1. Introduction to System Reliability
Section 3: System Reliability & Weibull Analysis
Introduction to System Reliability
The mathematics that form the foundation of reliability analysis has been presented. Several methods have been discussed to evaluate the reliability of a single component with complex limit states. The next issue is how does one analyze a system with two or more sub-components.
A fundamental principle of reliability analysis is that as the complexity of a system increases, the system reliability decreases.
To see this consider a system with n mutually independent subcomponents (think of a statically determinate truss). The reliability of this non-redundant system is:
The dramatic deterioration of system reliability is illustrated in the figure to the right. All subcomponent reliabilities are considered equal.

2. Section 3: System Reliability & Weibull Analysis
There are three basic categories of systems based on how the overall reliability is computed. They are
Series Systems (example just presented)
Parallel Systems (indeterminate trusses, composite material systems)
Systems with active redundancies (r out of n systems)
The brittle failure behavior of ceramic materials is best modeled by a series system. The next section provides the motivation for the expression found on the previous overhead.

3. Reliability – Series Systems
Section 3: System Reliability & Weibull Analysis
Reliability – Series Systems
Consider a system composed of two independent sub-components. If Failure of either sub-component results in system failure, then the system can be represented by the following block diagram:
In a structural application the block diagram should not be interpreted as a free body diagram, i.e., both blocks do not necessarily have the same load applied to them in order to maintain static equilibrium.

4. Section 3: System Reliability & Weibull Analysis
The probability of failure for this simple system is
This last result is based on the assumption that both events are statistically independent. With
Then

5. Section 3: System Reliability & Weibull Analysis
In general, for a system on n sub-components
Finally, note that we can alternatively compute the reliability of a series system as

6. Series System Example – Statically Determinate Truss
Section 3: System Reliability & Weibull Analysis
Series System Example – Statically Determinate Truss
(overheads)

7. Two Parameter Weibull Distribution and Size Effects
Section 3: System Reliability & Weibull Analysis
Two Parameter Weibull Distribution and Size Effects
In the ceramic and glass industry the Weibull distribution is universally accepted as the distribution of choice in representing the underlying probability distribution function for tensile strength. A two-parameter formulation and a three-parameter formulation are available for the Weibull distribution. However, the two-parameter formulation usually leads to a more conservative estimate for the component probability of failure. The two-parameter Weibull probability density function for a continuous random strength variable, denoted as S, is given by the expression
for s > 0 , and
for s  0 .

8. Section 3: System Reliability & Weibull Analysis
The cumulative distribution is given by the expression
for s > 0 , and
for s  0 . Here a (a scatter parameter, or Weibull modulus) and b (a central location parameter, or typically referred to as the Weibull scale parameter) are distribution parameters. Note that in the ceramics and glass literature when the two-parameter Weibull formulation is adopted then "m" is used for the Weibull modulus a, and either s0 or sq (see the discussion in the parameter estimation section regarding the difference between s0 and sq) is used for the Weibull scale parameter. Here the (a, b) notation is used somewhat interchangeably with the typical notation adopted in the ceramics literature. There is a tendency to overuse the "s" symbol (e.g., sq, s0, si-failure observation, and st-threshold stress, etc.). Throughout this presentation the use of the symbol "s" should be apparent to the reader given the context in which it appears.

9. Section 3: System Reliability & Weibull Analysis
If the random variable representing uniaxial tensile strength of an advanced ceramic is characterized by a two-parameter Weibull distribution, then the probability that a uniaxial test specimen fabricated from an advanced ceramic will fail can be expressed by the cumulative distribution function
Note that smax is the maximum normal stress in the component. When used in the context of characterizing the strength of ceramics and glasses, the central location parameter is referred to as the Weibull characteristic strength (sq). The characteristic strength is dependent on the uniaxial test specimen (tensile, flexural, pressurized ring, etc.) used to generate the failure data. For a given material, this parameter will change in magnitude with specimen geometry, i.e., ceramics exhibit a size effect. The Weibull characteristic strength typically has units of stress. The scatter parameter m is dimensionless.

10. Section 3: System Reliability & Weibull Analysis
Ceramic materials typically exhibit rather high tensile strength with a relatively low fracture toughness (typically quantified by KIC). Lack of ductility (i.e., lack of fracture toughness) leads to low strain tolerance and large variations in observed fracture strength. When a load is applied, the absence of significant plastic deformation or micro-cracking causes large stress concentrations to occur at microscopic flaws. These flaws are unavoidably present as a result of fabrication or in-service environmental factors.
To date non-destructive evaluation (NDE) inspection programs with sufficient resolution have not been successfully implemented into fabrication operations. The combination of high strength and low fracture toughness leads to relatively small critical defect sizes that can not be detected by current NDE methods. As a result, components with a distribution of defects (characterized by various crack sizes and orientations) are produced which leads to an observed scatter in component tensile strength. The distribution of defects leads to an apparent decrease in tensile strength as the size of the component increases. This is the so-called strength-size effect, i.e., with bigger components there is an increased chance that larger more deleterious flaws are present. Design methods for ceramic components must admit size dependence. This is accomplished through the use of reliability concepts where the component is treated as a system, and the probability of failure of the system must be ascertained.

