Simulation of fatigue crack growth with a cyclic cohesive zone model

Transcription

1 Article published in International Journal of Fracture (24) 88:23 45 The final publication is available at DOI:.7/s Simulation of fatigue crack growth with a cyclic cohesive zone model Stephan Roth Geralf Hütter Meinhard Kuna Received: 2 December 23 / Accepted: 2 March 24 / Published online: 8 April 24 Abstract Fatigue crack growth is simulated for an elastic solid with a cyclic cohesive zone model (CZM). Material degradation and thus separation follows from the current damage state, which represents the amount of maximum transferable traction across the cohesive zone. The traction-separation relation proposed in the cyclic CZM includes non-linear paths for both un- and reloading. This allows a smooth transition from reversible to damaged state. The exponential tractionseparation envelope is controlled by two shape parameters. Moreover, a lower bound for damage evolution is introduced by a local damage dependent endurance limit, which enters the damage evolution equation. The cyclic CZM is applied to mode I fatigue crack growth under K I -controlled external loading conditions. The influences of the model parameters with respect to static failure load K, threshold load K th and Paris parameters m, C are investigated. The study reveals that the proposed endurance limit formulation is well suited to control the ratio K th /K independent of m and C. An identification procedure is suggested to identify the cohesive parameters with the help of Wöhler diagrams and fatigue crack growth rate curves. Keywords Cyclic cohesive zone model Fatigue crack growth Damage mechanics Boundary layer model S. Roth G. Hütter M. Kuna Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, 9596 Freiberg, Germany stephan.roth@imfd.tu-freiberg.de Introduction The cohesive zone model (CZM) was developed by Dugdale (96) and Barenblatt (962). Its basic idea is to concentrate the entire fracture process to a thin cohesive zone. Inside this zone, the material behaviour is described by a local constitutive law, which relates the traction transferred across the cohesive zone to the displacement jump also denoted as separation. Considering a crack tip as sketched in Fig., the stress singularity expected from linear elastic fracture mechanics is replaced by a more realistic distribution along the cohesive zone. Damage initiation is determined by reaching a strength parameter called cohesive strength, t. Below t, i.e. ahead of the cohesive zone, the constitutive law captures reversibility. Due to the involved transition from reversible to damaged state, no explicit fracture criterion is needed. Instead, local failure is indicated by the vanishing local strength. The presence of an incipient crack is not necessary. Thus, CZMs allow to simulate crack initiation, crack propagation and final fracture in a unique manner. Within the classical finite element (FE) framework, the CZM is implemented as cohesive zone element. This requires in those regions of the structure, where cracks might occur, to prepare the mesh with cohesive zone elements. Nevertheless, the CZM is well applicable and widely accepted in order to simulate crack propagation along symmetry lines in specimens, at interfaces and for delamination problems. Beyond that, the extended finite element method (XFEM) allows the

2 2 St. Roth et al., Int J Fracture (24) 88:23 45, DOI:.7/s cohesive zone Fig. Sketch of a cohesive zone ahead of a crack tip (traction t, separation δ, cohesive strength t, corresponding separation δ ) t t δ x δ combination of CZMs and arbitrary crack propagation (Xu and Yuan, 29a). Separation processes under monotonic loading conditions are covered by monotonic CZMs. The formulation of monotonic CZMs, underlying model assumptions as well as applications are well documented, e.g. by Brocks et al (23). In most of the models, the cohesive laws are described by a scalar traction-separation relation and its curve. While the cohesive strength and the fracture energy density characterise the maximum traction and the area under the traction-separation curve, respectively, the shape of the curve remains as a free model assumption. There are, amongst others, bilinear, trapezoidal, polynomial and exponential laws reported in the literature. For numerical reasons, continuously differentiable approaches (polynomial, exponential) are preferred. A versatile and quite often used cohesive law is the exponential one proposed by Xu and Needleman (993). It is motivated by atomistic considerations and allows a closed form formulation in a thermodynamical framework by introducing a cohesive potential function (e.g. Bouvard et al (29); Ortiz and Pandolfi (999); Roychowdhury et al (22)). Modifications concerning the shape of the traction-separation curve are suggested e.g. by Kroon and Faleskog (25) or Goyal et al (24); Lucas et al (28), where the brittleness of the material is controlled by a shape parameter. However, these models lack a closed form cohesive potential. A further model assumption addresses the unand reloading behaviour. For undamaged states, unand reloading is restricted to the reversible branch (linear or not) of the traction-separation curve. After damage initiation, the literature offers different approaches of the unloading behaviour with respect to linearity and destination of the unloading paths. While unloading is assumed towards the origin in (Bouvard et al, 29; Camacho and Ortiz, 996; Goyal et al, 24; Lucas et al, 28; Nguyen et al, 2; Xu and Yuan, 29a; Yang et al, 2), which is associated with quasi-brittle material behaviour, some residual separation after unloading is admitted in (Roe and Siegmund, 23; Scheider and Mosler, 2; Xu and Yuan, 29b). Linear unloading paths are assumed (Bouvard et al, 29; Nguyen et al, 2; Roe and Siegmund, 23; Scheider and Mosler, 2; Xu and Yuan, 29a,b; Yang et al, 2) as well as non-linear ones (Goyal et al, 24; Lucas et al, 28). Considering the cohesive zone as the area where micro defects evolve, residual separation may result from surface mismatches at closing. Plasticity-induced surface mismatches (crack closure) can be incorporated including plastic matrix material in the surrounding of the cohesive zone. Concerning the question of the linearity of unloading paths, the onset of damage at the apex of traction-separation curves should be regarded. In case of linear unloading, the unloading behaviour changes abruptly when coming from the non-linear reversible branch of the traction-separation curve to the softening one. Even an infinitesimal step over this border results in a finite hysteresis with associated dissipation, which could be hardly justified physically by an infinite small change of the damage state. This effect is demonstrated in (Liu et al, 2; Roe and Siegmund, 23). Cyclic CZMs provide the capability to simulate subcritical crack growth. In these models, irreversible damage accumulation is controlled by an explicit damage evolution equation where an endurance limit can be incorporated (Bouvard et al, 29; Nguyen et al, 2; Roe and Siegmund, 23; Xu and Yuan, 29b; Yang et al, 2). While in monotonic CZMs the damage state is uniquely defined by the maximum separation attained during the loading history, cyclic CZMs need a more general damage variable. In the literature, stiffness-type (Nguyen et al, 2), separation-type (Yang et al, 2) and micromechanically motivated damage variables (Roe and Siegmund, 23) are sug-

