J Mater Sci: Mater Med (2012) 23:217–228 DOI 10.1007/s10856-011-4481-6
An innovative multi-component variate that reveals hierarchy and evolution of structural damage in a solid: application to acrylic bone cement Gang Qi • Ming Fan • Gladius Lewis Steven F. Wayne
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Received: 20 February 2011 / Accepted: 31 October 2011 / Published online: 10 November 2011 Springer Science+Business Media, LLC 2011
Abstract A major limitation of solid mechanics is the inability to take into account the influence of hierarchy and evolution of the inherent microscopic structure on evaluating the performance of materials. Irreversible damage and fracture in solids, studied commonly as cracks, flaws, and conventional material properties, are by no means descriptive of the subsequent responses of the microstructures to the applied load. In this work, we addressed this limitation through the use of a novel multi-component variate. The essence of this variate is that it allows the presentation of the random damage in the amplitude spectrum, probability space, and probabilistic entropy. Its uniqueness is that it reveals the evolution and hierarchy of random damage in multi- and trans-scales, and, in addition, it includes the correlations among the various damage features. To better understand the evolution and hierarchy of random damage, we conducted a series of experiments designed to test three variants of a poly (methyl methacrylate) (PMMA) bone cement, distinguished by the methods used to sterilize the cement powder. While analysis of results from conventional tension tests and scanning electron microscopy failed to pinpoint differences among these cement variants, our multi-component variate allowed quantification of the multi- and trans-scale random damage events that occurred in the loading process. We tested the statistical significance of damage states to differentiate the
G. Qi (&) M. Fan G. Lewis S. F. Wayne Department of Mechanical Engineering, The University of Memphis, Memphis, TN 38152, USA e-mail:
[email protected] S. F. Wayne Propane Education and Research Council, Washington, DC 20036, USA
responses at the various loading stages and compared the damage states among the groups. We also interpreted the hierarchical and evolutional damage in terms of the probabilistic entropy (s), the applied stress (r), and the trajectory of damage state. We found that the cement powder sterilization method has a strong influence on the evolution of damage states in the cured cement specimens when subjected to stress in controlled mechanical tests. We have shown that in PMMA bone cements, our damage state variate has the unique ability to quantify and discern the history and evolution of microstructural damage.
1 Introduction The quantification of irreversible damage is a challenge and key aspect of understanding the performance of materials. This may be attributed to the lack of means to quantify irreversible damage. An event of such damage, whether a crack or in other forms, resides in the material’s inherent microscopic structure, and the structure is hierarchical, so is the damage. The occurrence of the damage, thus, corresponds to the variation of the inherent microscopic structure. Hence, damage and variation of inherent microscopic structure complement to each other. To date, most contributions have focused on investigating specific mechanisms of physical damage, i.e. cracks or flaws in terms of their formation [1–6], propagation and propagation rate [5–11], orientation [12, 13], locations [14–20], fracture energy [21–23], to name a few. Others were focused on material imperfections, such as potential sources to develop a crack, linear defects (dislocations) and interfacial defects (grain boundaries) in most metals [24–26], or vacancies, interstitial atoms and ions, and chain-folded surface layers in polymers [27–30]. However,
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the understanding of the influence of evolving inherence microscopic structure on material macroscopic performance is not clear [31, 32]. The variables of damage such as timing, quantity, magnitude, rate, and location—aspects that are critical to the issue of ‘‘multiphysics and multiscale coupling’’—is not clear either. To address this issue, we have developed an innovative approach to determine the probabilistic characteristics of random damage events and a quantification algorithm [33, 34]. In the present work, we chose a commercial brand of a poly (methyl methacrylate) (PMMA) bone cement as a sample material. We sterilized the cement powder using three different methods, resulting in three variants of cement specimens. We then applied our new approach to these three variants by (1) revealing the hierarchical and evolutional damage state in the amplitude spectrum and the probability distribution; (2) testing the statistical significances at different loading stages according to the probabilistic entropy, and comparing the damage stages for the three variants; and (3) determining the variations of probabilistic entropy with respect to the applied stress. Finally, we revealed a unique ability of our method to capture the damage state in the absence of availability of the applied stress.
2 Damage field and its quantification 2.1 Damage field and its descriptors Damage field is a special physical quantity associated with each point of spacetime of irreversible damage that occurs in the inherent microscopic structure of a solid. Irreversible damage can be quantified by a set of variables, such as timing, quantity, rate of occurrence, amplitude, energy, location, orientations, patterns, rate, and direction of propagation of various damage events of various scales. The variations of these variables are obviously reflective of those of the inherent microscopic structure. Because damage and the microstructure are complementary, the growth of damage implies the deterioration of the integrity of the inherent microscopic structure. There are limited means of acquiring damage in a material in real time, with acoustic emission (AE) technique being the most well known. This is because irreversible damage releases strain energy in the form of stress/strain waves. These waves travel at the speed of sound and become detectable signals, which can be captured by AE sensor [35–37]. The physical characteristics of these signals parallel those of the aforementioned variables, thereby establishing a nearly oneto-one correspondence with the irreversible damage events [35, 38–49]. Depending on the sensitivity of an AE sensor, the detected signals of a damage event may contain information about damage even at the nanoscale level
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[34, 50–53]. For simplicity, we refer to all the irreversible damage as random microscopic events (RMEs). The use of all the aforementioned variables to define the damage field is nearly impossible. We may, however, define a limited damage field using one or a combination of a few of these variables, such as the amplitude, pattern, and source location fields. For instance, the damage source location field may be determined using the time differences from the acoustic signals of four or more sensors [17–19, 54–57], crack orientations and kinetics revealed by the time difference, signal energy, waveform of multi-channel acoustic signals [58–61], crack growth by examining acoustic signal waves [8, 21, 40, 48], and crack tip damage by accumulative acoustic signatures in terms of amplitude [42–45]. A dilemma is created because the understanding of a limited damage field is insufficient to establish the entire damage field and, therefore, limiting to address the response of the inherent microscopic structure of a material to the applied stress. To overcome this dilemma, we take a novel approach in which the damage state is quantified in terms of a deterministic subset of the aforementioned variables: timing, quantity, rate of occurrence, and amplitude. The first three are the measurements primarily in the domain of spacetime and their coupling, while the last one is proportional to the magnitude of the corresponding random damage events [34, 36, 62–64]. In our previous work, we established a means to integrate these four variables into a multi-component variate, and defined it as damage state variate [33, 34]. In the present work, we use this variate to quantify the damage states and to reveal the hierarchical and evolving microstructures due to various physical damage mechanisms subjected to applied stress in conventional mechanical tests. 2.2 A multi-component damage state variate Let D be a variate that integrates the four variables in the aforementioned subset such that [33]: D :¼ bij MN ð1Þ where M is the index that depends on the means to obtain the RME statistics and N is number of subintervals that divide the RME amplitude bandwidth. In this work, we used N = 10. Let bij be the measured quantity of RME from 0 - i whose amplitude falls in the jth sub-interval observed up to a measured load level, such that bij ¼
i X
xmj
for i ¼ 1; . . .; M and j ¼ 1; . . .; N
m¼1
where xi is an event of detected RME, normalized by the volume of the gauge section of the specimen measured in
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the interval of (i – 1, i). The corresponding Gibbs probabilities of bij are approximated by bij fij ¼ Li
for i ¼ 1; . . .; M
where Li ¼
i X N X
xmj
for i ¼ 1; . . .; M
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Z1 S :¼
qðxÞ lnð1=qð xÞÞdx
0
where q(x) is the Gibbs probability density function, and is estimated on each sub-interval of the x-axis by its average over that sub-interval. In the discrete space, S is approximated as [66]:
m¼1 j¼1
Replacing bij in Eq. 1 with fij, leads to the following expression for D : fij D ð2Þ MN where fij denotes the fraction of the acoustic signals whose amplitude falls in the jth sub-interval observed up to a is only a measured load level or a measured time. Since D as well. form of normalized D, the features of D apply to D Depending on bij, D may be determined per increment of either the nominal stress (r) or time-to-fracture (t). It is useful to note that, depending on the loading history, D may be continuous or non-continuous. Non-continuous measurements using time increments may be of particular importance and, thus, deserve further study. This is underway in the authors’ laboratory. In the present work, we focused on continuous measurements. share the same raw measurements, which Both D and D are made from a volume of the test body. The size of the volume is dependent on the AE sensor or sensor array. in the above equations Parameter N used to define D and D is, as stated, the number of subintervals to characterize the probabilistic features of the detected signals. We suggest that these subintervals are reflective of the so-called multiand trans-scale problems. Hence, knowing a damage state implies knowing an amplitude spectrum or a probability distribution of the detected RME, which corresponds to a respectively. The uniqueness of D or row vector of D or D, is that it reveals random damage not only in multi- and D trans-scales but also in the correlations among the random damage events. 2.3 Algorithm of Gibbs probabilistic entropy are two-dimensional data matrices, it is Since both D and D challenging to test the statistical significance of a specific damage state that is either in amplitude spectrum (D) or the In our previous work, we introprobability distribution (D). duced the concept of probabilistic entropy as one way of meeting this challenge and presented an algorithm for computing it [33, 34]. Thus, we recall here only the salient features. The probabilistic entropy based on the Gibbs formula is [65]:
ð3Þ
S Si :¼
10 X
fij ln 0:1=fij
for i ¼ 1; . . .; T
ð4Þ
j¼1
as, si is then the numerical indicator of D and D therefore, the damage state. The maximum possible value of si is zero and is achieved when the observations are equally distributed over the ten sub-intervals (i.e. fij = 0.1 for i = 1,…,t, and j = 1, 2,…,10). Its minimum possible value is ln(0.1) = -2.3 and is achieved when all the observations fall in the same sub-interval. Note that in Eq. 3, fij is a quantity measured independently; thus, whether or not information about the stress field is available, the value of fij is not affected. Therefore, fij has a unique role in quantifying damage state and in testing the statistical significance of damage states.