11. Section 3: System Reliability & Weibull Analysis
The failure behavior of ceramic materials is sudden and catastrophic. This type of behavior fits within the description of a series system, thus a weakest-link reliability approach is adopted. For a component fabricated from a ceramic material this requires that the probability of failure of a discrete element must be related to the overall probability of failure of the component. Finite element software is used to describe the state of stress throughout the component. Thus a component is discretized, and the reliability of each discrete element must be related to the overall reliability of the component. If failure occurs in one element, the component has failed.
We define the reliability of a continuum element as
The volume of this arbitrary continuum element is identified by V. In this expression s0 is the Weibull material scale parameter and can be described as the Weibull characteristic strength of a specimen with unit volume loaded in uniform uniaxial tension. This is a material specific parameter that is utilized in the component reliability analyses that follow. The dimensions of this parameter are stress×(volume)1/m.

12. Section 3: System Reliability & Weibull Analysis
The requisite size scaling is introduced by the previous equation. To demonstrate this, take the natural logarithm of this expression twice, i.e.,
Manipulation of this expression yields
with

13. Section 3: System Reliability & Weibull Analysis
then it is apparent that the double log equation has the form of a straight line, i.e., y = m1x + b, and plots as follows.
Consider several groups of test specimens fabricated from the same material. The specimens in each group are identical with each other, but the two groups have different gage sections such that V (which is now identified as the gage section volume; or DA which is identified as the gage area) is different for each group of specimens. Estimate Weibull parameters m and s0 from the pooled failure data obtained from each group (parameter estimation and pooling techniques are discussed in a later section).

14. Section 3: System Reliability & Weibull Analysis
After the Weibull parameters are estimated the line for y = m1x + b can be plotted that corresponds to say (i.e., the 50th percentile). This value for should correlate well with the median values in each group, if the number in each group is large.
Plot this specific curve as a function of the gage volumes (DV) or gage area (DA) for each group, as is done to the right. If no size effect is present, then median failure strengths of the groups will fall close to a horizontal line. This would indicate no correlation between gage geometry and the median strength value. A systematic variation away from a horizontal line approaching the curve y = m1x + b indicates that a size effect is present. The figure to the right appears in Johnson and Tucker (1991)

15. Component Reliability
Section 3: System Reliability & Weibull Analysis
Component Reliability
The ability to account for size effects of individual elements was introduced through the expression appearing in the previous section, i.e.,
Now a general expression for the probability of failure for a component is derived based. Under the assumptions that the component consists of an infinite number of elements (i.e., the continuum assumption) and that the component is best represented by a series system, then

16. Section 3: System Reliability & Weibull Analysis
substitution leads to
The limit inside the bracket is a Riemann sum. Thus
Weibull (1939) first proposed this integral representation for the probability of failure. The expression is integrated over all tensile regions of the specimen volume if the strength-controlling flaws are randomly distributed through the volume of the material, or over all tensile regions of the specimen area if flaws are restricted to the specimen surface. For failures due to surface defects the probability of failure is given by the expression

17. Section 3: System Reliability & Weibull Analysis
Over the years a number of reliability models have been presented that extend the uniaxial format of the previous two integral expressions to multiaxial states of stress. Only models associated with isotropic brittle materials are presented here. Anisotropic reliability models are beyond the scope of this presentation. The reader can consult the work of Duffy and co-workers (1990 and 1990) for information pertaining to reliability models for brittle composites. The monolithic models highlighted here include the principle of independent action (PIA) model and Batdorf’s model. A brief discussion is presented for each.

18. Multiaxial Reliability Models
Section 3: System Reliability & Weibull Analysis
Multiaxial Reliability Models
A number of reliability models have been presented in the literature that assess component reliability based on multiaxial states of stress. The two monolithic models highlighted here include the principle of independent action (PIA) model and Batdorf’s model. A brief discussion is presented for each. In order to simplify the presentation of each model component probability of failure is expressed in the following generic format
where y is identified as a failure function per unit volume. What remains is the specification of the failure function y for each reliability model.

