Modified Dynamic Reliability Model for Damage Accumulation Yair Shai Faculty of Industrial Engineering Technion – Institute of Technology Haifa, Israel
[email protected]
Dov Ingman Faculty of Industrial Engineering Technion – Institute of Technology Haifa, Israel
[email protected]
Following are the definitions of the parameters and components in (1). r(x,t) is the RDF defined as the probability density function, for finding an element functioning at the moment of time t and having strength in the interval (x,x+dx), x being a generalized strength parameter. The conventional reliability function, R(t), can be expressed through the RDF as follows:
Abstract— the earlier suggested model for the dynamics of element reliability distribution over a generalized strength (capacity) space [2] reflects the roles of both strength deterioration and failure processes. The present model is a modification of the previously suggested model and considers the interaction of the instantaneous stress (demand) with the instantaneous strength (capacity). Modified Smoluchowski’s probability of transition equation plays the role of the process generator allowing an iterative simulation for the processes of strength deterioration and failure from the initial strength distribution. A practical illustration of the suggested concept is demonstrated by a rather general simulation procedure, which avoids the need for any special assumptions regarding the transition rates within the processes in question. While keeping all the benefits and the attributes of the previously suggested model, the model addressed in this paper exhibits a more realistic matter-environment system behavior. The resultant damaged strength distribution shows tendency to the Weibull model, in which all the parameters change along the process. The calculated data show physically meaningful effect of the particular material parameters and environmental conditions.
( x)
The integral of the RDF over the strength space expresses the probability of finding an element functioning at time t, while being on a non-failed strength level. The hazard rate function H(t) may also be described as a function of the RDF [2], however both the reliability function and the hazard function cannot be derived prior to solving the balance equation. Hence, these functions are consequential to the damage accumulation process. The Smoluchowski’s probability of transition from the strength level x, to a weaker state x', should satisfy the normalization condition:
1. INTRODUCTION ................................................................ 1 2. MODEL MODIFICATION .................................................. 2 3. SIMULATION PROCESS AND RESULTS ............................. 3 4. DISCUSSION ..................................................................... 4 5. EXAMPLES ....................................................................... 4 6. CONCLUSIONS.................................................................. 6
∫ p ( x → x ')dx ' = 1
The rate of change of the RDF is determined by the rates of three processes: (a) The failure rate at the strength level x: λ(x,t)·r(x,t), where λ(x,t) is the rate of events with a stress level above the element strength x, and leading, therefore, to the element’s failure; (b) The rate of the strength deterioration: ω(x,t)·r(x,t), where ω(x,t) is the rate of events below the element strength at time t; (c) The integral rate of transition to the strength level x from all higher strength states x'. While the first term in (1), consists of (a) and (b), tends to reduce the RDF, the second term, equal to (c), tends to increase it. The function r(x,t) in the last term accounts for the initial values r(x,0), and should satisfy the requirement:
Damage accumulation process, be it the manifestation of the aging phenomenon, has fundamental impact on the reliability of materials. A standard reliability function R(t) serve only for explicit evaluation of the element probability not to fail [1] and enables the representation of the corresponding failure rate by the hazard rate function H(t). These however could not be used to describe the ongoing element deterioration under stress. The simultaneous effect of both processes – the element deterioration and failure, was modeled by an integro-differential transfer equation for the reliability distribution function (RDF) over the strength space [2-4]:
R ( 0 ) = ∫ r ( x, 0 ) dx ≡1 x
Solving (1) analytically is not trivial since the RDF is a nonstationary non-ergodic function. Some assumptions [2] about the transfer rates and the transition probability have shown a Weibull form solution for the RDF. Assuming constant failure rate and choosing ω ( x ) = Ω ⋅ xγ and:
= − ( λ ( x , t ) + ω ( x , t ) ) ⋅ r ( x, t ) +
∫ ω ( x ', t ) p ( x ' → x ) r ( x ', t ) dx ' + δ ( t ) ⋅ r ( x, t )
(1)
x '> x
978-1-4799-5380-6/15/$31.00 ©2015 IEEE
(2)
x '≤ x
1. INTRODUCTION
∂t
∫ r ( x, t ) dx
R(t ) =
TABLE OF CONTENTS
∂r ( x, t )
Ephraim Suhir Portland State University, Portland, OR, USA, and Technical University, Vienna, Austria
[email protected]
1
P ( x ' → x) =
β ⋅ x β −1
( x ')
The following chapters suggest a modification of the original theorem to include the matter-environment interaction in terms of strength deterioration under stress. In addition, a simulative approach is developed to represent the full dynamics subject to initial conditions and predetermined model parameters that stand for some intrinsic characterizations of the specimen under test.