3 Materials and methods 3.1 Study material and fabrications of specimens The sample material selected was SmartSet MV PMMA bone cement (DePuy Orthopaedics, Warsaw, IN, USA). The constituents of its powder are PMMA, methyl methacrylate (MMA)-styrene copolymer, BaSO4 particles, and benzoyl peroxide, while the constituents of its liquid are MMA, N, N-dimethyl-p-toluidine, and hydroquinone. The three variants of this material were obtained by sterilizing the cement powder (prior to molding the specimens) using three different methods; namely, exposure to c irradiation at a minimum dosage of 2.5 Mrad (High-c group); exposure to c irradiation at a minimum dosage of 1.5 Mrad (Low-c group); and exposure to ethylene oxide gas (ETO group). The test specimens were fabricated by mixing the cement powder and the liquid monomer in a polymeric bowl open to the ambient laboratory air (hand/manual mixing), and then pouring the mixture into a silicone specimen mold to form rectangular-cross-sectioned dogbone-shaped specimens (ASTM D638-98 Type IV). The specimens were cured in the mold, then removed, and aged for at least 48 h, with all these steps being carried out in ambient laboratory air. All the as-fabricated specimens were examined visually and those with surface voids and/
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or other defects with diameter [0.5 mm were discarded. The final numbers of accepted specimens were 14, 17, and 19 specimens for the High-c, Low-c, and ETO groups, respectively. The morphologies of the as-fabricated microstructures (Fig. 1) were obtained using a scanning electron microscope (SEM) (Model XL30; Philips, Achtsewed, The Netherlands), operated at an accelerating voltage of 30 kV. The specimens in the three study groups displayed structural complexity at different length scales and interphase interactions. For instance, there was no significant difference in either microstructure or quasi-static tensile properties (strength and modulus) [67–71] between the groups. An indication of the extent to which the powder sterilization treatment affected the physical properties of the cement was reflected in its polydispersity index (PDI): 2.43 ± 0.05, 2.67 ± 0.10, and 3.23 ± 0.07 for the High-c, Low-c, and ETO groups, respectively [72]. Also, there was
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a significant decrease in the estimated fatigue limit of the cement with decrease in PDI [72]. 3.2 Acquisition of acoustic signals To acquire the acoustic signals, we bonded an array of three piezoelectric sensors (Nano 30; Physical Acoustics, Inc., Princeton, NJ, USA) to the surface of the specimen at positions that would cover the entire gauge length. This type of sensor, whose resonant frequency and operating frequency range are 140 and 125–750 kHz, respectively, has high sensitivity to the signals from PMMA bone cement specimens [54, 73]. We subjected each specimen to quasistatic tension using a screw-driven universal materials testing machine (Model 4465; Instron, Inc., Canton, PA), at a cross-head displacement rate of 1 mm/min, until fracture. The resultant acoustic signals were pre-amplified by a 40 dB pre-amplifier, having a bandpass filter between 2.5 kHz and 3.8 MHz (AEP4; Vallen-Systeme GmbH, Munich, Germany), before being fed to an AE system (ASMY-5; Vallen-Systeme GmbH). A data acquisition threshold was selected to minimize the noise from the materials testing machine, this being done by installing the specimen on the testing machine with near zero loading and finding that there were no detectable acoustic signals at or below 35.5 dB. Thus, this value was used as the threshold. 3.3 Statistical analysis For a given parameter, test of significance between the results from the three study groups was conducted using one-way ANOVA, with the Bonferroni correction (SASVersion 8.02; SAS Institute Inc., Cary, NC, USA), with significance being denoted when P \ 0.05.
4 Results 4.1 Damage states: the amplitude spectrum
Fig. 1 The morphology of as-fabricated SmartSet MV PMMA bone cement specimens a 5009; b 20009. The polymer beads (dark circles) are bonded by the MMA matrix and the BaSO4 particles (white particles). Of note is that we did not find differences in morphology between the specimens in the three study groups
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The results for the amplitude spectrum, D variate, are presented in Fig. 2. In this figure, the connected points outline the measurements. Each outlined amplitude spectrum quantifies a damage state of the corresponding stress level, and the pattern of each spectrum is an important determinant of the correlations between random damage events that occurred under the corresponding stress level. For instance, when r = 4 MPa, the amplitude spectrum ranges from *35 to *70 dB for each of the c specimens (Fig. 2a, b), whereas it ranges from *35 to *55 dB for the ETO specimens (Fig. 2c). There are two common features among these results. First, initially, low-amplitude RMEs are dominant. As the
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the quantity of RMEs varies monotonically in each subinterval, with a pattern of the greater the amplitude, the less the quantity. The rates of RME occurrence can be obtained by the difference between the applied stress levels over a specified average amplitude subinterval. For instance, the rates were 5.4, 4.9, and 0.9 No./MPa for the High-c, Low-c, and ETO groups, respectively, when 4 MPa B r B 8 MPa and the subinterval of the average amplitude of interest was *38 dB; whereas, they were, respectively, 1.2, 1.0, and 0.1 No./MPa over the same stress range in the subinterval of average amplitude *62 dB. Of note is the fact that the rate of RME occurrence decreased sharply once the entire bandwidth of the spectrum was populated. This step was followed by a transition region that occurred at *8 MPa for the specimens in both of the c groups, and at *16 MPa for those in the ETO group. Also, the quantity of the RME in either of the c groups was significantly higher than that in the ETO group. 4.2 Damage states: probability distribution Figure 3 shows the same results given in Fig. 2 but now Of note presented using Gibbs probability distribution, D. is the fact that when r \ 12 MPa, the probabilities of the occurrence of low-amplitude RMEs were significantly greater than those of high amplitude. With one exception, we found that, for each study group, when the applied stress increased, i.e. r [ 16 MPa, these probabilities merged into one general trend that is in the form of an exponential function [33]. 4.3 Damage states: probabilistic entropy
Fig. 2 The amplitude spectrum defined by the D variate for the three study groups. Note each spectrum curve quantifies a damage state at the corresponding stress level
applied stress increases, RMEs with higher amplitude occur. As soon as RMEs with high amplitudes present, the entire spectrum bandwidth is quickly populated. Second,
Figures 4, 5, 6 are the same results of damage states as those in Fig. 3 but now quantified by the corresponding probabilistic entropy s1,…,s7. We point out three key trends observed in these results. First, for each study group, the damage states are significantly different initially from each other (P \ 0.001), but not subsequently (P [ 0.06) (Fig. 4). Close to the point of final fracture (s [ -0.95, equivalent to r [ 28 MPa), however, the damage states become significantly different again in specimens in each of the c groups (P \ 0.001), while no such difference was seen in the specimens in ETO group (P [ 0.07) (Fig. 4). Second, at r = 6 MPa, the probabilistic entropies increase in the order ETO [ High-c [ Low-c (Fig. 5a). At r C 8 MPa, however, the order is ETO [ Low-c [ High-c (Fig. 5b–f). Third, the applied stress needed to reach a given entropy level, or a damage state, is markedly higher for ETO specimens compared to that from each of the c groups,
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Fig. 3 The approximated Gibbs probability distributions of the data variate given in Fig. 2 re-presented by the corresponding D
whereas, there is essentially no difference in each given entropy level for specimens from the two c groups (Fig. 6). 4.4 The variations of s with r We recognize the need for a parameter that is capable of revealing the unique changes in damage states as a function of stress. We met this need by defining the slope of the entropy-versus- applied stress curves as the trajectory of damage states (TDS) (Fig. 7). Based on this slope, four key observations are highlighted.
