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Reliable Optimal Design in Contact Mechanics 39

Consider another case when a system of four pointed forces

()

iii

qQxxyy

=−−, 1, 2, 3, 4i = is applied outside contact domain at the points

()

2, 0xay==,

()

0, 2xyb==,

()

2, 0xay=− = ,

()

0, 2xyb==−. Using (27), (29), (32), (33) we will obtain

()

()()

fpDxy

xx yy

⎡⎤

⎢⎥

−+−

⎣⎦

∑

(39)

In dimensionless variables (37) we have the following expression

−1

−0.5

0.5

−1

11.5

12.5

13.5

14.5

Fig. 5. Optimal shape f under loading of four pointed forces

()

2, 2xayb==,

()

2, 2xayb=− = ,

()

2, 2xay b==−,

()

2, 2xayb=− =− applied outside

contact domain, when

= .

40 N.V. Banichuk, S.Yu. Ivanova, and E.V. Makeev

()

fDxy

⎛⎞⎛⎞

−+−

⎜⎟⎜⎟

⎝⎠⎝⎠

∑





(40)

depending on parameters

ba.

The optimal shape (40) is shown in fig. 4 for

5.0

= ,

1ba= . This optimal

shape is also symmetrical with respect to coordinates

x and y but in comparison

with the shape shown in fig. 1 it is more truncated.

−1

−0.5

0.5

5.5

6.5

7.5

8.5

Fig. 6. Optimal shape f for the only force applied outside contact domain at the point

()

2, 2xayb==, when 5

= .

The optimal shape presented in fig. 5 corresponds to the case of four pointed

forces

()

iii

qQxxyy

=−−, 1, 2, 3, 4i = applied outside contact domain at

the points

()

2, 2xayb==,

()

2, 2xayb=− = ,

()

2, 2xay b==−,

Reliable Optimal Design in Contact Mechanics 41

()

2, 2xayb=− =− for 5.0

= ,

1ba= . The shape shown in fig. 6 cor-

responds to the case of the only pointed force applied outside contact domain at

the point

()

2, 2xayb== for 5.0

= ,

1ba= . As it is seen from the pre-

sented figures, the depth of punch penetration is increased, if the outside contact

domain force is increased.

5 Conclusions

In this paper we formulated and investigated the problem of contact pressure op-

timization for the rigid punch interacted without friction with an elastic medium.

The forces and moments applied to the punch and the external loads acted at the

outside regions were taken into account. All considerations have been performed

in the frame of quasistatical loading of the system: rigid punch – elastic founda-

tion. It was shown that the application of the proposed approach gives us the pos-

sibility to decompose the formulated optimization problem to two successively

solved problems. The first problem is concerned with the finding of the pressure

distribution realized the minimal value of considered discrepancy functional under

constraints on the equilibrium equations and positive pressure condition. The sec-

ond problem is concerned with the finding of the optimal punch shape and is re-

duced to determination of the normal displacements arising as a result of solution

of the first problem. The shape of optimal punch has been found in the form of

“simple layer potential” corresponding to the case of linear distributed contact

pressure.

It was obtained analytical solution of the shape optimization problem for the

punch having rectangular form in plan. Note, that the proposed approach can be

applied to the case of a system of contacted punches. The main results obtained in

this study are illustrated using examples of normal loads application in the some

given domain. Of course, there are many other cases with tangential loads action,

which can be effectively represented.

Acknowledgement

The paper is performed under support of RFBR (grant 08-08-00025-a), RAS Pro-

gram 13, Program of Support of Leading Scientific Schools (grant 169.2008.1).

