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Interpretation of optical caustic

patterns obtained during unsteady

crack growth: an analysis based on a

higher-order transient expansion

Cheng Liu, Ares J. Rosakis

Cheng Liu, Ares J. Rosakis, "Interpretation of optical caustic patterns

obtained during unsteady crack growth: an analysis based on a higher-order

transient expansion," Proc. SPIE 1554, Second International Conference on

Photomechanics and Speckle Metrology, (1 December 1991); doi:

10.1117/12.49532

Event: San Diego, '91, 1991, San Diego, CA, United States

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Interpretation of optical caustic patterns obtained during unsteady crack growth:

An analysis based on a higher-order transient expansion

Cheng Liu and Ares J .

Rosakis

California Institute of Technology, Graduate Aeronautical Laboratories

Pasadena, California 91125

ABSTRACT

The optical caustic method is re-examined considering the presence of transient effects. Based on the higher-order

asymptotic expansion provided by Freund and Rosakis' ,

regarding

the stress field near a non-uniformly propagating

crack tip, the caustic mapping and the initial curve equations are derived. The dynamic stress intensity factor, Kf(t),

is related to experimentally measurable quantities of the caustic pattern by an explicit expression. It is shown that

the classical analysis of caustics is a special case of the new interpretation method. The Broberg problem is used as

an example problem to check the feasibility of analysing caustics in the presence of higher-order transient terms. It

is shown that the caustic patterns are sensitive to transient effects, and that use of the classical analysis of caustics

in the interpretation of the optical patterns for this problem may result in large errors in the value of the stress

intensity factor, especially at short times after initiation.

1. INTRODUCTION

The dynamic stress intensity factor, K!(t), plays a pivotal role in dynamic fracture mechanics since it forms

the basis of the formulation of crack growth criterion. It is often postulated that for dynamic crack growth at a

specific velocity, v, the instantaneous value of the dynamic stress intensity factor, K, should be equal to a unique,

material dependent function of crack tip velocity, called the dynamic fracture toughness and denoted by Kjc(v).

This forms the basis of the dynamic crack growth criterion. While K} is determined from the initial/boundary value

problem and is a known function of velocity v(i), the dynamic fracture toughness Kc(v) can only be determined

by experiment for a specific solid.

In the last two decades, extensive experimental effort has gone into the verification of the above postulate and

the measurement of the fracture toughness. However, experimental investigators seem to disagree on whether K

is a unique function of velocity for each material or whether it depends on other parameters such as crack tip

acceleration2. For the most part, all experimental techniques that have been used to obtain information about the

dynamic fracture toughness, assumed that the region from where the information was obtained, was K1-dominant.

The assumption of I(f-dominance states that the stress field at a finite region near the crack tip can be approximated

accurately by the asymptotic singular solution (to within some acceptable error). Recent experimental evidneces

by the bifocal caustic method3 and by the coherent gradient sensing (CGS) method4 have demonstrated that the

assumption of K!-dominance is often violated during dynamic crack growth. As a result, many of the experimental

measurements obtained in the last two decades based on this assumption may be inaccurate. This could explain

the source of disagreement and the debate over the validity of the dynamic fracture criterion. Freund and Rosakis1

have suggested that under fairly severe transient conditions, a representation of the crack tip field in the form of

higher-order expansion should be used to interpret the experimental observations.

In this paper, the optical caustic method is re-examined by considering the non-uniform crack growth history.

The caustic mapping equation and the initial curve equation are derived based on the higher-order expansion given

by Freund and Rosakis' for the transient crack growth which allows both the crack tip speed and dynamic stress

intensity factor to be arbitary differentiable functions of time. It is shown that the coefficients of this expansion

depend on time derivatives of K1(t) and v(t). here an explicit relation between the dynamic stress intensity factor

and the caustic diameter is established. The dynamic effects such as the acceleration of the crack tip and the time

derivative of the dynamic stress intensity factor are taken into account in the new interpretation method, and the

classical analysis (K1-doininant) of the caustic patterns is a special case of the new method. Finally, the Broberg

problem is employed to check the feasibility of accurately analysing caustics in transient crack growth situation.

