Recently proposed modifications to ASTM E399 would provide a new size-insensitive approach to analyzing the force-displacement test record. The proposed size-insensitive linear-elastic fracture toughness, KIsi, targets a consistent 0.5mm crack extension for all specimen sizes by using an offset secant that is a function of the specimen ligament length. The KIsi evaluation also removes the Pmax/PQ criterion and increases the allowable specimen deformation. These latter two changes allow more plasticity at the crack tip, prompting the review undertaken in this work to ensure the validity of this new interpretation of the force-displacement curve. This paper provides a brief review of the proposed KIsi methodology and summarizes a finite element study into the effects of increased crack tip plasticity on the method given the allowance for additional specimen deformation. The study has two primary points of investigation: the effect of crack tip plasticity on compliance change in the force-displacement record and the continued validity of linear-elastic fracture mechanics to describe the crack front conditions. The analytical study illustrates that linear-elastic fracture mechanics assumptions remain valid at the increased deformation limit; however, the influence of plasticity on the compliance change in the test record is problematic. A proposed revision to the validity criteria for the KIsi test method is briefly discussed

Allen, Phillip

James, Mark

Wallin, Kim

Wells, Doug

NASA Technical Reports Server

17th International ASTM/ESIS Symposium on Fatigue and Fracture Mechanics  
(41st National Symposium on Fatigue and Fracture Mechanics) 
 May 10-12, 2017, Toronto, ON Canada 
A Review of the Proposed KIsi Offset-Secant Method for Size-Insensitive Linear-
Elastic Fracture Toughness Evaluation 
Mark James, Arconic Technology Center, New Kensington, PA, USA 
Doug Wells, NASA Marshall Space Flight Center, Huntsville, AL, USA 
Phillip Allen, NASA Marshall Space Flight Center, Huntsville, AL, USA 
Kim Wallin, VTT Nuclear Safety, VTT, Finland 
 
 
 
 
 
Recently proposed modifications to ASTM E399 would provide a new size-insensitive approach 
to analyzing the force-displacement test record. The proposed size-insensitive linear-elastic 
fracture toughness, KIsi, targets a consistent 0.5mm crack extension for all specimen sizes by 
using an offset secant that is a function of the specimen ligament length. The KIsi evaluation also 
removes the Pmax/PQ criterion and increases the allowable specimen deformation. These latter 
two changes allow more plasticity at the crack tip, prompting the review undertaken in this work 
to ensure the validity of this new interpretation of the force-displacement curve. This paper 
provides a brief review of the proposed KIsi methodology and summarizes a finite element study 
into the effects of increased crack tip plasticity on the method given the allowance for additional 
specimen deformation.  The study has two primary points of investigation: the effect of crack tip 
plasticity on compliance change in the force-displacement record and the continued validity of 
linear-elastic fracture mechanics to describe the crack front conditions. The analytical study 
illustrates that linear-elastic fracture mechanics assumptions remain valid at the increased 
deformation limit; however, the influence of plasticity on the compliance change in the test 
record is problematic. A proposed revision to the validity criteria for the KIsi test method is 
briefly discussed. 
 
 
 
https://ntrs.nasa.gov/search.jsp?R=20170005328 2019-08-31T08:22:31+00:00Z
1 
 
INTRODUCTION 
To determine KIc, ASTM E399 [1] uses an offset secant to the force-displacement record to 
identify the load at which crack extension occurs during a test. This secant construction line, 
offset to 95% of the slope of the linear portion of the test record, corresponds to the compliance 
change when crack extension equals approximately 2% of the specimen’s original crack length 
under the assumption that all compliance change is due to crack extension. When the material 
exhibits stable tearing with a rising R-curve—corresponding to a Type I curve per ASTM E399-
12e3, Figure 7—this method provides a toughness result that is specimen size dependent. 
 
