Methods for the prediction of fatigue delamination growth in composites and adhesive bonds: A critical review

TL;DR: In this paper, an overview of the development of methods for the prediction of fatigue driven delamination growth over the past 40 years is given, and four categories of methods are identified: stress/strain-based models, fracture mechanics based models, cohesive zone models, and models using the extended finite element method.

About: This article is published in Engineering Fracture Mechanics.The article was published on 2013-11-01 and is currently open access. It has received 225 citations till now. The article focuses on the topics: Delamination & Extended finite element method.

Summary

1. Introduction

• Over the past fifty years the quest to produce ever lighter structures has led to a greatly increased use of both composite materials and adhesive bonding.
• Fitting parameter σ Stress (MPa) τ Shear stress (MPa) φ Potential ψ Mode-mix angle (deg) Subscripts a Amplitude c Critical eff Effective eq Equivalent I Mode I II Mode II III Mode III m Mean max Maximum min Minimum th Threshold tot Total philosophy will allow the design of lighter structures.
• Possible avenues for further research will also be identified.
• When examining the literature on delamination growth an obstacle faced by the researcher is that delaminations can grow both due to quasi-static loads and due to cyclic loads.
• Vice versa, fatigue growth models are generally not capable of predicting static delamination growth.

2. Classification of the Methods for Prediction of Delamination Growth

• Many methods and models have been proposed for the prediction of delamination growth.
• They can be roughly grouped into four classes: 1. Stress/strain based methods.
• These relate the delamination growth to the stress or strain in the material.
• XFEM is a technique which allows discontinuities to exist within a finite element, rather than only at the boundaries, by using so-called enrichment functions.
• The fracture mechanics based methods were first proposed for use in delamination problems in the 1970s [6, 7], following their successful introduction to deal with fatigue crack growth in metals a decade earlier.

3. Stress/strain based methods

• Stress/strain based methods have long been used to determine the strength of adhesive bonds.
• Generally stress/strain methods are only used to find a fatigue life, and not predict delamination growth.
• This equation was successfully used to predict S-N curves for a number of specimens.
• Thus equation 1 is equivalent to using the stress intensity factor to correlate with delamination growth rate when m = 0.5n.
• This condition does not hold for all geometries.

4. Fracture Mechanics Based Methods

• The fracture mechanics based methods link delamination growth to the fracture mechanics concepts of stress intensity factor (SIF) and strain energy release rate (SERR).
• It is important to note that these two parameters are equivalent and that using one rather than the other does not provide more or different information.
• During experiments the SERR can relatively easily be obtained by measuring the change of compliance with crack length (dC/da).
• A concise overview of these early developments has recently been provided by Jones et al. [25].
• In essence, most fracture mechanics methods for delamination growth are based on the Paris relation, written using the SERR, and modified to a greater or lesser extent.

4.1. Early development

• The first to apply the Paris relation to fatigue delamination growth were Roderick et al. [6, 29], who studied what would now be called fibre metal laminates, and Mostovoy and Ripling [7], who worked on adhesively bonded joints.
• Other contributions include those of Jablonski [39] and of Brussat and co-workers [40, 41].
• Furthermore Mall et al. [82], and Rezaizadeh and Mall [37] found that the use of Gmax produced different curves for different R-ratios, whereas when they used ∆G, the curves collapsed into a single line regardless of R-ratio.
• This shows that keepingGmax or ∆G constant and changing the R-ratio implies changing Gm. Thus Gmax and ∆G by themselves do not give sufficient information on the applied stress cycle to be able to fully characterise the delamination growth.
• In reference [2] delamination growth rates were plotted against ∆K for three different R-ratios.

4.3. Mode Mix

• The delamination growth behaviour of a material is known to depend upon the mix of opening modes.
• Gustafson and Hojo [50] suggested a similar relationship, but using ∆G rather than Gmax.
• Blanco et al. [97] analysed experiment data from the literature [107] that showed that the mode-mix dependence was non-monotonic.
• An understanding of the micro-mechanics may point the way towards understanding the macroscopic behaviour.

