full field study of strain distribution near the crack tip in the fracture of solid propellants via large strain digital image correlation and optical microscopy

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FULL FIELD STUDY OF STRAIN DISTRIBUTION NEAR THE CRACK TIP IN THE FRACTURE OF SOLID PROPELLANTS VIA LARGE STRAIN DIGITAL

IMAGE CORRELATION AND OPTICAL MICROSCOPY

Thesis by

Javier Gonzalez

In Partial Fulfillment of the Requirements for the Degree of

Aeronautical Engineer

California Institute of Technology Pasadena, California

1997

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© 1997 Javier Gonzalez

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iil

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ABSTRACT

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Abstract

TABLE OF CONTENTS LIST OF FIGURES 1 INTRODUCTION

2 LARGE DEFORMATION DIGITAL IMAGE CORRELATION METHOD (LD-DIC)

2.1 THE DIC PROGRAM 2 1 1 Optimization scheme

2.2 LIMITS IN THE DIC PROGRAM

2 2 1 Convergence depending on strain level 2.3 PROPOSED SOLUTION

2.4 ADDITION OF STRAIN AND DISPLACEMENT FIELDS METHODS

2.4 1 Method 1 2 4.2 Method 2 2 4 3 Method 3 2 4 4 Method 4

2.5 CALIBRATION OF METHODS FOR ADDITION OF FIELDS

3 APPLICATION OF THE LARGE DEFORMATION DIC METHOD TO THE CRACK OPENING PROBLEM IN A SOLID PROPELLANT

3 1 EXPERIMENTAL SETUP

3.2 SOLID PROPELLANT SPECIMEN 3.3 LOADING OF THE SPECIMEN

3 3 1 Straining stage 3 3 2 Translation stage

3 3 3 Translation stage controller

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3.4 OBSERVATION AND RECORDING OF THE PROCESS | 3 4 1 Optical microscope

3 4 2, CCD Camera

3 4 3 Frame grabbing unit and PC 3 4 4 Digital image processing 4 RESULTS AND DISCUSSION

4.1 INHOMOGENEITY OF THE MATERIAL

4.2 LAGRANGIAN STRAIN DISTRIBUTION AROUND THE CRACK TIP IN SOLID PROPELLANT TPH 1011

4.3 STRESS - STRAIN CURVE FOR THE SOLID PROPELLANT

5 CONCLUSIONS REFERENCES

APPENDIX A STRAINING STAGE

APPENDIX B GENERAL FEATURES OF THE FRACTURE

PROCESS OF THE SOLID PROPELLANT TPH 1011 RECORDED BY OPTICAL MICROSCOPY

APPENDIX C PROGRAM LISTING

APPENDIX D ALTERNATIVE ERROR ANALYSIS

APPENDIX E TRANSLATION STAGE CONTROLLER

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Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Mapping X

Convergence test for DIC program Large deformation DIC step Interpolation process

Calibration of strains for addition methods

Detailed calibration of strains for addition methods

Experimental setup schematic Stain stage

Positioning stage Stepping motor

Controller for the positioning stage Microscope

CCD camera

Volume fraction distribution of particles Solid propellant specimen

Straining of an uncracked specimen shown before and after loading

Lagrangian Eyy distribution for an uncracked solid propellant specimen

Distribution of Eyy along a line

Straining of a cracked specimen before and after loading

Tiff file of deformation at 0% strain Tiff file of deformation at 1% strain Tiff file of deformation at 2% strain

Tiff file of deformation at 3% strain

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Figure 24 Tiff file of deformation at 4% strain 45 Figure 25 Tiff file of deformation at 5% strain 46 Figure 26 Tiff file of deformation at 6% strain 47 Figure 27 Tiff file of deformation at 7% strain 48 Figure 28 Maximum principal strain distribution for step 1 52 Figure 29 Maximum principal strain distribution for step 2 54 Figure 30 Maximum principal strain distribution for step 3 55 Figure 31 Maximum principal strain distribution for step 4 37 Figure 32 Maximum principal strain distribution for step 5 59 Figure 33 Area around crack tip where more than 10% strain localizes

60 Figure 34 Evolution of strain level in high strain areas 62 Figure 35 Strain distribution along crack path 63 Figure 36 Strain distribution at selected positions in the crack propagation

path 64

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strains of 10% are reached, the Correlation algorithm fails to converge Strains of 50% to 100% are typical for crack propagation problems in solid propellants Accordingly we develop and examine here an incremental application that follows the deformation history This development is addressed first in Section 2, followed by a discussion of the experimental setup and arrangement to define fields around a slowly growing crack The method developed is used to quantitatively describe deformation on cracked and uncracked specimens of solid propellant TPH 1011 Particular interest is devoted to the inhomogeneity of the