Wallin [2] proposed modifications to E399 to provide a new approach to analyzing an ASTM 
E399 test record utilizing a secant construction line with an offset slope that is a function of the 
specimen size based on the remaining ligament, bo, thereby minimizing the influence of 
specimen size on the test result. The proposed size-insensitive linear-elastic fracture toughness, 
recently labeled KIsi, targets a consistent 0.5mm crack extension for all specimen sizes. The KIsi 
method also increases the allowable specimen deformation, and removes the Pmax/PQ criterion. 
These latter two changes allow more plasticity at the crack tip before a test result is deemed 
invalid.  
In the current version of E399, the deformation limit for ligament plasticity (also called the 
specimen size requirement) is expressed in terms of the ligament validity criterion as bo  MK (K 
/ys)2, where K is the linear elastic stress intensity factor, ys is the 0.2% offset engineering yield 
strength, and the constant MK = 2.5. Wallin [2] proposed that the deformation limit be extended 
to allow higher deformation with MK = 1.1 based on an evaluation of a variety of data sets 
available at the time of that paper. To make these limiting measures of deformation more 
tangible, note that with MK = bo*ys2 /K2, MK is simply the ratio of ligament length to plastic zone 
size (rp) with a proportional factor, such that MK ≈ 0.15bo/rp [3].  Using this engineering estimate, 
at MK = 2.5, rp is approximately 6% of bo, and at MK = 1.1, rp extends to 14% of bo. When 
evaluating the proposed changes for standardization of KIsi, concerns arose that with MK = 1.1, 
the contribution of crack tip plasticity to compliance change in the test record may be sufficient 
to influence the test result. In the E399 test method, without unloading compliance checks during 
the test, it is not possible to distinguish compliance change due to crack extension from that due 
to plasticity.  
This paper summarizes a finite element study with the following two objectives: first, evaluate 
the continued validity of linear-elastic fracture mechanics assumptions to describe the conditions 
at the crack tip using K at MK = 1.1 and, second, quantify the effects of crack tip plasticity on the 
compliance change in the force-displacement record and evaluate its influence on interpretation 
of the test record for identifying crack extension.  If required, the analytical study should provide 
results to formulate appropriate validity criteria to avoid detrimental plasticity.  
FINITE ELEMENT ANALYSIS APPROACH  
In this study, the common compact specimen (C(T)) with W = 50.8 mm (2.0 inch), a/W = 0.5, 
W/B = 4, and no side-grooves is used as the reference geometry. The C(T) analysis model was 
developed using the FEACrack [4] finite element modeling software and solved with WARP3D 
[5] v16.2.7. The model used a 1/4 symmetric mesh with 56863 nodes and 12305, 20-noded hex, 
small-strain elements. The crack tip was modeled using collapsed elements with untied duplicate 
nodes. The J-integral was calculated over 15 domains to gauge convergence while using the 
2 
 
WARP3D Type D domain evaluation for a bulk-average J-integral. Forces were applied at the 
center of the pin mesh, with pin rotation allowed, and elastic pin material. Figure 1 is an 
illustration of the finite element mesh. 
The material constitutive model used incremental plasticity and the isotropic Mises flow rule. 
The stress-strain relationship was modeled using linear behavior up to the proportional limit 
followed by a power law relation for plastic strain. Expressed in a normalized form with the 
proportional limit, σo = 1, this allows easy evaluation of a broad material space with stiffness to 
proportional limit ratio varying 100 ≤ E / σo  ≤ 1000 and strain hardening exponent varying 3 ≤ n 
≤ 20 with Poisson’s ratio, ν = 0.3. The calculation of MK, uses the engineering yield strength 
defined at 0.2% plastic strain. This material space covers practically all engineering alloys. 
ASTM E399 allows a variety of specimens for determining linear-elastic fracture toughness. 
This study focuses on the C(T) specimen with the understanding that specimens of different 
geometry will need to be evaluated to quantify the effects of specimen geometry on plasticity-
induced compliance change. The slender ligament aspect ratio, W/B = 4, is used to represent the 
E399 lower limit on specimen thickness and is the worst-case for allowing plasticity to influence 
the non-linearity in the force-displacement record.  
The KIsi offset secant 
For this work, the change in compliance (C) is defined as a percent increase in compliance (or 
percent decrease in slope) of the force-displacement record with respect to the initial linear 
portion, as defined by the first, fully-elastic analysis load step. See Figure 2. Consistent with 
Wallin [2], the offset compliance where fracture toughness is evaluated, C, is proposed to 
follow the convention yielding C in percent:  
 C = A/(W-a), (1) 
with A = 135 for the C(T) specimen for dimensions in mm, which corresponds to crack extension 
of 0.5mm.  For common E399 C(T) specimen sizes with a/W = 0.5: 
W = 25.4 mm (1 inch) C = 10.6%  W = 50.8 mm (2 inch) C = 5.3% 
W = 63.5 mm (2.5 inch) C = 4.25%  W = 101.6 mm (4 inch) C = 2.7% 
 