4.4. Normalisation of the SERR

• The existence of such a relationship must be questioned however, as it has been shown that there is no direct correlation between resistance to static delamination growth (i.e. fracture toughness) and resistance to fatigue delamination growth [7, 67, 86, 99].
• At first they proposed determining GR by a ‘re-loading’ approach, where after a certain amount of fatigue loading the specimen is quasi-statically loaded until delamination growth is observed [118].
• Giannis et al. [120] and Murri [121, 122] have proposed normalisation by GR as a method to account for fibre bridging during mode I fatigue testing.
• Recent fractographic evidence [123] suggests that this is in fact not the case.

4.5. Effect of environmental and testing conditions

• Rans et al. [130] reported non-monotonous behaviour.
• In their experiments the delamination growth rate at low temperature (-40 ℃) was lower than that at elevated temperature (70 ℃), but higher than the room temperature rate.
• Of the researchers discussed above, only Burianek and Spearing [129] attempted to find a quantitative relationship between temperature and growth rate.
• The effect of moisture on delamination growth was investigated by Chan and Wang [83] and Russell and Street [134].

4.6. Full SERR range models

• This relation generally holds over a certain range of values of Gmax or ∆ √ G. However already during the early investigations of Mostovoy and Ripling [7] it was known that the full fatigue behaviour has a sigmoidal shape, as shown in figure 1.
• Abdel Wahab et al. [144] also used a relation similar to equation 37 to find the fatigue life of adhesive bonds, but included a mode-mix dependence.
• Giannis et al. [120] also present a model of the form of equation 37, but again the experimental data does not clearly show a threshold or unstable growth region, making it difficult to evaluate the performance of the model outside the log-linear region.
• To generate a full SERR range model, the value of the fatigue delamination threshold Gth, i.e. the SERR value below which no delamination growth occurs, is required.
• A true understanding of the physics of the delamination problem should allow the formulation of a model that covers the entire range of delamination behaviour from the growth threshold through to unstable delamination growth.

4.7. Variable Amplitude Fatigue

• Delamination growth due to variable amplitude (VA) fatigue has so far not received much attention.
• Investigations have been reported by Schön and Blom [147] and Bathias and Laksimi [49] who studied FRPs, and by Marissen [73, 148] and Khan et al. [149] who examined VA delamination growth in FMLs.
• This investigation indicated that history effects were relatively small, but suggested that further work is necessary to check the robustness of the results to variations in material and lay-up.
• As delamination growth interacts with other damage modes, such as transverse cracking [64, 65], it seems somewhat unlikely that linear damage accumulation would hold for all materials and load cases, since that would imply the absence of any history or interaction effects.
• More research is therefore needed on this topic.

4.8. Other Approaches

• A number of researchers have suggested fracture mechanics based approaches that do not fit into the categories described above.
• Schön claims that this proposed approach will greatly reduce the amount of work needed to characterize the fatigue delamination growth [151].
• Furthermore, Schön’s model does not account for the effect of the R-ratio (possibly due to using ∆G rather than ∆ √ G) or mode-mix, making the value of this approach limited.
• Wimmer and Pettermann employed the growth relation proposed by Dahlen and Springer in a semianalytical model of delamination growth [157].
• The SERR is computed at each node along the delamination front.

4.9. Concluding remarks on the fracture mechanics based methods

• Starting with the efforts of Roderick et al. [6], and Mostovoy and Ripling [7], the fracture mechanics based methods have undoubtedly been successful at describing fatigue delamination growth behaviour.
• However this success can be largely ascribed to the presence of “constants” that are to be empirically derived in these methods.
• As a result, a wide range of models have been proposed to deal with a variety of factors such as the R-ratio effect, mode-mix and environmental influences.
• Only rarely are these models based on a consideration of the physics of the problem however.
• Fracture mechanics based models have shown great potential to predict delamination growth, but they need to be tied into the physics to allow true understanding of the material behaviour.

5. Cohesive Zone Models

• Like the VCCT, the CZM approach is a finite element method.
• When employing the VCCT in a predictive model, remeshing is required as the crack advances.
• In the CZM approach this is avoided, by modelling the interfaces along which delaminations are expected to grow using cohesive zone elements.
• These elements are not linear elastic, but follow a prescribed traction-displacement relation.
• Often some kind of damage parameter is used to progressively reduce the stiffness, simulating damage growth within the element.

