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Procedía Engineering 167 (2016) 97 - 102

Procedía Engineering

www.elsevier.com/locate/procedia

Comitato Organizzatore del Convegno Internazionale DRaF 2016, c/o Dipartimento di Ing. Chimica, dei Materiali e della Prod.ne Ind.le

Correlations Between Damage Accumulation and Strength Degradation of Fiber Reinforced Composites Subjected to

Cyclic Loading

Alberto D'Amore*, Luigi Grassia, Angelo Ceparano

The Second University of Naples-Sun, Department of Industrial and Information Engineering, Via Roma 19, 81031 Aversa (CE) Italy

Abstract

Composite materials subjected to cyclic loading degrade their strength and/or stiffness due to the accumulation of different overlapping damage mechanisms. Three substantial regimes of damage are commonly recognized. At the early stage of the loading a diffuse matrix cracking and matrix/fiber debonding (Stage I) precipitate in a saturation state termed "characteristic damage state" (CDS) that represents the initiation of fibers rupture and incipient delamination phenomena (Stage II). The final collapse is debited to the coalescence of macro cracks with resulting ply ruptures and delamination (Stage III). In this paper, summarizing the capabilities of a recently developed two-parameter phenomenological model based on residual strength, it is shown that the strength degradation kinetics can be described by three distinct functions associated to the sequence of damage mechanisms. Despite the phenomenological prerogative of the model, from the analytical approach it results that the multiple damage mechanisms develop simultaneously even with different kinetics and manifest their effectiveness at different time scales, accordingly. This highlights the hierarchical nature of damage accumulation in composites, from diffuse matrix cracking, to fiber/matrix interface failure to fiber and ply rupture and delamination.

© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-reviewunder responsibilityoftheOrganizingCommitteeofDRaF2016

Keywords:

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Organizing Committee of DRaF2016

doi: 10.1016/j.proeng.2016.11.674

1. Introduction

The phenomenological models developed for the prediction of fatigue properties of fiber composites do not provide any information about the damage development. They involve macro-stress components, the cycle-by-cycle change in stiffness and/or strength being predicted on the basis of empirical criteria. None of them shows any correlation with the physics behind the phenomenology of fatigue, namely the sequence of damage mechanisms that, through the degradation of stiffness/strength drive the materials to the final collapse. Indeed, most of them are largely unreliable in predicting simultaneously the fatigue life and the strength/stiffness degradation kinetics of composite materials while few [1] are occasionally more fit to purpose. Nonetheless, the phenomenological approaches constitute a viable solution for use in structural design reality, until the deeper understanding of fatigue damage mechanisms supports the development of more efficient and -most important- applicable design tools [1,2]. On the other hand, mechanistic/hierarchical models are based on specific failure criteria depending on the length scales and the sequence/interaction of the damage mechanisms. However, due to the complexity of the phenomenon and the diversity of material combinations they seem not sufficiently mature to describe the mechanical degradation arising from damage accumulation kinetics despite the effort spent on this task up to date. In a comprehensive review on the subject, Sevenois and Van Paepegem stated that "mechanistic models are currently not mature enough for commercial use. However, their promise of general applicability makes them very attractive for further development" [2]. Therefore, modelling the fatigue life and residual strength on the basis of simple and safe rather than complicated models remains open to new developments. The major difficulty to the development of new models arise from the prediction of residual strength, that is the strength measured on a sample subjected to cyclic loading under given loading condition. The reason is that the residual strength shows a progressive degradation depending on the cyclic loading severity, until it reaches the value of the maximum applied stress where failure occurs. However, the strength degrades smoothly in the first cycle's decades with a sudden drop occurring within a narrow cycle interval of about one or two decades. Moreover, fatigue is a stochastic phenomenon depending on both the statistical distribution of pre-existing defects, and the sequence of different damage accumulation mechanisms occurring at different length scales. Due to the pre-existing defects, even the static strength of composites is described statistically, normally adopting a two-parameter Weibull distribution function. The aspect related to preexisting defects is rarely tackled and requires to be better inspected when both phenomenological and mechanistic/hierarchical models are developed.

