.

Implementation of the Energy Domain Integral method in Ansys for calculation of 3D J-integral

of CT-fracture specimen.

Siva Shankar Rudraraju

June - August 2004

1

INDEX

I. Theoretical Background

II. J-Integral: Calculation Approaches

III. Contour Integral Method

IV. Weight Function

V. Finite Element Model

VI. ANSYS for Contour Integral Calculatio

VII. Implementation in ANSYS

VIII. ANSYS Macro’s

IX. Theoretical Solution

X. Experimental Results

XI. ANSYS Simulation Results

XII. Results Comparison

XIII. Conclusions and Suggestions

XIV. References

o.

4

7

8

9 12 n 15 17 18 21 22 23 25 26 27

Page N

3

Theoretical Background Fracture is a problem that society has faced since ages. It is one of the more catastrophic means of material failure. And just to get a grasp of the magnitude of this catastrophe, an economic study estimated the cost of fracture in the United States in 1978 at $119 billion. Further more this study estimated that the annual cost could be reduced by $35 billion if current technology were applied. Fortunately, the field of fracture mechanics has made rapid strides since then. And now there are many methodologies and technologies in place, to help us understand and prevent fracture. In this section a brief explanation of the theory of fracture mechanics is presented to provide the necessary theoretical background. How fracture failure is different from conventional tensile/brittle failure? In case of brittle/tensile failure, the material is assumed to be a continuum, and the material strength is calculated by taking into account the combined stress bearing capacity of the continuum. But in case of fracture, there are sharp discontinuities in this material continuum, which locally magnify the stresses, and hence locally exceed the strength of the material, thus giving rise to local failure initiation, which later spreads across the continuum, thus leading to material failure. So we can say that fracture is a micro level process, which destroys the macro load bearing capacity of the material. An attempt to understand and characterize such local stress magnifications is an important component of fracture mechanics. Fig 1.Stress flow in a plate near the vicinity of the elliptical crack.

4

5

As seen in Fig.2, the stress in the vicinity of a sharp crack is very high. It can be shown that the stress field in any linear elastic cracked body is given by.

( ) othertermsfr

kijij +⎟

⎠

⎞⎜⎝

⎛= θσ

Where ijσ is the stress tensor, r and θ are the distance from the crack tip and

angle about the crack tip. The above equation states that the stress is infinite at the crack tip (r=0). These locally high values of stress near the crack tip may exceed the strength of the material and lead to failure initiation. This failure once initiated, grows furthermore in a stable(ductile) or unstable(brittle) mode. Thus a small discontinuity (crack), in a continuum can lead to failure of the entire structure. So a description of critical stress states is of utmost importance for designers. Now, the various fracture parameters, which are used to characterize these stresses, are discussed below. UFracture parametersU: The most widely used fracture parameters are:

1. Stress intensity factor (K) 2. Elastic energy release rate (G) 3. J-integral (J) 4. Crack tip opening displacement (CTOD)

Fig 2. Stress magnification in a plate near the vicinity of the sharp crack.

6

UStress Intensity Factor (K)U: The stress fields ahead of a crack tip in an isotropic linear elastic material can be written as.

( )θπ

σ ijijrf

rK2

lim0

=→

where the proportionality constant, K, is referred as the stress intensity factor. But

the above equation is only valid near the crack tip, where the r

1 singularity

dominates the stress field. Thus, the stress intensity factor, which represents the proportionality constant, gives an idea about the level of stress magnification around the crack tip. UElastic Energy Release Rate (G)U: According to the first law of thermodynamics, a system goes from a non-equilibrium state to equilibrium, when there will be a net decrease in energy. Now when a material is loaded, its strain energy increases and hence the net energy also increases. Thus the system is moving away from equilibrium as the net energy is increasing. But always, the natural tendency of any system will be to jump to a state of greater equilibrium by decreasing its net energy, thus the system is constantly in want of a means to unload this excess strain energy. And the continuum discontinuities (cracks) provide a means to dump the strain energy as surface energy, by the creation of new crack surfaces during crack formation or propagation. Thus the energy release rate is an important parameter in understanding fracture tendency. It is a measure of the energy available for an increment of crack extension.

dAdG π

−=

It is defined as the change in potential energy per unit change in crack area. The previous two fracture parameters, K and G, are valid within the limits of linear elasticity, or with in the frontiers of Linear Elastic Fracture Mechanics (LEFM). But many materials (e.g. steel) have elastic plastic behavior, though the magnitude of plasticity may vary depending on the material and loading conditions. So, to characterize the crack conditions to a sufficient degree of accuracy for real materials, we need a fracture parameter, which can take into account the material plasticity. This parameter is known as the J-integral.