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Image Correlation (DIC) program cannot be applied in a straightforward manner, as is done for small strains In this section a method is presented by which the total deformation is subdivided into smaller deformation increments, each of which can be processed by DIC The results of the DIC program over the small deformation increments are then combined to compute the strain distribution for the large deformation

2.1 THE DIC PROGRAM

Developed by Sutton [Sutton, 1986] and improved by Vendroux and Knauss [Vendroux and Knauss, 1994], the Digital Image Correlation (DIC) program is used to measure the displacement field and its gradients from images of an undeformed and deformed body These images are gray levels images consisting of a grid of pixels, (typically 640 by 480) with gray level ranging from 0 to 255 In this way the images represent a surface in which the heights at grid points represent an associated gray level distribution

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significantly modify the gray level, i.e f(x,y)~g(x,y ) If one assumes that the deformation is such that the topology (profile pattern) after deformation is uniquely related to that before the deformation, one may determine the deformations (displacement and their gradients) through a correlation between the two pattern images Let a material point be represented by G(x,y), where x,y are its coordinates in the undeformed configuration Similarly the same point is represented by G (x,y) in the deformed configuration, where x,y are the coordinates of the material point in the deformed configuration (Fig 1)

s S e »< Figure 1 Mapping X

Define X as the mapping from the undeformed to the deformed configuration

X: R*~>R?

G->G=X(G)/ 2(%,7)= f(x y) (1)

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Gy be the image of Gp through the deformation X Let S be a neighborhood of Gp that is mapped onto the set S such that S is a neighborhood of Gp - Considering this neighborhood S to be small, the two configurations of the deformation are related by

V G(X, ), 3 G(x, y) such that

x =x + u(X9,¥o) + Ux |x, V9 * — Xo) + Uy | x9, ¥o 6 ¥ ~ Yo)

7 =y + v(Xq,¥9) + Vy (x9, Vo * — Xo) + Vy Ìtxe, vạ Ÿ — Yo): (3)

These equations define a new local mapping X, around Gj) At this point we introduce the least square correlation coefficient C

-_ SIG@-acxcayras

[[P@as 7

(4)

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Ug = U(X, Yo) Vo = V(X9.¥o)

Ou a

Upsx = GB (%02¥o) Yoox = GB (odo)

Ou Oa

Uory = Bo» ¥o) "ov B (x9 ¥o) (5)

are exact, i.e C is then identically zero Thus the displacements and displacement gradients of the deformation at the point of interest are obtained in the process of minimizing C In the present application (Pixel location), the definition of the least square coefficient is discretized and integration signs are replaced by

summation signs, so that one has

> ,Ư(G,)- s(X(G,))Ÿ

C= Duc est (Ge) : (6)

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parameters define a six-dimensional space D such that

6

D={P ER | P(u, VQ >Uoox> Ugsy> Vooxs Vooy )H- (7)

IfP, isa vector in D and P is the vector solution that minimizes (6), C(P) can be written as a truncated Taylor series around P,

C(P)=C(P,)+ V C(P,) ‘ (P-P,)+1/2(P-P,)’ V V C(P,)(P-P,) (8)

Since P makes C a minimum, it follows that V C(P)=0, thereby taking the gradient of (8) results in

V VC(P,)(P-P,) = - V C(P,) (9)

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2.2 LIMITS IN THE DIC PROGRAM

Before attempting to apply the DIC program to determine the strains and displacements, it was tested on known deformations in order to establish the largest strain which the program allows A test was performed on silicone specimens splattered with microscopic speckles to provide the random gray level distribution for the DIC program to identify The speckles were generated with an airbrush to match the scale of the surface fractures in the solid propellant specimens to be studied later

2 2 1 Convergence depending on strain level

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8 E $ S0E +4 © E È 50Ƒ © 5 © E o WE 3 L 5 SOF 2 20 F 10 E E ee °F 19 Straln Level (56)

Figure 2 Convergence test for DIC program

2 3 PROPOSED SOLUTION

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the overall deformation Four schemes for adding strain and displacement fields for the deformation are investigated to single out the most accurate These schemes are presented in the following section

2.4 ADDITION OF STRAIN AND DISPLACEMENT FIELDS METHODS

The general problem can be outlined with the help of Figure 3 For a simple deformation process we establish three pictures of the body, each associated with the configurations 1, 2 and 3 of the sequential deformation