Failuitf

Percent of life 100

Fig. 1. Schematic of damage evolution during fatigue

On the other hand, the damage accumulation during fatigue follows the complex path as reported in Fig.1 [3,4]. Depending on the severity of loading, a diffuse damage of matrix is observed in the first cycles decades (Stage I). The damage develops rapidly to great extent, as measured, for instance, by penetrating liquids. Nonetheless, the

strength degrades only slightly in this same cycles interval and starts showing substantial degradation when different damage mechanism come to play. It is frequently reported [3,4] that at the end of Stage I the crack density saturates and the composite material reaches a "characteristic damage state" (CDS). Under these conditions fiber/matrix debonding, fiber breaking and local delamination are observed. The damage within Stage II develops for a larger number of cycles decades and is responsible of a measurable strength degradation. Finally, at Stage III, the coalescence of the diffuse damages gives rise to large extent of fiber rupture, delamination and ply rupture, resulting in the characteristic sudden drop of strength until it reaches the maximum applied stress when the final collapse occurs. From the above description it seems that the residual strength cannot be a metric of the amount of damage, so that models based on residual strength should not reflect the above features. In this paper we discuss the potential of a recently developed two-parameter model, based on strength degradation, showing its capabilities in capturing the sequence of damage accumulation, despite its phenomenological prerogatives.

Fig. 2. The residual strength derivative (open circles) and the absolute residual strength degradation with cycles (filled circles). The lines are theoretical predictions of the strength degradation curve and its derivative for a sample with initial strength <7ioJV = 470 MPa for T300/5280 [±35]2S graphite/epoxy laminates subjected to cyclic stress with amax-200MPa and R=0.1

2. Model synthesis

In this section we summarize the principal features of the wear-out model already proposed in [5,6]. The model is expressed by the following equation:

a0N = ao[l + (np - 1)(1 - R)cc\amax (1)

where omax{ 1 — R) = Aa= (amax — amin)is the cyclic stress amplitude, R = ^^ is the loading ratio, N is

the number of cycles to failure, and a and P are the model parameters. u0N represents the "virgin strength" of samples fatigued until failure and coincides with the experimentally determined static strength statistics, o0, represented by a two-parameter Weibull distribution as follows:

FCTo(x) = P(ct0 <x) = 1- exp[-(x/r)s] (2)

1 I W!-—'HHBJM 0,8

oy 0,6 0,4 0,2 0

Fig. 3. The three degradation mechanisms acting simultaneously.

1 10 1 00 1 000 1 04 1 05 1 06 1 07

where Fao (x) is the probability of finding a a0 value<x and y and 8 are the scale and shape parameters of the distribution function. By means of Eq.1, from the experimental number of cycles to failure, N, the virgin strength , a0N, of the fatigued samples can be calculated, given the loading condition, omax and R. Therefore, the parameters, y and 8, of Eq.2 can be derived from a series of fatigue life data. Further, from Eqs. 1 and 2 the cumulative distribution function of the number of cycles to failure under given loading conditions was derived as follows:

FN(ri) = P(N <n) = 1- exp\-

In [7,8], it was shown that, the new model of strength evolution with cycles can be represented by the following function:

—-= exp

From the model above the rate equation for the strength degradation can be derived in the following form:

Ktt amax)'

Yi(°i0N).

Yi(°i0N)

n(°i„N)

A = [l + (nP-1){1-R)o\amax ffj is the residual strength of a generic sample, n is the current cycle

is the scaling factor for the i-th sample with a "virgin" strength of aioN, a(y) is the reference strength (routinely taken at the upper tail of the static strength distribution) and y is the characteristic scale factor of the static strength distribution, already defined. It is worth mentioning that the model described so far has been tested on a large amount of experimental fatigue and residual strength data concerning different categories of composites. [7,8]. The model requires a limited set of fatigue data and reliably predicts the principal features of composites under fatigue, based on straightforward procedures. The two model parameters, a and p, can be easily optimized by best fitting the fatigue data through Eq. 1. The same equation can be used to recover the statistical distribution of the static strength, if not experimentally supplied, and thus, the scale and the shape factor of the Weibull distribution function, y and 8, respectively, can be obtained directly from fatigue data. Moreover, the experimental conditions, expressed by, amax and R, are known values, while aioN is the "virgin" strength of a generic sample, and as such can be arbitrarily fixed within the statistical distribution of strength. That is to say that the residual strength and its derivative, described by Eq. 4 and 5, respectively, can be easily obtained, the running variable being the number of cycles, n. In Fig. 2 the theoretical predictions in terms of residual strength and its derivative are reported for T300/5280 [±35]2S graphite/epoxy laminates [9] subjected to cyclic stress with umax=200MPa and R=0.1, a case already discussed in [8]. From the procedure outlined above the theoretical curves are obtained with the following set of parameters: a=0.11, p=0.18, y=426 MPa , 8=20 and aioN = 470 Mpa. To simplify, not to decrease the generality of the approach, we fix aioN = a{y) and thus, Yi(CTi0N)=Y . For the sake of highlighting the physical insights coming from the model (despite its phenomenological prerogative), from Eq. 5 three substantial mechanisms of strength degradation, acting simultaneously, can be isolated as follows:

A( n)=nP-i=:±.

f2(n) = exp

-1)(1 -R)a\Gn

Yi(Vi0N)

/3L J V rt(*low) >

3. Discussion

(8) (9)

The three functions, f1, f2 and f3, are reported in Fig. 3. The functions resemble three different damage mechanisms occurring at different timescales, roughly corresponding to the Stages I, II, III schematically described in Fig. 1. It can be argued that f1 is responsible of the diffuse damage that occurs to within the matrix. In fact, it appears that the damage associated to f1 evolves at the early stage of loading (Stage I) and vanishes very soon as frequently observed. As expected, from Fig. 2 it is observed that no substantial strength degradation can debited to the diffuse damage of the matrix associated with the function f1. Nonetheless, a clear manifestation of Stage I come from looking at the derivative of residual strength: for the loading conditions we illustrate, namely R=0.1 and amax=200MPa, the mechanism f1 vanishes around 1x103 cycles. Concerning f2 and its association to Stage II, again looking at the derivative of the residual strength, it is clearly visible that the effects of a second mechanism of strength degradation becomes effective around 1x103 cycles and extends its effectiveness, with measurable strength degradation, until roughly 1x106 cycles. Accordingly, f2 is a descriptor of the effect of damage on residual strength that can be associated to fiber rupture and incipient delamination. The third mechanism, f3, corresponding to Stage III of damage accumulation, takes place in the vicinity of failure and, for the case under observation, its evidence appears at 1x106 cycles as clearly visible. It is worth mentioning that the function f3 is un-effective until 1x106 cycles and this is proof of its physical meaning, as it can be associated to the sudden drop of strength until failure. The three functions described so far act simultaneously but become effective at different timescales witnessing the hierarchical sequence of damage accumulation.

4. Conclusions

The model described so far belongs to the category of phenomenological residual strength models. The model doesn't treat of damage mechanism of composite structure under cyclic loading. It only adopts macro-parameters to characterize its fatigue and residual strength properties straightforwardly. Nonetheless, the rate equation is composed of three simultaneous functions that appear well correlated with sequence of damage mechanism occurring in a composite subjected to fatigue loading. Thus, despite its phenomenological prerogative the model has the potential to reveal the physical aspects of damage sequences with association of analytical functions to the different Stages of damage.

References

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[2] R.D.B. Sevenois, W. Van Paepegem, Fatigue Damage Modeling Techniques for Textile Composites: Review and Comparison With Unidirectional Composites Modeling Techniques, ASME Appl. Mech. Rev, (2015) Vol. 67 pp 1-12.

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[4] Talreja R, Singh C V. Damage and failure of composite materials[M]. Cambridge University Press, 2012

[5] A. D'Amore, G. Caprino, P. Stupak, J. Zhou, L. Nicolais, Effect of stress ratio on the flexural fatigue behaviour of continuous strand mat reinforced plastics, Science and Engineering of Composite Materials 5, 1-8 (1996)

[6] G. Caprino, A. D'Amore, Flexural fatigue behaviour of random continuous-fibre-reinforced thermoplastic composites, Composites Science and Technology, vol. 58, (1998), p. 957-965, ISSN: 0266-3538

[7] A. D'Amore, M. Giorgio, L. Grassia, Modeling the residual strength of carbon fiber reinforced composites subjected to cyclic loading. Int. J. Fatigue 78, 31-37 (2015)

[8] A. D'Amore, L. Grassia, Constitutive law describing the strength degradation kinetics of fibre-reinforced composites subjected to constant amplitude cyclic loading, Mechanics of Time-Dependent Materials February 2016, Volume 20, Issue 1, pp 1-12 .

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