7

UJ-Integral (J):U Path-independent integrals have long been used in physics to calculate the intensity of singularity of a field without knowing the exact shape of this field in the vicinity of the singularity. They are derived from conservation laws. The singularity in the vicinity of a crack tip, thus presents a fit case for the application of the path-independent integrals. CherepanovP

3P and RiceP

4P were the first

to introduce path independent integrals in fracture mechanics. RiceP

4P showed that the nonlinear energy release rate, J, could be written as a path

independent line integral. Hutchinson P

5P, Rice and Rosengren also showed that the J

uniquely characterizes crack tip stresses and strains in nonlinear materials. Thus the J-integral can be viewed as a:

• Stress intensity parameter (like K) • Nonlinear energy release rate (like G)

Rather J is a dual equivalent of K and G, in Elastic Plastic Fracture Mechanics (EPFM)

J –integral is defined as:

Where, γ = any path surrounding the crack tip W = strain energy density (that is, strain energy per unit volume t Bx B = traction vector along x axis = σBx Bn Bx B + σBxy B n By B

t By B = traction vector along y axis = σBy Bn By B + σBxy B n Bx B

σ = component stress n = unit outer normal vector to path γ s = distance along the path γ As stated earlier, J is also equal to the nonlinear energy rate.

ΑΠ

−=ddJ

Until now the necessary brief introduction of the theoretical aspects of fracture mechanics has been presented .The reader can find the theory in a greater detail in any of the many available books on fracture mechanicsP

1P.

Now, the actual problem, the calculation of the J integral is presented. J-integral: Calculation Approaches The J-integral can be calculated by invoking either of the two definitions, i.e. the line integral or energy release rate definition. Over the years many approaches have been developed for numerical evaluation of the J-integral. The Global energy estimates method involves finding the rate of change in global strain energy of the fracture model with crack growth. This technique involves minimal post-processing but however this involves multiple solution calculations (considering different crack lengths), which can be very cumbersome if large fracture models are to be analyzed. An alternative method is the numerical evaluation of J-integral along a contour surrounding the crack tip. This method is discussed in detail in the following sections

Energy release rate

Path Independent Integral

2D

3D

Line Integral

Area Integral

Volume Integral

Surface Integral

Virtual Crack Extension Theory

Stiffness Derivative Formulation

Weight function

Energy Domain Integral

Global Energy Estimates

J - Integral

J-Integral Calculation Approaches Contour Integral Method: The J-integral can be evaluated numerically along a contour surrounding the crack tip. The advantages of this method are that it can be applied to linear and non-linear problems, and path independence enables the user to evaluate J at a remote contour, where numerical accuracy is greater. For problems that include path-dependent plastic deformation or thermal strains, it is still possible to compute J at a remote contour, provided an appropriate correction term (i.e. area integral) is applied. For three-dimensional problems, however the contour integral becomes a surface integral, which is difficult to evaluate numerically. Recent formulations of J apply area integration for 2D problems and volume integration for 3D problems. Especially for 3D, the volume integral is much better then the surface integral as it is more accurate and easier to numerically implement. Here the Energy Domain Integral method for calculation of area and volume integrals is implemented.

8

P

** PThe reader

assumed, t

Energy Domain Integral The energy domain integral is a preferred methodology, as it has a general framework for easy numerical analysis. This approach is extremely versatile, as it can be applied to both quasistatic and dynamic problems with elastic, plastic, or viscoelastic material responses. Shih P

8P, Moran and Nakamura gave detailed instructions for implementing the domain

integralP

**P approach in FEM. Their method is summarized below.

In the absence of thermal strains, path-dependent plastic strains, tractions on the crack faces and body forces within the integration volume or area, the discretized form of the domain integral is as follows.

p

pk

i

orVA

m

pi

iij wx

xqw

xuLJ

⎭⎬⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

=∆ ∑ ∑= ξ

δσ**

det1 1

11

__

(Eqn.1)

Where q is the weight function, m is the number of gaussian points per element and

pw is the weighting factor for the gaussian integration.