3 Simulation of fatigue crack growth with a cyclic cohesive zone model 3 gested. For visualisation purpose, Ortiz and Pandolfi (999) propose an energetically motivated conversion of a separation-type damage variable into the range between zero and one. In contrast to monotonic CZMs, cyclic CZMs allow to specify an endurance limit to indicate a cyclic load level below which there is no damage initiation, see (Roe and Siegmund, 23; Siegmund, 24; Xu and Yuan, 29a,b). Therefore, additional material parameters are introduced in the damage evolution equation of the cited models mostly with the help of Heaviside step functions. Fatigue crack growth (FCG) and related overload effects are investigated by Bouvard et al (29); Nguyen et al (2); Roe and Siegmund (23); Siegmund (24), and Xu and Yuan (29b). The three stages of FCG rate curves has been reproduced: near threshold region, Paris line and static failure. While the static failure load without any doubt depends on the fracture energy density of the cohesive model, both Paris exponent and fatigue threshold load are found to depend on the endurance limit parameter. Except for these general statements, there are no further studies that clarify the relation between the characteristics of FCG rate curves and the properties of CZMs. Moreover, although the shape of the cohesive law is investigated severalfold (e.g. in the context of crack growth resistance of elastic-plastic solids (Hütter et al, 2; Tvergaard and Hutchinson, 992)), its influence on the simulation of FCG has, to the authors knowledge, not been studied before. This also applies to the assumptions with respect to the unloading behaviour. The present publication gives a contribution to these questions. A versatile cyclic CZM is presented. Monotonic and cyclic properties of the CZM are discussed. The applicability of the model to cyclic homogeneous stress-based loading and FCG under mode I loading conditions is demonstrated. In an extensive parametric study the influences of the parameters of the cyclic CZM with respect to the FCG rate curves are investigated systematically. Finally, a parameter identification strategy is proposed. 2 Monotonic Cohesive Zone Model The key feature of a CZM is the traction-separationlaw (TSL). It relates the traction vector, t i, to the separation vector, δ i, accounting for the current damage state. Both vectors consist of one normal and two tangential components indicated by the coordinate indices i=n, r, s. In consistency with the thermodynamical framework proposed e.g. in (Bouvard et al, 29; Roychowdhury et al, 22), a cohesive potential, Γ, is introduced in order to obtain the traction coordinates by partial differentiation with respect to the separation coordinates, t i = Γ ), i=n, r, s. () t ( δi δ The cohesive potential is a measure of the normalised stored reversible energy and captures the specific unloading behaviour, both considered below. Introducing the effective normalised quantities effective traction, τ, and effective separation, λ, τ = t t 2 n +t2 r +t2 s, (2) λ = δ δ n 2 + δ 2 r + δ 2 s, (3) as well as a damage parameter, D, the vectorial TSL reduces to a scalar relation τ(λ,d). The normalising parameters t and δ are the cohesive strength and the corresponding separation, respectively, see Fig.. The MacAulay brackets δ n = 2 [ ] δ n + δ n δ n, δ n =, δ n < (4) are introduced to ensure positive contributions of normal separation. The response of the model in the compression case is addressed below. Under monotonic loading conditions, the TSL describes an envelope in the normalised λ-τ-space bounding all admissible states (traction-separation envelope, TSE). Here, a generalised exponential approach based on Xu and Needleman (993) is used by introducing two shape parameters, ε and ω, λ exp( λ), λ < [ τ TSE = [ ] ] ε ω λ exp( λ), λ. (5)

4 4 St. Roth et al., Int J Fracture (24) 88:23 45, DOI:.7/s Some TSEs τ TSE (λ) are depicted in Fig. 2. The ascending branch of the TSE before reaching the cohesive strength refers to reversible material behaviour (subscript rev ) and does not depend on ε and ω. At the apex, damage initiates and develops with increasing displacement driven load λ. Since the maximum transferable traction degrades with increasing damage state, the descending branch of the TSE is called damage locus (subscript DL ). The shape parameters ε and ω only influence the shape of the damage locus. While the latter defines the plateau width of the TSE, ε affects the slope of its softening branch. This is demonstrated in Fig. 2, too. The area under the TSE represents the normalised fracture energy density, Γ, dissipated during the formation of new surfaces (cohesive fracture energy or work of separation per unit area normalised by t δ ). In Fig. 2, its value is kept constant by sizing ε in dependence of ω. The normalised fracture energy density can be decomposed into a reversible part, Γ rev, and the area under the damage locus, Γ DL, Γ = Γ rev +Γ DL. (6) While Γ rev represents a constant value, Γ rev = = λ= λ= τ TSE (λ)dλ λ exp( λ)dλ = e with Euler s number e, Γ DL depends on ε and ω, Γ DL (ε, ω)= λ= (7) τ TSE (λ)dλ. (8) Numerical solutions of Eq. (6) are depicted in Fig. 3. Note that the five parameters of the model t, δ, Γ, ε, and ω are not independent of each other. For the special parameter set ε =, ω = and Γ = e, the model of Xu and Needleman (993) is recovered. As stated above, under monotonic loading conditions damage evolution takes place exclusively on the damage locus. Below this curve, i.e. in unloading and reloading situations, reversible material behaviour is assumed. For these reasons, the position on the damage locus can be seen as a measure of the current damage state defined by the shape of the TSE and either the current effective traction or separation, respectively. Here, the latter variable is chosen since it allows τtse D=, D= Γ ω ={, 2, 5,, 5} ε ={,.569, 2.46, 3.85, 4.95} D, D λ Fig. 2 Exponential TSEs with constant normalised fracture energy density Γ and variation of the shape parameters ε and ω Γ Γ (ε, ω = const.) Γ = e Γ rev = e 2 ω ={, 2, 5,, 5} ε Fig. 3 Normalised fracture energy density Γ in dependence of the shape parameters ε and ω; dashed line: reversible part Γ rev = e 2; dot-dashed line: Γ = e, compare Fig. 2 for an unambiguous description. In the following, the damage state is defined by the separation-type fundamental damage variable D. Moreover, in order to interpret the damage state as an effective surface density of microdefects (Lemaitre, 996), the damage variable D known from damage mechanics is introduced. The conversion of D into D is described in Sect. 4. Nevertheless, the set of equations of the CZM is formulated in terms of D for sake of reducing computational costs. Note that the conversion is more or less a matter of postprocessing and does not affect the damage state. Besides the shape of the TSE, the cohesive potential is mainly determined by the unloading behaviour. Here we restrict to unloading towards the origin without any remanent separation. For undamaged

