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Polymer 53 (2012) 4525e4532

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Polymer

journal homepage: www.elsevier .com/locate/polymer

Use of poly (methyl methacrylate) in the study of fracture of randomly damagedsurfaces: I. Experimental approach

Hector Medina*, Brian Hinderliter 1

Virginia Commonwealth University, 401 West Main Street, Room E2232, Richmond, VA 23284-3015, USA

a r t i c l e i n f o

Article history:Received 28 May 2012Received in revised form31 July 2012Accepted 4 August 2012Available online 10 August 2012

Keywords:PMMARoughnessFracture

* Corresponding author. Tel.: þ1 804 827 3517, 1 80804 827 7030.

E-mail addresses: medinahe@mymail.vcu.ed(H. Medina), bhinderliter@vcu.edu (B. Hinderliter).

1 Tel.: þ804 827 3517; fax: þ804 827 7030.

0032-3861/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.polymer.2012.08.012

a b s t r a c t

Experimental approach and statistical theory are applied to broaden knowledge of materials failure byfracture, from initial service surface conditions and during damage accumulation, using PMMA beamswith repeated random surfaces, which were developed by computer controlled laser ablation, and thenbent to fracture. The Central Limit Theoremwas used to simulate dynamic degradation. Replicas allowedthe study of probability of failure location (PFL) versus parameters of the surface topography. Resultsshow that, depending on the auto correlation length (ACL), even at early stages of degradation, there ishigh correlation of PFL with RMS and average roughness, and two other proposed parameters. Thiscorrelation shows an asymptotic limit behavior that agrees with fracture mechanics results for brittlematerials. As important, a Sensitivity Transitional Region (STR), dependent on the ACL, was observed,below which the PFL is uncorrelated to the parameters studied; The STR’s dependence on temperature iscurrently being studied.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Poly methyl methacrylate (PMMA) has been successfully used inmany applications due to its excellent mechanical properties andlow cost [1]. Except for micro scale applications, PMMA can beeasily ablated without causing secondary mechanical effects on thesurface [2]; and it has been reported that PMMAhas a better cuttingedge surface quality than, for example, polypropylene and poly-carbonate [3]. It can be found commercially with relatively lowsurface roughness (10e200 nm) [1], so evenminute damage causedon it could be distinguished. Also, at room temperature it behavesbrittle and ductility starts to show above approximately 35 �C (seeFig. 2) [18]. Based on those characteristics, in this study, PMMA isused to broaden the knowledge ofmechanical failure ofmaterials atinitial service surface conditions and during random surfacedamage accumulation and environmental degradation. This isaccomplished by laser ablation of repeated random rough surfaceson PMMA beams and their mechanical analysis.

Early predictions of mechanical failure on surfaces that roughenrandomlywill be beneficial to severalfields includingDentistry [5,6],

4 245 7441 (mobile); fax: þ1

u, hedmedina@yahoo.com

All rights reserved.

Microelectronics, Nano Electromechanics [7e9], Coatings [15,25,26],Mineralogy [27,28], etc. Particularly, understanding the fracture ofpolymer coatings is vital to optimizing their design and servicing,and thus reducing the costs involved [10]. Muchwork in the fracturemechanics of coatingshasbeenadvancedbyNichols et al. [4]. Indeed,failure of coating films takes place after the loss in barrier integritydue to the accumulated damage of small scale weather-induceddegradation events [15]. These events imprint a random roughdamage on the coating surface and, under some proper type ofloading, it leads to chain scission, and macro failure [17].

In general, since constitutive models for deformation of amor-phous material failure have been developed for fracture of welldefined notches only, the present study couples both experimentaland statistical theory to extend knowledge of materials failure byfracture, from initial service surface conditions and during randomsurface damage accumulation and environmental degradation bymeans of correlating profile and surface statistical parameters[22e24] of a dynamic surface that moves from a quasi-flat (man-ufactured) [20,21] to a full Gaussian (early stages of wear) withrespect to fracture location probabilities.

2. Experimental procedure

Random surfaces were developed using the direct convolutionmethod and program by Bergstrom [11] to generate a randomrough surface after initial surface flatness has lost its memory.

Fig. 1. Cartoons showing dimensions of PMMA specimens: (Top) top view; (bottom)side view. (All units are in mm).

