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Yu. G. Matvienko, Frattura ed Integrità Strutturale, 49 (2019) 36-43; DOI: 10.3221/IGF-ESIS.49.04                                                                   
 

36 
 

Focused on Crack Tip Fields 

  
 
 
  
Comparison of the constraint parameters in  
elastic-plastic fracture mechanics  
 
 
Yu. G. Matvienko  
Mechanical Engineering Research Institute of the Russian Academy of Science. 4 M. Kharitonievsky Per.,101990 Moscow, 
Russia.  
matvienko7@yahoo.com, http://orcid.org/0000-0003-2367-0966  
 
 
 
ABSTRACT. To describe the effect of crack-tip constraint on the stress field 
for finite cracked bodies, the following most widely used constraint 
parameters can be employed in elastic-plastic fracture mechanics, namely, 
local triaxiality parameter, constraint parameter Q, the second fracture 
mechanics parameter A using a three-term elastic-plastic asymptotic 
expansion. To establish the relationship between different constraint 
parameters, the crack-tip stress fields are employed. Relationship between the 
crack-tip constraint parameters A, A2, Q and pS of elastic-plastic fracture 

mechanics is investigated. 
  
KEYWORDS. Constraint parameters; Elastic-plastic fracture mechanics. 
 

 

 
 

Citation: Matvienko, Yu. G., Comparison of 
the constraint parameters in elastic-plastic 
fracture mechanics, Frattura ed Integrità 
Strutturale, 49 (2019) 36-43. 
 
Received: 01.03.2019  
Accepted: 23.04.2019  
Published: 01.07.2019  
 
Copyright: © 2019 This is an open access 
article under the terms of the CC-BY 4.0, 
which permits unrestricted use, distribution, 
and reproduction in any medium, provided 
the original author and source are credited. 

 

 
 
INTRODUCTION 
 

t present time, it becomes obvious that classical one-parameter fracture mechanics based on traditional 
parameters, such as the stress intensity factor, the J-integral, etc., is not sufficient for describing the stress field in 
the vicinity of the crack tip for finite cracked bodies. It is necessary to take into account the crack-tip constraint 

effect, i.e. to employ models and criteria of two-parameter fracture mechanics. To describe the effect of crack-tip 
constraint in elastic-plastic two-parameter fracture mechanics, the following possible constraint parameters should be 
introduced into basic equations and criteria of fracture mechanics in the case of small scale yielding and large scale yielding 
conditions, namely, local triaxiality parameter [1], constraint parameter Q [2, 3], the second fracture mechanics parameter 
A of a three-term elastic-plastic asymptotic expansion [4-6]. Some aspects of the relationship between most widely used 
crack-tip constraint parameters have been discussed [7, 8]. 
Solutions of the parameter A for 2D test specimens are obtained by extensive finite element analysis [7]. Moreover, an 
engineering method of estimating the second parameter A in three-term elastic–plastic asymptotic expansion is developed 
to cover small-scale yielding to large-scale yielding loading conditions [9]. In particular, it was shown that the parameter A 
can be calculated through the elastic non-singular T-stress in the case of small-scale yielding. Some aspects of application 
of the J-A approach in elastic-plastic fracture mechanics as well as solutions of the parameter A for 3D test specimens are 
discussed in Refs. [8, 10, 11].  

A 

http://www.gruppofrattura.it/VA/49/2416.mp4


 

                                                               Yu. G. Matvienko, Frattura ed Integrità Strutturale, 49 (2019) 36-43; DOI: 10.3221/IGF-ESIS.49.04 
 

37 
 

The constraint parameter Q in the J-Q approach proposed by O’Dowd and Shih [2, 3] has been widely used in engineering 
applications. This parameter can be estimated as the deviation of the crack-tip stress field from that based upon possible 
reference fields. Relationship between the parameter Q and the non-singular T-stress was also established. 
To quantify in-plane and out-of-plane constraints and their interaction, a unified constraint characterization parameter Ap 
was proposed [12-14]. This parameter is based on the plastic area surrounded by the equivalent plastic strain isoline in the 
vicinity of the crack tip. A sole linear relation between the normalized fracture toughness in terms of the J-integral and 
√𝐴 was observed.  
The present paper concentrates on relationship between some crack-tip constraint parameters of elastic-plastic fracture 
mechanics.  