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Scaling of Strength and Lifetime Distributions of

Quasibrittle Structures



Zdenˇek P. Baˇzant and Jia-Liang Le

Northwestern University

{z-bazant,jialiang-le}@northwestern.edu

Abstract. Engineering structures such as aircraft, bridges, dams, nuclear contain-

ments and ships, as well as computer circuits, chips and MEMS, should be designed

for failure probability < 10

−6

–10

−7

per lifetime. The safety factors required to en-

sure it are still determined empirically, even though they represent much larger and

much more uncertain corrections to deterministic calculations than do the typical er-

rors of modern computer analysis of structures. The empirical approach is sufﬁcient

for perfectly brittle and perfectly ductile structures since the cumulative distribution

function (cdf) of random strength is known, making it possible to extrapolate to the

tail from the mean and variance. However, the empirical approach does not apply to

structures consisting of quasibrittle materials, which are brittle materials with inho-

mogeneities that are not negligible compared to structure size. This paper presents a

reﬁned theory on the strength distribution of quasibrittle structures, which is based

on the fracture mechanics of nanocracks propagating by activation energy controlled

small jumps through a nano-structure and an analytical model for the multi-scale

transition of strength statistics. Based on the power law for creep crack growth rate

and the cdf of material strength, the lifetime distribution of quasibrittle structures

under constant load is derived. Both the strength and lifetime cdf’s are shown to be

size- and geometry- dependent. The theory predicts intricate size effects on both the

mean structural strength and lifetime, the latter being much stronger. The theory is

shown to match the experimentally observed systematic deviations of strength and

lifetime histograms of industrial ceramics from the Weibull distribution.

1 Introduction

In the design of various engineering structures, it is of paramount importance to

understand and ascertain the types of probability distributions of structural strength

since an experimental veriﬁcation of design strength for the tolerable failure proba-

bility P

< 10

−6

[21, 38, 36] is virtually impossible. For perfectly brittle structures



This paper is an authorized republication, with minor modiﬁcation, of a previous paper in

CD proceedings [6].

44 Z.P. Baˇzant and J.-L. Le

failing at initiation of a macro-crack from one representative volume element (RVE)

with a negligible size, the strength distribution is known to be Weibullian, based on

the inﬁnite weakest link model. For perfectly ductile structures, the failure load is a

weighted sum of the strengths of the RVEs along the failure surface. According to

the Central Limit Theorem, the strength distribution must follow the Gaussian (or

normal) distribution, except the remote tail.

It has been shown that the failure behavior of quasibrittle structures varies from

quasi-plastic to perfectly brittle as the structure size increases [14, 2, 3]. Conse-

quently, it must be expected that the type of probabilistic distribution of strength of

quasibrittle structures will vary with the structure size and geometry.

This study deals with quasibrittle structures of positive geometry, which is a

broad class of sturctures that fail at the macro-crack initiation from one RVE. Exten-

sive experimental data showed that the strength histograms of various quasibrittle

materials, such as concrete [56], industrial and dental ceramics[37, 34, 46, 47, 52],

and ﬁber composites [55, 54], consistently deviate from the two-parameter Weibull

distribution. Recently, the three-parameter Weibull distribution with a non-zero

threshold has been adopted as a remedy [27, 49, 22].

It is observed, however, that the three-parameter Weibull distribution still shows

a systematic deviation from sufﬁciently broad experimental histograms in the high

probability regime (Fig. 10 in [13]). Furthermore, the three-parameter Weibull dis-

tribution implies a vanishing size effect on the mean strength for large size struc-

tures, which does not agree with the experimental observations [32, 39]. In fact, the

experimental size effect curve on concrete and ﬁber composites [16, 18] at large size

limit implies that the threshold must be zero [39].

Recent studies [12, 13] showed that the problem actually lies in the assumption of

inﬁnite weakest-link model which underlies the Weibull statistics of strength. One

must consider a ﬁnite weakest-link model because of the fact that the size of one

RVE is not negligible compared to the structure size, which is the salient feature of

all quasibrittle structures. A probabilistic theory was recently developed to model

the strength distribution of quasibrittle structures [12, 13, 8].

The theory is derived from the strength statistics of a nano-structure considered as

a nano-scale block of either a regular lattice or a disordered nano-structure, which is

based on the probabilistic fracture mechanics of random jumps of the crack tip prop-

agating through the nano-structure [7, 8]. The transition of strength statistics from

the nano-scale to the RVE scale can be statistically represented by a hierarchical

model consisting of bundles and chains [12, 13]. Based on the asymptotic proper-

ties of strength cdf’s of bundles and chains, it has been shown that the strength cdf

of one RVE can be approximately modeled as a Gaussian distribution onto which a

power-law tail is grafted at the failure probability of about P

≈ 10

−4

–10

−3

With the crack growth rate law, the theory is further extended to model the life-

time cdf of quasibrittle structures under constant loads [8, 33, 5]. The model agrees

well with the strength and lifetime histograms of various quasibrittle materials such

as concrete, industrial and dental ceramics, and ﬁber composites [13, 39, 32].