H. Medina, B. Hinderliter / Polymer 53 (2012) 4525e45324526

Fbf ¼ 2ffiffiffiffip

p,N,ACL

L,1N

X½DFTðzÞ,DFTðf Þ�,ei2pkn

N (1)

N�1

k¼0

Where:

f�x; y;ACL

�¼ e�

r2

ACL2 ; (2)

ACL¼ auto correlation length, Fbf ¼ Gaussian random surface (finalsurface), and DFT is the Discrete Fourier Transform (actual calcu-lations were performed using Fast Fourier Transforms) [19].

The dynamic evolution of surface damage was considered bystarting with an engineered (flat) surface which grows intoa Gaussian surface (i.e. height or depth is normally distributed) bysteps.

Fb�D� ¼ Fb : Fbij � D (3)

D was then changed from 0 to 45 in steps of 5, such thatFb(45) ¼ Fbf. In other words, Fb(0) represents the initial engi-neering surface and Fb(45) represents a Full Gausian, final surface.For ACL ¼ 45, D was studied beyond these values, i.e. D ¼ 90, 135.When D ¼ 45 the maximum true depth of ablation is about58 microns.

ACLwas varied from 10 to 90microns in steps of 10microns. Thesurfaces were designed in a computer code and developed (etched)on rectangular beams (Fig. 1) of commercial PMMA (Methyl Meth-Acrylate) containing 99.5% Poly methyl Methacrylate and 0.5%Methyl Methacrylate. Fig. 2 gives a perspective of the size of thespecimens. The experiments were run at room temperature and

Fig. 2. Actual picture of PMMA specimen to put dimensions in perspective.

stressestrain curves were compared with data previously pub-lished for PMMA (Fig. 3). The plot of Fig. 3 shows the same brittlebehavior as data previously reported at 20 �C [18]. Acrylic beamswere laser-cut to dimensions of 5” � 0.5” � 0.118”. A central areawas reserved for ablation of the random rough surfaces. The ablatedareawas kept always at 0.5”� 0.5”, (Fig. 1). Ablationwas performedusing a 30-WMini Epilog laser in 3-D rastermode, which interpretsgrayscale in a nonlinear way for the lightest and darkest values.However for values of gray between around 20% and 80% (of black)the behavior is completely linear. This linearity was used to cali-brate our gray surfaces to actual desired depth. Profiling was per-formed using a stylus profiler Ambios XP-1. Actual surfaces wereprofiled and compared with the theoretical ones, and found to bewithin 95e97% in similarity (Fig. 4). Surfaces from arrays given byEquation (3) are shown in Fig. 5 for differentD and ACL. Fig. 6 shows2D pictures to compare the mathematical model with the actualsurfaces laser ablated.

Fracture toughness of this material was found tabulated [18] inthe range of 0.7e1.6 MPa-m1/2. The elastic modulus was calculatedto be around 2.9 GPa and Poisson’s ratio was taken from tables [18]to be about 0.4.

Next, a total of 3000 etched specimens were bent on an MTSmachine, using a 3-point bending method of the type “ASTM D7903-point flex”, using 3 inches of span length and a loading rate of 2tenths of an inch per minute. These broken shims were photo-graphed and cropped, and the distance to the fracture line wasmeasured.

3. Analysis

Scanned images were produced and used to measure the loca-tion of fraction, see Fig. 7. Let the function (xi) be the number offractures that take place at the location corresponding to the line atxi, and let be the total number of fractures. So that, ¼ (xi)/can represent both density and the probability of fracture at loca-tion xi. Then, and hence, are directly proportional to the stressconcentration factor, Kt.

Statistical calculations were performed for surfaces and profilesat every location xi. Consider Fig. 7, showing cross-sectional profilesat a particular location for initial and final conditions. The y-axis ofthat profile plot is the ratio of true ablated depth to total specimenthickness (t).

Fig. 3. PMMA brittle behavior from present study. It has similar behavior whencompare this plot with previously published data of PMMA mechanical tensilebehavior from experiments reported by ASTM (after Carswell and Nason 1944 [20]).

Fig. 4. Profile showing actual depth compared to theoretical depth of ablation.

H. Medina, B. Hinderliter / Polymer 53 (2012) 4525e4532 4527

Statistical analysis was performed using [12,22e24]:

m ¼ 1N

XNi¼1

Zi (4a)

Fig. 5. (Top) top view, (middle) side view, (bottom) perspective view of surface plots for (frACL ¼ 10, D ¼ 0; (middle): ACL ¼ 45, D ¼ 0; (bottom): ACL ¼ 45, D ¼ 45.