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Fig. 4 Re-presentation of the results given in Fig. 3, with the damage states quantified by probabilistic entropy as s1,…,s7. A damage state that is significantly different from each of the other damage states is marked by an asterisk
First, there are generally four evolving stages of damage states, these being the initial, transition, stable, and final stages. The initial stage is characterized by nearly linear increase in s with increase in r. A significantly reduced slope characterizes the transition stage. The stable stage is characterized by a nearly-constant s with increase in r. The final stage is characterized by a slight decrease in s with increase in r.
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Fig. 5 Comparison of damage states among specimens at fixed stress levels. A damage state that is significantly different from each of the other damage states is marked by an asterisk
Second, for 0 B r B *7 MPa, the extent of the damage states, as indicated by s, is about the same for the two c groups, whereas it is significantly lower for the ETO group. At r [ *10 MPa, the damage states in the High-c specimens are significantly higher than those in the Low-c specimens while the damage states of the ETO group specimens are the lowest. This finding is consistent with the fractographic results: extensive microcrack damage in High-c and low-c specimens but no perpendicular cracks in ETO specimens, indicating the latter’s high resistance to damage (Fig. 8). Third, the stress values at the mid-point of the transition stage (or the ‘‘knee stress’’) are *10 MPa for specimens in each of the c groups and *17 MPa for ETO specimens. These stresses are *30 and *50% of the tensile strength for the c-irradiated and ETO-sterilized cements, respectively [74].
Fourth, the estimated slope of the initial stage is markedly lower for the ETO group compared to that for either of the c groups. Specifically, these slopes and corresponding 95% confidence intervals are *0.13 (0.11, 0.15) MPa-1, *0.11 (0.06, 0.15) MPa-1, and *0.07 (0.06, 0.09) MPa-1 for the High-c Low-c, and ETO groups, respectively. Hence, while Figs. 2 and 3 reveal the evolutional multiand trans-scale random damage event (RDE) of damage states in a greater detail through the probability space and multiscales, TDSs, by contrast, are more interpretive, thus allowing for direct visual comparison as seen in Fig. 7. 4.5 The variations of s with time-to-fracture The pattern of the variation of TDS with respect to time-tofracture of the specimen in the quasi-static tension test (t) (Fig. 9), is similar to that of the variation of TDS with
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Fig. 7 Variation of probabilistic entropy (s) with applied stress
Fig. 6 Comparison of the spectrum of damage state and applied stress at selected entropy
applied stress (Fig. 7) in the sense that the same four stages can be distinguished. Thus, similar observations can be made as those from Fig. 7. There are no significant differences among the three test groups in the early stages (i.e. within the first 40 s). For each test group, the transition stage has no obvious correlation with any property of the cement, as was the case for the knee stress in the s–r variation (Fig. 7). We note that, with absence of information on the applied stress, the s-versus-t variation reveals mainly the intensity and trends of the evolving damage states.
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Fig. 8 Morphological features at localized pore damage sites on the fracture surfaces in the specimens. Note the absence of microfractures in an ETO specimen
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Fig. 9 The variation of probabilistic entropy with time-to-fracture of a specimen in quasi-static tension tests
5 Discussion 5.1 The significance of multi- and trans-scale damage analysis The amplitude spectrum and probability space presented in Figs. 2, 3, 4, 5, 6 are an innovative means to reveal the irreversible damage in a specimen. This approach has many advantages because the timing, quantity, rate of occurrence, and amplitude of damage events are parallel to the hierarchical microstructure of the material. For our selected sample material (PMMA bone cement), which is a compound of multiphase of copolymer powder and liquid monomer together with micro-sized additives, the identified major failure modes of the hierarchical inherent microscopic structure comprise void nucleation, microfracture, crazing, fibril breakage, debonding [34, 38, 75–77], and fracture in craze development [34, 38, 75–80]. It is necessary and important to the understand the physics of each mode in terms of its occurring conditions and its potential role in failure process. However, the understanding of the correlative influences among these physical failure modes in the entire failure process is equally importance. It is biased to assume that the understanding of a particular physical failure mode would be an ultimate achievement. The correlations among the RMEs are revealed by the pattern of the amplitude spectrum (Eq. 1) and probability space. This argument is supported by the correlated growth patterns of the amplitude spectrum in each subinterval (Fig. 2) and the probability spaces (Fig. 3). 5.2 The evolution of damage states The TDS is expressive of the responses of the inherent microscopic structure of a solid to the applied stress in terms of the presence of irreversible damage. As evidenced in Fig. 7, although the absolute quantity of irreversible
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damage continuously grows monotonically (Fig. 2), the responses may be linearly, nonlinearly, and/or be independent of the applied stress. The approximate linear response stage is indicated by ds/dr & constant [ 0 (r \ *12 MPa for the each of the c groups, and r \ *16 MPa for ETO study group; Fig. 7). We speculate that this linear dependence may have played a significant role in setting the course for the development of the subsequent damage. Our argument is based on the statistical significance of the damage state for each study group. This significance suggests that this linear portion of TDS is dependent on the variations of microstructure due to the applied stress and its slope indicates the rate of the onset and development of the nucleation of tiny voids and their coalescence (Fig. 4a–c). There was evidence of these features in our previous work, where a different commercially available PMMA cement brand (VersaBond; Smith & Nephew, Inc., Memphis TN, USA) was used. We were not, however, able to obtain these features in the present cement specimens. This may be because: (1) damage initiation and nucleation in the present specimens are different from those in VersaBond specimens; and/or (2) the initiated and nucleated damage events in the present specimens were too few to be observed and characterized by SEM. Nonetheless, the damage events in this low stress level were detectable under AE, and, thus, cannot be overlooked. The aforementioned differences between the two PMMA bone cement brands (VersaBond and SmartSet MV) may be of fundamental importance from a solid mechanics perspective, and, thus, deserve further study. As the responses of inherent microscopic structure to the applied stress become nearly constant, the TDS becomes nearly flat, indicated by ds/dr & 0 (Fig. 7), meaning that damage state is independent of the applied stress. It appears that this finding contradicts the coupling of stress-damage obtained by analytic approaches [76, 78, 80]. Our argument is as follows. ds/dr & 0 indicates that a maximal damage state (at least an intermediate maximum) has been achieved. It does not mean that the absolute quantity of irreversible damage stops growing, although damage state remains nearly constant. To the contrary, the quantity of RMEs grows significantly, but the growth in each subinterval is nearly parallel (Fig. 2a–c). By our definition of damage state, no new damage modes occur in this stage despite the growing total quantity of RMEs. In other words, the absolute quantity of damage events grow continuously, while the mechanism of these events were the same or similar. Noteworthy is the stage of the formation of selfsimilar craze for general PMMA polymers, as established in both theoretical work and experimental measurements [78, 80]. Therefore, the ‘‘no new damage mode’’ scenario suggested by our data is consistent with the self-similar crazes and fibril breakage in polymeric materials.
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The transition from linear to independent responses is indicated by ds/dr [ 0 and d2s/dr2 \ 0 (Fig. 7). The stress level at the midpoint of this transition zone, designated as the ‘‘knee stress,’’ suggests the end of damage initiation and nucleation. Of particular note is that this ‘‘knee stress’’ is comparable to the estimated fatigue limit of the cement [72], which was observed similarly in our previous work using VersaBond cement [34]. Further work is warranted on determining the generality of the association between the ‘‘knee stress’’ and the fatigue limit (meaning whether it applies universally to all classes of solids) as well as its significance in characterizing the mechanical behavior of materials. When ds/dr \ 0, as shown in Fig. 7, occurred at the highest levels of applied stress, it strongly suggests imminent failure of the specimen. This aspect needs again to be examined in future studies. The decreasing trend of TDS is not so obvious in the present work, but it is more evident when non-continuous measurements are used to obtain D without knowledge of the loading history. This topic will be the subject of a forthcoming report from our laboratory. The similarity between the patterns of the s–r and s–t variations (Figs. 7, 9) suggests that the responses of the inherent microscopic structure to the applied stress are selfruled; that is, they do not depend on knowledge of the stress field. This is another appealing aspect of our method. 5.3 Final remarks on probabilistic entropy We point out that the probabilistic entropy used in the present study is not energy based. Thus, it is different from its classical counterparts in thermodynamics (Clausius) and statistical mechanics (Boltzmann) in a number of ways, and one in particular; namely, probabilistic entropy can decrease. In this study, we have shown that decreasing probabilistic entropy may have the potential to be used in denoting final failure of the material. There are a number of similarities between classical entropy and probabilistic entropy; for example, both are a measure of material utility, with the greater the entropy value, the lower the utility. We have introduced this idea in the case of probabilistic entropy in a previous work, where we divided the RME into ten subintervals and assume the amplitudes of detected RME are distributed over a subset of the entire bandwidth equally, numbering K’ in all. Then, the probability of each subinterval is 1/K’, yielding Eq. 5 [personal communication with Professor Oliver Penrose, Heriot Watt University, Edinburgh, Scotland; 7 February, 2009], 0 1 K S ¼ ln ¼ ln ð5Þ 0 10K 10 Thus, in essence, s is the logarithm of a fraction of the probability space on which damage events are concentrated.
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It follows: the low-valued entropy is associated with the occurrence of relative clustered RME in a few numbers of subintervals defined in Eq. 1, and vice versa. The patterns in the TDS curve evidenced in the current study are in complete agreement with our previous findings [33, 34]. The increasing entropy is associated with greater involvement of various damage modes; namely, void nucleation, localized random damage, bead decohesion, microcracks, void coalescence, and, ultimately, widespread random damage and fracture [34]. Therefore, probabilistic entropy retains a crucial feature of its classical counterpart: increasing entropy indicates decreasing material utility.
6 Conclusions We have addressed the responses of inherent microscopic structures of a material to applied stress (r) by utilizing our newly developed concept of multi-component damage state variate. This was done by analyzing acoustic signals obtained from specimens of three variants of a PMMA bone cement, subjected to quasi-static tension. The multicomponent variate takes into account multi- and trans-scale random damage events that occurred in the loading and presents them in terms of amplitude spectrum, probability space, and probabilistic entropy (s). While analysis of results from conventional mechanical tests and scanning electron microscopy examinations failed to distinguish between these variants, our results revealed significant differences whether in the various loading stages of one study group, or in the comparison among the groups. The damage state trajectory is a potent means of interpreting the hierarchical and evolving random damage which occurs under applied stress and, hence, of quantifying the damage state and different responses to the loading. We also noted an interesting observation; namely, the stress at the midpoint of the transition zone, which is a section of reduced slope in the s–r plot, is comparable to the fatigue limit of the cement. The similarity between the patterns of the variations of s-versus-r and s-versus-time-to-fracture reveals that the responses of random damage events can also be obtained in the absence of knowledge of the applied stress. Acknowledgments We thank Prof. Y. L. Bai and his group at The State Key Laboratory of Non-linear Mechanics, Institute of Mechanics, Chinese Academy of Sciences and Dr. W. B. Luo, Xiangtan University, Hunan Province, China, for many discussions and suggestions; and Mr. Jinke Mo, Mr. Bin Zhang, Mr. Rick Voyles, Mr. Robert Jordan, Mr. Srikanth Thota, Ms. Sundari Vankamamidi, and Mr. Rajesh Muthireddy, all of The University of Memphis, for their many contributions to various experimental and computational aspects of the work. Funding was partially provided by NIH/NIAMS (Grant Number AR051119) and The University of Memphis (Faculty Research Grant).
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References 1. Szolwinski MP, Farris TN. Mechanics of fretting fatigue crack formation. Wear. 1996;198(1–2):93–107. 2. Bao Y, Wierzbicki T. A comparative study on various ductile crack formation criteria. J Eng Mater Technol. 2004;126(3):314–25. 3. Lawn BR. Partial cone crack formation in a brittle material loaded with a sliding spherical indenter. Proc R Soc Lond Ser A Math Phys Sci. 1967;299(1458):307–16. 4. Arndt PF, Nattermann T. Criterion for crack formation in disordered materials. Phys Rev B. 2001;63(13):134–204. 5. Sinnett-Jones P, Browne M, Moffat A, Saffari N, Buffiere J-Y, Sinclair N. Crack initiation processes in acrylic bone cement. J Biomed Mater Res A. 2008;89:1088–97. 6. Sinnett-Jones PE, Browne M, Ludwig W, Buffie‘re J-Y, Sinclair I. Microtomography assessment of failure in acrylic bone cement. Biomaterials. 2005;26:6460–6. 7. Gilbert CJ, Ritchie RO. Fracture toughness and fatigue-crack propagation in a Zr–Ti–Ni–Cu–Be bulk metallic glass. Appl Phys Lett. 1997;71(4):476–8. 8. Ng ET, Qi G. Investigation of fatigue crack propagation behavior of bone cement by wavelet based acoustic emission analysis. Eng Fract Mech. 2001;68(13):1477–92. 9. Brown EN, White SR, Sottos NR. Fatigue crack propagation in microcapsule toughened epoxy. J Mat Sci. 2006;41(19):6266–73. 10. Qi G, Ng ET. A new method to evaluate material fatigue behavior. J Mater Sci. 2000;36:2483–9. 11. Roques A, Browne M, Thompson J, Rowland C, Taylor A. Investigation of fatigue crack growth in acrylic bone cement using the acoustic emission technique. Biomaterials. 2004;25(5): 769–78. 12. Go´mez-Mancillaa J, Sinou J-J, Nosova VR, Thouverezb F, Zambrano A. The influence of crack-imbalance orientation and orbital evolution for an extended cracked Jeffcott rotor. Comptes Rendus Mecanique. 2004;332(12):955–62. 13. Lu Y, Ye L, Su Z, Huang N. Quantitative evaluation of crack orientation in aluminium plates based on Lamb waves. Smart Mater Struct. 2007;16(5):1907. 14. Qi G, Li J, Mouchon WP, Lewis G. Defect-induced fatigue microcrack formation in cement mantle. J Biomed Mater Res A. 2005;75(2):414–24. 15. Suresh S, Omkar SN, Ganguli R, Mani V. Identification of crack location and depth in a cantilever beam using a modular neural network approach. Smart Mater Struct. 2004;13:907. 16. Rizos PF, Aspragathos N, Dimarogonas AD. Identification of crack location and magnitude in a cantilever beam from the vibration modes. J Sound Vib. 1990;138(3):381–8. 17. Ge M. Analysis of source location algorithms overview and noniterative methods, part I & II. J Acoust Emiss. 2004;21:14–51. 18. Qi G, Li J, Mann KA, Mouchon WP, Hamstad MA, Salehi A, Whitten SA. 3D real time methodology monitoring cement failures in THA. J Biomed Mater Res A. 2004;71(3):391–402. 19. Qi G, Pujol J, Fan Z. 3-D AE visualization of bone-cement fatigue locations. J Biomed Mater Res. 2000;52(2):256–60. 20. Garcimartin A, Guarino A, Bellon L, Ciliberto S. Statistical properties of fracture precursors. Phys Rev Lett. 1997;79(17):3202–5. 21. Evans AG, Dalgleish BJ, He M, Hutchinson JW. On crack path selection and the interface fracture energy in bimaterial systems. Acta Metall. 1989;37(12):3249–54. 22. Mecholsky JJ, Rice RW, Freiman SW. Prediction of fracture energy and flaw size in glasses from measurements of mirror size. J Am Ceram Soc. 1998;57(10):440–3. 23. Mann KA, Edidin AA, Ordway NR, Manley MT. Fracture toughness of CoCr alloy–PMMA cement interface. J Biomed Mater Res. 2000;38:211–9.
227 24. Sham-Tsonga S, Sanboh L. A thermodynamic approach to the interaction between dislocation and crack and its applications. Eng Fract Mech. 1985;22(6):1105–15. 25. Ballarini R, Denda M. The interaction between a crack and a dislocation dipole. Int J Fract. 1988;37(1):61–71. 26. Zhu T, Li J, Yip S. Atomistic study of dislocation loop emission from a crack tip. Phys Rev Lett. 2004;93(2):025503. 27. Wunderlich B. A thermodynamic description of the defect solid state of linear high polymers. Polymer. 1964;5:125–34. 28. Fleischer RL, Price PB, Walker RM. Ion explosion spike mechanism for formation of charged-particle tracks in solids. J Appl Phys. 1965;36(11):3645–52. 29. Selinger Jonathan V, Bruinsma RF. Statistical mechanics of defects in polymer liquid crystals. J Phys II Fr. 1992;2:1215. 30. Trachenko K, Dove MT, Salje EKH. Structural changes in zircon under a-decay irradiation. Phys Rev B. 2002;65:180102. 31. Bai YL, Wang HY, Xia MF, Ke FJ. Statistical mesomechanics of solid, linking coupled multiple space and time scales. Appl Mech Rev. 2005;58:372–88. 32. Bai YL, Wang HY, Xia MF, Ke FJ. Trans-scale mechanics: looking for the missing links between continuum and micro/ nanoscopic reality. Acta Mech Sin. 2008;24:111–26. 33. Qi G, Fan M, Wayne SF. Measurements of a multi-component variate in assessing evolving damage states using a polymeric material. IEEE Trans Instrum Meas. 2011;60(1):206–13. 34. Qi G, Wayne SF, Penrose O, Lewis G, Hochstein JI, Mann KA. Probabilistic characteristics of random damage events and their quantification in acrylic bone cement. J Mater Sci Mater Med. 2010;21(11):2915–22. 35. Hamstad MA. A review: acoustic emission, a tool for compositematerials studies. Exp Mech. 1986;26(1):7–13. 36. McMaster RC, editor. Nondestructive testing handbook. Columbus: ASNT; 1982. 37. Battie A. Acoustic emission, principles and instrumentation. Sandia National Lab, Report SAND82-2825; 1983. 38. Kramer EJ. Craze fibril formation and breakdown. Polym Eng Sci. 1984;24(10):761–9. 39. Kander RG. A study of damage accumulation in unidirectional glass reinforced composites via acoustic emission monitoring. Polym Compos. 1991;12(4):237–45. 40. Dunegan HL, Harris DO, Tatro CA. Fracture analysis by use of acoustic emission. Eng Fract Mech. 1968;1:105–22. 41. Evans AG, Linzer M, Russell LR. Acoustic emission and crack propagation in polycrystalline alumina. Mater Sci Eng. 1974; 15(2–3):253–61. 42. Fang D, Berkovits A. Fatigue design model based on damage mechanisms revealed by acoustic emission measurements. J Eng Mater Technol. 1995;117(2):200–8. 43. Berkovits A, Fang D. Study of fatigue crack characteristics by acoustic emission. Eng Fract Mech. 1995;51(3):401–16. 44. Qi G, Barhorst AA. On predicting the fracture behavior of CFR and GFR composites using wavelet-based AE techniques. Eng Fract Mech. 1997;58(4):363–85. 45. Qi G, Barhorst AA, Hashemi J. Discrete wavelet decomposition of AE signals from CFR composites. Compos Sci Technol. 1997;57(4):389–403. 46. Bassim MN, Lawrence SS, Liu CD. Detection of the onset of fatigue crack growth in rail steels using acoustic emission. Eng Fract Mech. 1994;47(2):207–14. 47. Caprino G, Teti R. Quantitative acoustic emission for fracture behavior of center-hole GFRP laminates. J Comp Mat. 1994;28(13): 1237–49. 48. Harris DO, Dunegan HL. Continuous monitoring of fatigue-crack growth by acoustic-emission techniques. Exp Mech. 1974;14: 71–80.
123
228 49. Hardy HRJ. Acoustic emission microseismic activity, vol. 1: principles, techniques and geotechnical applications. London: Taylor and Francis; 2003. 50. Tymiak N, Daugela A, Wyrobek T, Warren O. Highly localized acoustic emission monitoring of nanoscale indentation contacts. J Mater Res. 2003;18(4):784–96. 51. Ma X-G, Komvopoulos K. Nanoscale pseudoelastic behavior of indented titanium–nickel films. Appl Phys Lett. 2003;83:3773–5. 52. Dimiduk DM, Woodward C, LeSar R, Uchic MD. Scale-free intermittent flow in crystal plasticity. Science. 2006;312(5777): 1188–90. 53. Siegmann A, Kander RG. In situ acoustic emission monitoring and mechanical testing in the scanning electron microscope. J Mater Sci Lett. 1991;10(11):619–21. 54. Qi G. Numerical assessment of the quality of AE locations in the presence of noises. J Acoust Emiss. 2000;17:111–20. 55. Qi G, Li j. Experimental–computational solution to fatigue induced microcrack source localization in cemented hip arthroplasty models. NDT & E Int. 2007;40(5):378–89. 56. Li H, Bao Y, Ou J. Structural damage identification based on integration of information fusion and Shannon entropy. Mech Syst Signal Process. 2008;22:1427–40. 57. Li J, Qi G. Improving source location accuracy of acoustic emission in complicated structures. J Nondestruct Eval. 2009;28: 1–8. 58. Ohtsu M. Acoustic emission theory for moment tensor analysis. Res Nondestruct Eval. 1995;6(3):169–84. 59. Ohtsu M, Okamoto T, Yuyama S. Moment tensor analysis of AE for cracking mechanisms in concrete. ACI Struct J. 1998;95(2): 87–95. 60. Thio HK, Kanamori H. Moment-tensor inversions for local earthquakes using surface waves recorded at TERRAscope. Bull Seismol Soc Am. 1995;85(4):1021–38. 61. Dahm T. Relative moment tensor inversion based on ray theory: theory and synthetic tests. Geophys J Int. 1996;124:245–57. 62. Colombo IS, Main G, Forde MC. Assessing damage of reinforced concrete beam using ‘‘b-value’’ analysis of acoustic emission signals. J Mater Civil Eng. 2003;15(3):280–6. 63. Morscher GA. Modal acoustic emission of damage accumulation in a woven SiC/SiC composite. Compos Sci Technol. 1999; 59(9):687–97. 64. Faudree M, Baer E, Hiltner A, Collister J. Characterization of damage and fracture processes in short fiber BMC composites by acoustic emission. J Compos Mater. 1988;22(12):1170–95.
123
J Mater Sci: Mater Med (2012) 23:217–228 65. Gibbs JW. Elementary principles in statistical mechanics. New York;: Dover; 1960. 66. Penrose O. Foundations of statistical mechanics—a deductive treatment. New York: Dover Publications; 2005. 67. Lewis G. Apparent fracture toughness of acrylic bone cement: effect of test specimen configuration and sterilization method. Biomaterials. 1999;20(1):69–78. 68. Lewis G, Mladsi S. Effect of sterilization method on properties of Palacos R acrylic bone cement. Biomaterials. 1998;19(1): 117–24. 69. Wright DD, Lautenschlager E. P., Gilbert JL. Bending and frcture toughness of woven serfl-reinforced composite polymethyl methacrylate. In: The 5th World Biomedical Materials Congress, Toronto, Canada; May 1996. p. 916. 70. Hughes KF, Ries MD, Pruitt LA. Structural degradation of acrylic bone cements due to in vivo and simulated aging. J Biomed Mater Res. 2002;65A:126–35. 71. Shen J, Chen C, Sauer JA. A numerical simulation for effective elastic moduli of plates with various distributions and sizes of cracks. In: International conference on polymers and plastics rubber. London; 1983. p. 6.1–6.10. 72. Lewis G, Li Y. Dependence of in vitro fatigue properties of PMMA bone cement on the polydispersity index of its powder. J Mech Behavior Biomed Mater. 2010;3:94–101. 73. Qi G. Attenuation of acoustic emission body waves in acrylic bone cement and synthetic bone using wavelet time-scale analysis. J Biomed Mater Res. 2000;52(1):148–56. 74. Qi G, Thota S, Vankamamidi S. Mechanical test reports— SmartSet MV bone cement. Memphis, TN 38152: Medical Acoustic Research Lab., University of Memphis; 2008. 75. Huang NC, Korobeinik MY. Interfacial debonding a spherical inclusion embedded in an infinite medium under remote stress. Int J Fract. 2001;107:11–30. 76. Argon AS, Salama MM. Growth of crazes in glassy polymers. Philos Mag. 1977;36(5):1217–34. 77. Koenczoel L, Hiltner A, Baer E. Crazing and fracture in polystyrene studied by acoustic emission. J Appl Phys. 1986;60(8):2651–4. 78. Bucknall CB. New criterion for craze initiation. Polymer. 2007;48(4):1030–41. 79. McCrum NG, Buckley CP, Bucknall CB. Principles of polymer engineering. Oxford: Oxford University Press; 1997. 80. Marshall GP, Culver LE, Williams JG. Craze growth in polymethylmethcraylate: a fracture mechanics approach. Proc R Soc Lond A. 1970;319:165–87.