For a KIc assessment per the E399 standard, the offset secant is fixed at C = 5% for all 
specimens.  Thus, to target 0.5mm crack extension, this corresponds to a C(T) with W = 54.0 
mm. At this specimen size, the KIc and KIsi methods are equivalent. Note that the coefficient A 
listed here is different from that in Wallin [2] because in that case, the coefficient was developed 
for the ASTM E1820 [6] specimen that measures displacement at the load line, whereas in the 
current paper the coefficient is developed for specimens that measure displacement at the 
specimen front face. 
In experiments for KIsi, the change in compliance throughout a test is estimated as a summation 
of crack extension, plasticity, and experimental error: 
C = Ccrack ext + Cplasticity + Cexperimental error (2) 
 
3 
 
Currently, experimental methods for KIsi assume Cplasticity and Cexperimental error are sufficiently 
small relative to Ccrack ext such that the offset secant will properly identify crack extension of 
0.5mm. The magnitude of Cplasticity is easily estimated in the finite element study where the 
crack length is fixed at a/W = 0.5, such that Ccrack ext ≡ 0. This study does not provide insight 
into the magnitude of Cexperimental error though this could be of importance as specimen size 
increases and the offset secant for KIsi gets smaller. 
Though not shown herein for brevity, the authors demonstrate that all calculations related to 
compliance change and deformation are fully scalable with geometry, thus one model provides 
results for all size specimens of the same proportion. 
RESULTS 
The main result of the analytical work is represented in a plot of change in compliance, C, 
versus specimen deformation, MK = bo*ys2 /K2 ≈ 0.15bo/rp.  Figure 3a illustrates this plot and its 
interpretation. The abscissa axis, MK, is plotted log scale and reversed, left to right—decreasing 
values of MK correspond to increasing force and deformation. The green box indicates the 
compliance and deformation requirements for KIc fixed at C = 5% and MK = 2.5, respectively.  
The solid line marked with open circles represents a typical result from the analysis, in this case 
representative of aluminum alloy 2219-T8.  Recall the analysis reflects only the C contribution 
from plasticity—compliance increases non-linearly with plastic zone size. In an experiment, the 
measured C includes all sources as considered in Eq 2, though for the current discussion, we 
will consider experimental error contributions negligible. The dashed line in Figure 3a represents 
a typical experimental result, in this case, one that is valid according to the deformation limit for 
KIc per E399, i.e., MK ≥ 2.5. Prior to the onset of crack extension, Cplasticity is the only source of 
compliance change. Once crack extension begins, Ccrack ext contributes strongly to the total C.  
In this example, the KIc test is valid for deformation because C = 5% is reached while MK ≥ 2.5.  
The pertinent detail is that, in this example, nearly 2/5ths of C is due to plasticity, not crack 
extension.  
 
Figure 3b illustrates the key findings of the analysis effort needed to answer two questions 
regarding the consequence of allowing the deformation measure to extend to MK = 1.1 as first 
proposed for KIsi: 1) Does the increased deformation invalidate LEFM assumptions for the use of 
K; and 2) Can plasticity alone create sufficient C to reach the KIsi limit before crack extension 
begins? Either case is undesirable, leading to a non-relevant underestimate of linear-elastic 
toughness.  In Figure 3b, the analysis result of Cplasticity remains shown by the solid line with 
open circles. The contribution to KJ from plasticity (KJ-plastic, given in percent) is shown by the 
curve marked with solid circles and the proposed deformation limit of MK = 1.1 is shown by the 
vertical dotted line. Recall that analysis results plotted here are examples representative of Al 
2219-T8 material; however, the findings are fully supported by results from the larger material 
space. 
 
The KJ-plastic result answers question 1.  In this example, KJ-plastic = 4.4% at MK = 1.1.  In the full 
material space of the analysis set, at MK = 1.1, KJ-plastic ranges from 3% and 6.5%.  In the LEFM, 
E399 test method, the plastic contribution to fracture energy is not captured (KI computed from 
only force, not absorbed energy in the specimen). The LEFM toughness measurements that 
ignore the small plastic contribution are low, or conservative, by this percentage.  This range of 
4 
 
conservative error would generally not be grounds to consider the result invalid for LEFM 
assumptions; therefore, the use of MK = 1.1 appropriately maintains LEFM assumptions.   
 
The Cplasticity result answers question 2.  The C limits for KIsi are shown for various size 
specimens by horizontal dashed lines in Figure 3b.  It is clear that the Cplasticity result crosses 
many of these lines before reaching MK = 1.1.  In the example given by Figure 3b, only the 
smallest specimen of W = 25 mm avoids having Cplasticity cause the test record cross the KIsi 
offset secant prior to MK = 1.1. If non-linearity in the test record due to Cplasticity causes the test 
record to cross the KIsi offset secant prior to crack extension, the result is not a measurement of 
toughness, but merely a measure of yielding in the specimen.  Additional limits on specimen 
deformation are needed to control plasticity to ensure crack extension has occurred at the secant 
crossing.  The proposed remedy is to make MK a function of specimen size, for example, MK ≥ 
bo/12.5mm, which is equivalent to a fixed maximum size for rp.  Using MK ≈ 0.15bo/rp, this 
criterion equates approximately to rp ≤ 1.9 mm for all specimen sizes. 
 