 
 

TWO-PARAMETER J-A CONCEPT 
 

The J-A asymptotic elastic-plastic crack-tip stress field and fracture criterion 
n the case of the elastic-plastic material deformed according to the Ramberg–Osgood power-law strain hardening 
curve 
 

0 0 0

n
  


  

 
   

 
              (1) 

 
the stress field in the vicinity of  the mode I plane strain crack is described by the three-term asymptotic expansion [5, 6]  
(Fig. 1) 
 

2
( 0 ) (1) 2 ( 2 )

0
0 0

( ) ( ) ( )
ij s t t s

ij ij ij
A

A A
A


        


               (2) 

 
Here, 0  σ0 is the yield stress,   is the hardening coefficient, n is the hardening exponent (n > 1), 0 0 / E  is the yield 
strain and E is Young's modulus, J is the energetic integral proposed by Cherepanov [15] and Rice [16], A is the second 
fracture parameter, ij are stress components r ,   and r  in the polar coordinate system r  with origin at the crack 

tip, ( )kij  are dimensionless angular stress functions obtained from the solution of asymptotic problems of order (0), (1) 

and (2). Angular stress functions ( 0 )ij  and 
(1)
ij  are scaled so as maximal equivalent Mises stress is equal to unity, i.e.  

( 0 ) (1)max max 1e e     . Exponent s has closed form expression 1/ ( 1)s n   . Exponent t is a numerically computed 
eigenvalue that depends on strain hardening exponent n. For materials with n=5 and 10 that are used in this paper values 

of exponent t are: t(5) = 0.05456, t(10) = 0.06977. Coefficient A0 is specified as 0 0( )
s

nA I . Dimensionless radius   is 
defined by the following formula 
 

0/

r

J



                (3) 

 
Comparing Eqn. (2) and the crack-tip stress field proposed by Hutchinson [17] and Rice and Rosengren [18] ((so-called 
the HRR field), it is easy to see that the first term of the asymptotic expansion (2) is exactly the HRR field. It can be seen 
that the HRR field does not describe correctly stresses in the region 01 / 5r J   (Fig. 1) that is significant for fracture 
process. Finite element solutions of elastic-plastic crack problems show that the J-A stress field is much closer to finite 
element results than the HRR field [8]. 
The three terms of expansion (2) are controlled by two parameters, namely, the J-integral and the parameter A. The 
parameter A is a measure of stress field deviation from the HRR field. Assuming that cleavage fracture of a specimen and 
a cracked structure occurs when stress fields near the crack tip are same it is possible to formulate two-parameter J-A 
fracture criterion. This assumption also justifies the use of A as a constraint parameter [8]. The two-parameter fracture 

I 



 

Yu. G. Matvienko, Frattura ed Integrità Strutturale, 49 (2019) 36-43; DOI: 10.3221/IGF-ESIS.49.04                                                                   
 

38 
 

criterion implies comparison of computed J-integral for a cracked structure and the experimental fracture toughness JC 
corresponding to computed value of the constraint parameter A 
 

 
 

Figure 1: Comparison of the J-A stress field and the HRR field for edge cracked plate. 
 

( )| ( )A CJ P J A               (4) 
 
Determination of the dependency JC(A) can be accomplished by testing cracked specimens with different constraint at 
fracture load. Change of the constraint conditions is achieved by varying crack length in the specimen. To avoid numerous 
experiments it is desirable to develop computational approaches for predicting the fracture toughness as a function of the 
constraint parameter A. Estimating change of the fracture toughness JC with variation of the constraint parameter A can 
be based on an assumption that fracture under different constraint conditions corresponds to constant fracture probability 
Pf [19]. 
 
Numerical estimation of the J-integral and parameter A 
The equivalent domain integral method (EDI) [20, 21] is employed for computing the J-integral. Determination of the 
constraint parameter A is based on stress calculation in the vicinity of the crack tip by means of the finite element method. 
If stresses are known at points ( ),i i   then the value of the parameter A at ith point is found from the following 
quadratic equation [6] 
 

2

2
( 2 ) (1) ( 0 )

0
0 0

0

( , )
( ), ( ), ( )

i i i

t s FEM
t si i i

i i i i i i i i

a A b A c

a b c A
A

   
       





  

      
    (5) 

 

where FEM  is stress calculated by the finite element method. Solution of Eqn. (5) produces different A values at 
different points due to deviation of actual stress field from the three term asymptotic expansion. 
Better estimate of A for the set of points is obtained by minimizing sum of squares of deviations of the J-A asymptotic 
field from the finite element results. Application of the least squares method leads to a cubic equation for the parameter A 
[6] 
 

 
3 2

3 2 1 0

2 2
3 2 1 0

0

2 3 2, , ,i i i i i i i i

d A d A d A d

d a d a b d a c b d b c

   

       
     (6) 

 

Usually values of the stress   at finite element integration points inside region 1 4  , 0 45
   are used for 

estimation of the parameter A. 
 
