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

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 Gonzale

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 (submitted January 14, 1997)

© 1997 Javier Gonzalez

All rights reserved

iil

To my parents, my sister Ana and my brothers Pedro, Alejandro and Santiago

I wish to thank Professor W G Knauss for his support and advice and acknowledge the valuable help of my classmates: Demirkan Coker, Pradeep Guduru, Bibhuti Patel, Samudrala Omprakrash, Sangwook Lee, Eduardo Repetto, Sandeep Sane and Raul Radovitzky for the valuable insight they gave me through discussions on various segments of my research topic It has been of great help to discuss also my project with the Post doctoral scholars Dr Allan Zhong, Dr Anne Gelb and Dr Alfons Noe I would like to thank Professor Ravichandran and Professor Shepherd for being part of the thesis committee and giving me valuable advice in my research topic Finally I would like to thank the doctoral student Hongbing Lu for teaching me the digital image correlation procedure.

ABSTRACT

A full field method for visualizing deformation around the crack tip in a fracture process with large strains is developed A digital image correlation program (DIC) is used to incrementally compute strains and displacements between two consecutive images of a deformation process Values of strain and displacements for consecutive deformations are added, this way solving convergence problems in the DIC algorithm when large deformations are investigated The method developed is used to investigate the strain distribution within 1 mm of the crack tip in a particulate composite solid (propellant) using microscopic visualization of the deformation process

2.2 LIMITS IN THE DIC PROGRAM

2 2 1 Convergence depending on strain level

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

VI Vill

Vil

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

PROCESS OF THE SOLID PROPELLANT TPH 1011 RECORDED

BY OPTICAL MICROSCOPY

Convergence test for DIC program

Large deformation DIC step

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

ix

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

Figure 37 Minimum principal strain distribution for step 1 66 Figure 38 Minimum principal strain distribution for step 2 67 Figure 39 Minimum principal strain distribution for step 3 68 Figure 40 Minimum principal strain distribution for step 4 69 Figure 41 Minimum principal strain distribution for step 5 70 Figure 42 Stress-strain curve for solid propellant TPH 1011 72

filled with small particles, while other rigid polymers are toughened through the addition of rubber particles Solid propellant rocket fuels are physical mixtures of ammonium perclorate and aluminum powder often in multimodal size distribution, bonded together by a rubber phase called the matrix Failure in all these materials is heavily dependent upon the interaction between the particles and the matrix, specifically on the separation of particles and binder Failure is also dependent upon the volume ratio of particles to matrix, which is typically close to 75% In the sequel we examine the failure progression in a solid propellant Triokol TPH 1011 Application of continuum mechanics to the stress/strain analysis of structures made of these types of materials typically invoke macroscopically homogeneous material performance, even though at the scale level of the particles deformations are anything but homogeneous We shall see that inhomogeneous deformations occur at a size scale substantially larger than the largest particle, and that the failure process is directly dependent upon these micro-structural deformations Measuring large deformation strains over small domains of tens to hundreds of microns is not a trivial matter Imprinted grids tend to serve well at a size scale just above what is required here Determining the micromechanical deformation with the aid of optical microscopy, e.g on the tip of a macroscopic crack, implies the need to extend the presently available tools of strain measurements In principle the digital image correlation method [Sutton, 1986; Vendroux/Knauss, 1994] is ideal for this purpose except that it is not suitable if the deformations are too large For deformations where

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

material

To compute strain and displacement fields in large deformations, the Digital 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

Let X be the mapping of the undeformed configuration onto the deformed configuration so that a material point is represented in the undeformed configuration at the coordinates (x, y) and has an associated gray level value f(x,y) The same material point in the deformed configuration is represented at (x,y ) with a gray level of g(x,y ) It is assumed that the deformation does not

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)

where u and v are the displacements of G in the Lagrangian reference frame Let

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

The present correlation method minimizes this correlation coefficient C It will be

a minimum when the parameters of the mapping X ,

Ug = U(X, Yo) Vo = V(X9.¥o)

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

the deformation parameters of X, , i.e Ug, Vo, Uy>,> Uorys Yoox Vooy- Let these parameters define a six-dimensional space D such that