This paper, representing both a digest and an extension of [8], reviews the re-

cently developed theory, and derives some key asymptotic properties of strength

Scaling of Strength and Lifetime Distributions of Quasibrittle Structures 45

distribution of bundles with softening stress-strain behavior, which further validates

the grafted distribution of RVE strength underlying the weakest-link model of struc-

tural failure. The theory is then veriﬁed by optimum ﬁts of some recent tests of the

strength and lifetime histograms of industrial ceramics.

2 Failure Statistics of Nano-structure

The failure of a structure originates from the failure of its nano-structures, either

atomic lattice blocks or disordered nano-structures. Naturally, the statistics of struc-

tural failure at macro-scale must be related to the statistics of breakage of nano-

structures. In the present theory, a nano-crack is considered to propagate by random

jumps through either an atomic lattice block or through a disordered nano-structure.

These jumps are governed by the activation energy barriers separating a series of

metastable states on the surface of the free energy potential of the nano-structure.

When the nano-crack advances by one atomic spacing or one nano-

inhomogeneity, the energy release due to fracture must correspond to the change

of activation energy barrier. Based on the equivalent linear elastic fracture mechan-

ics, the energy release can be explicitly related to the remote stress applied on the

nano-structure [7, 8].

Since the interatomic separation across the crack line increases by only a small

fraction of the atomic spacing during each jump, the activation energy barrier for

the forward jump differs very little from the backward jump. Therefore, the jumps

of the state of nano-structure must be happening in both directions, though with

slightly different frequencies.

By virtue of the transition rate theory [30, 40], the ﬁrst-passage time for each

transition can be calculated by Kramer’s formula [45], which gives the net frequency

of the propagation of nano-crack. After a number of jumps of the nano-crack tip,

the crack loses its stability and propagates dynamically, which leads to the break of

nano-structure.

It may reasonably be assumed that each jump of the nano-crack tip is history

independent [31]. Therefore, the failure probability of the nano-structure is propor-

tional to the sum of frequencies of all the jumps needed to reach a certain critical

crack length. The failure probability has thus been found to follow a power-law

function of the remote stress with a zero threshold (e.g. [7] and [8]).

3 Multi-scale Transition of Strength Statistics

To relate the strength cdf of an RVE at the macro-scale to the strength cdf of a nano-

structure, a certain statistical multiscale transition framework is needed. Though var-

ious stochastic multi-scale numerical approaches have been proposed [53, 26, 57],

the capability of these approaches is always limited due to incomplete knowledge

of the uncertainties in the information across all the scales. Instead, for the sole

purpose of statistics, the multi-scale transition of strength statistics has been char-

acterized by a hierarchical model, which consists of bundles and chains shown in

Fig. 1 [12, 13].

46 Z.P. Baˇzant and J.-L. Le

chainslong

Fig. 1. Hierarchical model for multiscale transition of strength statistics.

For a chain of n

elements where all of the elements have a strength cdf with a

power-law tail of exponent p, the strength cdf of the entire chain has also a power-

law tail and its exponent is also p. If the tail exponents for different elements in the

chain are different, then the smallest one is the tail exponent of the cdf of strength.

For a bundle of n

elements (or ﬁbers) of random strength, the strength cdf of

a bundle depends on the load-sharing mechanism of the ﬁber. Various load sharing

rules have been discussed in the literatures [20, 41, 42, 43, 35]. A more realistic

model is to derive the load redistribution rule based on a mechanical model. Con-

sider that all the elements (ﬁbers) have the same elastic stiffness and are subjected

to the same displacement.

Two limiting cases are now well understood: 1) the brittle failure, in which the

stress in the ﬁber suddenly drops to zero once it reaches the peak stress, and 2) the

plastic failure, in which, after the ﬁber reaches the peak stress, the stress remains

constant as the displacement increases. Two asymptotic properties are of interests

here: the tail behavior of strength cdf of a bundle and the type of strength cdf for a

large bundle.

