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Crack-tip deformation field

measurements using coherent

gradient sensing

Hareesh V. Tippur, Sridhar Krishnaswamy, Ares J.

Rosakis

Hareesh V. Tippur, Sridhar Krishnaswamy, Ares J. Rosakis, "Crack-tip

deformation field measurements using coherent gradient sensing," Proc. SPIE

1554, Second International Conference on Photomechanics and Speckle

Metrology, (1 December 1991); doi: 10.1117/12.49513

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

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Crack Tip Deformation Field Measurements Using

Coherent Gradient Sensing

Hareesh V. Tippur1, Sridhar Krishnaswamy2 and Ares J. Rosakis3

Abstract

A real time, full field, lateral shearing interferometry -

coherent

gradient sensing (CGS) -

has

recently

been developed for investigating fracture in transparent and opaque solids. The resulting interference

patterns are related to the mechanical fields by means of a first order diffraction analysis. The method has

been successfully applied to quasi-static aiid dynamic crack tip deformation field mapping in homogeneous

and bimaterial fracture specimens.

Introduction

In recent times, optical methods have greatly assisted in understanding the fracture mechanics of solids

through local crack tip field measurements [1]. Photoelasticity [2], geometric moire methods [3,4], and

caustics [5,6] are some of the commonly used incoherent optical methods in fracture mechanics. Moire

interferometry [7] ,

holography

[8], Twyman-Green interferometry [9] fall into the category of coherent

optical methods used in experimental fracture mechanics. These techniques can be readily applied to

quasi—static, elastic and/or elasto-plastic crack growth investigations. For measuring crack tip fields near

dynamically growing cracks, however, photoelasticity [10,11] and caustics [12} are generally used. Optically

birefringent models are required for dynamic photoelastic studies while the method of caustics can be used

with both opaque and transparent solids. Caustics, however, is not a full field optical method and hence

finer details of the crack tip fields may not be apparent. There has been a need for a simple, full field

technique for dynamic fracture mechanics which can be used with either opaque solids or optically isotropic

transparent solids.

A relatively new optical method, coherent gradient sensing (CGS) has been developed for quasi-static

and dynamic fracture mechanics investigations. It is a lateral shearing interferometry wherein a pair of

gratings is used as a wave front shearing device. CGS has several advantages. It provides full field interfer-

ence patterns corresponding to the deformation fields in real time. The method can be applied to crack tip

field measurements in both transmission mode and reflection mode. This feature allows one to use CGS to

study either optically isotropic transparent solids or opaque solids. The relative insensitivity to vibrations,

controllability of resolution of measurements and simplicity are some of the additional advantages. The

purpose of this article is to present a review of the method and provide examples which demonstrate the

feasibility and applicability of CGS to the study of quasi-static and dynamic fracture of homogeneous and

bimaterial solids.

Coherent Gradient Sensing

Experimental Set-up

In Fig.(1) the schematic of the experimental set up used for transmission CGS is shown. A transparent,

optically isotropic specimen is illuminated by a collimated bundle of laser light. The transmitted object

wave is then incident on a pair of chromium-on-glass master gratings, G1 and G2, separated by a distance

A. The field distribution on the G2 plane is gathered by a lens L1 and its spatial frequency content is

1Auburn University, Auburn. AL

2Northwestern University, Evanston, IL

3California Institute of Technology, Pasadena, CA

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displayed on its back focal plane. Information around either the 1 diffraction orders is filtered in order

to obtain interference patterns representing stress gradients on the image plane.

Figure (2) shows the modification of the above set up for measuring surface deflections of opaque solids

when studied in reflection mode. In this case, the specularly reflecting object surface is illuminated by

a collimated beam of laser light using a beam splitter. The reflected beam, as in the previous case, gets

processed through the optical arrangement which is identical to the one shown in Fig.(1).

A first order diffraction analysis is presented in the next sections will demonstrate that the interference

patterns obtained are related to local crack tip deformations namely, gradients of in—plane stress or gradi-

ents of out-of-plane displacement.

Working Principle

The process of object wave shearing by a pair of line gratings is shown schematically in Fig.(3). For

the sake of simplicity of representation, the line gratings are assumed to have a sinusoidal transmittance.

Let the grating lines of G1 and G2 be parallel to, say, the x1-axis. A plane wave transmitted through or

reflected from an undeformed specimen and propagating along the optical axis, is diffracted into three plane

wave fronts E0, E1 and E1 by the first grating G1 .

The

magnitude of the angle between the propagation

directions of P20 and E± is given by the diffraction equation 0 =

sin1(A/p),

where A is the wave length

and p is the grating pitch. Upon incidence on the second grating G2, the wave fronts are further diffracted

into E(o,o), E(o,1), E(1,_1), E(i,o), E(1,1) etc. These wave fronts which are propagating in distinctly different

directions, are then brought to focus at spatially separated diffraction spots on the back focal plane of

the filtering lens. The spacing between these diffraction spots is directly proportional to sin 0 or inversely

proportional to the grating pitch p.