6

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,

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

The problem in applying the DIC program to compute strain distributions in a large deformation process is the failure of convergence of the DIC algorithm if the deformation is too large To test the actual limits on the strain level for convergence of the program, a test on a homogeneous silicone rubber stretched in the y direction was performed The resultant undeformed and deformed images for stretches from 0% to 40% were compared by the DIC program For each deformation the strains and displacements were computed at 300 points The fraction of points at which the minimization process converged is presented in Figure 2 as a function of the Lagrangian strain For deformations larger than 10% there is a serious decrease in the successful points of convergence For the purpose of studying cracked solid propellants, where strains in excess of 30% need to be measured, the applicability of the standard DIC method is therefore seriously compromised Thus a new analysis tool must be developed

The largest deformation for which the DIC program provides an acceptable result

is for a deformation corresponding to a principal strain of about 10% For larger deformations, we apply the method incrementally through a set of deformations defined in consecutive deformation stages such that a strain greater than 10% is not reached in any increment Once the strain and displacement maps for the incremental deformations are obtained, continuum concepts are used to construct

10

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

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:

The first method proposed to add the strain and displacement maps from consecutive deformations (A and B) in order to reconstruct the global deformation uses the fact that for large strain deformations, one can compute the deformation gradient tensor for the global deformation by multiplying the deformation gradient tensors of the other two deformations (A and B), giving the expression

12

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

Figure 4 illustrates the process

Configuration 1 Configuration 2 Configuration 3

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

14

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

2 4 4 Method 4

In the last scheme the interpolation process for the displacements is changed to a least square fitting with respect to 16 neighboring points The biquadratic surface obtained in this way is evaluated at a pixel point The surface is differentiated to evaluate the required derivatives at the same point

16

2.5 CALIBRATION OF METHODS FOR ADDITION OF FIELDS

A specimen of an homogeneous silicone rubber without a crack and coated with microscopic speckles was strained sequentially in the y direction up to a Lagrangian strain of 80% A sequence of 15 images of the process was taken so that the global deformation was divided into 14 sub deformations of 3-4% Lagrangian strain each The resultant deformations were added by using the four methods, and the results were compared with the optically measured Lagrangian strain at every step The results of this test series are presented in Figure 5, which shows very similar characteristics for each of the four methods Figure 6 shows the difference of the results of each method and the optically measured Lagrangian strains From this figure we can conclude that method 1 deviates the least from the optically measured strain The maximum deviation occurs at a strain value of 40%, and yields 1% strain difference between the optical and the DIC Lagrangian strain

UE11S uegiBue1BeT 2Iq-G Figure 5.

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

images are associated with six different deformation configurations They also define five intermediate deformations Using the Large Deformation Digital Image Correlation (LD-DIC) method, the displacement and displacement gradient fields for the global deformation are constructed from fields corresponding to the intermediate deformations A schematic of the experimental setup is shown in Figure 7 The equipment used to prescribe the desired deformation consists of the following: a strain stage (Figure 8), a positioning stage (Figure 9), a stepping motor (Figure 10), and a controller for the positioning stage (Figure 11) The instruments used for the visualization of the process are a Nikon Metallurgical microscope (Figure 12), a CCD camera (Figure 13) and a personal computer with

20

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

Figure 8 Strain stage.

22

Figure 10 Stepping motor

24

Figure 11 Controller for the positioning stage

25

Microscope igure 12

F

26

igure 13 CCD Camera F

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

Figure 14 Volume fraction distribution of particles

For structural analysis, the material is modeled as a viscoelastic particulate composite material with grain size between 10 microns and 600 microns in

28

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

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

The applied strain is imposed by a straining stage developed at GALCIT (Figure 8) Using set screws, the user can control the displacement of the upper aluminum

30

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

34PM-C101 It is capable of applying a torque of 300 in/oz It is shown in Figure 10.

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

An important feature of the material that determines the length scale of interest is the particle size of the solid propellant Since this is of 10 - 400 microns in diameter, most of the important characteristics in the fracture process, Le inhomogeneous distribution of strains, void formation etc., occurs at this length scale In addition, the area around the crack tip where significant damage appears during the fracture process has a diameter of about 5 mm to 1 mm The detection

of these features is of main importance for the experiment, and therefore proper