19. PIA – A Phenomenological Model
Section 3: System Reliability & Weibull Analysis
PIA – A Phenomenological Model
For the principle of independent action (PIA) model:
where s1 , s2 and s3 are the three principal stresses at a given point. Recall that s0 is the Weibull material scale parameter and can be described as the Weibull characteristic strength of a specimen with unit volume, or area, loaded in uniform uniaxial tension.
The PIA model is the probabilistic equivalent to the deterministic maximum stress failure theory. The PIA model has been widely applied in brittle material design. However, the PIA model does not specify the nature of the defect causing failure. In essence the model is phenomenological, which does not imply that the model is not useful. The simplicity of phenomenological models can often times be a strength, not a weakness

20. Weibull Analysis – Closed Form Examples
Section 3: System Reliability & Weibull Analysis
Weibull Analysis – Closed Form Examples
(overheads)

21. Batdorf’s Theory - Mechanistic Model
Section 3: System Reliability & Weibull Analysis
Batdorf’s Theory - Mechanistic Model
The concepts proposed by Batdorf are important in that the approach incorporates a mechanistic basis for the effect of multiaxial states of stress into the weakest-link theory. Here material defects distributed throughout the volume (and/or over the surface) are assumed to have a random orientation. In addition, the defects are assumed to be non-interacting discontinuities (cracks) with an assumed regular geometry. Consider a finite volume of material where the stress state is uniform throughout. Batdorf postulated that the probability of failure for this finite volume can be expressed by the following joint probability
Here  represents the uniform multiaxial state of stress throughout the finite volume. In addition, P1 is the probability that a crack exists in V (the finite volume) with an associated critical stress (i.e., fracture stress) in the range of cr to cr+dcr. Note that cr is defined as the critical far field normal stress for a given crack configuration under mode I loading. The second marginal probability, P2, denotes the probability that a crack with an associated critical stress of cr will be oriented in a direction such that e equals or exceeds cr.

22. Section 3: System Reliability & Weibull Analysis
These two probabilities can be expressed as
and
The function N(cr) is referred to as the Batdorf crack density function. This function is independent of the multiaxial stress state . Typically the function is expressed in a power law format, i.e.,
where m is the Weibull modulus and kB is a material parameter, both of which can be estimated or computed from failure data (see discussion below).

23. Section 3: System Reliability & Weibull Analysis
The solid angle  is mapped onto a unit sphere in the principle stress space and represents all crack orientations for which
Failure is assumed to occur when a far field effective stress e associated with the weakest flaw reaches a critical level, cr. The effective stress e is a predefined combination of the far field normal stress and the far field shear stress. It is also a function of the assumed crack configuration, the existing stress state, and the fracture criterion employed (hence the claim that the approach captures the physics of fracture). Accounting for the presence of a far-field shear stress reduces the far-field normal stress needed for fracture for special variations of the Batdorf model.

24. Section 3: System Reliability & Weibull Analysis
The Batdorf model is defined by taking
Substituting for N(cr) yields
thus
for Batdorf’s model.

25. Section 3: System Reliability & Weibull Analysis
The solid angle is defined by the following expression
where a and b are azimuthal angles in the principal stress space, and
if and if

26. Section 3: System Reliability & Weibull Analysis
This leads to the following expression for the component probability failure
assuming that flaws are distributed uniformly throughout the volume of the component. A similar development would lead to the following expression for failure due to flaws spatially distributed along the surface of the component, i.e.,

27. Section 3: System Reliability & Weibull Analysis
The crack density coefficient kB that appears in the previous expressions is obtained from failure data. The methodology for obtaining estimates of the Weibull parameters m and sq was outlined (refer to the section on maximum likelihood estimators). Once these quantities are estimated kB can be computed since
However, note that kB is a material specific parameter in much the same manner that s0 (the Weibull material scale parameter) is a material specific parameter. The specimen geometry incorporated in the expression above is compensated by the fact that sq is also dependent on specimen geometry. Thus the form of the expression above changes from specimen type, but the estimated value of sq will also change, and the result is that the computed value of the crack density coefficient should stay consistent from test geometry to test geometry for a given material.

28. Section 3: System Reliability & Weibull Analysis
Numerous authors have discussed the stress distribution around cavities of various types, under different loading conditions, and proposed numerous criteria to describe impending fracture. However, several issues must be pointed out:
No prevailing consensus has emerged regarding a best probabilistic fracture theory. Most current models predict somewhat similar reliability results, despite the divergence of initial assumptions.
Moreover, one must approach the mechanistic models with some caution. The reliability models based on fracture mechanics incorporate the assumptions made in developing the fracture models on which they are based.
One of the fundamental assumptions made in the derivation of fracture mechanics criteria is that the crack length is much larger than the characteristic length of the microstructure. This is sometimes referred to as the continuum principle in engineering mechanics. For ceramic materials that characteristic length is the grain size (or diameter). If one contemplates the fact that most ceramic materials are high strength with an attending low fracture toughness, then the critical defect size can be quite small. If the critical defect size approaches the grain size of the material, then the Batdorf models discussed above should be based on the physics of cleavage fracture through single crystal microstructures.

29. Section 3: System Reliability & Weibull Analysis
Given the words of caution on the previous overhead, it is noted that a fracture based criterion lends itself well to temporal or time dependent analyses, otherwise known as the subcritical crack growth method.

30. Section 3: System Reliability & Weibull Analysis