ME - 414 : Fracture Mechanics

Innovative Project Report

on
Case study of stress intensity factor solutions

By

Ayush Kumar 2K17/ME/70

Ayush Raj 2K17/ME/71

For

Bachelor of Technology
in
Mechanical Engineering
Under the Guidance
of
Prof. A.K. Agarwal
Submitted to
Department of Mechanical Engineering

Delhi Technological University

(Formerly Delhi College of Engineering)
Government of NCT of Delhi
Shahbad Daulatpur, Bawana Road
Delhi-110042

 DECLARATION
This is to declare that the innovative project report entitled " Case study of
stress intensity factor solutions " is carried by me at Delhi Technological
University, under the supervision of Prof. A.K. Agarwal. The matter
embodied in this project has been copied as well as submitted earlier for the
award of a degree or diploma to the best of our knowledge and belief.

Ayush Kumar (2K17/ME/70)

Ayush Raj (2K17/ME/71)

 CERTIFICATE
This is to certify that the above declaration made by Ayush Kumar (2K17/
ME/70) and Ayush Raj (2K17/ME/71) is correct to the best of my
knowledge and belief.

Prof. A.K. Agarwal

Department of Mechanical Engineering
Delhi Technological University
Delhi-110042

 ACKNOWLEDGEMENTS
It gives us immense pleasure to take this opportunity to acknowledge our
Project mentor Prof. A.K. Agarwal, (Mechanical Engineering), DTU, who
is providing us with his guidance.

We thank each other for our keen interest, moral support, invaluable
suggestions and guidance.

We also appreciate the help of Mechanical Department and all the faculty
members along with the officials of Delhi Technological University for
assisting us in the realization of this project.

We are thankful to my colleagues and other staff members for their suport

We take the opportunity to thank my parents and relatives for their moral
support to complete project work

At last not the least we are thankful to Almighty for their blessings.

Ayush Kumar (2K17/ME/70)

Ayush Raj (2K17/ME/71)

 ABSTRACT

This paper deals with the problem of calculating stress intensity factors in two ways.
Theoretical and Numerically in two ways: virtual crack closure technology and J-integral
method. Three types of shapes Consideration: Plates with central / boundary cracks and
notch bending specimens. Numerical simulation covered mesh Sensitivity studies of three
mesh sizes and linear / nonlinear material models. With the exact solution. The
numerical test has been completed.



 INDEX

S. No. Title Page

No

1 Introduc on 7

2 Methodology 8

3 Theore cal Solu ons 9

4 Numerical Model 10

5 Results and Discussion 11

6 Conclusions 13

7 References 14
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 Introduction
Fracture mechanics is a field of science, focusing on the formation and propagation of
cracks in structures. From civil engineering (concrete) to the aircraft industry (airplane).
It is based on the rules of elastic waves. It is used to determine the stress distribution
near the crack tip. Fracture mechanics is linear and Behavior of non-linear materials.
The beginning of linear fracture mechanics dates back to 1920, when Griffith. We have
begun studying the damage and crack distribution of brittle materials and have
established criteria based on: Energy release rate. This is based on Inglis's study, which
states that stress levels approach infinity. The ellipse flattened and cracks formed. This
assumption does not meet other mechanical laws that assume a yield zone. Emphasized
material.

The most important parameters of fracture mechanics used to determine the state of stress
caused by The external force applied to the area near the crack tip is the stress intensity
factor (SIF). It usually refers to elastic materials Used to predict breakage criteria for
brittle materials. This parameter is highly geometry dependent (Sample or structure
shape and dimensions) and load conditions (load values and methods). The actual
application of SIF was limited to cases where the analytical solution could be calculated.
Numerical value Calculation of stress intensity factors opens up new opportunities for
scientific and engineering applications. Fundamental The work provided by Parks and
Hellen , Rice , Rybicki, Kanninen laid the foundation for SIF's new technology.
Calculation. However, the numerical algorithm for SIF calculation Commercial FE
software. Until then, some researchers have made efforts to implement SIF calculation
algorithms. Commercial software by adding user subroutines. Currently, the SIF
calculation function is standard. Commercial software options. In addition, the user can
choose the implemented method. Description application. The method is mainly related
to the problem of delamination of agglomerates and composites. The purpose of this
white paper is to compare the effectiveness of the virtual rhagades closure method and
the J-integral. A method of SIF calculation in linear and nonlinear fracture mechanics.
The main purpose is to estimate the error Obtained from simplification of analysis to
linear problems and verification of usage of linear VCCT method Non-linear problem
with different mesh sizes. This study was conducted with examples using known
theoretical solutions. We compared two different cases of crack shape. Mesh sensitivity
studies have been conducted and stress distribution. The cross section (crack surface)
was compared. The MSC.Marc commercial code was used as the numeric solver.