5 Simulation of fatigue crack growth with a cyclic cohesive zone model 5 states, un- and reloading paths follow the non-linear reversible branch of the TSE. In order to establish a smooth transition from reversible to damaged states, a non-linear approach is needed, which guarantees little changes of unloading behaviour in consequence of little changes of the damage state, as discussed in Sect.. Beyond that, the (non-)linearity of the unloading paths influences the amount of reversible energy, stored in the cohesive zone at a certain damage state. This is expressed in terms of the area under the path and the particular value of the cohesive zone potential, respectively. Besides the requirement that the unloading path and the reversible branch of the TSE coincide for the undamaged state D =, all paths have to be monotonically increasing. Furthermore, they are bounded by the TSE. A cohesive zone potential covering all these properties is found as Γ(λ, D)= Gκ F [ ] e [+λf]exp( λf), (9) wherein the functions F( D) and G( D) provide the specific shape of the TSE and the damage dependent unloading behaviour in terms of the fundamental damage variable D,, D< F( D)= D ( W ) e G κ, D, (), D< G( D)= τ TSE ( D), D. () The further parameter κ enters to describe the nonlinearity of the unloading behaviour. The Lambert function W(x) (also known as product logarithm) is defined implicitly by x=w(x)exp ( W(x) ) (2) and can be evaluated numerically. The use of W indicates, that F is more or less an inverse function of Eq. (5). Several unloading paths for different damage states and ε =, ω =, κ = are depicted in Fig. 4. Note that with increasing damage the non-linearity decreases. The dependence of F on the fundamental damage variable D is depicted in Fig. 5 for some combinations of the shape parameters ε, ω and κ =. The influence of κ is illustrated in Fig. 6. As shown there, the parameter κ determines the non-linearity of τ D= λ D Fig. 4 TSE (ε =, ω = ) and unloading paths (κ = ) from the damage locus at various damage states D = {,.,.5, 2, 2.5, 3, 4, 5, 6} F ω ={, 2, 5,, 5} ε ={,.569, 2.46, 3.85, 4.95} D Fig. 5 Functions F( D) for κ = and various combinations of ε and ω with Γ = e=const. the unloading paths and thus the amount of stored energy and dissipation, respectively. With decreasing κ the non-linearity decreases. For κ = the unloading paths intersect the damage locus horizontally leading to the largest cohesive zone potential and the lowest dissipation. Evaluating Eqs. () and (9), the normal and tangential coordinates of the traction vector result from partial differentiation of Γ with respect to the coordinates of the separation vector, t n = δ n G κ F exp( λf), (3) t δ t r/s t = δ r/s δ G κ F exp( λf). (4)

6 6 St. Roth et al., Int J Fracture (24) 88:23 45, DOI:.7/s τ D=2.5 κ ={,,.5, } λ Fig. 6 TSE (ε =, ω = ) and unloading paths from the damage locus ( D=2.5) for various κ With the help of Eq. (2), the relation between the effective quantities is obtained as τ(λ, D)=λG κ F exp( λf), (5) plotted in Fig. 4. Both branches of the TSE are recovered by Eq. (5), τ(λ, D=), λ < τ TSE (λ)= τ(λ, D=λ), λ, (6) which emphasises the key role of the cohesive potential. Note that the monotonic CZM is uniquely defined by Eqs. () (3) and Eq. (9). Comparably to the exponential CZM presented here, a bilinear one is deducible by replacing Eq. (9) with Γ bilinear (λ, D)= λ 2 (λ f D) 2 D(λ f ) (7) where λ f denotes the normalised effective separation at failure. In order to complete the monotonic CZM, the compression case is considered. Negative normal separations δ n < are penalised by a contact formulation by steeply rising the stiffness, t n t = µe [ ( exp Gκ F µ δ n δ ) ]. (8) For reasons of simplicity, friction under contact conditions has been neglected, compare (Roe and Siegmund, 23). The penalisation factor is chosen to be µ =. throughout all analyses presented in the following. 3 Cyclic Cohesive Zone Model In the monotonic CZM presented above damage evolution is restricted to the damage locus. Under cyclic loading with constant separation amplitude the model would therefore allow an infinite number of repetitions without any accumulation of damage. In order to simulate fatigue, the model has to be augmented by a damage evolution equation covering the following features: i) Un- and reloading paths do not coincide for D>. While damage keeps constant during unloading, it must evolve during subsequent reloading. ii) The maximum transferable traction decreases with increasing damage. iii) The damage locus still forms an envelope of all admissible states. There is damage evolution below the damage locus. Once the damage locus has been left, it is never reached again at constant amplitude displacement driven loading. This would require sufficient increase of the amplitude. So, the damage locus remains the upper bound of damage evolution. iv) The monotonic CZM forms a special case of the cyclic CZM. v) Comparable to the cohesive strength for the undamaged state, there is a local damage dependent endurance limit providing a lower bound for damage evolution. The behaviour of such a cyclic CZM is qualitatively depicted in Fig. 7. The damage evolution equation may depend on the damage state, the current separation, its rate, and the shape of the damage locus, D= D( D, λ, λ,...). Since damage evolution is an irreversible process, D is non-negative, D. Furthermore, a smooth transition from unloading to reloading is desirable. Therefore, damage evolution is assumed to scale with the distance to the damage locus. While damage evolution tends to the monotonic CZM with increasing load, it diminishes in the vicinity of the origin. In order to implement this, the ratio between the current separation and the fundamental damage variable seems to be an appropriate measure. It ranges between zero at the origin and unity at the damage locus. A one-parametric power-law approach is chosen as scaling function, which leads to the following dam-

7 Simulation of fatigue crack growth with a cyclic cohesive zone model 7 τ τ e I D= E λ e II III λ D Fig. 7 Initial(I), un(ii)- and reloading (III) paths described by the cyclic CZM including an endurance limit (E) (dashed line: damage locus, monotonic CZM) age evolution equation, [ ] ρ D= λ. (9) λ D The damage exponent ρ controls the influence of the current state, the MacAulay brackets ensure the positiveness of D, and restrict damage evolution to loading conditions only. Equation (9) features no endurance limit yet. In order to modify Eq. (9) to include an endurance limit, let us consider again the monotonic CZM. Here, the cohesive strength t and the respective separation δ define the endurance limit for the initial undamaged state. Below t, there is reversible material behaviour described by the ascending branch of the TSE. Once damage has initiated, the respective endurance limit decreases according to the damage state at the damage locus. Note that it is the immanent characteristic of monotonic CZMs that the endurance limit does only change as a consequence of an increasing cyclic load level or due to single overloads. In contrast, within a cyclic CZM a lower endurance limit exists that indicates a cyclic load level below which there is no damage initiation. The endurance limit enters the damage evolution equation via a Heaviside step function, which prevents damage initiation and accumulation below the endurance limit. The initial endurance limit τ e is introduced as a further material parameter. Once damage has initiated, the argumentation with respect to the monotonic CZM leads to the necessity of a suitable approach to describe the damage dependency of the endurance limit. Moreover, the postulated smoothness of the model at damage initiation used above to motivate the non-linear unloading behaviour, ends up in the assumption that a local state dependent endurance limit must exist, as well. Starting at the initial endurance limit, indicated by the initial endurance limit τ e at the TSE, the expected endurance limit vanishes for D. In the following, the endurance limit is assumed to depend solely on the damage state. The endurance limit can either be expressed in terms of effective traction τ e or effective separation λ e, see Fig. 7. Again, a simple power-law approach is assumed scaling the damage locus (compare Eq. (5)), τ e ( D)=τ e Gα (2) with the initial endurance limit τ e and the endurance exponent α. The dependency of τ e with respect to the new model parameter α is illustrated in Fig. 8. The respective endurance separation is found with Eqs. (5) and (2) from τ e ( D)=λ e G κ F exp( λ e F) (2) as λ e = ( ) F W τe e Gα κ. (22) Respective curves λ e ( D) are depicted in Fig. 8, too. Note that the endurance separation behaves nonmonotonic during damage evolution. The endurance exponent α determines the relation between the maximum transferable traction at the specific damage state and the minimum load to continue damage evolution. This consideration directly leads to the endurance locus τ e (λ e ), which envelops all endurance states comparably to the damage locus. It is found by eliminating D in Eq. (2) with the help of Eq. (22). A parameter plot is depicted in Fig. 9 (parameter D ). Each of the depicted curves forms a lower bound of damage evolution. All states to the left of the endurance locus are endurable. Damage may only evolve between endurance locus and damage locus. For α = and τ e = the endurance locus and the damage locus coincide. This describes the special case of the monotonic CZM where damage evolution is restricted to the damage locus. The endurance limit is embedded in the damage evolution equation Eq. (9) by multiplying a Heaviside

8 8 St. Roth et al., Int J Fracture (24) 88:23 45, DOI:.7/s τ e, λ e τ e α ={.,.5, 2., 5.} τ D.5 D D λ λ e = λ res λ 2 λ Fig. 8 Endurance limit τ e ( D) (solid) and endurance separation λ e ( D) (dashed) for various endurance exponents α and τ e =.8, ε =, ω =, κ = Fig. Initial monotonic loading from the origin to λ = D at the damage locus, subsequent unloading with constant damage down to λ and reloading with damage evolution until λ = λ 2, D= D 2 ; damage evolution starts at the endurance locus, λ res = λ e ( D ) τ e τ e τ e =.8 α ={,.,.2,.5, 2, 5, } Fig. 9 Endurance locus τ e (λ e ) for various endurance exponents α and τ e =.8, ε =, ω =, κ = (solid lines); dashed line: TSE with damage locus step function, H(λ λ e ), which restricts damage evolution to cases of λ > λ e, [ ] ρ D= λ H(λ λ λ D e ). (23) This evolution equation allows an analytic integration once the evaluation of the MacAulay brackets reveals λ and the applied λ exceeds the current endurance separation, λ > λ e ( D). When starting at D = D, the fundamental damage D 2 reached after loading from λ to λ 2 is calculated by D 2 = ρ+ λ ρ+ 2 + D ρ+ λ ρ+ res (24) λ e with λ res being the effective separation at which damage accumulation is resumed, λ res = max ( λ, λ e ( D ) ). (25) Here, the index indicates known quantities at the beginning of the loading process and the index 2 at the end, respectively. By means of Eq. (25), the starting point of damage evolution is shifted to the endurance separation. This situation is illustrated in Fig.. Herein, an unloading step from the damage locus down to λ precedes the considered load step. Note, since dλ e dλ does not apply in the whole range of D and for all parameter combinations, λ λ e ( D 2 ) has to be checked after analytical integration of Eq. (23). In case of λ < λ e ( D 2 ), D 2 is recalculated with the inverse of Eq. (22), D 2 = D(λ e = λ 2 ), i.e. by mapping on the endurance locus. Note that within the cyclic CZM, the initiation value of the fundamental damage variable is shifted from one to λ e ( D = ). For D <, damage accumulates according to Eq. (23) but without any effect on neither the endurance limit nor the un- and reloading path, compare Eqs. () and (). 4 Energetic Considerations In damage mechanics, damage states are commonly characterised by a damage variable D that ranges between zero and one indicating the undamaged state and material failure, respectively. Since the presented

9 Simulation of fatigue crack growth with a cyclic cohesive zone model 9 cyclic CZM is formulated in terms of the fundamental damage variable D, an appropriate conversion D( D) is needed. This relation distributes D at the damage locus. Ortiz and Pandolfi (999) propose an energetically motivated approach. According to this, damage is defined as the ratio between the cohesive potential evaluated for the maximum attained separation and the fracture energy density. In the view of the authors, the application of this approach to the present CZM is problematic because of two reasons. Firstly, in this definition damage is not limited to the damage locus but rather distributed at the whole TSE including the reversible branch. Secondly, the meaning of the measure seems questionable. While in the CZM of Ortiz and Pandolfi (999) the cohesive potential amounts the work done in a monotonic loading process starting at the origin up to the current maximum separation (by following the TSE), the cohesive potential in the present CZM is interpreted as a measure of the reversible or stored energy. Both quantities are not directly related to damage. Since damage accumulation is an irreversible and dissipative process, the ratio between the dissipation density D and the fracture energy density D/Γ (both normalised) is suggested as a more convenient damage measure. Unfortunately, under cyclic loading conditions the amount of dissipated energy becomes history dependent and material failure does not necessarily occur at D = Γ. However, both conditions are fulfilled under monotonic loading conditions, which leads to the modified definition D := D mon Γ. (26) Thus, the damage state after arbitrary loading history is associated with a monotonic loading process that leads to the same damage state (i.e. the same D) by dissipating D mon. Moreover, the maximum load of the comparable monotonic process ˆλ exactly equals the fundamental damage for which the conversion into D is searched for. Consequently, the relation D( D)= D mon(ˆλ = D) Γ (27) is to be evaluated. The respective distribution of D at the damage locus is depicted in Fig.. In order to determine D, an arbitrary reloading process including damage evolution is considered first τ D=. D.3.5 Γ D.7.9 D λ Fig. Distribution of damage variable D at the damage locus (ε =, ω = ) and respective unloading paths (κ = ); highlighted areas refer to D=.5 and cover the same area demonstrating D=D/Γ =.5 (λ λ 2, D >, depicted in Fig. 2). The area under the loading path between λ and λ 2 covers the normalised total work, W 2 tot, done in this step, compare Fig. 2: A-C-D-B. Furthermore, the cohesive potential Γ( D, λ) introduced in Sect. 3 describes the normalised stored energy density of a damaged state at a given deformation. This amount of energy could be recovered by total unloading. In λ-τ-space, Γ is identified by the area under the unloading path, Eq. (5). Regarding the loading process considered, the material point possesses the stored energies E rev = Γ( D, λ ) (Fig. 2: O-A-B) and E2 rev = Γ( D 2, λ 2 ) (Fig. 2: O-C-D) at initial and final states, respectively. According to Fig. 2: E-D-B, the energy dissipated during the loading step is found as the difference between the sum of stored energy at the beginning and total work on the one hand, and the stored energy at the end on the other hand, D = E rev +W tot 2 Erev 2. (28) While the cohesive potential is defined analytically by Eq. (9), there is no closed solution available for the integral over the loading path, λ2 ( W 2 tot = τ λ, F ( D(λ) ), G ( D(λ) )) dλ. (29) λ=λ Instead, Eq. (29) is solved numerically. Note that dissipation is intrinsically associated with damage evolution. In contrast, for un- and reloading processes at constant damage between points E and D, the total

