PLASTIC ZONE SIZE AT CRACK TIP MODE I BY USING THE
TRESCA YIELD CRITERIA
Ton That Hoang Lan
Department of Civil Engineering, Ho Chi Minh City University of Architecture, Vietnam
Email: lanksxd78@yahoo.com.vn
ABSTRACT
Research at the plastic zone crack tip problem has been made in further expertise. Scope of
application is very large throughout the engineering. Statistics shows that there have been many
studies focusing on the relationship between shapes, plastic zone size as well as the formation
and development of fatigue cracks. On this basis, the crack tip plastic zone shape have been
derived for a semi-infinite crack in an isotropic elastic-perfectly plastic solid under both plane
stress and plane strain state. A yield criteria have been applied: Tresca yield criteria in invariant
form. The solutions have been developed for crack modes I.
Keywords : Plastic zone; Tresca Yield Criteria; Crack; Stress condition; Invariant
1. Introduction
Research at the plastic zone crack tip and
shape as well as size of its has generated a
lot of interest in the mechanical field. A lot
of research focus on the relationship
between shape, size and the plastic fatigue
crack growth. The first theoretical work
mentioned on the size and shape of the
plastic zone has been provided by Irwin and
Dugdale [1], [2]. In this paper, we will
redefine the plastic zone at the crack tip
based on Tresca Yield Criteria when it is
expressed in invariant form. Calculation
process will be supported by mathematical
Maple_software.
Zone at crack tip
Figure 1. Crack tip mode I
2. Calculation
Here we present the formula in polar
coordinates
Stress field under mode I
σx 
σy 
 θ 
 θ   3θ 
cos  1  sin  sin  
2πr
 2 
 2   2 
KI
 θ 
 θ   3θ 
cos  1  sin  sin  
2πr
 2 
 2   2 
KI
1
τ xy 
KI
 θ   θ   3θ 
cos sin  cos 
2r
2 2  2 
0

z  
 x   y 
Plane stress state
Plane strain state,  z  0
τ xz  τ yz  0
Tresca yield criteria in invariant form
The expression is
4
6
4J 32  27J 32  36k 2 J 2
2  96k J 2  64k  0
in which
J2 
k=0/2


1
σ x  σ y 2  σ y  σ z 2  σ z  σ x 2  6 τ 2xy  τ 2yz  τ 2zx
6
2σ x  σ y  σ z
and
τ xy
3
J3 
τ xz
2σ y  σ x  σ z
τ yx
τ yz
3
τ zx

2σ z  σ x  σ y
τ zy
3
3. Determination for plastic zone at crack tip
Application of mathematical theory
r 3  aθ r 2  bθ r  cθ   0
 p θ    qθ  
D
 
 0
 3   2 
3
with
in which
pθ  
3bθ   aθ 
3
2
Solutions of this equation
with
2
qθ   cθ  
and
r1 θ  
 aθ 
2
3
pθ 
r2 θ  
 aθ 
2
3
pθ 
r3 θ  
 aθ 
2
3
pθ 
cosφ  
3
3
3
qθ 
2aθ  aθ bθ 

27
3
3
φ
cos 
3
φπ 
cos

 3 
φπ 
cos

 3 
 pθ  

2 

 3 
3
The results obtained with plane stress state when presented as invariant form
2
2

cos16 θ/2 23sin θ/2   9sin 3θ  K12
I 0
2
D
6
6912σ12
0 π
Results by using MAPLE
1
 θ 
*
r1 θ     cos 2    5  3 cosθ 
3
 2 
1
 662  1035 cosθ   378 cos2θ   27 cos3θ   
 
cos 2  cos 1 
3

6




4

5

3
cos
θ






θ 
 7  4 cosθ   3 cos2θ   4 cos 2    5  3 cosθ 

2






1
*
r2 θ  




24   1 








662

1035
cos
θ

378
cos
2
θ

27
cos
3
θ

1
 
   
3
 cos 3  π  cos 

4 5  3 cosθ 

   
  



θ 
 7  4 cosθ   3 cos2θ   4 cos 2    5  3 cosθ 

2






1
*
r3 θ  



24   1 








662

1035
cos
θ

378
cos
2
θ

27
cos
3
θ

1
 
   
3
 cos 3   π  cos 





4

5

3
cos
θ

   
  
Figure 2. Plastic zone with solution r*1
Figure 3. Plastic zone with solution r*2
3
The results obtained with plane strain state when presented as invariant form

θ 
2
2
 1  2ν  cos12   1  cosθ 1  8 1  ν ν  cosθ  
2


   7  8 1  ν ν  9 cosθ 2 K 12

I
 0
D
12 6
13824σ 0 π
Results by using MAPLE
r1 θ  
1 2
sin θ 
2
1
 θ 
*
r2 θ     cos 2    3  8 1  ν ν  cosθ  
4
 2 
*

 θ    θ  
 1  2ν  cos 2    sin   
 2    2  

1
 θ 
*
r3 θ     cos 2    3  8 1  ν ν  cosθ  
4
 2 

 θ    θ  
 1  2ν  cos 2    sin   
 2    2  

Figure 6. Plastic zone with solution r*1
Figure 7. Plastic zone with solution r*2
4
4. Comparison and conclusions
The results are entirely based on
mathematics and its solutions are accurate,
so that significant improvements compared
with the results given by Irwin and Dugdale.
Figure 10. Plastis zone around the crack tip
mode I with ABAQUS’s simulation
References
[1] Irwin, G. R., “Analysis of Stresses and
Strains near the End of a Crack Traversing
a Plate,” J. Appl. Mech., 24, pp. 361−364
(1957).
[2] Dugdale, D. S., “Yielding in Steel
Sheets Containing Slits,” J. Mech. Phys.
Solids, 8, pp. 100−104 (1960).
[3] Lankford, J., Davidson, D. L. and
Chan, K. S., “The Influence of Crack Tip
Determination plasticity zone under
invariant form provides a direct approach to
solve the solutions of a third order equation
and then superposition of them to find the
outer boundary of the plastic zone around
the crack tip mode I.
Figure 11. Plastis zone around the crack tip
mode I with XFEM’s simulation
Plasticity in the Growth of Small Fatigue
Cracks,” Metallurgical Transactions A, 15,
pp. 1579−1588 (1984).
[4] Sahasakmontri, K. and Horii, H., “An
Analytical Model of Fatigue Crack Growth
Based on the Crack-Tip Plasticity,”
Engineering Fracture Mechanics, 38(6), pp.
413−437 (1991).
[5] Sadananda, K. and Ramaswamy, D.N. V., “Role of Crack Tip Plasticity in
5
Fatigue Crack Growth,” Philosophical
Magazine A, 81(5), pp. 1283−1303 (2001).
[6] Kelly, P. A. and Nowell, D., “ThreeDimensional Cracks with Dugdale-Type
Plastic Zones,” International Journal of
Fracture, 106, pp. 291−309 (2000).
[7] Jing, P., Khraishi, T. and Gorbatikh,
L., “Closed-Form Solutions for the Mode II
Crack
Tip
Plastic
Zone
Shape,”
International Journal of Fracture, 122(3-4),
pp. L137−L142 (2003).
6