(3)
β
The parameters β,γ,Ω should be found from experimental information of failure and aging mechanisms. For β=1 and γ= -0.5 the solution obtained was:
rx0 ( x, t ) = β
ω ( x0 ) x0
β
⋅ x β −1 ⋅ t ⋅ e −Ω⋅ x
β
⋅t
+ δ ( x − x0 ) ⋅ e −ω ( x0 )⋅t
2. MODEL MODIFICATION The kernel of the model is the probability of transition from the strength level x, to a weaker state x', p(xx'). Equation (3) refers to the model parameter β as a constant power factor. This constant is empirically fitted to experimental data. However, one may expect the routes of dynamics to differ for an element's strength with specified initial conditions when exposed to different stresses. Namely, the higher the stress relative to the strength, the greater is the chance for strength deterioration. In that case, β should stand for the specific experiment stress conditions as well as for some intrinsic material characteristics. Hence, the power factor can be modified to include the influence of the instantaneous stress. In that spirit, an alternate probability of transition is suggested:
Here, β is a shape parameter of Weibull distribution, not to confuse with β in (3) and x0 is the initial strength r ( x, 0 ) = δ ( x − x0 ) . Note that an initial delta function brings no lack of generality since, for any given initial distribution f ( x ) = r ( x,0 ) , the RDF may be constructed based on the solution rx0 ( x, t ) as follows: ∞
r ( x, t ) = ∫ dx0 rx0 ( x,0 ) f ( x ) 0
In that case,
r ( x, t ) = W (t ) β ⋅ Ωx β −1 ⋅ t ⋅ e−Ωx
β
⋅t
+ f ( x) ⋅ e −Ωx
β
⋅t
(4)
x b b −1 1 x x ' y for th ≤ y ≤ x (5) p( x → x' y) = x ⋅ y ⋅ x for y > x δ ( x ')
Where
W (t ) =
ω ( x0 ) 1 dx0 ∫ Ω ( x0 ) x0 β
where x is the current strength, y is the instantaneous induced stress, x' is some strength value 0≤ x'< x. b and th are model parameters which stand for the intrinsic characteristics. Note that th in (5) is the threshold mentioned above in the introduction. The Item fails whenever the stress is greater than the strength, y>x, and its strength drops to zero. This probability density over the strength space (5) satisfies the condition (2), since:
The second term in (4) represents the exponentially decaying arbitrary initial strength distribution f(x). One should notice that the solution (4) yields a constant shape parameter whereas the scale parameter changes in time. The RDF (4) explains the phenomenon of bimodality of the strength distribution in materials under different doses of load. In [5], experimental bimodality in the strength distribution of optical fibers is explained not as an intrinsic feature of the fibers, but as a combination of the original strength distribution of the "virgin" (undamaged) material and the secondary one, developed in the process of damage accumulation under load. Yet, another attribute is involved; the term fatigue limit (or endurance limit) in S_N curves [6] is the value of the stress below which a material can presumably endure an infinite number of stress cycles. In analogy, there exists a threshold for stress values, under which no deterioration occurs from a certain strength state to a lower one. This attribute is well understood by the elastic energy potential representation of damage accumulation process, heuristically represented in [3,4]. However, the RDF (4) does not relate its own evolutionary behavior to stress. In fact, the most significant drawback of the described theorem is the lack of representing the influence of stress fluctuation on the damage accumulation process. From a physical aspect, the stress, whether it is a simple tensile stress, internal thermal fluctuations or any other load, is the driving force of the dynamics.