WeigThe geneinter

whe

poin

Fig.3 Example of closed contour around a crack a front. SB0 Band SB1 Bare inner and outer surfaces, which enclose VP

*

9is referred to the book by T.L.AndersonP

1 Pfor a detailed explanation of this method. Here it is

hat the reader is familiar with this method, and just the FEM implementation is presented.

ht Function (q) weight function q, is merely an mathematical concept that enables the ration an area or volume integral. But for the sake of understanding, q can be preted as a normalized virtual displacement.

max)()( aqa ∆=∆ ηη

re η is any point along the crack front, a∆ is the crack displacement at that

t, and maxa∆ is the maximum crack displacement along the crack front.

Fig.4 Example of a q function defined locally along the crack front. Shih, et al. have also shown that the computed value of J-integral is insensitive to the assumed shape of the q function. So we are free to assume any crack extension shape, and hence any arbitrary smooth q function. But care should be taken such that the Q-function should have the correct values on the domain boundaries.

q1=1

A

q=0 q1=0

q2=1

q2=0

B

q2=0

q2=1 q2=1

q2=0

q2=1

q2=0

C D E

1

q = q1 x q2

Z

Y

X Fig.5. Representation of various possible q functions.

A: q1 plot (XY plane) B-E: q2 plot (YZ plane)

10

11

Another important benefit of the weight function is that it allows the calculation of local values of J-integral within a model. From this we can estimate the variation of the J-integral and hence the fracture tendency at different positions along the crack front. If the point-wise value of the J-integral does not vary appreciably over L∆ , an approximation of )(ηJ is given by:

∫∆

∆=

L

drqLJJ

ηηη

),()(

0

__

⎟⎠⎞⎜

⎝⎛

∆∆

⎟⎠⎞

⎜⎝⎛ ∆

=∴

max

__

)(

aA

LJJ

C

η (Eqn.2)

Thus, from (Eqn.1) and (Eqn.2), )(ηJ can be calculated at any point along the crack front. Now the task is to numerically calculate (Eqn.1) and then )(ηJ from (Eqn.2). Hereafter, the Finite Element implementation of the J contour integral (Eqn.1) is presented.

The Finite Element Model The ASTM documentations refer to four standard fracture specimen configurations. Among them the one of the most widely used model is the Compact Tension (CT) Specimen. Here the J-integral calculations are performed on a standard CT specimen with w=50mm.

B

AA

w = 50mm a = 0.50*w b = 0.50*wh = 1.20*wg = 1.25*ws = 0.55*wd = 0.25*wn = 0.10*wl = 0.25*w

Fig.6. (A) – The Standard CT Specimen, (B)-The parametric FEM model in ANSYS.

To simplify the computational complexity, the full FEM model is reduced to the quarter symmetry model (Fig.7.) and the appropriate symmetry boundary conditions and constraints are applied.

C B A

l

Fig.7. (A) Full Model (B) Half Symmetry Model (C) Quarter Symmetry Mode

12

To prevent concenan intermediate celastic modulus th MESH DESIGN (CrAs discussed earlithe element mesh

• In Elassolid bmidside

r1 s

• In plas

So the midside

• The mo“spiderelemenare destress acrack tmesh a

)

I.M

Fig.9. “Spider near the

Fig.8. Load applied through intermediate material (I.M

13

trated loading, loads are uniformly applied over an area through ylindrical wedge shaped material (shown in fig.8.) with higher en the specimen material.

ack Tip) er, the crack tip is an area of stress singularity. So while designing around the crack tip the following points are to be considered:

tic problems, the crack tip is an area of stress singularity, so the rick elements are to be degenerated down to wedges, and the nodes (if any) are moved to ¼ points. Such a model results in a

ingularity.

tic problems, the r

1 singularity no longer exists at the crack tip.

elements are degenerated to wedges (like in elastic case), but the node (if any) positions are unchanged. st efficient mesh design for the crack tip has proven to be the

web” configuration, which consists of concentric rings of four sided ts that are focused towards the crack tip. The innermost elements generated to wedges. Since the crack tip region contains steep nd strain gradients, the mesh refinement should be greatest at the

ip. The spider web design facilitates a smooth transition from a fine t the tip to a coarse mesh remote to the tip.

web” mesh around the crack tip, with degenerated wedge elements crack tip.

14

Element Type For selecting the element type, a compromise has to be made between computational accuracy and computational time. We have a choice between the linear 8-node brick element and the quadratic 20-node brick element. Some important comparisons between these elements are:

The 20 node brick element is formulated with quadratic polynomials so it can yield results with greater accuracy, especially in regions of high stress gradients like that exhibited near the crack tip. The 8-node brick element is formulated with linear equations, so it is less suited for regions with high stress gradients.

The computational time required to process a 20 node element model is many times more then a similar 8 node element model

The Ansys documentation states, “In TnonlinearT structural analyses, you will usually obtain better accuracy at less expense if you use a fine mesh of these linear elements rather than a comparable coarse mesh of quadratic elements.”

Both element types can take the degenerated wedge shapes required at the crack tip.

Taking into consideration the above points, and noting the absence of stress singularity at the crack tip for elastic-plastic material, the CT specimen was finally modeled with 8-node linear brick elements, with very fine meshing around the crack tip.

15

Ansys for Contour Integral Calculation The J-integral calculation involves a lot of post processing calculations. But some of the results required for these calculations are not readily available in ANSYS.A description and solution of these limitations is presented below. Limitations 1.The simulation results like displacement, stress and strain energy density are to be

obtained at the integration points of each element enclosed within the selected contours. Unfortunately, Ansys results can only be obtained at the element nodes rather then the integration points.

2.Ansys strain energy density results are available only as element solutions. But the

strain energy gradient varies across the element, and the available Ansys element value is an average across the element. But for J-integral calculation, the strain energy density values are required at each element integration points.

3.The integration point locations are available only through the PRESOL command,

and hence to obtain the integration point locations the data should be dumped into an output file and then read into the ANSYS database.

Solutions The above three limitations can be overcome by the following two methods: Method-I Step 1: Store the nodal displacement and stress results for the elements within the

contour. Step 2: Deduce the Strain energy density values by the following method

Store the element strain energy (SENE) and element volume (VOLU) values in the element table (ETABLE command) for all the elements within the contour.

Calculate the element strain energy density values by dividing (SEXP) the element strain energy by the element volume values in the element table.

Define infinitesimally small paths (using PATH and PMAP commands) at the nodes of the elements within the contour, such that the corresponding node is the last point of the path.

Interpolate the strain energy density values (PDEF command) from the element table to the defined paths at the nodes.

Store the strain energy density value at each of the nodes by reading in the value at the last point of the defined paths. (*GET command)

Now we have the nodal values of displacement, stress and strain energy density. Step 3: Derive the values at the integration points by interpolating the nodal values,

using the shape functions and local element coordinates of the integration points.

Thus, we obtain all the required results at the integration points. Method-II Step 1: Follow the first three points of Step 2 of Method-I above.

16

Step 2: Printout the integration point locations (PRESOL command) into an external file. Then read the x, y and z locations of each integration point into an array (*VREAD command).

Step 3: Define infinitesimally small paths (using PATH and PMAP commands) at the

integration points of the elements within the contour, such that the corresponding integration point is the last point of the path.

Interpolate the displacements, stresses and strain energy density values to the defined paths at the integration points. (For strain energy density, the values must be first stored in an element table as in Method-I)

Store the results at each of the integration points by reading in the value at the last point of the defined paths. (*GET command).

Comparison The results obtained by the above two methods differed negligible. And the computational time required was also comparable. So the Method-I is adopted hereafter, as it better fits into the general architecture of the J-integral macros.

17Macro “Module3.Mac”

Macro “Qfunc.Mac”

Macro “Shape. Mac”

Assign Element attributes and Material Properties

Input all problem parameters

Define Plasticity Curve

Build Parametric Geometric Model

Build 2D FEM model

Extrude 2D model to 3D model

Modify Crack Tip wedge elements

Set Solution Controls

Solve

Apply Loads and Constraints

Select elements within the Contour

Sort selected elements

Deduce required results at nodes

Interpolate Nodal values to get Integration point Values

Deduce Q-Function values at nodes

Deduce the gradients of displacement and Q-Function

Calculate J-Integral

Store all required values

End of File

Plastic Material

Title

Macro “Module1.Mac”

Macro “Plasticity. Mac”

Macro “Module2.Mac”

Macro “Modify. Mac”

Macro “Parameters. Mac”

Implementation in ANSYS The following Block Diagram represents the general structure of the problem and functions of the macros involved.

