Statistics and thermodynamics of fracture

Vol. 22, No. 8-10,

pp.

989-997, 1984

OOZO-7225/84 $3.00 + .OO @ 1984 Pergamon Press Ltd.

STATISTICS AND THERMODMVAMICSOF FRACTURE ALEXANDERCHUDNOVSKY Departmcm of Civil Enginaring Case Western Reeve University, Cleveland, OH44106

Abstract A probabilistic model of the fracture processes unifying the phenomenological study of fracture mechanics and statistical approaches to fracture long term strength of materials, The general framework of irreversible thermodynamics is employed to is briefly outlined. The stochastic calculus is used to model the deterministic side of the failure phenomenon. account for the failure mechanisms controlled by chance; particularly, the random roughness of fracture surfaces.

Introduction We start example,

tive

with Fig.

scale hierarchy using an amorphous polymer (polystyrene) On a macroscale one can make use of the conventional 1.

as an illustramodel of con-

t inuum medium.

REPRESENTATIVE VOLUME

%e

properties

of

a point

properties of the actual size of the representative [l,Zl, which

V* should i.e., play an essential

within

the

material volume be large role in

f’.‘)

continuum

averaged

are

over

defined

a certain

A FRAGMENT

OF At<

It~~lVf~UAL

CRAZE

a$ the

corresponding

representative

volume

V*.

The

from statistical homogeneity condition V * is determined in comparison with any distinct morphological aggregates the properties under consideration.

In the example (Fig. 1) we choose the microscale to be correspondent to craze size since crazing is the leading mechanism of irreversible deformation and fracture. Consideration of submicroscopic scale helps one to understand and model an elementary event such as single craze formation in our example. Obviously more complex hierarchy should be introduced for poly crystalline materials, composites, etc. Such scale hierarchy is necessary to Iink the mechanics of continuum and materials science.

989

A. CHUDNOVSKY

990

Using scale hierarchy one can distinguish three stages of the fracture pr~sess under The first stage consists of nucleation and growth of microcreep and fatigue conditions. Then a macrocrack nucleon until a critical density of microdefects is reached. defects, This process is illustrated in Fig. 2 appears as a manifestation of a local instability. The halo of crazes that have been formed in the vicinity of the notch taken from [31. larger then the representative volume Then the nucleation of the macrocrack, (Fig. 2a). size concludes the first stage (incubation period of the fracture process, Fig. 2b).

FIGURE 2 Slow crack growth constitutes growing crack is accompanied from [31.

the second by an array

For most engineering materials the slow stage. of microdefects as is illustrated in Fig. 3 taken

FIGURE 3 At-a certain loading condition the growth of the crack and its surroundina damage becomes The avalanche-like fracture propagation constitutes the third itage vhich occurs unstable. Therefore, in a very short time. the lift time of a solid consists mainly of the period of time t, before the crack nucleates and the period tp of slow crack propagation. Stress concentration induced by a crack usually determines crack growth. For this reason crack propagation is a well determined and reproducible phenomenon. The variations, although they exist, are within a reasonable range. Crack nucleation, on the contrary, is extremely sensitive to morphological fluctuation. Therefore, the prediction of the location and time of the crack nucleation can be made in probabilistic terms only. Even when the nucleon location is predetermined by stress concentration induced by a notch, the time of the nucleation still varies significantly (orders of magnitude). The brief description of the fracture process given above explains the existence of following main directions in studies of long term strenth: Phenomanological theories of local long term strengths, i.e., crack nucleation in 1. a contemporary context. Crack stability and slow crack growth studies. 2. Statistical theories of failure. 3. Each of these reflects certain features of the fracture process. Together they reconstruct a rather complete picture of the fracture phenomenon. The last statement has served as the main motivation for an attempt to unify the three approaches [2,4] of the Statistical Thermodynamics of Fracture resulting from such an attempt is presented belov. the

Statistics and thermodynamfcs of fracture Probabilistic

Model

of

991

Failure

Fracture occurs by formation of new surfaces separating an initially continuous solid into parts. The fracture surfaces are random in the following sense. Let us consider an assemblage of macroscopically identical specimens (Fig. 4a). For simplicity the specimens chosen are very thin (0.21mn x 2Onss x 80 mm). Thus, one-dimensional crack trajectories represent the fracture surfaces. A trial set 0, of fatigue crack trajectories is shown in Fig. 4b taken from 151. Although macroscopically identical test conditions were carefully performed, there were no microscopically identical trajectories observed. On the basis of macroscopic identity of the test conditions each trajectory from the setn, can be expected in any arbitrarily chosen sample from the assemblage.

of

all

Thus the set R, possible crack

statistical

analysis

of

trajectories

shown

trajectories of

G,,

the

in Fig.