10 St. Roth et al., Int J Fracture (24) 88:23 45, DOI:.7/s τ D D 2 D B E A C O λ λ 2 λ β, γ β(ε, ω, κ) ω ={, 2, 5} γ(ε, ω, κ) κ ={, } κ ={, } κ ={, } ε Fig. 2 Unloading at constant damage ( D = D ) and reloading (λ = λ... λ 2 ) with damage evolution D = D... D 2 ; energetic quantities: E rev O-A-B, E2 rev O-C-D, W 2 tot A-C-D-B, D E-D-B work equals the difference of the respective reversible energies. Pure changes in the direction of the traction and separation vector at constant effective quantities (and thus unchanged damage) are assumed to be reversible. The dissipation density D mon of a process loaded monotonically from the origin up to λ = ˆλ reduces to D mon = ˆλ λ= τ TSE (λ)dλ Γ( D= ˆλ, λ = ˆλ), (3) wherein Eqs. (5), (28) and (29) are used. The combination of Eq. (27) with Eq. (3) evaluated for ˆλ = D now enables the conversion from D to D. Due to the embedded integral again a closed form solution is not available. That is why D( D) is fitted with a two-parametric approach, D= exp ( ) D γ. (3) β Because the shape of the damage locus in Eq. (5) depends on ε and ω, and the κ-dependent unloading path, the distribution of the damage variable, D, and thus both of the fit parameters β and γ in Eq. (3), depend on ε, ω and κ. Some numerically obtained results for β(ε) and γ(ε) in the range of. ε are depicted in Fig. 3 for selected combinations of ω and κ. Obviously, ω and κ do not affect β. The latter decreases with increasing shape parameter ε. In contrast, γ increases with increasing ε, ω or κ, respectively. Fig. 3 Fit parameters β (solid) and γ in dependence of ε =..., ω ={, 2, 5} and κ ={(dot-dashed), (dashed)} 5 Model Response to Cyclic Homogeneous Stress-based Loading In order to investigate the response of the cyclic CZM to cyclic homogeneous stress-based loading, a large number of simulations was performed by varying the mean stress level and stress amplitude in the ranges of σ m /t.95 and <σ a /t, respectively. This covers load ratios in the range of R <. Comparable to (Roe and Siegmund, 23), normal loading (superscript N ) as well as shear loading (superscript S ) was considered. The parameters of the cyclic CZM were chosen as ε =, ω =, κ =, ρ =, τ e =.25 and α = 2. The predicted number of cycles to failure, N f, at a certain mean stress level are plotted in a semi-logarithmic Wöhler diagram (S-N diagram) shown in Fig. 4. The expected continuously decreasing S-N curves are obtained as known from experimental investigations, e.g. in (Suresh, 998). With increasing mean stress, the number of cycles to failure decrease at a constant load amplitude. For σ m =, σ a (N f = )/t = applies while the endurance load amplitude is found as σ a (N f = )/t = τ e =.25 as expected. Noticeable differences between normal and shear loading are only found for low mean stress levels. For higher mean stresses the curves for normal and shear loading can even not be distinguished in Fig. 4. Respective constant-life plots for normal and shear loading are depicted in the Haigh diagram in Fig. 5 for N f = {, 2, 5,, 2, 5,, 2, 5}. Between the upper and lower bounds, σ a (N f = )/t = σ m /t and σ a (N f = )/t = κ σ m /t, each of the constant-

11 Simulation of fatigue crack growth with a cyclic cohesive zone model life plots shows a characteristic shape. However, the curves differ from the Goodman relation where it is assumed that all curves are linearly leading to the point (,) in Fig. 5. The curves are dominated by a region defined by a mean stress sensitivity (after Schütz (967), M = [ σ a (R= ) σ a (R=) ] /σ m (R=)) of M=. From Haigh diagrams we know that this corresponds to an exclusive dependence on the upper load level of constant amplitude cyclic loading. Regarding the cyclic CZM, the deviation from the Goodman relation is caused by the following model assumptions: i) Under normal loading conditions there is no damage evolution in the compression case. ii) Unloading is assumed towards the origin. iii) Damage evolution requires a deformation or stress level beyond the local endurance limits, λ e or τ e, respectively. The first assumption applies only for normal loading, which causes the deviations of the corresponding S-N and constant-life curves for shear and normal loading in Fig. 4 and Fig. 5. In contrast, the assumption iii) leads, independently of the loading direction, to limit curves σ a /t = σ m /t ± τ e, which are both depicted in Fig. 5 (dot-dashed). Below the lower limit curve in the bottom right region, the constant-life curves can be approximated conservatively by a linear relationship of Goodman type. Thus, these curves are virtually bilinear. Such bilinear S-N curves are proposed in (Haibach, 989) to account for the mean stress influence of notched specimens. Nevertheless, Fig. 5 reveals that the present type of cyclic CZM is not able to predict convex constant-life curves (Goodman, Gerber,... ) due to its basic model assumptions. Note that this applies in the normal loading case even without the local endurance limit approach. Moreover, following (Smith, 942) such concave constant-life curves are associated with brittle metals. In contrast, the cyclic CZM proposed in (Xu and Yuan, 29b) captures the linear Goodman relation. It bases on the model presented by Roe and Siegmund (23) wherein unloading is assumed to be of elastic-plastic type leading to a residual separation (compare assumption ii). This enables non-negative separation in the compression case, i.e. no penetration at negative normal tractions. Thus, in this situation damage accumulates depending on the negative load ratio reducing the mean stress σ a N/S /t σ N/S m t ={,....9}. E+ E+ E+2 E+3 E+4 Fig. 4 Wöhler diagram: S-N curves for normal (superscript N, solid) and shear (superscript S, dashed) loading (τ e =.25, α = 2, ρ = ) for mean stress levels σm N/S /t = {,.,....9} σ a N/S /t N f N f ={, 2, 5,, 2, 5,, 2, 5} σm N/S /t Fig. 5 Haigh diagram: Constant-life plots for normal (superscript N, solid) and shear (superscript S, dashed) loading (τ e =.25, α = 2, ρ = ); dot-dashed: endurance limit curve: σ a /t = τ e σ m/t and bounds of region with mean stress sensitivity M = : σ a /t = σ m /t ± τ e (upper bound only for shear loading) sensitivities to M < (compare assumption i). Furthermore, in (Xu and Yuan, 29b) the Heaviside function, which is associated with the endurance limit, is not evaluated (compare assumption iii). Under these conditions, the linear Goodman relation is predicted. However, respecting the endurance limit leads to constant-life curves comparable to the presented ones of our model due to assumption iii). For that reason, the results presented above qualitatively match those in (Roe and Siegmund, 23) despite the different unloading behaviour.