For a bundle of n

ﬁbers, if the strength cdf of each ﬁber has a power-law tail of

exponent p

(i = 1,...,n

), then the strength cdf of the bundle has also a power-law

tail, with its exponent being p =

∑

i=1

. For the plastic bundle, this property can

simply be proven by asymptotic expansion of cdf [13] or through Laplace transform

of cdf. For the brittle bundle, this property was proven by induction based on the set

Scaling of Strength and Lifetime Distributions of Quasibrittle Structures 47

theory [28, 44]. A simpler proof was presented in [13] based on asymptotic expan-

sion of the recursive equation for the strength cdf of brittle bundle by Daniels [20],

who also showed that the cdf of a brittle bundle approaches Gaussian distribution as

→∞. For a plastic bundle, such a convergence is obvious by virtue of the Central

Limit Theorem.

However, the actual behavior of a ﬁber may exhibit gradual post-peak softening.

For this intermediate case, the proof of additivity of tail exponents and the conver-

gence to Gaussian distribution for a large bundle is still lacking. Here we present a

simple proof.

Consider a bundle with two ﬁbers having the same cross section area, although

a generalization to any number of ﬁbers is easy. Assume that each element has a

bi-linear stress-strain curve (Fig. 2a), which has an elastic modulus E and softening

modulus E

≤0). Let the only random variable be the peak strength

(i = 1,2).

Then the peak of average stress in the bundle can be written as:

= max

(

)+s

(

)]/ 2(1)

where

= strain in the ﬁber, and s

= stresses in ﬁbers 1 and 2. We seek the

critical strain

∗

at which the load on the bundle reaches its maximum. The critical

strain depends on the ratio

= −E

/E (

≥ 0). Two cases must be distinguished,

depending on whether the weaker element fails completely (1) before or (2) after

the stronger element reaches its peak.

Let the two ﬁbers be numbered such that

≤

. Then the peak stress of the

bundle,

, can be written as follows:

Case 1: 0 ≤

≤ 1

if (1 +

)

=[(1 +

)

+(1 −

)

]/2(2)

if (1 +

)

≤

/2(3)

Case 2:

> 1

if (1 +

)

(4)

if (1 +

)

≤

= max(

/2) (5)

Note that these results cover not only the softening bundles but also the limit cases

of both the plastic and brittle bundles. When

= 0, the element is plastic and the

peak average stress in the bundle is (

)/2 (which was statistically analyzed

in [13]). When

→ ∞, the element is brittle and the peak stress of the bundle is

max(

/2) [20].

If the average bundle strength is less than a prescribed value S,i.e.

≤ S, then,

based on Eqs. 2-5, the strength of each ﬁber must lie in the domain

(S),shown

in Figs. 2 b and c. Since the strengths of these two ﬁbers are independent random

variables, we may use the joint probability theorem to express the cdf of the average

bundle strength;

(S)=2



(S)

(

) f

(

(6)

48 Z.P. Baˇzant and J.-L. Le

2S/(1

D

)



)

Fig. 2. a) Mechanical behavior of softening ﬁber, b) Feasible region of strength of ﬁbers for

< 1, c) Feasible region of strength of ﬁbers for

≥ 1.

where f

= probability density function (pdf) of the strength of the ith element (i =

1,2). For the limiting cases of brittle and plastic bundles, Eq. 6 becomes equivalent

to Daniels’ formulation for the brittle bundle [20] and the convolution integral for

the plastic bundle becomes equivalent to the formulation in [13].

Now we assume that the strength of each ﬁber has a cdf with a power-law tail,

i.e. P

(

)=(

)

. Considering the transformation: y

/S, we can write the

cdf of bundle strength as:

(S)=2S

)



(1)

−1

(7)

where

(1) denotes the feasible region

(S) normalized by S. Since the integral

in Eq. 7 results in a constant, the cdf of bundle strength has a power-law tail whose

exponent is p

+ p

. Q.E.D.

By induction, the foregoing analysis is then easily extended to a bundle with n

ﬁbers, for which the cdf of average bundle strength can be written as:

(S)=n



(S)

∏

i=1

(

...d

(8)