5.1. Early CZM approaches

• The basis for CZMs were the cohesive zone formulations developed by Dugdale [166] and Barenblatt [167].
• Camanho et al. have also provided a review of the early developments of CZMs [170].
• To model fatigue delamination growth an irreversible stiffness reduction must be added to the CZM formulation [171].
• Roe and Siegmund [171] proposed a damage parameter that is applied to the traction-separation behaviour and thereby also governs the unloading and reloading behaviour.
• Here D is the damage parameter, which is used to reduce the stiffness of the cohesive element.

5.2. Further developments

• Robinson et al. [180] proposed a damage parameter that was split into two parts: one for the static portion of delamination growth and one for the fatigue portion.
• For the fatigue component, the model of Blanco et al. [97] (see equations 31 and 32) is used to predict the delamination growth rate.
• Harper and Hallet present a comparison between their model and experimental data reported by Asp et al. [107].
• Recently Landry and LaPlante have also proposed a model linking fracture mechanics and CZM [189], with the damage parameter depending on a modified Paris relation applied to GImax.
• The model was then validated by experiments on a second configuration.

5.3. Concluding remarks on the CZM

• The CZM approach suffers from the same shortcomings as the fracture mechanics based methods, i.e. a lack of grounding in an understanding of the physics underlying the delamination process.
• For that reason the tripping traction is often treated as a penalty parameter, i.e. an assumed value is used [190].
• Of these parameters only the values of Gic are determined experimentally, the others are chosen based on numerical considerations.
• Without comparing both the CZM and the fracture mechanics approach with experimental data it can not be ascertained whether the CZM approach or the fracture mechanics approach (based on the VCCT for the SERR calculation) produces a better delamination growth prediction.

6. Extended Finite Element Method

• The extended finite element method (XFEM) is a “meshless” finite element technique that allows more flexible modelling of crack growth.
• Thus crack growth can be simulated without the need to predefine a crack path or crack plane.
• As the experimental data had also been used to generate the model inputs.

7. A note on applicability to real structures

• The vast majority of the models mentioned in this paper were developed and validated in the lab at coupon level, often using standardised specimens such as the DCB [209], and mixed mode bending (MMB) [210] specimens.
• In addition the specimens under consideration in the developments of the discussed models are usually flat, whereas real structures, especially in aviation, are often singly or doubly curved.
• Some steps have already been taken towards bridiging the gap between laboratory specimens and real structures, with investigations on delamination initiation and growth in the presence of a hole [58], or following an impact [59].
• Others have investigated the growth of circular delaminations [60], or of stacking sequence or fibre orientation [71, 72].

8. Conclusions and recommendations for future research

• Over the past forty years a great variety of models has been proposed for delamination growth due to fatigue loading.
• Starting from the Paris relation for fatigue crack growth in metals, the first class of methods to be developed were those based on LEFM.
• This disguises the lack of understanding of the physical processes.
• Fractographic investigations may assist by elucidating which mechanisms are at work, and under what conditions other mechanisms may be activated, or become dominant.
• A more careful consideration of the energy balance during delamination growth may lead to a greater understanding of the delamination process and by what variables it is governed.

Figures

Figure 1: The sigmoidal behaviour of the delamination growth rate. Three regions can be observed. At low values of G there is threshold or sub-critical behaviour (region I), at intermediate values the behaviour is log-linear (region II) and at high values unstable growth occurs, ultimately resulting in (quasi-)static failure (region III).

Figure 2: A typical cohesive model. T is the traction, δn is the normal (mode I) displacement, and δs is the shear (mode II and III) displacement. The indicated quantities (initial stiffness K0, critical SERRs GIc and GIIc, and the loci of initiation and propagation) must be assumed or determined experimentally.

Frequently asked questions

Q1. What are the contributions in "Methods for the prediction of fatigue delamination growth in composites and adhesive bonds - a critical review" ?

Pascoe et al. this paper classified fatigue life models into three major categories: fatigue life, phenomenological models, and progressive damage models. 

Q2. What future works have the authors mentioned in the paper "Methods for the prediction of fatigue delamination growth in composites and adhesive bonds - a critical review" ?

As these models include several parameters that must be determined by curve fitting, a wide array of variations can be fit to experimental data, especially if a limited dataset is used. The authors suggest that future efforts should not focus not on adding yet more variations on the Paris relation to the collection of models that can be fit to experimental data. Fractographic investigations may assist by elucidating which mechanisms are at work, and under what conditions other mechanisms may be activated, or become dominant. An illustrative example is the work of Khan [ 211 ], where a quantitative examination of fractographic features suggested that the monotonic ( i. e. Gmax ) and cyclic ( i. e. ∆ √ G components of the SERR should be superimposed to predict the delamination growth rate. 

Q3. What has been used to investigate the effect of laminate strengthening?

Equation 7 with f (G) = Gmax has been used to investigate the effect of laminate strengthening by interlayer strengthening [53–55] and through the thickness reinforcement [56, 57]. 

Q4. How many cycles can be used to determine the threshold?

If a specimen is used in which the SERR reduces with increasing delamination length (e.g. a double cantilever beam (DCB) specimen), the threshold can be determined by continuing the fatigue test until no further growth is observed after a set number of cycles. 

Q5. What were the main problems studied in these investigations?

The main problems studied in these investigations were the stress (and/or strain) distribution in the adhesive joint and the definition of suitable quasi-static failure criteria. 

Q6. What is the traction displacement criterion used by Landry and LaPlante?

The traction-displacement behaviour was defined such that softening starts at a displacement valueδ0i where the matching peel or shear traction exceed the critical peel or shear values, i.e:max [ 〈tI〉 TI , tII TII ] ≥ 1 (59)Failure displacement δfi was defined using the Benzeggagh-Kenane fracture criterion [179] also used by Camanho et al [178]. 

Q7. What is the principle of the use of fracture mechanics to predict the delamination growth rate?

The use of fracture mechanics to predict the delamination growth rate is based on the similarity principle, i.e. the principle that if the SERR or SIF is equal for two delaminations, (whether in the same or different specimens), they are experiencing the same driving force and thus will have the same growth rate. 

Q8. Why is the prediction difficult to judge?

Due to the large scatter bands and low number of specimens in the experimental results the quality of the prediction is difficult to judge however. 

Q9. What is the main reason why the SERR is preferred for delamination growth modelling?

The difficulties encountered in calculation of the SIF for inhomogeneous layered materials, such as fibre reinforced polymers (FRPs), have made the SERR the preferred parameter for the modelling of delamination growth. 

Q10. What was the main reason for the change in the locus of failure of the adhesive bond?

This analysis suggested that moisture changed the locus of failure of the adhesive bond, thus causing a different failure mechanism (with a different SERR vs growth rate relationship) to be dominant. 

Q11. What is the reason for the lack of connection to the underlying physics?

This lack of connection to the underlying physics can lead to issues such as the observation of an R-ratio effect when an incomplete description of the stress cycle is used as a similarity parameter. 

Q12. What has been used to correlate delamination growth in the case of high cycle fatigue?

∆G has been used to correlate delamination growth in the case of high cycle (>108 cycles) fatigue [64– 66] and to investigate the influence of the matrix [67–69]. 

Q13. What can be done to help in elucidating the mechanisms at work?

Fractographic investigations may assist by elucidating which mechanisms are at work, and under what conditions other mechanisms may be activated, or become dominant. 

Q14. What is the main reason for the success of the fracture mechanics based methods?

Starting with the efforts of Roderick et al. [6], and Mostovoy and Ripling [7], the fracture mechanics based methods have undoubtedly been successful at describing fatigue delamination growth behaviour. 

Q15. What is the damage parameter used to predict the propagation of a crack?

After a crack is initiated, propagation is predicted using a damage parameter which is incremented based on the Paris relation (equation 7). 

Q16. Why is the XFEM approach less suitable for delamination growth within a composite?

Due to the difficulty of calculating the SIF in a composite material, such an approach may be less suitable for delamination growth within a composite. 

Q17. How can one model fatigue delamination growth using the XFEM?

By combining these formulations with a suitable damage parameter, as done in regular CZM approaches, it should be possible to model fatigue delamination growth using the XFEM.