b
1 x x' ⋅ x 0 x
∫ x ⋅ y
x b −1 y
x' dx ' = x
x y
b
x
=1 0
(6)
∀ x, y, b; b ≥ 0; x > y ≥ th The delta function for the case of y>x is normalized to unit as well. One consequent advantage of the suggested modification is the ability to track accelerating dynamics of strength deterioration along the process since as the strength decreases, the probability of transition to a further lower strength value increases for a certain given stress value. Figure 1 shows the probability density function of the transition (5) for different values of the stress, denoted as y, and the parameter b when the current strength is x=5. Lower stress values or higher values of the parameter b lead to narrower distributions, i.e. the strength is more likely to drop to nearer lower strength values.
2
simulation, we shall focus on the physical dynamics of RDF.
6
y=0.1x, b=1 y=0.063x, b=1 y=0.1x, b=1.4
2.a 2.b
4
probability
probability density
−3
6×10
2
−3
4×10
−3
2×10
0
3
3.5
4
4.5
0
5
0
200
x' - lower level strength
400
600
time to failure
Figure 1 – modified transition probability density
Figure 2 – Time to Failure distribution (simulated)
The formula (5) has no analytic solution; hence, a simulative approach is required. An initial strength level was set for a large population of material elements in a simulation program followed by iterative steps of the probabilistic response of strength level transitions under given stress distribution. This procedure, imposed on the whole population, one element at a time, entails no need for special assumptions about the rates of transitions; all characteristics of the two processes are both included based on (5) alone – the time to failure distribution and the strength distribution at any step of the process.
Following the solution for the original RDF model (4), a Weibull distribution was fitted to the strengths of un-failed items at each cycle of the simulation; here a three parameters distribution. These distributions best represent the deteriorating population after statistically disengaged from the original strength value, i.e., the daughter distributions. Figure 3 shows probability density distributions of strength after some different number of cycles and the fitted Weibull models accordingly. The bimodal behavior is noticed in Figure 3 on the right hand curve since the original strength value was set to 100 arbitrary strength units.
3. SIMULATION PROCESS AND RESULTS
0.08
Probability density
The simulation intends to find the dynamic characteristics of the strength distribution and the time to failure for a variety of initial conditions and stress distributions (Figures 2-3). Particularly, these characteristics are shown as the functionals of the model parameters b and th (Figures 4-5). The simulation consists of the following steps: 1. Introduce the stress distribution. 2. Set the initial strength distribution – without loss of generality and for simplicity of treatment, we choose initial stress to be single value, i.e. delta function. 3. Choose values of model parameters b and th. 4. Draw a random stress value for a specimen under test. 5. If stress > strength than note the failure time (cycle). 6. Otherwise, if stress > th than draw a random lower strength level for the specimen according to (6). 7. Repeat steps 4-7 for all items. 8. Aggregate the data at each stress cycle to statistical representation. The results of a representative simulation are presented hereinafter: A probability density distribution of the time to failure (TTF) is presented in Figure 2. A large population was set to an initial strength value and subjected to load cycles of certain stress distribution. Two simulations were executed with the same conditions for different values of the model parameter, b. The curve 2.a refers to the higher value and 2.b to the lower one. Though TTF is a simple feature of the
0.06
0.04
0.02
0
20
40
60
80
100
Strength (arbitrary units)
Figure 3 – strength distributions after some load cycles In the following figures, we refer to the simulation cycles as time. Figure 4 describes the statistical dynamics by following the trends of each Weibull parameter along the time axis and as a functional of the model parameter b.
3
scale factors
shape factor
160
b=0.95 b=1 b=1.05
140 120
10 8
80
6
60
4
60
80
100
120
140
b=0.95 b=1 b=1.05
12
100
40 40
Note that the specific values of b and th shown within Figures 4 and 5 merely represent the relative change in the dynamics while all other conditions were steady. They were arbitrarily set for the convenience of representation.