Ra ¼ 1N

XNi¼1

jZi �mj (4b)

s2 ¼ 1N

XNi¼1

ðZi �mÞ2 (4c)

Sk ¼ 1s3N

XNi¼1

ðZi �mÞ3 (4d)

R2q ¼ 1N

XNi¼1

ðZi �mÞ2þðmÞ2 (4e)

K ¼ 1s4N

XNi¼1

ðZi �mÞ4 (4f)

Z0�x� ¼ vZ

vx¼ 1

N � 1

XN�1

i¼1

Ziþ1 � ZiDx

(4g)

om left to right) ACL ¼ 45 and D ¼ 40, ACL ¼ 45 and D ¼ 0, ACL ¼ 10 and D ¼ 0, (top):

Fig. 6. (Left column): 2-D grayscale surfaces developed using mathematical model;(right column): Top view pictures of actual laser-ablated PMMA central region. Noticethe strong similarity.

H. Medina, B. Hinderliter / Polymer 53 (2012) 4525e45324528

�Z00�x� ¼ �v2Z

vx2¼ 1

N � 2

XN�1

i¼2

2Zi � Ziþ1 � Zi�1

ðDxÞ2(4h)

For profiles, Equation (4) give the arithmetic mean (m), averageroughness (Ra), RMS roughness (Rq), variance (s2), standard deviation(s), skewness (Sk), kurtosis (K), the slope (Z0), and the curvature (Z00).

Now, due to the facts that the actual fracture lines are relatively“thick”, and also that these lines are not straight, a partition of 20theoretical fracture locations was performed, at each 5% of Wa

(Fig. 7). Since N ¼ 600 in (1), then each theoretical fracture locationwill correspond to 600/20 ¼ 30 real profiles. Referring to Fig. 7, onecould easily visualize this by just imagining that in between everytwo of the 20 dotted lines, there are 28 actual profiles. Therefore,for the computation of the aforementioned profile’s parametersand moments, average values of 30 profiles were taken.

This is actually a very accurate approximation since each parti-tion only corresponds to a total length of 0.500/20 ¼ 0.025 inches or0.635 mm.

Fig. 7. (Left) Lines parallel to fracture locations, (right) profile along some line at Xi The dcorrespond to a final full Gaussian surface (D ¼ 45). (Note: These particular profiles corres

3.1. Stress concentration factor

The stress concentration factor, Kt, is defined [17] as the ratio ofthe local stress to the remote stress:

Kt ¼slocalsload

(5)

Fractures occur in locations where slocal is high. Kt is some typeof function dependent on the geometry of the random roughsurface, which is exactly what this study is trying to investigate. It isproposed that Kt can be written as,

Kt�x; y� ¼

Xni¼0

kiFiðx; y; iÞ (6)

Where F(x, y, i) is some kind of polynomial whose terms andrespective exponents must be investigated. Also, ki are constants ofthe series. Intuitively, it can be stated that F(x, y, i) is a functionrelated to the different statistical and derived parameters of therough surface. For a given point (x, y) on the surface, the greater Kt

is, the higher the probability fracture will occur at that particularpoint. So this study will focus on the correlation between statisticaland derived parameters with fracture location probability. V-sha-ped stress distribution due to the bending stress applied wasaccounted for.

3.2. Stress concentration at the interfaces

A rather significant (i.e. high) value of was noticed at bothflat-to-rough interfaces. This same trend was observed for mediumACL (25� ACL � 75, approximately). However, it was observed thatfor low ACL surfaces, (0) and (Wa) are not as significant. Ingeneral, these effects are due to high stress concentration caused bythe irregularities in the flat-to-rough surface interfaces. This lead tothe decision of investigating away from the ends, approximatelybetween 5%Wa and 95%Wa. This is actually in agreement with St.Venant’s Principle (St. Venant 1855) [13]. So from now on, fracturelocation probability function, , unless otherwise specified, isstudied away from the interfaces.

3.3. Correlation

Function (xi/wa) for different ACL and D was correlated withthe aforesaid moments and parameters defined in Equation (4) andthose given by (7) and (10) to be discussed in the next section.Bivariate correlations were used to find Pearson’sproductemoment correlation coefficient. The Pearson coefficient is

otted light profile correspond to an initial surface (D ¼ 0) and the solid thick profilepond to another specimen. It is only shown for visualization of the procedure).