12 2 St. Roth et al., Int J Fracture (24) 88:23 45, DOI:.7/s σ S a/t R= N f σa N /t Fig. 6 Biaxial fatigue envelopes for N f = {, 2, 5,, 2, 5,, 2, 5} with load ratio R = ; parameters: τ e =.25, α = 2, ρ = ; dashed: circle (σa N ) 2 +(σa N ) 2 = (τ et ) 2 for limit case N = ; dot-dashed: reference circle In order to compare both models also with respect to mixed-mode loading, simulations of homogeneous biaxial fatigue tests were performed. As proposed in (Roe and Siegmund, 23), in phase normal and shear loading was applied simultaneously with a constant load angle, φ = tan (σ S a/σa N ) and a load ratio of R= leading to zero mean stresses, σm N/S =. The resulting biaxial fatigue envelopes are depicted in Fig. 6. For N f = and N f = the envelopes are quarter circles with radii of σ a /t = and σ a /t = τ e, respectively. In between, the circular shape gets lost due to the minor damage accumulation under normal loading in the compression case as discussed above. The number of load cycles to failure in pure normal loading exceeds that of pure shear loading at constant load amplitudes. This applies to combined loading, too. In this case, the envelopes in Fig. 6 deviate from the circular shape looking compressed in direction of the shear component σa S. These results differ from those presented in (Roe and Siegmund, 23). As stated there, the resulting load amplitude for combined loading exceeds the load amplitudes for both pure shear or normal loading. This effect may possibly arise due to the different model assumptions concerning the unloading behaviour. 6 Mode I Fatigue Crack Growth This section addresses the application of the cyclic CZM to mode I fatigue crack growth (FCG) under K I - controlled external loading conditions. 6. Boundary Layer Model The crack is modelled with a boundary layer formulation. Details on the finite element implementation are summarised in (Roth and Kuna, 23). The geometry of the boundary layer is defined by the radius of the outer rim, R o = 8 δ (Fig. 7). The outer rim of the half model is meshed with 2-3 finite elements. The matrix material is assumed to be isotropic and linear elastic with Young s modulus E = t and Poisson s ratio ν =.3. At the ligament, cohesive zone elements are applied. A fine meshed region in the vicinity of the crack tip with a width of L a R o allows the evaluation of FCG under constant (far-field) loading conditions. For lower load levels, a fine meshed region of width L a = δ with a cohesive zone element size of L f =.δ is chosen. In order to save computational costs, increased values of L f =.5δ and L a = δ are applied at higher load levels. The boundary layer is loaded by K I -controlled displacement boundary conditions according to the first term of the Williams series expansion of the far field solution in linear fracture mechanics with K I being the stress intensity factor of opening mode I. The applied cyclic load K I is determined in terms of load ratio R=K min I /KI max and maximum load K max I, K I = KI max KI min = KI max [ R]. (32) Under plane strain conditions, the energy release rate, J, and the corresponding cyclic load read J = ν2 E K2 I, (33) J = ν2 [ ] 2 [ KI max R 2]. (34) E Note that for R=, J and K I can be directly converted into each other by use of Eq. (33). The reference load in normalised form is associated with the fracture energy density J = t δ Γ leading with Eq. (33) to K t δ = E t Γ ν 2. (35)

13 Simulation of fatigue crack growth with a cyclic cohesive zone model 3 a) b) R o y x, a L f u r/ϕ (K I ) y ϕ L a x, r, a Fig. 7 a) Boundary layer model with K I -controlled displacement boundary conditions; b) detailed mesh within the fine meshed region, x L a (taken from Roth and Kuna (23)) The finite element mesh of the boundary layer half model and the boundary conditions are depicted in Fig. 7. The height of the cohesive zone elements is just for visualisation purpose and has no influence on the numerical analyses. The presented cyclic CZM was implemented as FORTRAN subroutine using the ABAQUS UEL interface. For detailed information concerning the finite element formulation see e.g. (Ortiz and Pandolfi, 999). Within the FE calculations, the ABAQUS UAMP and URDFIL interfaces were used, too. The latter allows the controlling of time incrementation, output generation, as well as the simulation abort as described detailed in (Roth and Kuna, 23). 6.2 Fatigue Crack Growth Fatigue crack growth rate curves cover three characteristic stages with increasing load: the near-threshold stage (I), the Paris region (II), and static failure (III). Within the Paris region a power-law is presumed to relate FCG rates to the applied cyclic load (Paris and Erdogan, 963), ) d( a [ ] m δ dn = C KI. (36) K Here, two parameters are introduced: the Paris exponent, m, describing the slope of the straight FCG rate line in the double-log plot, and the (normalised) Paris coefficient, C, which defines its vertical position. In the following, cyclic load ranges are investigated, bounded by the assumed threshold load as well as the static failure load. Static failure occurs if the applied energy release rate J equals the fracture energy density J and KI max = K, respectively. Thus, the reference load forms the upper bound of KI max. The minimum value of KI max that still leads to damage accumulation depends on the endurance properties. As described above, damage initiation is characterised by the point λ = λ e ( D = ), τ = τ e at the TSE and thus by the initial endurance limit. The cohesive potential at this point (evaluated using Eq. (9)) describes the corresponding normalised energy density to be applied for damage initiation. For this reason, Γ init is presumed to by a threshold value Γ init (τ e )=Γ ( D=, λ = λ e ( D=) ). (37) With Eq. (33) the respective stress intensity factor is found as K init E Γ init = t δ ν 2. (38) t Note that these initiation quantities do neither depend on the shape parameters ε, ω, the unloading parameter κ, nor the endurance exponent α. About 3 FCG analyses were performed to cover the resulting load range K init (τ e) Kmax I < K with fixed parameters ε, ω, τ e, α, κ, ρ, and R. Depending on the load level, each analysis took computing times of at least one day up to a week using four CPUs. The parametric study presented in the following was performed with an HPC cluster and extensive parallel computing. Further details on the reduction of the computational costs are found in (Roth and Kuna, 23). The postprocessing of the analyses requires a measure to define the crack extension a. The increment of the crack extension within one load cycle, d a, determines the FCG rate. Regarding the centre of the boundary layer as the initial crack tip position, one suitable formulation defines the current crack tip position at the location where no tractions are transferred any longer and D =, respectively (Siegmund, 24). Because of the asymptotic behaviour of the present exponential cyclic CZM, this seems not to be appropriate. Instead,

