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Experimental investigation of

dynamic mixed-mode fracture

initiation

John M. Lambros, James J. Mason, Ares J. Rosakis

John M. Lambros, James J. Mason, Ares J. Rosakis, "Experimental

investigation of dynamic mixed-mode fracture initiation," Proc. SPIE 1554,

Second International Conference on Photomechanics and Speckle Metrology,

(1 December 1991); doi: 10.1117/12.49481

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

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EXPERIMENTAL INVESTIGATION OF DYNAMIC

MIXED-MODE FRACTURE INITIATION

J. Lambros, J.J. Mason and A.J. Rosakis

Graduate Aeronautical Laboratories

California Institute of Technology

Pasadena, CA 91125

Abstract

The use of a coherent gradient sensing (CGS) apparatus is explored in dynamic fracture

mechanics investigations. The ability of the method to accurately quantify mixed-mode crack

tip deformation fields is tested under dynamic loading conditions. The specimen geometry and

loading follow that of Lee and

who

give the theoretical and numerical mixed mode K

values as a function of time for the testing conditions. The CGS system's measurements of K1

and K11 are compared with the predicted results, and good agreement is found.

1. Introduction

The full-field method known as CGS,3'4 is investigated here for the measurement of mixed-

mode, dynamic crack tip deformation fields under plane stress conditions in optically transparent,

non-birefringent materials. The Coherent Gradient Sensor is a lateral shearing interferometer uti-

lizing two identical line gratings. The set-up was first proposed for measuring lens abberations,5'6

but, until recently, other possible applications of the CGS interferometer have been overlooked.

When used in fracture mechanics the method gives real time measurements of the in-plane stress

gradients for transparent materials or the in-plane gradients of the out-of-plane displacements

for opaque materials. With data taken at a wide range of points near the crack tip, it is possible

for the CGS method both to show whether or not the stress field at a crack tip is well described

by the dominant (v )

singular

term of the asymptotic expansion (K-dominance) and to find

accurate values of Kj and Ku, the stress intensity factors.

Tippur et. al.3'4 have demonstrated the accuracy of the CGS statically for mode I loading,

however, its accuracy in dynamic investigations, including dynamic mixed-mode loading, has

not been reported.

An analysis of the loading geometry shown in Figure lb by Lee and Freund1'2 shows that

both modes of deformation, mode I and mode II, can be expected for such a loading geometry.

/ SPIE Vol. 1554A Speckle Techniques, Birefringence Methods, andApplications to Solid Mechanics(1991) O-8194-0682-1/91/$4.OO

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Apart from the dominant Kj, a small negative K is also predicted. In this study it is desired

to establish the ability of CGS to accurately measure the dynamic, mixed-mode stress intensity

factors by comparing experimental results with the solution of Lee and Freund. Also, it is

realized that a positive agreement of the measurements of the CGS system with the Lee and

Freund solution will confirm the validity of that solution.

2. Theoretical Development

2.1 Analytical Model

One side of an elastic, half-space containing an edge crack is loaded dynamically by some

prescribed velocity, v(t). (See Figure lb.) An elastic plane strain solution to this problem is

reported1'2 for a step input velocity, v(r) = v0H('r). For this study, the analysis was adjusted

to reflect a plane stress 'field.

The normalization factor, K', for the stress intensity factors is given by

,ii Ev0

for plane stress

K' = V2c

(1)

Ii.

Ev0

for

plane strain

1% V 712c

(1_2)

The

time axis is normalized by the characteristic time, l/c' where c1 is the plane stress

dilatational wave speed, and 1 is the crack length. The results of the plane stress calculations for

the model material, PMMA (ii

.35), for both mode I and mode II stress intensity factors are

shown in Figures 7a and 7b, respectively.

Most of the qualitative features of the curves in these figures can be explained. Upon impact

a plane compressive wave is generated. It is followed by cylindrical unloading waves generated

at the corners of the impact area. The compressive wave gives rise to K11. The existence of

the unloading wave makes the increase in Kj progressively more gradual and forces the crack

faces to close, thus causing a smaller, but significant, negative IKE. As can be seen in Figures

7a and 7b there exist three regions in the solution. These correspond to the arrivals at the crack

tip of the first dilatational, shear and Rayleigh waves, respectively. The solution is valid up to

Cdt/i = 3, which corresponds to the arrival of a second dilatational wave that is reflected from

the impact surface.

2 .2 The Method of CGS

In contrast to Tippur et.al.,3'4 the theoretical development of CGS shown here follows the

more traditional approach of Murty7 for lateral shearing interferometers. The two approaches

are equivalent; the same assumptions are made and the same governing equations result. It

is hoped that the more traditional development will result in an easier understanding of the

method. A schematic of the set-up is shown in Figure 2. A coherent, collimated laser beam, 50

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mm in diameter, passes through a notched transparent specimen. After exiting from the deformed

specimen, the beam falls upon the first of two identical diffraction gratings (40 lines/mm). The

primary grating splits the beam into a direct beam and numerous diffraction orders. For the sake

of brevity, only the first diffraction orders (±1) and the direct beam are considered. The second

diffraction grating diffracts both the direct beam and the first diffraction orders into three beams

each, giving a total of nine beams behind the second grating. Of these nine beams the (O,±1)

and the (±1 ,O) orders are parallel—as can be seen in Figure 3.

An on-line spatial filter is used to isolate one of the two pairs of parallel beams. A lens

is placed a distance equal to its focal length behind the secondary grating as in Figure 2. The

Fourier Transform of the intensity distribution at the second grating is observed in the back-focal

plane of the lens where an aperture is placed on either the +1 or -

diffraction order spot. The

aperture filters all but the two desired parallel beams from the wavefront. Another lens is placed

at a distance equal to its focal length behind the aperture to invert the Fourier transformation.

It is assumed that the wave front before the first grating is approximately planar with some

phase difference, 6S(xi ,

x2). Deviations

of the propagation direction from the optical axis are

neglected. Thus, the two gratings shift one beam with respect to the other by a distance

= Ltan6

(2)

where L

the separation between the gratings, see Figure 2, and 9 is the angle of diffraction

(assumed small), given here by

6=sin_1z

—.

(3)

A is the wavelength of the illumination, and p is the pitch of gratings.

The two parallel, sheared wavefronts constructively interfere at a point if their difference

in phase is an integer multiple of the wavelength, i.e. if

6S(xi + E,X2)

— SS(xi,x2) = rnA

(4a)

where m is called "the fringe order." Dividing this equation by gives

S(xi +E,x2)—S(xi,x2)

(4b)

which, for sufficiently small ,

may

be approximated by

a(6S(x1,x2))

—

iii])

(5)

—

equation (5),

the

approximations in equations (2) and (3) have been used, and the result has

been generalized to include shearing in either the x1 or x2

direction,

= 1,

/ SPIE Vol. 1554A Speckle Techniques, Birefringence Methods, and Applications to Solid Mechanics(199 1)

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