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4. DISCUSSION
2 40
160
60
80
test cycle time
100
120
140
Damage accumulation process is vastly dealt with in the scientific literature. The variety of models and approaches, by itself, emphasizes the complexity of this behavior. The fact that pure physical approaches are incapable of describing the probabilistic nature of the process is also well established. The RDF model combines probabilistic failure occurrence and physical strength deterioration of which its statistical properties follow a diffusion-like pattern in time. By implementing the modification described above, the dynamics of actual damage accumulation finds expression in a fully dynamic probabilistic behavior, through which all three Weibull parameters of the characterizing strength distribution change in time. This feature is new to previous analyses; the change of the shape parameter, which was not derived in the analytical solution (4), shows continuous change of elasticity characteristics throughout the deterioration process. Moreover, the modified model is robust in the sense that on one hand, it considers the interaction of the item strength with the environment at any given time and on the other, it allows the researcher to tune the dynamics according to the material's characteristics and the nature of its unique aging mechanism. In other words, the model is generic and flexible for a variety of environments and materials suchlike brittle materials e.g. glass, ceramics, steel etc., silicon devices, fiber optics and others. The model is hardly applicable for plastic or viscoelastic materials. The model parameters, b and th, are not merely unique constants; it is obvious these parameters are influenced by many environmental characteristics, however these dependencies are a matter for separate research. In that case, materials under specific set of conditions are defined by a set of these parameters. The simulative approach enables one to fit the model parameters to empirical results. Hence, the model's generality necessitates no prior knowledge of the exact material's physical, mechanical or chemical characteristics nor the specific environmental measures. All that counts is the model's ability to be adapted to the strength distribution after different number of load cycles. It follows from the discussion that routes of damage accumulation cannot be assumed to be predictable according to other cases. Finally, the model may show new perspectives for the field of accelerated tests.
160
test cycle time
delay factor 80
b=0.95 b=1 b=1.05
60
40
20
0 40
60
80
100
120
140
160
time test cycle
Figure 4 – trend of the strength Weibull parameters for different values of the model parameter b Figure 5 presents each Weibull parameter along the time axis and as a functional of three different constant values of the model parameter th. In case that th is not constant, the simulation may be easily changed in accordance. scale factors
shape factor
400
30
th=3 th=3.3 th=3.6
300
th=3 th=3.3 th=3.6
20
200 10 100
0 40
60
80
100
120
140
160
0 40
60
80
100
120
140
160
testtime cycle
test cycle time
delay factor 250
th=3 th=3.3 th=3.6
200
150 100
50 0 40
5. EXAMPLES 60
80
100
120
140
160
A population of fiber optics was tested [5] for some behaviors of damage accumulation and reliability. By these experimental data, we demonstrate some of the important characteristics of damage accumulation offered by our
time test cycle
Figure 5 – trend of the strength Weibull parameters for different values of the model parameter th
4
modified model. As expected, a very good fit to the Weibull distribution was accepted while the dynamics of the damage accumulation process, in this example, is realized by the change in the Weibull parameters as follows:
Example I – the Dynamic Weibull Characteristics of the Strength Distribution A tensile test was carried out for comparative analysis of the initial and the post-loading distributions of fibers. In these experiments, 5 m long silica-glass fiber specimens of outer diameter 300 μm were divided into five groups. At least 60 specimens were prepared and tested. The first group was subjected to the standard test (with a constant strain rate), and the other specimens were preliminarily subjected to different “doses” of load (low, medium and high), and only after that were subjected to the tensile test. The initial fiber clearly showed a unimodal distribution of nearly a delta function. However, the two groups of fibers, exposed to medium and high load doses respectively, showed bimodal strength distributions as presented in Figure 6.