814

/ SPIE Vol. 1554A Speckle Techniques, Birefringence Methods, and Applications to Solid Mechanics (1991)

0-8194-0682-1 /91/$4.00

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2. CAUSTICS GENERATED BY NON-UNIFORMLY PROPAGATING CRACKS

2.1. Caustic method

Consider a plate specimen of uniform thickness, h, in the undeformed state. Let its mid-plane occupy the

(

x1,

x2) plane of an orthonormal Cartesian coordinate system. If the specimen is subjected to applied loads, light

rays transmitted through its thickness or reflected from its surface, undergo optical path changes. These changes

are related to stress induced gradients in refractive index and/or gradients in surface elevation. Both gradients of

refractive index and surface elevation are related to gradients in the stress state. If the stress gradients satisfy certain

condition, then a collimated light beam transmitted through the specimen or reflected from its surface, will form

a three-dimensional envelope in space, see Fig.1. This envelope, which is called the causiic surface, is the locus

Specimen

Virtual Screen

Specimen

Real Screen

Figure

1: Caustic formation in (a) reflection, (b) transmission.

of points of maximum luminosity in the reflected or transmitted light fields. If a screen is positioned parallel to

the x3 =

plane, and so that it intersects the caustic surface, then the cross-section of the caustic surface can be

observed on the screen as a bright curve (the cansic curve) bordering a dark region (the shadow spoi). Suppose that

the incident ray, which is reflected from or transmitted through point p(x1

x2)

on the specimen, intersects the screen

at the image point P(X1 ,

X2).

The (X' ,

X2)

coordinate system is identical to the (x1 ,

x2)

system, except that the

origin of the former has been translated by a distance z0

the screen (zo can be positive or negative). The position

of the image point P is given by5

X,a= X,o+Z{LS(Xi,X2)},a ,

(1)

where a =

2 ,

and

zS(xi, x2) is the optical path change. Relation (1) describes the mapping of points on the

specimen onto points on the screen.

If the screen intersects the caustic surface, then the resulting caustic curve on the screen is the optical mapping

of the locus of points for which the determinant of the Jacobian matrix of the caustic mapping equation (1) must

vanish on the specimen, i.e.

J(xi ,

x9;

zo)

= det

[Xa,J

dcl [5 + zo(zS),a II

(2)

Equation (2) is a necessary and sufficient condition for the existence of a caustic curve. The locus of points on the

specimen plane(xi, x2, x3 =

for which the Jacobian vanishes is called the initial curve whose geometry is described

by equation (2). All points on the initial curve map onto the caustic curve, and all points inside and outside this

curve map outside the caustic curve. It should be noted that the equation of the initial curve depends parametrically

on ZO,

which

is a variable of the experimental set-up.

SPIE

Vol. 1554A Speckle Techniques, Birefringence Methods, andApplications to Solid Mechanics (1991) / 815

D/2

Crack

Front

D/2

(a)

(b)

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2.2. Caustic mapping equation and initial curve equation in the presence of transient effects

For a cracked linear elastic plate of uniform thickness and finite in-plane dimensions, the optical path difference

zS, in general will depend on the details of the three-dimentional elastostatic or elastodynamic stress state that

would exist at the vicinity of the crack tip. This will be a function of the applied loading, in-plane dimentions and

thickness of the specimen. In the present work, we assume that the two-dimensional asymptotic analysis may provide

adequate appproximation for LS(xi, x2) and the initial curve is always kept outside the near tip three-dimensional

zone. Under these conditions, the optical path difference zS(xi ,

x2)

will be5

zS(x1,x2) =

ch[cii(xj,x2)

+ o22(x1,x2)} ,

(3)

where C is the stress-optical coefficient for reflection, or transmission, and o and o22 are thickness averages of the

stress components in the solid.

For a planar, mode-I crack that grows with a non-uniform speed v(t), along the positive x1 direction, where

(

£2)

iS a coordinate system translating with the moving crack tip, the asymptotic representation of the first stress

invariant is1

'iii + t722

3v2

2r,2

—Ao(t)r1 -

cos

—

—-A1()

2p(c1 —ce)

4c1

+ {-2A2(t)cos2 + D!{Ao(t)} [(i —

) cos

+ cosL]

v2\

O I

5v2\

39j

7O )

+Bi()

(4)

where

4 1+c

Ao() =

3p/;:

D(v)

R1()

Dt{A0(i)} =

_3;);

{v(t)Ao(i)}

3v2(t)

dv(i)

2v2(t)

1 + o d dv(i)

B1(t) =

2c$c?40(t)dt

1Lv/:;:c4c;1 D(v)

II(t)d

D(v) =

4cic — (1

+ a)2

I ai 3(t)x2

r?,(i) =

+ a?3(t)x ,

O,(t)

tan

a3(i)

and

10(t) is the dynamic stress intensity factor, p and p are the mass density and the shear modulus, and ci and c5

are the longitudinal and shear stress wave velocities of the elastic material.

By substituting the above expression for the first stress invariant into the optical path difference relation (3), the

mapping equation (1) becomes

X1 =

cos 0

+ zochp(c —

c;9)

[Y-Ao(t)ri1 cos

15v2

If

v2 \

v2

5O

-

{A2(i)cos

+ D{A0(l)} [1

) cos

-

cos

3v2\

50i

3v2

90l)

_.L1

—.Bi(t) 1 —

cos

—

cos

cos Ti

(5)

816

/ SPIE Vol. 1554A Speckle Techniques, Birefringence Methods, andApplications to Solid Mechanics (1991)

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