CONCLUSIONS 
This body of work has provided a concise and convenient way to visualize and quantify the 
relationship between compliance change in the C(T) specimen due to plasticity and the 
deformation level in the specimen.  By using non-dimensional metrics, the conclusions from the 
single analysis are valid for all specimen sizes of identical proportion.  Though not detailed 
herein, the material space covered by the analysis allows the evaluation to extend to all practical 
engineering alloys.  Two key conclusions were developed: 
1. An increase in the allowable limit for deformation to MK = 1.1 would not invalidate 
LEFM assumptions, rendering only small conservative errors in the evaluation of K.  
2. An increase in the allowable limit for deformation to MK = 1.1 would create sufficient 
compliance change in the force-displacement record due to plasticity to result in potential 
misidentification of toughness prior to crack extension for specimens larger than 
approximately W = 25mm. The proposed remedy is to make MK a function of specimen 
size, e.g. MK ≥ bo / 12.5mm, which equates to a fixed maximum plastic zone size for all 
specimens.    
 
REFERENCES 
[1] ASTM E399-12e3 “Standard Test Method for Linear-Elastic Plane-Strain Fracture 
Toughness KIc of Metallic Materials,” ASTM International, 2016. 
[2] Wallin, K. R. W., “Critical Assessment of the Standard ASTM E 399,” Fatigue and 
Fracture Mechanics: 34th Volume, ASTM STP 1461, S. R. Daniewciz, J. C. Newman and 
K.-H. Schwalbe, Eds., ASTM International, West Conshohocken, PA, 2004. 
[3] Wang, Yangyi, “A Two-Parameter Characterization of Elastic-Plastic Crack Tip Fields and 
Applications to Cleavage Fracture,” PhD diss., Massachusetts Institute of Technology, 
1991. 
[4] http://www.questintegrity.com/software-products/feacrack 
[5] http://www.warp3d.net 
[6] ASTM E1820 “Standard Test Method for Measurement of Fracture Toughness,” ASTM 
International, 2016. 
 
 
5 
 
 
 
Figure 1. Finite element C(T) mesh exploiting 1/4 symmetry (side and fracture-plane views). 
 
 
 
Figure 2. The change in compliance (ΔC) is defined as a percent increase in compliance (or 
percent decrease in slope) of the force versus displacement trace with respect to the initial linear 
portion as defined by the first analysis load step. 
 
6 
 
 
Figure 3a. Percent change in compliance versus normalized loading level, illustrating sources of 
compliance change. 
 
Figure 3b. Percent change in compliance and percent plastic contribution to KJ versus 
normalized loading level, illustrating issues with Cplasticity at high deformation. 


A Review of the Proposed KIsi Offset-Secant Method for Size-Insensitive Linear-Elastic Fracture Toughness Evaluation

17th International ASTM/ESIS Symposium on Fatigue and Fracture Mechanics  
(41st National Symposium on Fatigue and Fracture Mechanics) 
 May 10-12, 2017, Toronto, ON Canada 
A Review of the Proposed KIsi Offset-Secant Method for Size-Insensitive Linear-
Elastic Fracture Toughness Evaluation 
Mark James, Arconic Technology Center, New Kensington, PA, USA 
Doug Wells, NASA Marshall Space Flight Center, Huntsville, AL, USA 
Phillip Allen, NASA Marshall Space Flight Center, Huntsville, AL, USA 
Kim Wallin, VTT Nuclear Safety, VTT, Finland 
 
 
 
 
 
Recently proposed modifications to ASTM E399 would provide a new size-insensitive approach 
to analyzing the force-displacement test record. The proposed size-insensitive linear-elastic 
fracture toughness, KIsi, targets a consistent 0.5mm crack extension for all specimen sizes by 
using an offset secant that is a function of the specimen ligament length. The KIsi evaluation also 
removes the Pmax/PQ criterion and increases the allowable specimen deformation. These latter 
two changes allow more plasticity at the crack tip, prompting the review undertaken in this work 
to ensure the validity of this new interpretation of the force-displacement curve. This paper 
provides a brief review of the proposed KIsi methodology and summarizes a finite element study 
into the effects of increased crack tip plasticity on the method given the allowance for additional 
specimen deformation.  The study has two primary points of investigation: the effect of crack tip 
plasticity on compliance change in the force-displacement record and the continued validity of 
linear-elastic fracture mechanics to describe the crack front conditions. The analytical study 
illustrates that linear-elastic fracture mechanics assumptions remain valid at the increased 
deformation limit; however, the influence of plasticity on the compliance change in the test 
record is problematic. A proposed revision to the validity criteria for the KIsi test method is 
briefly discussed. 
 