18

Macros: The problem description is: “Building numerical parametric models of the CT fracture Specimen and calculation of the 3D J-Integral of standard fracture specimens by implementing the energy domain integral method in Ansys” This has been implemented through a sequence of 8 Macros (3 main Module macros and 5 auxiliary macros) as shown in the block diagram previously. These macros are described below. Macro “Parameter.Mac”: This is the first macro in the sequence, where the problem “TITLE”, Element attributes, material properties and other required parameters are defined. This macro acts as the user interface to the entire problem. Varying the parameters in this macro can control almost all problem characteristics. The only other parameters, which should be defined by the user, are the contour position parameters at the start of “Module2.Mac” and the plasticity parameters in “Plasticity.Mac” Macro “Module1.Mac”: This is the main macro of the problem, where the parametric solid model is build, then converted to FEM model and solved. This Macro encompasses the Pre-Processing and Solution Routines. Macro “Plasticity.Mac”: This is an auxiliary macro for defining the Multi-linear isotropic hardening (plasticity) curve as a data table. Macro “Modify.Mac”: Ansys documentation states that, “Generating a 3-D fracture model is considerably more involved than a 2-D model. The KSCON command is Tnot T available, and you need to make sure that the crack front is along edge KO of the elements” And when the 3D mesh is generated directly by extruding 2D mesh, the crack front is along the edge JN rather than KO (face KL-OP). So we need to reorder the node numbering to place the edge KO (face KL-OP) along the crack front This Macro performs two functions:

Reordering element node numbers. Creating new nodes to change the point crack tip to a circular crack tip,

with very small radius “CKTIP” Macro “Module2.Mac”: This is a Post-Processing macro. The macro involves selection of elements within the specified contours, and sorting them depending on the element centroid coordinates. Macro “Module3.Mac”: This is the J-Integral calculation macro. In this macro Equation 4 and Equation 5 are implemented. This Macro calls:

“Qfunc.Mac” to get the value of Q-function at any point (x, y, z). “Shape.Mac” to get the required displacement gradients, Qfunc

gradients, stresses and strain energy density values.

Macro “Qfunc.Mac”: This Macro passes on the Q-Function values at any point (x, y, z) to “Module3.Mac”

19

Macro “Shape.Mac”: This is the main calculation Macro, where the displacement gradients, Qfunc gradients, stresses and strain energy density values are computed for the integration points. Here the nodal values are transformed to integration point values using element shape functions and a series of matrix operations. The basic two matrices involved in this macro are JS, JP. From these two matrices the Jacobian matrix and then the inverse Jacobian matrix are formed. Then through a series of matrix operations involving the inverse Jacobian matrix, we obtain all the required values at the integration points. The sequence of matrix operations is described below. Input Matrices: Here, N-shape functions, u-displacements, q- Q Function, σ - Stress, w-strain energy density, (x, y, z): Global coordinates, (s, t, r): Local element coordinates

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

888

777

666

555

444

333

222

111

zyxzyxzyxzyxzyxzyxzyxzyx

JX

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

rN

rN

rN

rN

rN

rN

rN

rN

tN

tN

tN

tN

tN

tN

tN

tN

sN

sN

sN

sN

sN

sN

sN

sN

JP

87654321

87654321

87654321

[ ]87654321 NNNNNNNNJS =

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

38

28

18

37

27

17

36

26

16

35

25

15

34

24

14

33

23

13

32

22

12

31

21

11

uuuuuuuuuuuuuuuuuuuuuuuu

MU

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

38

28

18

37

27

17

36

26

16

35

25

15

34

24

14

33

23

13

32

22

12

31

21

11

qqqqqqqqqqqqqqqqqqqqqqqq

MQ

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

811

811

811

811

811

811

811

711

711

711

711

711

711

711

611

611

611

611

611

611

611

511

511

511

511

511

511

511

411

411

411

411

411

411

411

311

311

311

311

311

311

311

211

211

211

211

211

211

211

1113

123

112

133

122

111

wwwwwwww

MS

σσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσ

20

Resultant Matrices: Thus, as seen above the entire calculations are condensed to only 6 matrix multiplications per integration point. This matrix method is simple to implement and leads to faster and very efficient analysis.