IMAGE 1 DEFORMATION A IMAGE 2 GLOBAL | DEFORMATION DEFORMATION B IMAGE 3 › aA

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Between image 1 and 2 (deformation A) and between images 2 and 3 (deformation B), the program is successful in giving the deformation fields However, the strains between images 1 and 3 (Global deformation) are larger than those that lead to the convergence of the correlation We determine the deformation fields for the global deformation, corresponding to image 3 by using the results that the DIC program provides for the deformations A and B The DIC strain and displacement maps are discrete representations The strains and displacements are only computed at a set of pixels forming a grid over the undeformed configuration A point located in the undeformed reference frame at a pixel which lies in this grid while undergoing displacements in deformation A is not likely to end up as a pixel point in the grid over which the correlation process is performed in the beginning of the second deformation step (B) In order to refer both deformation increments to a common (Lagrangian) reference frame, it will be necessary to interpolate the positions of gray level features for the second deformation relative to the first pixel location One feature to emphasize in this discussion is that the DIC program calculates the large deformation parameters in a Lagrangian setting The methods used are:

2 4 1 Method 1:

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12

F ota =F,F, & (10)

A discrete set of particles, H,; , are represented on configuration 1 by the rectangular grid of points, G, Those particles are also represented after deformation A by the points G, in the configuration 2 The results that the DIC program yields for this process are the displacements u*, and v, and the displacement gradients u,’, v,*,, u,‘, and v,“, These values are presented in a Lagrangian setting, that is, with respect to configuration 1 On the configuration 2, the discrete set of material point, H,, are represented by

G=G,+u" (1)

During the second deformation, a different set of particles, J,, represented on configuration 2 by the rectangular grid of points K ; are mapped onto the set K j

in configuration 3 The set of points K , can be represented in a Lagrangian setting as

ì

“oO (12)

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Figure 4 illustrates the process

Configuration 1 Configuration 2 Configuration 3

G 8 /

ù >

Some

Figure 4 Interpolation process

Since the global deformation must be expressed in a Lagrangian frame, interpolation of the results of deformation B on the particles J; is required to obtain the displacements and displacements gradients u”,, v”,, u°;, Vy’, uy; and v,’, of the particles H, during deformation B This is done by fitting a bilinear surface to the four closest points G, to the point K ; and evaluating it at K i Then, invoking the tensorial relation (12) we can derive expressions for the displacement gradients of the global deformation by

global A B A, B BLA Uu, =u, +U, + u, H_ + My Vv

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lobal

tí g y _,, =u, TH +H, H, TH) Vy A B A,B B.A

giobal —_

Vv = Vv, TY, + Vy +Vy VY, A B A B B_ A (13)

while the displacements are simply

global _ A B

u / +

về“ —y4+v#, | (14)

The next step is to construct the Lagrangian strain tensor according to the definition:

_ global 1 global2 global

6 =u +— ju, +,

2

] 2 2

_ global , — global global

Ey =Y y + 2 {u, + y

_ 1 glabal + giobal + I global, global + global global

é,, = 2 uy v HỒ 1, V, vy (15)

2

2 4 2 Method 2

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2 — uÌ u.=— * Ax v3—vl v= Ay _ u3-ul u, = Ay _ v2—vi (16) Vv, = a

From which we compute the Lagrangian strains

2 4 3 Method 3

The third scheme is basically the same as the second except that when finding the displacements of the deformation B, the interpolation process is performed using six neighboring points instead of four This allows the appearance of second-order terms in the interpolating functions

2 4 4 Method 4

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2.5 CALIBRATION OF METHODS FOR ADDITION OF FIELDS

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3 APPLICATION OF THE LARGE DEFORMATION DIC METHOD TO THE CRACK OPENING PROBLEM IN SOLID PROPELLANT

The Large Deformation Digital Image Correlation method is used to obtain Lagrangian strain distributions within 1 mm of a crack tip in a solid propellant TPH 1011 specimen

3 1 EXPERIMENTAL SETUP

During this experiment, a cracked specimen of solid propellant TPH 1011 is loaded with a constant strain rate in the direction perpendicular to the crack The crack, initiated with a razor blade, opens with an increase in global strain level The crack opening process is monitored at a microscopic level Six digital images of 640 x 480 pixels, representing 3 mm x 4 mm of the specimen surface are obtained, one every 10 seconds These images of the specimen surface are taken

for far field Lagrangian E,, strains of 0%, 1%, 2%, 3%, 4% and 5% These six

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a frame grabber unit Finally the images of the experiment are processed by a Sun

workstation CCD ,| FRAME CAMERA GRABBER PC MICROSCOPE T SUN WORKSTATION | Specimen STEPPING —[ Tờ | MOTOR STRAIN STAGE TRANSLATION Ö STAGE / | ] | JOYSTICK DEVICE