Fig. 8. Superposition of several halves elliptical holes. The depth of each pit is denotedas ai.

H. Medina, B. Hinderliter / Polymer 53 (2012) 4525e4532 4529

a dimensionless index that can measure linear dependencebetween two variables [14], in order words it is invariant to lineartransformations of either compared variable.

It is noticed that the existence of points x0, such that (x0) ¼ 0,creates noise in the correlation. The influence of these values wastaken into account.

Now, let RHm be Pearson’s correlation coefficient betweenparameter “m” and function . So, for example, the correlationbetween kurtosis, skewness, and max depth with are repre-sented as RHk, RHSk, RHMD, respectively.

3.4. Derived parameters

Since the Pearson’s correlation investigates linear relationship,then other parameters resulting from the combination of thestatistical ones must be also investigated. This discussion will startwith a parameter proposed by Hinderliter et al. [15]:

sðTÞsð0Þ ¼

1þ RqðTÞ2�

Rqð0Þ�2!�1

4

(7)

Where, Rq(T) and Rq(0) are the surface RMS roughness at any time Tand initially, respectively, based on the Central Limit Theoremevolution of the surface. The ratio in the left side of (7) is thendirectly proportional to the toughness of the polymeric beam. Itsreciprocal then will be directly proportional to the fracture density,

. Two parameters based on the above equation are used in the

Fig. 9. Correlation behavior of fracture location probability as the

present study and referred to, in this study, as the RMS HM-factorand the AVG HM-factor.

Another parameter is proposed here based on the resultsdeveloped by Inglis [16]. It is known that the first quantitativeevidence for the stress concentration effect of flaws was providedby Inglis, who analyzed elliptical holes in flat plates [17]. Hisanalyses included an elliptical hole, 2a long by 2b wide, withapplied stress perpendicular to the major axis of the ellipse. Thestress at the tip of the major axis is given by:

sP ¼ s

�1þ 2a

b

�(8)

So the Stress Concentration Factor, in this case, is given by,

kt ¼�1þ 2a

b

�(9)

If an elliptical hole is cut in half parallel to the minor axis,a notch of depth “a” is obtained. Now extending this idea to amulti-crack arrangement, an array of cracks of ai depth is obtained (Fig. 8).It is proposed here an average stress concentration factor, definedas follows,

kt ¼�1þ 2

ARACL

�(10)

Where AR is the average roughness and ACL is the auto correlationlength. Despite the fact, that half the pit depth is being used, thefactor 2 is kept based on the approximation also proposed by Inglisfor a notch that is not elliptical except at the tip. In this study, thestress concentration factor given by Equation (10) is referred to as“MI factor” (Modified Inglis factor). Of course, secondary effects dueto the presence of several pits close to each other are yet to beconsidered in the proposed Equation (10); notwithstanding, this isjust an initial approximation. Besides, the secondary effects due tothe superposition of pits are essentially the purpose of this study.

The Griffith factor was taken as the squared root of p times theaverage surface roughness. This was done to compare our resultswith a hypothetical one for which the whole random roughness ofa profile was compared to an average single pit.

surface degrades, for auto correlation length ¼ 10 microns.

Fig. 10. Correlation behavior of fracture location probability as the surface degrades, for auto correlation length ¼ 45 microns.

H. Medina, B. Hinderliter / Polymer 53 (2012) 4525e45324530

4. Results and discussion

Following there are 3 plots of Pearson’s correlation coefficientversus D for several parameters that were correlated with . Thoseplots, shown in Fig. 9, Fig. 10, and Fig. 11 correspond to ACL ¼ 10, 45and 90, respectively, for the parameters indicated.

This discussion will be centered on the results for ACL ¼ 45(Fig. 10), and then compared with the results for ACL ¼ 10 and 90,

Fig. 11. Correlation behavior of fracture location probability as the

shown in Figs. 9 and 11, respectively. The first observation (Fig. 10)is that, for an initial engineering surface, is insensitive to allparameters considered. This means that prediction of failureprobability is rather difficult using the aforementioned statisticalparameters or factors. This is due to the fact the introducedroughness (in both type and amount) is not sufficient to competewith other factors like internal flaws, impurities, and undesiredscratches on the samples. However, it was noticed that roughness

surface degrades, for auto correlation length ¼ 90 microns.