Now, consider a plane wave normally incident on a deformed specimen surface. The resulting trans-

mitted or reflected wave front will be distorted either due to changes of refractive index or due to surface

deformations. This object wave front that is incident on G1 now carries information regarding the specimen

deformation, and consists of light rays travelling with small perturbations to their initial direction parallel

to the optical axis. If a large portion of such a bundle of light has rays nearly parallel to the optical axis,

each of the diffraction spots on the focal plane of L1 will be locally surrounded by a halo of dispersed light

field due to the deflected rays. The extent of this depends on the magnitude of the deformations. By using

a planar aperture at the filtering plane, information existing around one of the spots can be further imaged.

Analysis

Consider a specimen whose midplane, in transmission, or surface, in reflection, occupies the (x1, x2)

plane in the undeformed state. Let ej denote unit vector along x-axis, (i =

2, 3) [see Fig.(4)]. When

the specimen is undeformed, the unit object wave propagation vector d0 =

e3.

After deformation, the

propagation vector is perturbed and can be expressed by,

d0 =

a0e1

+ 30e2 + 0e3,

(1)

where Qo(Xi

, X2), i30(x

x2),

and 70(xi ,

x2) denote

the direction cosines of the perturbed wave front. This

upon incidence on G1, the grating lines of which are parallel to, say, the x1-axis, is split into three wave

fronts propagating along d0, d±i and whose amplitudes Eo(x'), E±i(x') can be represented by,

Eo(x') =

aoexp[ikdo

.x'],

Ei(x') =

a1exp[ikd1

.x'],

(2)

where a0 and a±i are constants and k =

2ir/,\

is the wave number. Now, the propagation directions of the

diffracted wave fronts can be related to the direction cosines of the incident propagation vector through

[See ltef.13 for detailsj,

d1 =

[c0e1

+ (3 cos C °

sin

O)e2 +

cos

C /3 sin C)e3],

(3)

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using the diffraction equation 9 = sin1(A/p). On the plane G2 (x = L\)

[see

Fig.(4)], the amplitude

distribution of the three diffracted wave fronts are,

I (L\

EoI' =

aoexp

[zk

—)

(4)

E1 J'

ai exp I ik

(5)

1 (70cos9i30sinO)j

The wave fronts E0, E±i will undergo further diffraction upon incidence on G2 into secondary wave

fronts E(o,o), E(o,i), E(1,_i), E(i,o), E(i,i) etc. Of these secondary diffractions, E(o,i) and E(i,o) have their

propagation direction along d1 ,

E(o,_i)

and E(_i,o) along d_1 and E(o,o), E(ii) and E(i,_i) along d0

[Fig.(3)]. If the information is spatially filtered by blocking all but

diffraction order, only the wave

fronts E(o,±l) and E(±lo) contribute to the formation of the image. Noting that the two wave fronts do not

acquire any additional relative phase differences after G2, the amplitude distribution on the image plane

is,

Eim

(Eo + E±i)I'

a0exp

[ik

()]

a1 exp [ik

] .

(6)

(70

cos

0 :F /3 sin 0)

Hence, the intensity distribution on the image plane, for small 0 and small defiections (y

1), is,

'im EimEm

a + 4

2a0a1 cos(kz/300),

(7)

where Em S the complex conjugate of Eim. Thus, 'im denotes an intensity variation on the image plane

whose maxima occur when

kz/300 = 2M2ir,

M2 = O,±1,±2

(8)

where M2 denotes fringe orders. Similarly, when the grating lines of G1 and G2 are parallel to the x2-axis,

it can be shown that,

kLa00=2A'I1ir,

M1 =O,±1,±2

(9)

Equations (8) and (9) are the governing equations for the method of CGS and they relate fringe orders

to the direction cosines of the object wave front. In the above two equations note that the sensitivity of

measurement can be easily controlled by either the grating separation distance L

the grating pitch p.

Direction Csines and Crack Tip Deformations

The direction cosines of the object wave front can be related to crack tip deformations for both transmis-

sion and reflection CGS.

(

a) Transmission

Consider a planar wave front normally incident on an optically isotropic, transparent plate of uniform

nominal thickness h and refractive index n. Now, if the plate is deformed, the transmitted wave front

acquires an optical path change S which is given by the elasto-optical equation [14],

fl/2

t5S(xi,x2)

= 2h(no —

1)]

d(x3/h) + 2h1

5n0 d(x3/h).

(10)

The first term represents the net optical path difference due to the plate thickness change caused by the

strain component €33. The second term is due to the stress induced change in refractive index of the

material. This change in the refractive index 5n0 is given by the Maxwell relation,

fl0(X1,X2)

=

Di(au + t722 + 733),

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