14 4 St. Roth et al., Int J Fracture (24) 88:23 45, DOI:.7/s D a) N d a δ x/δ a/δ K I τ b) d a δ x/δ Fig. 8 Profiles at the ligament with increasing number of cycles, N, and constant crack extension increment d a: a) damage distributions and b) respective effective traction distributions (taken from Roth and Kuna (23)) another damage based definition is applied, here. The crack extension is determined by the evaluation of the integral of the damage variable over the whole ligament, a= x= D(x)dx. (39) In Fig. 8 a) several damage profiles D(x) of subsequent load cycles are presented. Each profile consists of two parts, a transient region, D <, of constant width and a fully damaged region, D, whose width increases with d a = const. per load cycle indicating steady-state FCG. Since the damage process is situated at the transient region, it is known as active cohesive zone (Siegmund, 24). Note that its width depends on the particular definition of D (see Sect. 4). Other definitions of a (e.g. depending on the location of the maximum effective traction, see (Hütter, 23)) are possible but not considered here since they have no effect on the steady-state FCG rate d a. Profiles of distributions of effective tractions at the ligament τ(x) are shown in Fig. 8 b). In Fig. 9 stabilised FCG curves for increasing cyclic load levels are depicted. Independent of the load level, the cracks propagate with constant rates after some load cycles. Thereby, the crack extension in- N N Fig. 9 Fatigue crack extension, a/δ, in dependence of the number of load cycles N for increasing cyclic load K I (taken from Roth and Kuna (23)) crements increase with higher load levels leading to monotonically increasing FCG rate curves. In order to demonstrate the near-threshold behaviour, FCG curves at low load levels are depicted in Fig. 2. It shows that the crack growth retards before a stationary state is reached. The corresponding dependency of the crack extension rate d ( a/δ ) /dn with respect to the crack extension is plotted in Fig. 2. In the literature, such retardation is attributed to the short crack effect. Following e.g. Radaj and Vormwald (27), small cracks retard when they approach microstructural obstacles like grain boundaries or inclusions. Besides these general remarks, the ability of the cyclic CZM to predict FCG of microstructurally small cracks is to be addressed elsewhere. Furthermore, selfarresting cracks are found in Figs. 2 and 2 for loadlevels below a threshold value K th. In order to explain this effect and to formulate an applicable criterion for fatigue crack propagation, the J-integral proposed by Rice (968) is considered, J = t δ x= ( ) dλ(x) τ λ(x), D(x) dx. (4) dx It equals the applied energy release rate of the far field, provided that the matrix material is elastic. Note that the effective traction τ depends on the current damage state and thus on the history experienced by the particular material point. Under monotonic loading conditions, the fundamental damage variable D equals the effective separation λ. The integrand in Eq. (4) then becomes a total differential and the integration variable

15 Simulation of fatigue crack growth with a cyclic cohesive zone model 5 can be changed from x to λ. This allows the evaluation of Eq. (4) as the area under the traction-separationrelation at the crack tip, J mon = t δ λ t λ= τ TSE ( λ) d λ (4) with λ t = λ(x = ) being the opening at the initial crack tip. In cyclic loading situations, D is an independent state variable representing the loading history which has to be taken into consideration. Thus, Eq. (4) cannot be simplified to Eq. (4) but has to be evaluated as a line integral along the cohesive zone. It can be rewritten as J cyc = t δ Γ x= + x= λ(x) λ= τ D d D dx d λ dx, (42) wherein the first term inside the brackets describe the stored reversible energy at the initial crack tip. Once the initial crack tip has failed in the sense of the cohesive model, i.e. D(x = ), this term vanishes. Under constant amplitude loading conditions, steadystate FCG is reached and J cyc is uniquely defined by the (constant) damage profiles D(x) or D(x), respectively. Even more, note that the separation at the crack tip then is not determined by the current value of the far field energy release rate (i.e. J) but rather by its complete history. This effect is known as crack tip shielding in the context of plasticity, see (Ritchie, 988). In fact, the differences between Eqs. (4) and (42) arise from the dependence of the cohesive potential Γ on the damage state in terms of the fundamental damage variable D. In general, Eq. (4) can be interpreted as the normalised area under parametric traction-separationplots derived from τ-profiles and λ-profiles at the ligament (parameter x). The D-profile is taken into account indirectly via Eq. (5). Consequently, the present cyclic CZM offers the possibility to predict an arrest of the damaged zone provided that the τ-profiles and λ-profiles are completely covered by the respective endurance quantity profiles, i.e. whenever the whole ligament is marked as endurable. In these situations, further damage evolution requires an increase of the applied cyclic load. This effect is demonstrated in Fig. 22 where cyclic traction-separation curves are depicted for increasing cyclic load levels K I. In this a/δ K max I K max I < K th N Fig. 2 FCG curves in the near-threshold region: maximum load < K th causes crack arrest (non-propagating cracks) K max I d( a/δ)/dn E-2 E-3 E-4 E-5 K max I K max I < K th E-2 E- a/δ Fig. 2 FCG behaviour in the near-threshold region: crack extension rate vs. crack extension in dependence of the applied load level; for KI max < K th cracks arrest context, cyclic traction-separation curves are parametric traction-separation-plots generated with the help of profiles which have been evaluated at maximum applied load and steady-state FCG conditions. As can be seen, the τ-λ-curves approach the damage locus for K I K. On the other side, the lower limit curve representing the threshold value for FCG adhere to the endurance locus. It is found that the cyclic tractionseparation curves have to exceed this limit curve to establish steady-state FCG. Otherwise, the area at the ligament where the local endurance limit is not passed shields the damage zone from further evolution. The damage zone and thus the micro-crack arrest. These considerations demonstrate, that both the cyclic traction-separation-curves in the λ-τ-diagram and the respective enclosed area characterise the FCG