Table 1 – Fitted Weibull parameters (example I) Weibull parameters Medium damage Higher damage
Probability density
0.2
0.1
87.143
95.714
104.286
112.857
121.429
130
Strength, psi
Figure 6 – tensile strength of fibers, p.d.f. The remaining initial population retains its mean and standard deviation under the applied load whereas the damaged population, represented by the daughter distribution, becomes more pronounced for larger load dose. According to the analysis above, a three-parameter Weibull model was fitted to each of the populations of damaged fibers after being normalized to its share of the total p.d.f., as shown in Figure 7. The fitted Weibull distributions for both populations are plotted by the dotted curves.
Probability density
0.05
90
100
τ
yn Where τ is an empirical constant, representing the time to failure (rupture) under a unity stress level, and n is an empirical (material's) exponent. The results clearly showed an advantage for NPM technology in terms of better TTF for all stress levels. However, the suggested power law is not a physically based model; on the horizontal axis, the TTF approaches zero for infinite stress level whereas, in fact, the initial strength is a limit over which any stress will fail the specimen. On the vertical axis, the power law model determines there is a stress level, even infinitesimally small, which may fail the specimen after a very long period. Therefore, a more suitable model must refer to the initial strength and allow for minimal stress value (i.e. the threshold) under which the TTF becomes infinite. We refer the following as the threshold model:
high load dosed fibers fitted Weibull model medium load dosed fibers fitted Weibull model
80
33.022 32.265
TTF ( y ) =
0.1
0 70
delay
67.020 64.806
Fibers require long-term protection against moisture and oxygen, as well as mechanical and thermal protection. Most existing coatings for optical fibers are polymer-based. These are moisture-sensitive. A long-term healing technology uses nanoparticle-type coating materials (NPM's) to fill the gullies, micro-cracks and other defects of the plastic and thus prolongs its durability and enhances its reliability. In order to compare the mechanical and environmental characteristics of the NPM-based and "regular" fibers under different loading and environmental conditions an experiment was undertaken. Specimens from light guides consisting of silica cores and polymer coatings were used; regular (“reference” fibers) or modified by adding the NPM. In this experiment mechanical stresses were applied to the fiber specimens (two-point bending conditions) and the time-to-failure (TTF) was measured for each case under two relative humidity (RH) conditions – RH =35% (normal) and RH=100%. All specimens were subjected to the predetermined humidity for at least 48 hours. In order to compare the TTF for different specimens, a power law model was used to approximate the relationships between the TTF and the applied stress, y:
fibers exposed to medium load dose fibers exposed to high load dose
78.571
scale
16.449 12.260
Example II –Threshold Existence Evidence
0.3
0 70
shape
TTF ( y ) =
110
Strength, psi
Figure 7 – Weibull model fit for the normalized daughter distributions of tensile strength of fibers
5
y − SI ⋅τ y − 1 th
n
th ≤ y ≤ S I
environmental conditions and the induced stress, into a generalized system governed by a probabilistic strength deterioration process. A simulative solution was demonstrated offering traceability for the full dynamic probabilistic process of strength deterioration under given stress. The solution reveals new features previously unfamiliar due to the absence of general analytic solutions. The RDF tends to follow a three parameters Weibull distribution after departing from the original strength value, where all three parameters change in time in paths unique to each case. At the same time, the model is flexible for a variety of materials and conditions for which empirical data may be fitted by the model parameters.