H. Medina, B. Hinderliter / Polymer 53 (2012) 4525e4532 4531

introduced does confine failure location probability within theablated area. In this case, a more micro-scale research must becarried out which is beyond the scope of the present study.

The second observation is that as degradation progresses, frac-ture location probability becomes more and more dependentprimarily on average roughness, root mean squared roughness, andparameters derived from them. Likewise dependence of on otherparameters like kurtosis, maximum depth, slope and curvature arerather weak, as deepening increases, for 0 � D � 135. The fact thatmaximum pit depth shows a quite low correlation infers that localisolated deep valleys do not influence fracture probability ina significant way.

There is a transitional region (TR), D ( 20, below which theprobability of fracture cannot be correlated to any of the statisticalparameters studied, and above which there is a rather strongcorrelation.

Additionally, as degradation progresses even further (D > 45),Pearson’s correlation coefficient finds a normalized asymptoticlimit of around 0.9 (Fig. 10). This asymptotic limit agrees very nicelywith fracture mechanics results for brittle materials since itapproaches agreeably the Griffith’s crack criterion, though withmultiple fracture locations.

It can also be seen that the proposed approximation formulas inEquations (7) and (10) correlate very well with the fracture density.This could be a significant result, since it is based on an equationused for regular notches. There is a high probability, according tothis result, that an analytical solution will involve a polynomialfunction of pit depth.

A similar trend was observed for ACL ¼ 10 (Fig. 9). An importantdifference between ACL ¼ 10 and 45 is the lower gradient observedin Fig. 9 in comparison to that shown in Fig. 10. So the transitionalregion of the former is larger and takes place in a less drastic way.However, correlations at higher level of degradation are equivalentfor both (ACL ¼ 10 and 45).

For ACL ¼ 90 (Fig. 11) as degradation progresses, fracture loca-tion probability increases its dependence on the parameters shown,as well, but at a much lower rate. Also, the range of correlationfound was between 0.3 and 0.45, approximately. This range isclearly between those found for ACL ¼ 10 and ACL ¼ 45. Thisundoubtedly infers a scaling problem.

5. Conclusions

Due to the use of PMMA at room temperature the followingconclusions can be applied to brittle plastic materials in service.This is actually very realistic, for example, for coatings which tendto become brittle as they degrade. Moreover, these conclusionscould extend to metals at low enough temperatures, where theystop behaving in a ductile manner.

Quasi-flat surfaces (e.g. engineered surfaces) are unpredictableto define, in the sense of where failure takes place. However,prediction can be carried out even under random degradation atearly stages. Prediction of location failure is also dependent onthe damage pit size (auto correlation length) of the degradationprocess. For medium auto correlation lengths (with respect tooverall dimensions), fracture location probability rapidlyincreases its dependence on surface roughness as degradationprogresses. For lower auto correlation lengths this dependencegrows a little less rapidly. Moreover, for both low and mediumauto correlation lengths, prediction of fracture location can becarried out at very early stages of degradation. However, if the pitsize of the degradation process is even larger (for example,erosion pits which are considerably large with respect to theoverall dimension of the surface in question), then it takes

a larger penetration of degradation before prediction of thefracture location can be made.

Additionally, as degradation penetrates, and for low andmedium ACL, and roughness (height) becomes normally distrib-uted (according to the CLT), the 3-dimension multi-pit problem canbe very well approximated using a 2-dimension multi-pit problemwhere every 2-dimension pit depth is characterized by the corre-sponding RMS roughness of the respective profile. In this case, thedeepest corresponding RMS pit’s location will have the highestprobability of fracture.

The high correlation of fracture probability to the parameterstied to Griffith crack criterion suggests that the energy required toseparate the shim at the particular location is low. The fracturelocation probability distribution being clearly not exclusivelya deterministic failure repetitively at the same location shows thatthe failure criteria associated with the topology must possessa broad minima. Further investigation will lead to more insightsinto the failure function based on topological properties.

Finally, the transitional region observed depends, as expected,on the auto correlation length and the degree of degradation. Sincefor PMMAductility appears above approximately 35 �C [18], furtherstudy of this transitional region, for this polymer, is in progress inorder to find its dependence on temperature, as well.

Acknowledgment

This work was sponsored by NRC grant: NRC-38-10-929.

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