 



 

                                                               Yu. G. Matvienko, Frattura ed Integrità Strutturale, 49 (2019) 36-43; DOI: 10.3221/IGF-ESIS.49.04 
 

39 
 

 
RELATIONSHIP BETWEEN THE PARAMETER A AND OTHER ELASTIC-PLASTIC CONSTRAINT PARAMETERS  

 
he most widely used constraint parameters mentioned in the introduction of this paper can be employed in 
engineering applications. To establish the relationship between these parameters, the crack-tip stress fields can be 
very useful. Relationship between the constraint parameters A, A2 and Q was discussed in Refs. [7, 8]. A constraint 

characterization parameter pA [12-14] is also analyzed in connection with the constraint parameter A.  

 
Relationship between the parameters A and A2 
The well-known J-A2 approach for two-dimensional body with a mode I crack under plane strain conditions is based on 
the following expression of the three-term asymptotic expansion for stress in the vicinity of the crack tip in elastic-plastic 
material [22] 
 

           
1 2 3

1 2 32
1 2 2

0

s s s
ij

ij ij ij
r r r

A A A
L L L


     


              
       

        (7) 

 

Here, A1=    10 0 1/ ,  1/ 1
s

nI L J s n     and 3 2 12s s s  . Comparing Eq. (2) and Eq. (7), it should be noted that 

three corresponding dimensionless angular functions  kij are the same in Eq. (2) and Eq. (7). Moreover, there is the 

connection between exponents ks  and t in Eq. (2), namely, 1 2 3,  s ,  2s s t s t s    . Therefore, the following relationship 
between the constraint parameters A and A2 in the three-term asymptotic expansions of elastic-plastic crack-tip stress field 
can be written  
 

 2 0
0

s t
s

n
J

A I A
L





  

   
 

             (8) 

 
Thus, J-A and J-A2 approaches are mathematically equivalent. The difference of these approaches is in distance scaling of 
r according to Eqs. (2) and (7). Comparison of parameters A and A2 is shown in Fig. 2. Dependencies of both parameters 
are plotted for 3PB specimen with cracks of same aspect ratios a/W. In contrast to the parameter A, the parameter A2 
does not have its small-scale yielding value. It can be seen that the parameter A2 tends to infinite value with load decrease. 
It is less convenient for constraint interpretation and engineering application. 
 
Relationship between the parameters A and Q 
The constraint parameter Q in the J-Q approach [2, 3] is defined by the deviation of the crack-tip stress field under 
consideration from a reference stress field. Possible reference fields can be the HRR field or field corresponding to small-
scale yielding conditions 
 

 
0 0

ijij SSYQ


 
  or 

 
0 0

ijij HRRQ


 
                                                       (9) 

 
The following relationship between the parameters A and Q can be obtained from Eq. (2) and Eq. (9)  
 

     
2

1 2 2

0

t t sAQ A
A

      
     at 0,  2.                                                      (10) 

 
Fig. 3 presents dependencies of A and Q as functions of the applied load for cracks of different depth in 3PB specimen. 
Constraint parameters A and Q demonstrate similar behavior and can be used in engineering applications. But, the Q 
parameter can be considered as a qualitative measure of crack-tip constraint, because small errors in the crack-tip stress 

T 



 

Yu. G. Matvienko, Frattura ed Integrità Strutturale, 49 (2019) 36-43; DOI: 10.3221/IGF-ESIS.49.04                                                                   
 

40 
 

obtained by finite element method lead to the significant error in the value of Q [23]. The J and A parameters allow 
describing the elastic-plastic stress field near the crack tip with sufficient accuracy.  
 

 
 

Figure 2: Comparison of the parameters A and A2 as functions of applied load P. 
 

 
 

Figure 3: Comparison of the parameters A and Q as functions of applied load P. 
 

Relationship between the parameters A and p pS A  

A parameter /p PEEQ refA A A is introduced into consideration as the unified constraint parameter to characterize in-

plane and out-of-plane crack-tip constraint [12-14]. The parameter PEEQA is the area surrounded by the equivalent plastic 

strain contour with specified value of strain p , the parameter refA is PEEQA with same strain p  measured in a standard 
specimen test. Finite element method based on Gurson-Tvergaard-Needleman damage model was used to calculate the 
equivalent plastic strain p distribution ahead of the crack tip for specimens with different in-plane and out-of-plane 

constraints. The following plastic strains p =0.1, 0.2 and 0.3 were employed for determination of plastic contours. It was 

found that p =0.2 and 0.3 provides approximately linear dependencies /C refJ J versus pA [13]. This is not a surprising 

result since the area inside specified strain p is proportional to
2J . So, the J-integral is compared to the quantity that is 