4b represents a trial set from a set surfaces) for any single specimen. Dsir

(fracture set

fi can

be

fully

characterized

Q

[2,41.

FIGURE 4 The model of a homogeneous continuum implies that all parameters are averaged over the representative volume V* mentioned above. Hence, a homogeneous continuum model does not allow for the microscopical roughness of a crack trajectory w to be distinguished. Thus, a bandAC of trajectories(quasi intervals in Wiener’s terminology) rather then an individual The “thickness” of the band is apparently assocrack trajectory should be considered. Based on statistical analysis of the ciated with the size of the representative volume V*. one can introduce a probabilisitic measure p on n as an additional microscopic roughness, characteristic of fracture process complementary to the conventional parameters of the homogeneous continuum. Under creep and fatigue conditions only one fracture surface get formed within a speone can conclude that the appearances of two or more fracture surfaces From that, c imen. within an individual specimen are usually exclusive events. Therefore, the probability of failure {F} not later then at a certain moment of time t can be represented as a sum: P{F,t}

= &

P[w.tlp

(mk)

where p[w,t] stands for the conditional probability of failure not later then at the moment wcASlk and surmsation is performed over a decomposition t along a certain fracture surface A$l,fIACl, = $ (i.e., for all k, 1 and k # 1). of Q into a union gAQk without overlappings. Therefore, taking limit in (1) one can For a continuum medium R is an innumrable set. obtain [2,4] P{F,tl

= I P[w,tl

du(Q)

(2)

The evaluation of the continual integral (2) has been well defined since the late fifties prediction problem is reduced into three interrelated tasks. 161. Thus the failure Defining the set R of all possible fracture surfaces; 1. Defining the probabilistic measure b on 8, reflecting the morphology of fracture 2. surfaces, and Defining the conditional probability functional p[w,tj. 3. The formation of a fracture surface begins from the first local failure, i.e., a crack the fracture process appears as a sequence of local failures forming evennucleon. ‘Then, Thus the local failure is the key problem for the model. tually a fracture surface.

A. CHUDNOVSKY

992

Local

three

Characterization tasks: Introduction a) b) Suggestion Formulation c) We consider

of

Local

failure

Failure

on the

of a damage parameter. of constitutive equations of local failure criteria.

these

tasks

macrolevel

for

in

a continuum

its

turn

with

can be decomposed

into

damage.

briefly.

a) Damage parameter is called upon to model the defects within the representative various defects could be of imporDepending on hierarchical level of observation, volume. the discontinuities localized on Z-d manifolds such For most engineering materials, tance. are the leading mechanisms of failure on microscoas microcracks, crazes, sliplines, etc. we introduce damage parameter P as a paring of a s2alar damage denTherefore, pic level. sity pwhich is the area of discontinuity surfaces per unit volume, [PI= %. ) and damage Experimental studies of damage basedmdn this parameter orientation parameter 0: P =Io,O}. are reported 114,161. Damage accumulation usually manifests itself as a b) Constitutive Equations. Therefore one can make use of the general framework of irreversible dissipative process. we introduce a list of thermodynamic parameters Following the framework, thermodynamics. by incorporating the damage parameter P with the conventional parameters of an elastic Using the firs& stress tensor g and absolute temperature T. medium such as, for instance, principles of thermodynamics and the following list of parameters @,T,P) one can derive the entropy production for an elastic medium with damage f4,?1:

T$ = 33:g -’

3(h+n)

- -

:P

. P

_

jQ.7,

‘P for a portion of the dissipated work spent on damage nucleation and where nU:E _ _ stands growth (the remaining dissipated work(l-a)o:$ is converted into heat), CP is the difference between the total (small) deformatio; tensor E and elastic defo;mation ge;handn are the densities of the enthalpy and elastic potential energy correspondingly, the fast term reflects the entropy production due to heat transfer (jQ is the heat flux vector and VT is the temperature gradient). Bilinear form (3) suggests thermodynamic forces (causes CXU.a(h+n)12p,VT reciprocal to the thermodynamic fluxes). -I---z-The second law of thermodynamics, taken as anvariational principle (minimum entropy production, for instance) instead of inequality (si 2 O), suggests a certain relationship between thermodynamic fluxes and forces, particularly, the eonstitutive equation for damage parameter P[8i: r; =

@@T,P,:Tj

(4)

c) Conditions of local failure. Various strength criteria have been listed by the strength of materials theory. The list of the criteria is long, most likely because none of them was experimentally confirmed for wide enough range of loading conditions. The failure of classical strength criteria can be understand in light of an analogy between local failure and melting proposed by M. Born [9,101. Born’s proposition is based on an analysis of perfect crystalline structure instability. It suggests two conditions of instability (necessary and sufficient ones) as a criterion. One can argue about the similarity between the melting of a perfect crystal and a local failure of engineering of the local failure as a sort of instability with materials. However, consideration respect to damage perturbation can be justified phenomenologically [4]. Then using the list of thermodynamic parameters {o,T,Pj proposed above, one can deduce from necessary conditions of instability: a) the existence of the damage parameter critical values P* and b) the dependency of P* on stress, temperature and specific energy of damage y*:p*=p* ($Qy*). The sufficient condition of instability conventionally yields an energy requirement (latent energy of a transition). The use of the latent energy as a failure criterion is not convenient because of its strong dependency on stresses and temperature (such dependency is well documented in phase transition studies). At the same time the invariancy of the entropy jume during a phase transition with respect to stresses and tern-perature was experimentally discovered 1111. It suggests the hypothesis of the entropy jump constancy for the local failure (hypothesis was proposed in [12,4]). Sumaerizing, we propose the pairing of the critical damage P*=p*($Gr*) and an entropy jump AS*requirement as a local failure criterion AS as well as y*=TAS* are believed to be material constants which can vary.for a different mode of failure [4].

the

Integrating local failure

equation (4), one obtains may occur (to stands for

the equation the initial

for the moment)

period

of

time

t*-to

before

Statistics and thermodynamics of fracture

it* _ SC”(t),

i

t

At the delays

T(t),

P(t).

pT)dt

= P*(@t*,.

993

T( A

(5)

J

moment t* the local failure either until this requirement is met.

if

occurs,

the

entropy

requirement

is

met,

or

Lnstability phenomena, the Local failure is not an exception, are extremely sensitive Therefore, P* and AS* should be considered as random fields. to morphological ffuctations. the mathematical expectations of P* Macroscopic stresses and temperature then determine The fluctuations of P* are caused by the fluctations of the specific energy of only. characterization of submicromechanisms of fracdamage v: which we use for phenomenological is the most crucial for The experimental examination of a highly fluctuated Y-field ture. An experimental method of statistical y-field characthe proposed framework problem. terization below.

based

on statistical

analysis

Thermodynamic A damage zone damage is distinctly

similar to Localized

Lt is convenient to macroscopic entity which

of

the

of

fracture

Slow

Crack

surfaces

(see

Fig.

6)

is

described

Growth

3 usually that shown in Fig. within a layer surrounding

accompanies the

the

cra$k.

The

crack.

consider a crack and surrounding array of microdefects (CL, Fig. 5). is referred to as Crack Layer

as one

FIGURE 5 One can distinguish

active

and inert

zones

within

the CL.

The active

CL adjacent to the crack tip where the damage keeps growing under the concentration (Fig. 5). When the crack propagates through the active released and consequentIy the process of damage growth at this region Thus, the inert zone appears as a trace of the active zone propagation. is part of CL complimentary to the active zone VA-

zone

VA is

a part

of

influence of stress zone the stresses are practically stops. The inert sane Vi

Since the stress distribution in the vicinity of a crack has an invarient shape with respect to the crack length (the crack length just scales the stresses by the stress intensity factor k), the critical level of damage is maintained constant during an isothermal process of crack propagation. Due to that, the crack layer propagation can be visualired as an active zone (i.e., damage distribution) movement. The Latter can be decomposed into translation and rotation as a rigid body and deformation (i.e., damage dissemination). In order to model modymamics again. The solid from the entropy entropy production for

CL propagation we make use+of the framework of irreversible therglobal entropy production Si is the integral over the volume of production(3). Providing the integration one can find the global isothermal conditions in the following bilinear form [13,7]:

+

Xtr a

2 - Y* {

tr

. ,

;;

rot

w

=L-Y*Rrot;Xexp

m

xrot + ; xexpc i m

the

dev Kk ‘KE

= >{ _ ‘{ * gexp;

:;oev

_ *‘:_&

rlev

.

Here are the rates of translations, rotation, isotropic expansion vkrmmretdkl distorsion of the active zone correspondingly. The reciprocal thermodynamic forces cribed by the following differences Ftr = J-y*Rtr;Xrot=L-y*Rrot;XeXP=;~-y*Rexp.Xdev,N . where J, L and ?4 are well known in the linear Fracture Mechanics path indepenbent

and des+&ev

integrals

A. CHUDNOYSKY

994

r:prssrntinq en,irgy i*aCropLc zupansi;rn indeprndency; ., * i.;

rcltsse rates with rrspect to active zitni’ tr:lnslation, r+spectiveiy; 3 has si.nilar physical ma.xnin+ but dors _:;cspecirTic energy of damage which hss been Intraduced

rotation not posses

1114 path

in the previous chapter; “R” labels the resistance moments (translationa:, rotational, etc.) such that y+R represents an amount of energy which is required for the active zone propagation. D stands for part of dissipated work spent on damage growth within the active zone. Using the minimum entropy production principle and an analysis of the CL configuration stability, one derives the constitutive equations for CL extention, expansion, etc. [13, 71. Complete set of constitutive equations describes the ctaek trajectory as well as the law of the active zone propagation, along the trajectory. Recently the model described have been intensively applied to the fatigue crack propagation in various materials [3, 14, 15, 161. A typical example of the applications to the retilinear fatigue crack growth in polystyrene is shown in Fig. 6, taken from 131. Obviously, theoretical prediction is in a good agreement with experimental data within more than four orders of magnitude range of crack speed (from the nucle,lrisn us LO avalanche-like crack propagation).

Fi:O.2 v=0.2Hr I o;rF$G.OMPa a U&lO.?MPa

15

lo

5 Energy

20X103

ReIeese Rate (Jni’] FlGURE

6

ahigher level of damage of applied stress i5m=II(omaxturnin) yields the lower level Notably, larger citical energy release rate Jc which is, in agreement with density and consequently theory suggests a decomposition of the conventional equal to y*Rt :r . Thus,the the theory, of two quantities. One, i.e., y*, is a true toughness parameter J, into the product the second, Rtr, is aloading history dependent parameter. The constitumaterial parameter; tive equation for the evolution of R is proposed as well [13, 71. The Crack Layer Theory briefly outlined above models fairly well the macroscopic features of slow crack propagaOur next task is to model some microscopic features of the fracture process, partion. ticularly the formation of fracature surfaces (trajectores for 2-D>, reflecting the fluctuations of y-field on the microscopic level. Diffusion

We start heterogeneity sidered (Fig.

Model

from a deterministic problem A .“conical” following [131. 7;+

of

Crack

Growth

of a crack trajectory deviation by y-field heterogeneity in front of the crack rip is

con-

Statistics and thermodynamicsof fracture

995

FIGURE 7 The dashed circles in the Fig. 7 corresponds to constant levels of y-field. of the crack trajectory equation are shown by the continuous lines: the correspond tov*heterogeneity larger and smaller then ‘I’ average (stronger heterogeneity) respectively.

The solutions lines 1 and 2 and weaker

The described trajectory deviation has been used as an elementary event for the problem of crack propagation within a continuum characterized by a random Y-field. One can employ a white noise type random field to model yron macroscopic level since Y-field fluctuates vithin the representative volume. Then, a system of stochastic differential equations describes Brownian type crack movement (see Fig. 4) [5]. Using correlative approximation one can deduce from the stochastic equations the relationship between the average value of the critical crack length , the varience of crack tranjectory ordinates V(y(x),x) and applied stress o on one side, and the mathematical expection and varience V(v* of the Y-field on the other side. The last two are not directly observable. This result suggests the conducting of the experiment depicted in Fig. 4. for various load level;. The sets of the fatigue crack traiectorics in flat titan alloy samples for three different levels of u dare shown in Fig. 8 taken from 151, Statistical analysis of the data is shown in Fig.m9 151.

FIGURE 8

A. CHUDNOVSKY

996

D(y)lan

FIGURE 9

Tn described experiments an optical observation of a damage zone surrounding the crack The distinct correlation between a core of damage sane in each individual has been made. from the corresponding assemblage was specimen and the trial set of crack trajectories It suggests the substitution of very expensive and time consuming statistical noted. experiments by damage zone characterization following the CL pattern. Using the diffusion model of crack growth and the Then, evaluate and V(p), not observable directly. local failure theory (see chapter I), one can evaluate crack nucleation and the location of the nuclei.

data presented plugging these in probabilistic

in Fig. results terms

9, one can into the the time of

Conclusion The framework, lifetime estimation

which unifies various as the ultimate goal,

facts described above is shown in Fig. 10.

yielding

the

structure

RANDOM

EXPERIMENT

LAW OF DAMAGE

FIGURE

LAW OF CRACK PROPAGATION

10

Acknowlegement The studies Research Center Marvin Hirshberg ciated.

within the framework described above is supported by the NASA Lewis under Grant No. NAG 3-223. The encouragement and useful discussions with and Bernard Gross and the assistance of Pita Tessmann are deeply appre-

Statistics and thermodynamics of fracture

997

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