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

rx

rx

rx

tz

ty

tx

sz

sy

sx

JACO

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

zr

zt

zs

yr

yt

ys

xr

xt

xs

JACOINV _

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

zN

zN

zN

zN

zN

zN

zN

zN

yN

yN

yN

yN

yN

yN

yN

yN

xN

xN

xN

xN

xN

xN

xN

xN

MC

87654321

87654321

87654321

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

zu

zu

zu

yu

yu

yu

xu

xu

xu

DU321

321

321

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

zq

zq

zq

yq

yq

yq

xq

xq

xq

DQ321

321

321

[ ]wMRES 132312332211 σσσσσσ=

UMatrix Operations: JACO = (JP)*(JX) INV_JACO = (JACO)P

-1

MC = (INV_JACO)*(JP) DU = (MC)*(MU) DQ = (MC)*(MQ) MRES = (JS)*(MS)

21

pleltotal JJJ +=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

⎟⎠⎞

⎜⎝⎛ −

+=

432

2/3 60.572.1432.1364.4866.01

2)/(

wa

wa

wa

wa

wa

wa

waf

Theoretical Solution: The Electric Power Research Institute (EPRI) J estimation scheme provides a means for computing the J-Integral in a variety of configurations and materials. A fully plastic solution is combined with the stress intensity solution to obtain an estimate of the elastic-plastic J. The EPRI scheme is presented here: And the stress intensity factor, KB1 B is given by: And for a CT specimen These above theoretical solutions are used to verify the simulation results. However, these theoretical solutions assume plane stress or plain strain conditions, whereas in case of a real specimen neither pure plane stress or plane strain conditions exist.

'

21

EKJ el =

1

0100

+

⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

pl PPbhJ σαε

Where 2

'

1 ν−= EE 00 455.1 σηBbP = , Plain Strain

EE =' 00 072.1 σηBbP = , Plain Stress

b - characteristic length hB1 B- geometric factor P - characteristic load PB0 B- reference load Other parameters are flow properties defined by Ramberg-Osgood fit:

n

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

000 σσα

σσ

εε

wBwaPfK )/(

1 =

⎟⎠⎞

⎜⎝⎛ +−++⎟

⎠⎞

⎜⎝⎛= 12242 2

ba

ba

baη

22

Experimental Results: The main objective of this FEM simulation was to verify the previously conducted experiments for calculation of J-Integral using the unloading compliance technique, as described in “ASTM Standard E1820-01: Standard Test Method for Measurement of Fracture Toughness, ASTM 2001”

3D J-integral

020

4060

80100

120140

160180

0 20 40 60 80

Load (kN)

J-in

tegr

al (k

N/m

)

Experimental Data

The experiments were conducted at 100P

0 PC. And hence a tensile test using a video

extensometer was conducted to obtain the stress-strain curve at this temperature.

Stress - Strain Curve

0

100

200

300

400

500

600

700

800

0 0.5 1 1.5 2 2.5 3 3.5 4

total strain [ - ]

stre

ss [M

Pa]

It should be noted that, there exist small difference in the specimen and FEM model structure. The test specimen is side grooved to avoid tunneling and maintain a straight crack front. However in the FEM model, the model width was reduced to account for the side groove. ( )

BBBBB N

eff

2−−=

Fig.10.Experimental Results

Fig.11.True stress strain curve Fig.12.Side grooved Specimen

FEM Simulation Results The simulation results are presented below.

1. J-Integral: The following (Load – J Integral) plot was obtained for the CT specimen, with width=24.44mm.The experimental values are also shown for comparison.

3D J-integral

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80

Load (kN)

J-in

tegr

al (k

N/m

)

Experimental Data

Ansys Results

CT Specimenw=50.81 mmB=24.44 mma=25.5 mm

2. Path independent J-Integral: This plot shows contours at various sections along the width of the specimen. (Width of the specimen, w=B/2).These results are in line with the expected path independent nature of J-Integral.