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Microscope igure 12

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3.2 SOLID PROPELLANT SPECIMEN

The material under study is the solid propellant TPH 1011 This material contains particles of ammonium perclorate, which acts as oxidizer embedded in a rubber matrix that provides the carbon for the combustion In order to control the rate of burning, the material also contains aluminum particles

0.0055 0.005 0.0045 0.004 0.0035 0.003 89% 0.0025 0.002 TTTTTTTTTTT TT TT {1T TT TT TT TT TT TTTTTTỊ 0.0015 Volume Fraction Distribution 8% 0.001 0.0005 ol 0 100 200 300 Radius (microns)

Figure 14 Volume fraction distribution of particles

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diameter The matrix is a very soft rubber with a Young’s modulus of elasticity of 0.1MPa, while the Young’s modulus of elasticity of the aluminum particles is 70GPa The Young’s moduli of elasticity of the ammonium perclorate and the aluminum are suficiently large as to model the particles as rigid when they are compared with the rubber matrix The material is a filled elastomer containing solid particles on a microscopic scale The volume fraction of the particles is close to 70% The grain size of the particles is of great importance in interpreting the results obtained by the subsequent experiments

Upper tab a † a» 4 + † a ‡ + a — “—1/8 in | _ V - 6n | ` © đồ C) Ỉ 2in 1,=.8 in ———— | ——> +— 1/8in Ỷ Lower tab kj——_————31n Vv (Fixed)

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To the naked eye, the material looks like dark gray rubber, the texture of which is very similar to erasers at the end of pencils To study the mode I fracture behavior of this material, 1/8 in thick sheets of the solid propellant are cut into 3 in x 2 in rectangular pieces Aluminum tabs are attached to the ends to provide a constant displacement boundary condition The aluminum tabs also ensure that both sides of the specimen remain parallel to each other throughout the deformation (Figure 15) By using a razor blade, a 1 in initial crack is cut in the specimen This crack opens as the experiment and the crack progress The surface of the specimen is very irregular under microscopic observation (Figure 1b in Appendix B) Small dimples of the order of 200 microns in diameter are seen in numbers of 3 to 5 per square millimeter These dimples are generated during the manufacturing process of the solid propellant sheets These features play a key role in the fracture process of the material Most of the damage generated around the crack tip is localized around these dimples

3.3 LOADING OF THE SPECIMEN

The strain is applied to the specimen by a prescribed displacement at the boundaries in a manner such that the aluminum tabs always remain parallel The devices used to load apply the loads to the specimen are:

3 3 1 Strain stage

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tab while the lower aluminum tab remains in its original position (Figure 14) Appendix A contains a set of drawings with the dimension of the strain stage With the help of a stepping motor’ (Figure 10), the upper tab velocity can be precisely controlled and therefore the strain rate is accurately prescribed For the present experiment, the upper tab velocity was set to 0.0008 in/sec For 1, equal to 0.8 in (Figure 14), it corresponds to a far field strain of 1%, 2%, 3%, 4% and 5%

at times of t = 10 sec, t = 20 sec, t = 30 sec, t = 40 sec and t = 50 sec These five

states are called in what follows steps 1 to 5

3 3 2 Translation stage

As the load is applied to the solid propellant specimen, the position of the observation region relative to the microscope changes In order to track the same area of the specimen, the position of the specimen under the microscope is controlled by two movable platforms The first one enables the movement of the specimen in the x and y direction by 10 mm in each direction (Figure 9) It is used to position the crack tip under the microscope before the experiment This position stage is a Newport Model 405 A second translation stage moves the specimen during the experiment also in the x and y directions This second position stage is a Newport Model 462 The shift distance of the second stage is 25 mm ( 1 in ) in each direction As the specimen is loaded, the position of the crack tip moves fast relative to the objective lens of the microscope For fast and accurate movement of the second translation stage, two electric motors drive this stage as controlled by a GALCIT built joystick device that can be easily used by

' The stepping motor used to drive the strain stage is a ASTROSYN Miniangle stepper motor type

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the operator The combination of the second translation stage, electric motors and

joystick controller are depicted in Figure 11

3 3 3 Translation stage controller

The second translation stage is powered by two 12V electrical motors They turn two millimetrized screws that control the position of the translation stage in the x and y directions The motors are controlled by an electronic device operated by a joystick that can prescribe the velocity and direction of the movement of the second stage The translation stage controller can be seen in Figure 11 and is presented to a greater detail in Appendix E

3.4 OBSERVATION AND RECORDING OF THE PROCESS

The process is monitored using an optical microscope, a CCD camera and a frame grabbing unit installed in a PC

3 4 1 Optical microscope