​​

 Methodology
As mentioned earlier, the stress intensity factor has decisive implications in fracture
mechanics. Accurate solution calculations are possible only with simple shapes and
general load conditions. For more complex shapes with many irregular cracks, the
numerical method should be used. These techniques are based on the Finite Element
Method (FEM) or Ore Boundary Element Method (BEM) solutions. Among the FEM
methods, the most commonly used computer algorithms for SIF calculations are:

• Virtual Crack Closure Technology (VCCT) – Based on energy release rate. The main
assumption is that the energy required to separate the surface of the crack is the same as
the energy required to close this surface. Consider three modes of cracking.
• J-Integral is A method of calculating the strain energy release rate (work) per unit
fracture surface area around the crack tip. This method is very important in elasto-
plastic fracture mechanics. The parameters calculated using this method for materials
that are linearly elastic and completely brittle are directly related to fracture toughness.
The accuracy study of the computational method chosen for crack propagation
modeling was based on two different shape cases. The criterion for selection was the
method of calculating the stress intensity factor described below.
• Uniaxial Tensile Test (A) – Stretching of a plate with edge cracks is achieved by
applying a uniformly distributed tensile stress σ to one of the free surfaces of the
sample. The KI calculation does not take into account sample thickness and length. The
load conditions described allow simulation of plane strain model problems, but in this
paper we use a 3D model to reduce the effects of calculated stresses and the 3D (depth)
of KI. I decided to set the thickness to 1.
• 3-Point Bending Notch Test (B) – The loading conditions for this sample are slightly
different from the previous case – the bending is applied to the top surface of the
sample at half the distance between the supports and is evenly distributed acting
throughout Sample depth achieved using force. This t-problem is treated as three-
dimensional and samples of all lengths are used to calculate the stress intensity factor
KI. This means that there must be multiple individual elements throughout the depth of
the model.

Figure 1 shows a drawing of the scheme using the sample dimensions of both cases
described, with boundary conditions (constraint method and load application) applied.
SIF value, gradual increase of load has been implemented. Due to the linear behavior of
the material under small, linear and completely reversible forces, only two loads in this
range were applied: 1N and 2448N (A) / 5184 (B) N. The initial load determines the force
at which yield begins (the stress gained near the crack tip has been scaled to reach a yield
point of 235 MPa in this region). In the plastic range, four different monotonically
increased loads were used. A summary of the applied forces is shown in Table 1.

THEORETICAL SOLUTION

On the introduction section it was mentioned, that beginning of fracture mechanics is

related with research of Griffith and Inglis, which are based on energy balance near the
crack tip. After determination of work related to the opening of crack and the balancing
work necessary to create a free surface inside the body, after several mathematical
operations and substitutions it can be concluded, that the crack propagation occurs when
the following condition is met:

 NUMERICAL MODEL
Before beginning the numerical simulations, a discrete models have to be prepared. As
mentioned above, two different cases were investigated: plate with edge crack and three
point bending notch test. In both cases a geometry were modeled using 3D eight – nodes
brick elements. Due to the difference in an anticipated stress elds two way of meshing
were applied. First case – model consisted only with one element layer (one element
through the thickness) in order to simulate a plane strain behavior (in which a thickness
does not in uence for the results). The second case – model contained 10 layers of
elements representing the third dimension. As the basic size of elements a 1 mm edge size
was adopted. In order to check a mesh sensitivity for computed SIF, around a crack tip a
10 mm and 20 mm offsets were circled. First zone was fully lled with elements
characterized by uniform edge length according to Table 3, while the second zone was
used to model a transition region, which was used to connect a 1 mm grid with a reduced
mesh without losing the quality of elements (surface deviation option with growth rate
equal to 1.2 was adopted). According to theory, a densi cation of elements should
improve the results. Only a small region was remeshed in order to reduce the number of
degrees of freedom and decrease the simulations times. Mesh sensitivity study were
performed only for plate sample, and the elements with best accuracy to simulation time
ratio was adopted for the second geometry – the size was 0.25 mm.
Modelling of cracks in MSC.Marc software uses a function Toolbox, in which two
described methods were implemented. Both: VCCT and J-Integral requires a de nition of
crack tip as a set of nodes (the last path of nodes which are merged through the thickness
of sample). Because the propagation of cracks (losing a continuity) was not investigated
in this article, the rest of input parameters does not in uenced results. On the basis of
numerical simulations authors proven, that both used methods can be used together in the
same simulation without in uencing on results. The loading was realized linearly in 5
increments using Point Load function and a force was equally distributed for each node.
The degrees of freedom of the xed ends of the sample applied according to Fig. 1.
Because three dimensional elements were used, a rotational degrees of freedom doesn’t
play any role.
For modeling an elastic – plastic constitutive model with linear hardening rule was
adopted. Steel material with Yield Strength equal to 235 MPa was used. Hardening
tangent modulus was determined as T = 280 MPa and Poisson ratio v = 0.3.
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 RESULTS AND DISCUSSION
The Stress Intensity Factor (SIF) obtained from the numerical simulations using two
methods: J-Integral and VCCT were compared with theoretical results for each
geometry / mesh size / force. In Table 4. values of SIF for plate sample with edge crack
were listed. Results for 1 N are in very good agreement with theory – the stress and strain
caused by this load were linear due to the lack of plastic region in the sample.
Densi cation of mesh grid enclosed the results to theoretical value up to 1.16 % error for
VCCT method. For small forces when the deformation is only elastic, decreasing a mesh
size causes an approach to exact solution. It should be noticed that convergence from
below was observed – it means that numerical results were generally lower than
theoretical. The next magnitude of load was 2448 N, when the plasticity begins. Due to
determination of this value based on 1 mm model, plastic strain occurred for models with
element size < 0.5 mm. This phenomena is related with the displacement of nodes – for
smaller elements a global displacement of sample remains similar, while computed
strains are larger casing a higher stresses and initiating plasticity a bit earlier for densi ed
meshes. As a result the calculated values of SIF are still generally smaller, but error
increases as the element dimension decreases up to 11.31% for VCCT method and 4.22%
for J-Integral for 0.1 mm model. For loads: 3000, 4000, 6000 and 8000 N Stress Intensity
Factor obtained from VCCT method increases when the mesh decrease, while for J-
Integral it remains on relatively constant level. Values for large element size are
underestimated, therefore densi cation of elements near the crack tip increase a plastic
zone and value of SIF thus reducing the error. This situation takes a place for loads up to
6000 N for which a maximum error was 18.67%, while for 8000 N a SIF still increase,
and value for 0.1 mm elements are very overestimated (82.22% error). For J – Integral
method the results for plastic range are slightly better – the difference between successive
results and theoretical solution are smaller but for 8000 N the difference is still large and
amounts to 36.72%.