 

                                                               Yu. G. Matvienko, Frattura ed Integrità Strutturale, 49 (2019) 36-43; DOI: 10.3221/IGF-ESIS.49.04 
 

41 
 

proportional to the J-integral. In our opinion the parameter p pS A  is not suitable for using as a constraint parameter 

because plastic zone areas and J values are proportional. To avoid direct dependence of these parameters on each other, it 
is proposed to replace coordinate r (as well as x and y) with dimensionless coordinate  0/ /r J  [8]. In this case, the 
area inside equivalent plastic strain becomes dimensionless 
 

 20/ /PEEQ PEEQA A J               (11) 
 

Therefore, a normalized constraint parameter pA  can be introduced into consideration by means of the following 

expression 
 

p pS A , where /p PEEQ refA A A .            (12) 

 

The parameter pA  is denoted as pS . The modified constraint parameter pS should be less dependent on the J-integral. 

In this case, the separation of the constraint parameter and the J-integral improves the quality of a fracture criterion that is 
based on these parameters. 
Behavior of the parameter pS as a function of A/ASSY is investigated using the J-A elastic-plastic asymptotic field. The 

area surrounded by the equivalent plastic strain contour with specified value of strain p is estimated with the following 

algorithm [8]. Stress p corresponding to specified plastic strain p can be represented as  
 

1/ 1/

0 0 0

n n
p p pE   

  
   

    
   

             (13) 

 
The rectangular area in the vicinity of the crack tip is divided into square cells. Equivalent stress is computed at each cell 
by means of the J-A elastic-plastic asymptotic field. If this stress is larger than p , the cell area is added to PEEQA . 

Computation of the plastic area is performed in coordinates 0 /x J and 0 /y J , so the resulting area is PEEQA . 
Fig. 4 summarizes the computed results for plastic zones inside plastic strain contours corresponding to p =0.1, 0.2 and 

0.3 and relationships between parameters pS and A for a material with 1   and 5n  . Smallest plastic zone corresponds 
to small scale yielding value of the constraint parameter A (ASSY (n=5) = 0.3803). Intermediate plastic zone is computed 
for the value of  max / 2SSYA A . Largest plastic zone is related to Amax (shown in Fig. 4). It can be seen that there is a 
unique relationship between the constraint parameters pS  and A in a wide range of specified values of strain p . 
 
 
CONCLUSIONS 
 

elationship between the crack-tip constraint parameters A, A2, Q and pS of elastic-plastic fracture mechanics is 

based on analysis of the stress fields in the vicinity of the crack tip. The following conclusions can be drawn. 
It was demonstrated that J-A and J-A2 approaches are mathematically equivalent. The difference of these 

approaches is in distance scaling r. But, in contrast to the parameter A, the parameter A2 does not have its value under 
small scale yielding conditions. Constraint parameters A and Q show similar behavior. At the same time, it should be 
noted that small errors in the crack-tip stress obtained by finite element method lead to the significant error in the value of 

Q. The constraint parameter p pS A  based on the area surrounded by the equivalent plastic strain p contour ahead of 
the crack tip is not suitable as a crack-tip constraint parameter because plastic zone areas and J values are proportional. To 

avoid direct dependence of these parameters on each other, the constraint characterization parameter pA is modified as 

R 



 

Yu. G. Matvienko, Frattura ed Integrità Strutturale, 49 (2019) 36-43; DOI: 10.3221/IGF-ESIS.49.04                                                                   
 

42 
 

follows p pA S . In this case, the separation of the constraint parameter and the J-integral improves the quality of a 

fracture criterion in two-parameter elastic-plastic fracture mechanics. It was shown that there is a unique relationship 
between the constraint parameters pS  and A in a wide range of specified values of strain p . 
 

 
a) 

 
b) 

 
c) 

 

Figure 4: Equivalent plastic strain isolines for A/ASSY with minimal, average and maximal values (left); the constraint parameter pS  as 

a function of the parameter A (right): n=5, p =0.1 (a), p =0.2 (b) and p =0.3 (c). 



 

                                                               Yu. G. Matvienko, Frattura ed Integrità Strutturale, 49 (2019) 36-43; DOI: 10.3221/IGF-ESIS.49.04 
 

43 
 

ACKNOWLEDGEMENTS 
 

he authors acknowledge the support of the Russian Science Foundation (project N 18-19-00351). 
 
 

 
 
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    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice