. Therefore, DD models are very suitable tools to analyze plasticity in nanostructured materials and to relate macroscopic quantities such as fracture toughness to underlying material lengths scales and microstructures. In plane strain two dimensional (2D) DD models, a dislocation is represented by a point singularity in a linear elastic body. The problem of a finite body with dislocations is formulated by superimposing the singular solutions for the elastic fields, known for a single dislocation in the infinite space, with an auxiliary solution that corrects for the boundary conditions. For the auxiliary solution, a boundary value problem has to be evaluated numerically. A number of problems concerning micro scale plastic deformation has been investigated based on the 2D-DD method developed Van der Giessen et al. [2]. In this work, a technique is introduced where the boundary value problem (BVP) of discrete dislocations is transformed into a weak formulation and then solved within a finite element (FEM) approach. In later work [3,4] a cohesive surface formulation was implemented in the DD method to study fracture accompanied by plastic deformation. A similar approach was used by Chng et al. [5,6] to study interfacial fracture and delamination of thin films. Other workers followed alternative approaches based on Green’s functions to incorporate the evolution of a dislocation microstructure under different boundary conditions into DD models. For example, a procedure was developed to evaluate contact stresses emerging in the wake region of growing fatigue cracks [7], similar methods were used to study nanoindentation [8], DD simulations were performed to study constrained diffusional creep in polycrystalline thin films [9,10]. In the present paper, a combined DD/CZ model is used to investigate the transition behaviour of the fracture toughness of a semi-brittle layered material. The problem formulation and the method for its solution have already been described in [1]. The model has been developed to study geometrical confinement effects on fracture in ductile layers. An analytical analysis of the problem based on certain simplifying assumptions is given by Hsia et al. [11]. In this work, the effect of pile-up formation at dislocation barriers on fracture in a metal layer is investigated. The analysis is based on the elastic energy balance and shows that the fracture toughness increases proportional to the square-root of layer thickness. After briefly revisiting the numerical method used in the present work, the results of a detailed parametric study on the factors influencing plasticity and fracture toughness of the sandwich structure are presented. From these results, finally, the dominating factors defining the fracture toughness are derived. 2. Dislocation dynamics–cohesive zone model Fig. 1 shows a plane strain mode I crack propagating through a thin elastic–plastic (ductile) layer embedded in an elastic matrix. Shown is only one near-tip-region of a slit-like crack that is chosen so long that its other crack tip does not interfere with that under consideration here. Ahead of the crack tip a cohesive zone is situated in the projection of the crack line. The cohesive zone is characterized by a traction–separation law with theoretical strength rth and the cohesive energy C. The plastic layer and the matrix can have different elastic properties. Dislocations nucleate at randomly placed Frank–Read like sources within the plastic layer. The intrinsic fracture toughness of the material (i.e. its fracture toughness in the absence of plasticity) is specified by the theoretical strength rth and the cohesive energy C. The plastic deformation around the crack tip develops from a distribution of dislocation sources with density qsrc and strength rsrc. The strength of the source is the critical resolved shear stress at which the sources operate and thus corresponds to the yield
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KI
Blunting
h ρ src , σ src , B
σ th , Γ
Fracture 2D Frank-Read Source
Shielding
Fig. 1. Schematics of the model problem. An initial crack is located within a ductile layer. The layer thickness h is the characteristic length scale which controls the fracture behaviour. See text for notations.
strength of the material. The 2D analogon of a dislocation loop is a dislocation dipole. Hence, to simulate a Frank–Read source we place two test dislocations with anti-parallel Burgers vector at the source position and calculate the total force on the dislocations. If this force is positive, i.e. it drives the dislocation dipole apart, the dislocation loop emanating from a Frank–Read source would also expand and thus both dislocations are nucleated. If the dipole collapses under the acting shear stress or by the mutual attraction of the dislocations, a Frank–Read source at this position could not have generated a dislocation loop, and dislocation nucleation is disregarded. The separation of the dislocations in the dipole is calculated such that dislocation start to nucleate if the local resolved shear stress exceeds rsrc. To take into account that a Frank–Read source needs a finite time to generate a dislocation loop, we require that the force on both test dislocations be positive over a given initiation time interval. During the simulation a crack opening tensile stress is applied giving rise to a mode I stress intensity factor K1 I , which is calculated by a J-integral (see [1]). The development of a plastic zone shields the crack tip from the applied load. Dislocation sources subject to the high stresses near the crack tip generate a large number of dislocations and as long as the plastic strain rate at the crack tip is large enough to keep the load on the crack tip below the critical value, the crack is stable. At some stage during the loading the crack starts to propagate. In particular at the beginning of this phase crack propagation is not always critical, but the crack may be stopped again due to groups of dislocations that locally provide a strong elastic shielding. Finally the crack propagates throughout the domain and the applied load at this stage defines the fracture toughness. A cohesive zone segment of 2000 nm ahead of the initial crack tip is discretized with a high resolution of about 8 nm node spacing. This is the predefined maximum distance of allowed crack advance which was found to be sufficient to resolve the stable crack growth prior to ultimate failure. In all simulations with a crack tip source, the stable crack advance occurred within a region of less than 1200 nm. The energy dissipation due to the breaking of atomic bonds defines the lower bound for fracture toughness in absence of any dislocation activity, the intrinsic fracture toughness kIc that corresponds to the Griffith energy. If dislocations are nucleated and provide elastic shielding as well as dissipation of mechanical energy, the fracture toughness increases. For the class of semi-brittle materials considered here the fracture toughness of the bulk material is determined by the competition between the loading rate and the expansion rate of the plastic zone, and sets an upper bound for the fracture toughness. In the case of a thin layer investigated here, the geometrical confinement limits the size of plastic zone. In this case dislocations pile up at the interface to the elastic matrix resulting in a back stress that eventually prevents further dislocation nucleation in the crack tip-region. The critical layer thickness ^ h at which the crack cannot be shielded sufficiently and the material becomes brittle is investigated by the DD simulation. 3. Boundary value problem and constitutive rules In the fully deterministic discrete dislocation model that is considered here, the driving force for dislocation motion is assumed to be completely described within the framework of elasticity. Within this mesoscopic
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approach, the microstructure of a solid and the internal elastic fields are associated with a number of discrete defects such as point defects, dislocations and cracks. In response to the thermodynamic force acting on each individual defect the entire microstructure evolves in time and space. To evaluate the interaction between dislocations and crack tip the corresponding mechanical boundary value problem needs to be formulated and solved numerically. The evolution of the microstructure is computed from a chosen initial configuration based on an explicit time integration scheme by successively updating the dislocation position and the displacements in the cohesive zone. The original dislocation-boundary value problem for a semi-infinite body with embedded dislocations and its decomposition is shown in Fig. 2. The decomposition leads to an unbounded problem with homogenous fields, an unbounded problem with singular fields, and a corrective problem with the image fields. In order to reduce the computational costs the model problem is assumed to be mirror-symmetric applying the corresponding boundary conditions at the crack line. The crack is modeled by distributing virtual dislocations along the crack line. Normal and tangential tractions are related to the virtual dislocation distribution in terms of boundary integrals. The system of boundary integral equations is evaluated numerically using the boundary collocation method [1]. The constitutive rules that account for material separation, dislocation motion and nucleation are described next. Material separation near the crack tip is modeled with a cohesive zone that specifies the cohesive strength and the cohesive energy of the ductile material corresponding to the intrinsic fracture toughness or Griffith energy, respectively. The traction separation law, relating the cohesive traction N to the crack opening D, is of trapezoidal shape [12], see Fig. 3. The fracture (Griffith) energy is defined as C ¼ 0:5rth Dc ð1 d1 þ d2 Þ;
ð1Þ
where rth is the cohesive strength, Dc is the crack opening separation above which the cohesive interaction vanishes and d1, d2 are the shape parameters that define the corners of the trapezoid. The corresponding intrinsic plane strain fracture toughness for a purely elastic material is obtained as rffiffiffiffiffiffiffiffiffiffiffiffiffi EC k Ic ¼ ; ð2Þ 1 m2 where C is the cohesive energy, E is the Young’s modulus and m is the Poisson’s ratio. In the presence of plastic deformation in the vicinity of the loaded crack tip, higher fracture toughness values are expected. The Peach–Koehler force (per unit length) which drives the dislocation in the slip direction is expressed as fi ¼ eij3 rjk bk ;
ð3Þ
where eijk is the permutation tensor, bk is the Burgers vector with norm b = 0.25 nm in the examples presented later. The velocity of dislocation (k) is assumed to be in the overdamped viscous glide regime and follows the form
Fig. 2. The boundary value problem of a semi-infinite body B with boundary oB and dislocations. Part of oB with normal vector n is traction-free and the other part is subjected to cohesive tractions N(u). The decomposition leads to an unbounded problem with homogenous fields ()1, an unbounded problem with singular fields ð~Þ, and a corrective problem with the image fields ð^Þ.
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N (Δ)
σ th 1
δ1
δ2
1
Δ Δc
Fig. 3. Traction separation law (N-D-law) with characteristic parameters including the theoretical strength rth and the critical crack tip opening displacement Dc. d1 and d2 are shape parameters giving the fracture energy C.
ðkÞ
vgd ¼
1 ðkÞ f ; B
ð4Þ
where the drag coefficient B = 104 GPa ns expresses the lattice resistance to dislocation glide. Two slip planes at angles of a1 = 45° and a2 = 45° with respect to the crack line are considered. Initially a randomly distributed source configuration is generated keeping a minimum distance of 10b between neighboring slip planes that are associated with the sources. At each time step, a nucleation criterion is evaluated by considering the stability of a test dislocation dipole with the average nucleation distance, as described above. For the near-field dislocation–dislocation interaction, a critical distance of 8b is used as an onset radius below which formations of obstacles or dislocation annihilation is triggered [13]. Obstacle formation occurs when two dislocations approach on different intersecting slip planes. When their distance falls below the critical distance for dislocation–dislocation interaction, their motion is stopped. Obstacle formation leads to a macroscopic hardening effect in the material. The obstacles act as barriers to the glide of further dislocations such that dislocation pile-ups form behind the two crossing dislocations, exerting an increasing driving force on them. An obstacle strength is assumed to be about 50% of the materials cohesive strength. This quantity controls the highest possible magnitude of pile-up stresses within the bulk material due to the formation of locks. Two locking dislocations are disengaged from each other when the driving force acting on them exceeds the obstacle strength. If two dislocations of opposite Burgers vectors are found to be approaching each other on parallel slip planes within the critical distance, they annihilate and are subsequently removed from the simulation. When a single dislocation travels across the crack line it is mirrored, which means that the dislocation is transferred to a slip plane of the alternative type at the same location. This consequently requires that its Burgers vector be rotated accordingly. Thus its shielding effect with respect to the crack tip changes the sign, which in most situations is accompanied by an update of the system matrix. Oscillations by successive back and forth mirroring of a dislocation must be avoided since that could tremendously slow down the solution algorithm. Hence, a driving force that pushes the dislocation across the crack line is required to be present over the initiation time period before mirroring is actually performed. Furthermore, for the dislocationboundary interaction, a onset distance of 40b is considered. The process region has a dimension of 2000 nm in the direction of the crack line. In this region, the moving dislocations are expected to interact strongly with the crack. They may be absorbed if they run into the traction-free crack surface or mirrored where the material is still coherent. 4. Results The objective of this work is to carry out parametric studies over a wide parameter range and to examine the characteristics of this generic fracture problem. The results obtained are presented and discussed in two parts. First, the influence of the source distribution on the toughness–thickness dependency is shown. Second, the correlation of the toughness–thickness dependency with cohesive strength and source strength indicated by the results will be pointed out. The source strengths, equivalent to the yield stresses of the model materials, are within the range of 0.05 GPa–1.5 GPa. The cohesive strengths are varied within the range of 0.8 GPa–
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2.4 GPa. The setup of the geometry follows Fig. 1. The simulations start at zero applied stress with an initially dislocation free material and the load is increased monotonically with a constant loading rate of pffiffiffiffiffiffiffi 3 K_ 1 ¼ 10 GPa nm =ns. The material is taken to be elastically homogeneous with shear modulus I G = 26 GPa and Poisson ratio m = 0.33. 4.1. Influence of the microstructural surrounding In this section the influence of the local source distribution as well as that of the source density within the bulk material are investigated. For fracture of thin layers the availability of sources close to the crack tip rather than the overall source density becomes important. In Fig. 4 simulation results for four 1 lm thick layer materials A, B, C and D with different source distributions are plotted. The other material parameters are chosen as rsrc = 0.05 GPa and rth = 0.8 GPa. Fig. 4a and b show clearly the effect of dislocation emission right at the crack tip. The source density of the bulk material in both simulations has been chosen to be 80 lm2 with an overall identical distribution, but in one case (shown in Fig. 4b) a source with a slip plane angle of 45° has been placed at position (x, y) = (40 nm, 50 nm). This crack tip dislocation source is a bulk source that is placed close to the crack tip. The anti-shielding dislocation of a nucleated dipole is absorbed by the free crack surface right after its nucleation. Note, that the serrated shape of the local stress intensity factor in Fig. 4b is a result of the strong dislocation–interface interaction and reflects single nucleation events of the near crack tip source. The consequence of the existence of a crack tip source is that the number of nucleated dislocations has been doubled. The formation of a compact dislocation pile-up of 20 dislocations exerts a strong shielding effect and causes an increase in fracture toughness of about 20%. The results for a higher bulk source density of 120 lm2 are shown in Fig. 4c and for only one crack tip source in Fig. 4d. Another observation in our simulations is that due to the formation of a dislocation array at the crack tip the fracture propagation mechanism depends on the film thickness. For thicker films, in most cases the crack temporally arrests due to the elastic interaction of the growing crack with an existing group of dislocations ahead of the crack. In very thin layers, however, the dislocation microstructure degenerates to a single dislocation array on a single slip plane at the crack tip and, correspondingly, no crack arrest is observed. The resulting fracture toughnesses for the different source distributions are shown in Fig. 5 as a function of film thickness. It is seen that for thicknesses below 1000 nm the crack tip source is dominant, because the curves for the materials with the crack tip source (B, C and D) do not deviate from each other in this regime. Above this thickness the fracture toughness of the material D with only on source close to the crack tip remains approximately constant. The curves of materials B and C start to deviate from each other at a thickness of 2000 nm. Interestingly the material with the higher source density gives the lower fracture toughness, which is explained by the higher dislocation density in this case, hindering the free expansion of the shielding dislocations emanating from the near crack source. Thus the material with a higher source density exhibits a more pronounced work hardening. The comparison of material A and B shows that the bulk fracture toughness of a material with a bulk source density of 80 lm2 increases by about 15% when additionally a crack tip source is considered. The influence of the bulk sources on the fracture toughness of a layered material is noticeable only above a thickness of 1000 nm. The bulk toughness of a material with a crack tip source increases by about 20% (10%) when in addition to the crack tip source a distribution of bulk sources with a density of 80 lm2 (120 lm2) is considered. 4.2. Influence of source strength and cohesive strength We have conducted extensive parameter studies to investigate the effect of cohesive strength, intrinsic fracture toughness as well as source strength on the resulting fracture toughness. Fig. 6 shows toughness–thickness curves for different sets of parameters for a layer material with qsrc = 80 lm2 with crack tip source. Fig. 6a–d reveal the effect of cohesive strength and intrinsic fracture toughness. Both parameters enhance the macroscopic fracture toughness significantly. Fig. 6c shows simulation results for two different intrinsic fracture toughnesses for two different source strengths rsrc. A 40% increase of the intrinsic fracture toughness enhances the bulk toughness by about 10%.
N.C. Broedling et al. / Engineering Fracture Mechanics 75 (2008) 3743–3754 Shielding
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b kI / kIc
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1.0 0.9 0.8 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.1 0.0
total 24 absorbed 2
mirrored 3 dissociated obstacles 138
total 58 absorbed 20
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c
d
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2
KI / kIc
KI / kIc
1.0 0.9 0.8 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.1 0.0
total 21 absorbed 19
mirrored 0 dissociated obstacles 0
Fig. 4. Comparison of four different source distributions for a 1 lm thick layer. Red and green dots denote dislocation sources associated with the two different slip systems. Shown is shielding and final microstructure for each distribution together with the corresponding variation of the opening stress component: (a) qsrc = 80 lm2 without crack tip source, (b) qsrc = 80 lm2 with crack tip source, (c) qsrc = 120 lm2 with crack tip source, (d) only a single crack tip source.
We define the critical layer thickness to be that thickness where the fracture toughness of the layer material reaches 95% of the materials bulk toughness, see Fig. 7a. The bulk toughness as well as the critical layer
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1.8
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1.4
ρsrc = 80, no CT source ρsrc = 80, with CT source ρsrc = 120, with CT source
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only CT source
1
0
5000
10000
h [nm]
Fig. 5. Normalized fracture toughness as a function of layer thickness, cf. plots of four different source distributions for a 1 lm thick layer in Fig. 4. The lines connecting the data points are mere visual guides.
b 2.4 2.2 2 1.8 1.6 1.4 σ th = 0.8, kIc= 0.37 σ th = 1.6, kIc= 0.52 σ th = 2.4, kIc= 0.64
1.2 1
0
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Norm. fracture toughness KIc/kIc
Norm. fracture toughness KIc/kIc
a
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1
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σsrc= 0.05, k Ic=0.52 σsrc= 0.10, k Ic=0.52 σsrc= 0.15, k Ic=0.52
1.2 0
Layer thickness [nm]
d
2.5 σ th = 0.8, σ th = 0.8, σ th = 1.6, σ th = 1.6, kIc= 0.37
2.25 2
σ src = 0.10 σ src = 0.15 σ src = 0.10 σ src = 0.15
1.75 1.5 1.25 1
0
5000
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Layer thickness [nm]
Norm. fracture toughness KIc/kIc
Norm. fracture toughness KIc/kIc
c
5000
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Layer thickness [nm]
2.5 σ th = 1.6, σ th = 1.6, σ th = 1.6, σ th = 0.8,
2.25
σ src = 0.10, kIc =0.37 σ src = 0.15, kIc =0.37 σ src = 0.10, kIc =0.52 σ src = 0.15, kIc =0.52
2 1.75 1.5 1.25 1
0
5000
10000
Layer thickness [nm]
Fig. 6. Toughness–thickness dependencies for a layered material with qsrc = 120 lm2. In the legends, the stresses rth and rsrc are given in units of 1 GPa, fracture energies kIc have the unit 1 MPa m1/2. (a) Three different sets of cohesive parameters and constant source strength of 0.1 GPa. (b) Influence of source strength for a constant cohesive strength of 1.6 GPa. (c) Influence of cohesive strength for two different source strengths of 0.1 GPa and 0.15 GPa. (d) Influence of cohesive energy for two sets of source and cohesive strengths.
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Critical layer thickness [nm]
Norm. bulk toughness KbulkIc /k Ic
2.5 2.25 2 1.75 1.5
σsrc
ρsrc
1.25
0.05 0.05 0.1 0.1 0.15 0.15
80 120 80 120 80 120
1
0
0.5
1
1.5
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2.5
4000 3500 3000
σsrc
ρsrc
0.05 0.05 0.1 0.1 0.15 0.15
80 120 80 120 80 120
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2500 2000 1500 1000 500 0
3
Cohesive strength [GPa]
0
0.5
1
1.5
2
2.5
3
Cohesive strength [GPa]
Fig. 7. Bulk fracture toughness (a) and critical layer thickness (b) as a function of cohesive strength for different source strengths. Curves correspond to two source densities qsrc = 80 lm2 (filled symbols), qsrc = 120 lm2 (open symbols) and three different source strengths rsrc 0.05 GPa, 0.1 GPa and 0.15 GPa, respectively.
thickness is increasing with the cohesive strength, see Fig. 7b. A 100% increase of the cohesive strength enhances the bulk toughness about 10% and the critical layer thickness about 20%. More significant is the influence of the source strength, corresponding to the initial yield strength of the material. When dislocation nucleation is harder (100% higher yield strength), the bulk fracture toughness drops about 20% and the critical layer thickness decreases about 80%. 5. Discussion
1 80 with CT ; σ th = 0.8 80 with CT ; σ th = 1.6 80 with CT ; σ th = 2.4
0.75
0.5
0.25
0
0
2500
5000
7500
Layer thickness [nm]
10000
Norm. max. tensile stress on interface
Norm. max. tensile stress on interface
The semi-brittle bulk materials considered here show a rather low fracture toughness determined by the competition of loading rate and expansion rate of the plastic zone. Due to this rate effect the layer materials reach their bulk fracture toughness at a layer thickness of a few micrometers at the high loading rate applied. In this model the dislocations nucleated close to the crack tip and piling up against the interface to the elastic region do not cause decohesion at the ductile/elastic interface, nor cracking of the brittle layer. Fig. 8 shows that the stresses at the dislocation pile-up stays below the theoretical strength of the cohesive zone model even for the smallest layer thicknesses. The results show that crack initiation at the ductile/elastic interface is only expected if the interface strength is less than 50% of the bulk cohesive strength. For numerical purposes dislocations pile-up at a minimum distance of 40b to the interface, see [1].
1 80 with CT; 80 with CT; 80 with CT;
σ src = 0.05 σ src = 0.1 σ src = 0.15
0.75
0.5
0.25
0
0
2500
5000
7500
10000
Layer thickness [nm]
Fig. 8. Maximum tensile stress at the ductile/elastic interface as a function of layer thickness for a source distribution with qsrc = 80 lm2 and a crack tip source. The stress is normalized with respect to the cohesive strength employed in the cohesive zone model. Curves are plotted for different cohesive strengths (a) and for different source strengths (b).
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In the following we analyze the importance of dislocation nucleation right at the crack tip for the overall fracture toughness of the layer materials. Various simulations have been performed with the present DD model suggesting that for thick layers a propagating crack temporarily arrests when it comes close to a source. This is caused by the sudden increase in the local stress at the source due to the approaching crack tip, which causes the source to emit a number of dislocations within a short period of time. These dislocations suddenly exert a strong shielding effect on the crack tip, thus causing its arrest. This mechanism has also been described by Cleveringa et al. [3], where crack growth in a bulk material was studied. However, we observe that for thin layers the back stress from a dislocation pile-up at the elastic–elastic/plastic interface not only inhibits further dislocation generation at the source, but also increases the local stress intensity factor at the crack tip. Therefore, crack arrest seems to be more difficult in thin layers. The crack tip source implemented in this model in order to enable dislocation activity right at the crack tip nucleates complete dislocation dipoles, representing dislocation loops in three dimensions. However, the antishielding dislocations emitted from this source are absorbed from the free crack surface right after each nucleation event and hence the results are identical to the monopole source model used in other work [14,15]. The strongest effect of enabling dislocation nucleation right at the crack is that the materials fracture toughness increases more rapidly with the layer thickness. At a thickness of about 300 nm the materials fracture toughness can already reach 120% of the intrinsic fracture toughness, whereas a material without crack tip source still exhibits an almost brittle response. The analytical ‘‘super dislocation” model in [11] considers the total dislocation density and hence the origin of the shielding factor to be concentrated at a point where the slip plane that is associated with the crack tip intersects the ductile/elastic interface. The super dislocation model underestimates the shielding effect of a compact dislocation pile-up right at the crack tip that we found to be the reason for a jump-like increase of fracture toughness with the layer thickness. Furthermore, in our multi source model the back stress originating from dislocation pile-ups at the interface triggers source operation close to the interface and thus remote to the crack tip. If the source generates dislocations on a slip plane parallel to that of the piled-up dislocations, the newly created dislocations move back to the crack tip and increase the local stress at the source that generated the piled-up dislocations. Thus the secondary dislocations reduce the back stress of the pile-up and enable further generation of shielding dislocations. The activation of the second slip system furthermore produces dislocation groups that travel in front of the crack tip and exert a strong local shielding effect on the approaching crack tip. Consequently the crack tip is trapped by the group of dislocations until the external load again exceeds a critical value, which causes a jump-like mechanism of crack advance. This unsteady crack motion becomes even more pronounced if a nearby dislocation source can be activated. In any case, dislocation nucleation at the interface enhances the toughness either by reactivating crack tip sources or by immediately shielding the advancing crack. Hence the present numerical model predicts a rapid transition at smaller layer thicknesses whereas the analytical super dislocation model predicts a rather gradual toughness–thickness dependency. Moreover, the super dislocation model does not account for rate effects due to the competition between loading rate and expansion rate of the plastic zone and does not indicate saturation of fracture toughness at large layer thicknesses. Fig. 9a shows that in the multiple source model the number of total dislocations that participate in the system increases strongly with increasing thickness. However, not all of the nucleated dislocations contribute equally to the overall shielding factor, in fact some dislocations do not contribute at all, as can be seen by comparing the large differences in the total number of dislocations plotted in Fig. 9a to the relatively small difference in the fracture toughness of the different source distributions. The comparison reveals that the crack tip source provides the most efficient shielding and thus the best ratio of fracture toughness versus number of dislocations. In Fig. 9b it is shown that the number of blunting dislocations gives a much better correspondence to the fracture toughness than the total number of dislocations. Blunting dislocations are defined as those dislocations that are absorbed by the open crack flanks, their number is consequently also related to the magnitude of the net Burgers vector within the plastic zone and hence to the density of geometrically necessary dislocations. In all cases where dislocation generation at the crack tip is considered, the shielding is mainly caused by the dislocations emanating from that source. If dislocation nucleation directly at the crack tip is not enabled, the net Burgers vector at failure is lower, which means that the shielding has to be produced by non-blunting dislocations. Our results show that fracture toughness in the regime below the critical layer thickness is strongly dependent on the availability and activity
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2
Norm. fracture toughness KIc/kIc
Total dislocations at failure
1400 120 with CT source 80 with CT source only CT source 80 without CT source
1200 1000 800 600 400 200 0
0
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7500
Layer thickness [nm]
10000
1.75
1.5
1.25
1
120 with CT source 80 with CT source only CT source 80 without CT source
0
10
20
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40
Blunting dislocations
Fig. 9. Comparison of four different source distributions (see text) (a) Number of dislocations present at failure as a function of layer thickness. (b) Normalized fracture toughness as a function of blunting dislocations. The values in the legends are the densities of the bulk source distribution per lm2.
of dislocation sources right at the crack tip. Consequently it can be argued that since a propagating crack tip can be stopped when it comes close to a dislocation source this configuration, which yields the maximum fracture toughness, dominates the fracture process in layered structures. 6. Conclusions A combined two dimensional dislocation dynamics (DD)–cohesive zone (CZ) model is employed to study fracture of a layered composite material. Frank–Read like dislocation sources are distributed homogenously in an elastic–plastic layer sandwiched within elastic regions. It is assumed that two slip systems are active in the elastic–plastic material. A cohesive zone model of trapezoidal shape, which needs fracture energy and maximum tensile strength of the interface as input parameters, describes crack propagation. The competition between nucleation and motion of shielding dislocations and the propagation of the crack determines the amount of plastic energy dissipation and thus the fracture toughness of the layered structure. A general result of the numerical model is that fracture toughness scales with the size of the plastic zone. This leads to an embrittlement of the layered structure due to geometrical confinement. However, even in the absence of geometrical confinement it is found that the size of the plastic zone, now determined by the yield strength and the cohesive strength of the material, decides on its bulk fracture toughness. The results of the numerical model show that fracture of the layered structure undergoes a transition from ideally brittle to bulk behaviour with increasing layer thickness. The critical layer thickness at which the bulk fracture toughness of the elastic–plastic material is reached as well as the bulk fracture toughness itself increase with the cohesive strength of the interface, but become smaller for higher yield strengths. Fracture toughness as a function of layer thickness saturates gradually if dislocation activity is dispersed, dilute and not compact around the crack tip and increases abruptly within the layer thickness range of 300–1000 nm when dislocation activity right at the crack tip is possible and a compact pile-up of shielding dislocations forms near the crack tip. While the slip localization is beneficial for the fracture toughness at small layer thicknesses, the bulk fracture toughness is found to be slightly lower compared to a model with a homogeneous source distribution. Nucleation of dislocations on sources away from the crack tip increases the density of shielding dislocations in front of the crack and thus shields the crack after it starts to propagate. This leads to a jump like crack advance, with periods of crack arrest in regions where groups of dislocations elastically shield the crack from the applied load. While in thicker layers crack growth is thus stable over some distance, crack arrest in very thin layer materials seems to be more difficult. This is attributed to the back stress from the dislocation pile-up at the layer interface that adds to the applied load and to the comparatively small density of shielding dislocations in front of the crack tip in very thin films.
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Acknowledgement AH and NB gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) under contract number HA 2925/8-1. References [1] Broedling NC, Hartmaier A, Gao H. A combined dislocation–cohesive zone model for constraint plastic deformation at crack tips in layered structures. Int J Fract 2006;140:169–81. [2] Van der Giessen E, Needleman A. Discrete dislocation plasticity: a simple planar model. Model Simul Mater Sci Engng 1995;3:689–735. [3] Cleveringa HHM, Van der Giessen E, Needleman A. A discrete dislocation analysis of mode I crack growth. J Mech Phys Solids 2000;48:1133–57. [4] Deshpande VS, Needleman A, Van der Giessen E. Discrete dislocation plasticity modeling of short cracks in single crystals. Acta Mater 2003;51:1–15. [5] Chng AC, O’Day MP, Curtin WA, Tay AAO, Meng Lim K. Fracture in confined thin films: a discrete dislocation study. Acta Mater 2006;54:1017–27. [6] Chng AC, Curtin WA, Tay AAO, Lim KM. Effect of residual stress on fracture in confined thin films: a discrete dislocation study. Model Simul Mater Sc 2008;16:1. [7] Riemelmoser FO, Pippan R, Kolednik O. Cyclic crack growth in elastic plastic solids: a description in terms of dislocation theory. Comput Mech 1997;20:139–44. [8] Kreuzer HGM, Pippan R. Discrete dislocation simulation of nanoindentation: the effect of moving conditions and indenter shape. Mater Sci Engng A 2004;387–389:254–6. [9] Hartmaier A, Buehler MJ, Gao H. A discrete dislocation plasticity model of creep in polycrystalline thin films. Defect Diffus Forum 2003;224–225:107. [10] Hartmaier A, Buehler MJ, Gao H. Multiscale modeling of deformation in polycrystalline thin metal films on substrates. Adv Engng Mater 2005;7:165–9. [11] Hsia KJ, Suo Z, Yang W. Cleavage due to dislocation confinement in layered materials. J Mech Phys Solids 1994;6:877–96. [12] Tvergaard V, Hutchinson JW. The influence of plasticity on mixed mode interface toughness. J Mech Phys Solids 1993;41:1119–35. [13] Essmann U, Mughrabi H. Annihilation of dislocations during tensile and cyclic deformation and limits of dislocation densities. Philos Mag A 1979;40:731–56. [14] Roberts SG, Booth AS, Hirsch PB. Dislocation activity and brittle–ductile transitions in single crystals. Mater Sci Engng 1994;A176:91–8. [15] Hartmaier A, Gumbsch P. Thermal activation of crack-tip plasticity: the brittle or ductile response of a stationary crack loaded to failure. Phys Rev B 2005;71:024108-1–024108-11.