In Fig. 3 a stress distribution over a cross section position of plate sample were presented
on graph. A at parts of stress curves over 235 MPa called “plateau” marks a region,
where plastic deformation appeared. The length of it depends of loads and increase
nonlinearly as a tensile stress increase.
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CONCLUSIONS
This paper deals with a problem of calculation a Stress Intensity Factor using numerical
methods for two different cases of geometries with an initial edge crack – treated as 3D
models. Two material models were adopted – fully elastic and elasto-plastic. Both
samples were loaded above the elastic region to begin a plastic deformation. For
computation a SIF two numerical method were used: VCCT and J – Integral. Results
were compared with known values. For loads in which the deformation was elastic or the
plastic region was insigni cant (1 N and 2448 N) results are in good correlation.
Decreasing of mesh size increased a computed SIF and depend on applied load it
approached or got away from the exact solution. Finally it can be deduced, that J-Integral
method handle with a plasticity better than VCCT, but still the results for large forces
(wide plastic zone) are far from theoretical solution.

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 REFERENCES
1. A. A. Grif th, The Phenomena of Rupture and Flow in Solids" in Philosophical
Transactions, Series A, 221, pp. 163-198 (1920).

2. C. E. Inglis, Stresses in Plates Due to the Presence of Cracks and Sharp Corners.
Transactions of the Institute of Naval Architects, 55, pp. 219-241 (1913).

3. D. M. Parks, A stiffness derivative nite element technique for determination of

crack tip stress intensity factors, International Journal of Fracture, 10, pp. 487–
502 (1974).

4. T. K. Hellen, On the method of virtual crack extension, International Journal for

Numerical Methods in Engineering 9, pp. 187–207 (1975).

5. J. R. Rice, A Path Independent Integral and the Approximate Analysis of Strain

Concentration by Notches and Cracks, ASME. J. Appl. Mech. 35 (2), pp. 379-386
(1968).

6. E. F. Rybicki and M. F. Kanninen, A nite element calculation of stress intensity

factors by a modi ed crack closure integral, Eng. Fract. Mech. 9, pp. 931–938
(1977).

7. A. Leski, Implementation of the virtual crack closure technique in engineering FE

calculations, Journal of Finite Elements in Analysis and Design 43 (3), pp.
261-268 (2007).

8. M. M. Shokrieh, H. Rajabpour, M. Heidari and M. Haghpanahi, Simulation of

mode I delamination propagation in multidirectional composites with R-curve
effects using VCCT method, Comput. Mater. Sci., 65, pp. 66–73 (2012).

9. V. N. Burlayenko and T. Sadowski, FE modeling of delamination growth in

interlaminar fracture specimens, 2, pp. 95–109 (2008).

10. P. S. Valvo, A revised virtual crack closure technique for physically consistent
fracture mode partitioning A revised virtual crack closure technique for physically
consistent fracture mode partitioning, No. February, 2016.

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