Where SI is the initial strength, th is the threshold, n and τ are model parameters. The results of the described experiment were presented [5] by the power law curves fitted to the output data, shown by the solid line curves in Figure 8. In this example, the alternative threshold model was used. It is plotted on the same graph by the dotted curves. Figure 8 compares the population of reference fibers and the NPM modified fibers in the case of RH =35%, each fitted by the two described models. 1.5×10
refference sample - power law refference sample - threshold model NPM based sample - power law NPM based sample - threshold model
3
REFERENCES
TTF, sec
1×10
3
[1] K. C. Kapur and L. R. Lamberson, Reliability in Engineering Design: John Wiley, 1977. [2] D. Ingman and L. A. Reznik, A dynamic model for element reliability. Nuclear Engineering and Design, 1982, 70 , pp. 209-213. [3] D. Ingman and L. A. Reznik, Dynamic reliability model for damage Accumulation Processes. J. Nucl. Technol, 1986, 75, pp. 261-282. [4] D. Ingman and L. A. Reznik, Dynamic Character of failure State in Damage Accumulation Processes. Nuclear Science and Engineering, 1991, 107, pp.284290. [5] D. Ingman, T. Mirer and E. Suhir, “Dynamic Physical Reliability in Application to Photonic Materials”, in E. Suhir, C.P. Wong, Y.C. Lee, eds. “Micro- and OptoElectronic Materials and Structures: Physics, Mechanics, Design, Packaging, Reliability”, Springer, 2007. [6] B. Boardman, Fatigue Resistance of Steel. ASM Handbook, Volume 1: Properties and Selection: Irons, Steels, and High-Performance Alloys, ASM Handbook Committee, 1991, pp.637-688.
500
0 4.4
4.9
5.4
5.9
6.4
stress, GPa
Figure 8 – TTF vs. bending stress, comparison of the power law (solid) and the threshold (dotted) models The model parameters for the cases described in Figure 8 are detailed in the following table: Table 2 – Fitted parameters for two models (example II) SI Power law model Reference sample Threshold model Reference sample Power law model NPM based sample Threshold model NPM based sample
6.222
6.209
Model parameters th τ 1.02E14 3.437
3.446
n 16.691
9.992
3.679
1.55E14
16.866
11.066
3.694
BIOGRAPHIES Yair Shai received a B.Sc. in Electrical Engineering from the Technion – Institute of Technology, Haifa, Israel, in 1990 and a M.Sc. in Quality assurance & Reliability engineering from the Technion, in 2005. Currently he is studying for Ph.D. at the Technion – Institute of Technology, Haifa, Israel, and his thesis subject is 'Reliability of Technologies'. Shai had served as an electrical and electronics engineer since 1990 in various government positions. Currently, he is an active system Reliability & Safety engineer.
Within the experiment's stress range, the agreement of the two models is impressive. However, the predicted threshold, satisfactorily similar for the two populations, is clearly not negligible, evidently justifying our theoretical assumptions. In addition, the fitted initial strength values comply with the actual data.
6. CONCLUSIONS A modification of the dynamic reliability model for damage accumulation was demonstrated. The kernel of the RDF dynamics, i.e. Smoluchowski’s transition probability from strength state x to a lower state x', was redefined as part of the integro-differential model for RDF. The modification combines the material's physical characteristics, the
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Prof. Dov Ingman, Technion, Israel, is staff member of Industrial and Management Eng. Department, and member of graduate studies group (program) in Quality Assurance and Reliability at Technion. His research interests include element and system reliability, damage accumulation processes, physical kinetics, pattern recognition, information theory, neural nets, measurement theory and instrumentation, desalination technology, nondestructive testing, and quality control.
Microelectronics Packaging Society (IMAPS) and the Society of Plastics Engineers (SPE). He is Distinguished Lecturer of the IEEE CPMT (Components, Packaging and Manufacturing Technology) Society, Associate Editor of the IEEE CPMT Transactions on Advanced Packaging, Member of the ECTC (Electronic Components and Technology Conference) Applied Reliability Subcommittee, the IEEE CPMT award committee, and the IEEE Fellow nomination committee. Dr. Suhir has authored about 350 technical publications (patents, papers, book chapters, books), including monographs “Structural Analysis in Microelectronics and Fiber Optics”, Van-Nostrand, 1991, and “Applied Probability for Engineers and Scientists”, McGraw-Hill, 1997. Dr. Suhir organized many successful conferences and symposia, presented numerous keynote, invited talks worldwide, and received many professional awards.
Prof. Ephraim Suhir is Fellow of the Institute of Electrical and Electronics Engineers (IEEE), the American Physical Society (APS), the Institute of Physics (IoP), UK, the American Society of Mechanical (ASME), Society of Optical Engineers (SPIE), International
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