3D J-integral

020406080

100120140160180200

0 20 40 60 80Load (kN)

J-in

tegr

al (k

N/m

)

Section ,0< Z<B/10Section ,(2B/10)< Z<(3B/10)Section ,(4B/10)< Z<(5B/10)

23

3. Variation of local J-integral: The advantage of contour integral method is that it allows the evaluation of J-integral locally at any location within the specimen width. The following plot shows the comparison of the local J-integral values and experimental values for different contours along specimen width.

Variation of local J-integral along specimen thickness

0

20

40

60

80

100

120

140

0 5 10 15

Contours along Specimen Thickness

J-In

tegr

al (k

N/m

) Load=58 kN

Load= 58kN (ExpValue)Load=34 kN

Load=34 kN (ExpValue)Load=42 kN

Load=42 kN (ExpValue)

24

25

Results Comparison Now the simulation results obtained are compared with the experimental results and theoretical solutions described earlier.

Results Comparison

0

50

100

150

200

250

0 20 40 60 80

Load (kN)

J-in

tegr

al (k

N/m

) Experimental Results

Ansys Results

Theoritical (pl. stress)

Theoritical (pl. strain)

We can infer the following from the above plot: 1.The experimental and Ansys results are almost identical. 2.The results lie in between the theoretical solutions for plain stress and plain strain

conditions. 3.Within the elastic range of deformation (load less than 40 kN), the results and

theoretical solutions are identical. The slight deviation in experimental and simulation results may be due to the following differences:

• The experimental specimen was side grooved to maintain a straight crack front, But there is some deformation at the crack front boundaries in the FEM model.

• The material properties of the specimen material and the FEM model vary slightly. The values inputted to the FEM model where through the Ramberg- Osgood relation.

0

100

200

300

400

500

600

700

800

0 0.05 0.1 0.15 0.2 0.25 0.3

Stre

ss [M

Pa]

Strain [-]

experimental dataramberg-osgood fit

Fig 13.Plot of experimentally determined stress strain curve and Ramberg-Osgood fit

26

Conclusions

Experimental and FEM simulation results identical, hence this implementation in ANSYS very effective in calculation of 3D J-Integral Local J-Integral evaluation by domain integral method leads to a better

understanding of the crack front behavior.

The specimen geometry independent nature of the macros allows their usage for J-Integral evaluation of other standard fracture specimens.

Suggestions During the course of this work, many observations have encouraged in thinking beyond the domains of this project and resulted in ideas for further extension of the present work. So the following suggestions regarding possible future work in this field are summarized below.

Finite Element calculation of K, G, J and CTOD and their mutual comparison through available theoretical relations. Comparison of different methods for calculation of J-Integral.

• Element Crack Advance Method

• Domain Integral Method

• 3D Line Integral Extending J-integral calculations to dynamic crack length models.

References 1. T.L. Anderson., Fracture Mechanics: Fundamentals and Applications, CRC Press, Boca Raton, FL, 1991. 2. Hertzberg, Richard W., Deformation and Fracture Mechanics of Engineering Materials, John Wiley & Sons, 1996. 3. Prashant Kumar., Elements of Fracture Mechanics, Wheeler Publication, New Delhi. 3. Cherepanov,C.P., Crack propagation in continuous media, Appl.Math.Mech.31 (1967),476-488 4. Rice,J.R., A path independent integral and the approximate analysis of strain

concentrations by notches and cracks,J.Appl.Mech.35(1968),379-386. 5. Hutchinson,J.W., ”Singular Behavior at the End of a Tensile Crack tip in a

hardening material.” Journal of the Mechanics and Physics of Solids, Vol. 16, 1968, pp.13-31 6. Atluri, S.N., Energetic Approaches and Path-Independent Integrals in Fracture Mechanics., Computational Methods in the Mechanics of Fracture, Chapter 5, S.N. Atluri, Ed., pps. 121-165, 1986 7. Brocks.W, Scheider.I, Numerical Aspects of the Path-Independence of the J -Integral in Incremental Plasticity.,GKSS-Forschungszentrum Geesthacht, October 2001. 8. Shih, C.F., Moran. B, Nakamura. T., Energy Release Rate along three dimensional

crack front in a thermally stressed body, International Journal of Fracture, Vol.30, 1986, pp.79-102.

9. Chandrupatla, T. R. and Belegundu, A. D., Introduction to Finite Elements in Engineering, Prentice-Hall India, New Delhi, 2003.

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