 
 
https://ntrs.nasa.gov/search.jsp?R=20170008063 2019-08-29T23:24:52+00:00Z
1 
 
INTRODUCTION 
To determine KIc, ASTM E399 [1] uses an offset secant to the force-displacement record to 
identify the load at which crack extension occurs during a test. This secant construction line, 
offset to 95% of the slope of the linear portion of the test record, corresponds to the compliance 
change when crack extension equals approximately 2% of the specimen’s original crack length 
under the assumption that all compliance change is due to crack extension. When the material 
exhibits stable tearing with a rising R-curve—corresponding to a Type I curve per ASTM E399-
12e3, Figure 7—this method provides a toughness result that is specimen size dependent. 
 
Wallin [2] proposed modifications to E399 to provide a new approach to analyzing an ASTM 
E399 test record utilizing a secant construction line with an offset slope that is a function of the 
specimen size based on the remaining ligament, bo, thereby minimizing the influence of 
specimen size on the test result. The proposed size-insensitive linear-elastic fracture toughness, 
recently labeled KIsi, targets a consistent 0.5mm crack extension for all specimen sizes. The KIsi 
method also increases the allowable specimen deformation, and removes the Pmax/PQ criterion. 
These latter two changes allow more plasticity at the crack tip before a test result is deemed 
invalid.  
In the current version of E399, the deformation limit for ligament plasticity (also called the 
specimen size requirement) is expressed in terms of the ligament validity criterion as bo  MK (K 
/ys)2, where K is the linear elastic stress intensity factor, ys is the 0.2% offset engineering yield 
strength, and the constant MK = 2.5. Wallin [2] proposed that the deformation limit be extended 
to allow higher deformation with MK = 1.1 based on an evaluation of a variety of data sets 
available at the time of that paper. To make these limiting measures of deformation more 
tangible, note that with MK = bo*ys2 /K2, MK is simply the ratio of ligament length to plastic zone 
size (rp) with a proportional factor, such that MK ≈ 0.15bo/rp [3].  Using this engineering estimate, 
at MK = 2.5, rp is approximately 6% of bo, and at MK = 1.1, rp extends to 14% of bo. When 
evaluating the proposed changes for standardization of KIsi, concerns arose that with MK = 1.1, 
the contribution of crack tip plasticity to compliance change in the test record may be sufficient 
to influence the test result. In the E399 test method, without unloading compliance checks during 
the test, it is not possible to distinguish compliance change due to crack extension from that due 
to plasticity.  
This paper summarizes a finite element study with the following two objectives: first, evaluate 
the continued validity of linear-elastic fracture mechanics assumptions to describe the conditions 
at the crack tip using K at MK = 1.1 and, second, quantify the effects of crack tip plasticity on the 
compliance change in the force-displacement record and evaluate its influence on interpretation 
of the test record for identifying crack extension.  If required, the analytical study should provide 
results to formulate appropriate validity criteria to avoid detrimental plasticity.  
FINITE ELEMENT ANALYSIS APPROACH  
In this study, the common compact specimen (C(T)) with W = 50.8 mm (2.0 inch), a/W = 0.5, 
W/B = 4, and no side-grooves is used as the reference geometry. The C(T) analysis model was 
developed using the FEACrack [4] finite element modeling software and solved with WARP3D 
[5] v16.2.7. The model used a 1/4 symmetric mesh with 56863 nodes and 12305, 20-noded hex, 
small-strain elements. The crack tip was modeled using collapsed elements with untied duplicate 
nodes. The J-integral was calculated over 15 domains to gauge convergence while using the 
2 
 
WARP3D Type D domain evaluation for a bulk-average J-integral. Forces were applied at the 
center of the pin mesh, with pin rotation allowed, and elastic pin material. Figure 1 is an 
illustration of the finite element mesh. 
The material constitutive model used incremental plasticity and the isotropic Mises flow rule. 
The stress-strain relationship was modeled using linear behavior up to the proportional limit 
followed by a power law relation for plastic strain. Expressed in a normalized form with the 
proportional limit, σo = 1, this allows easy evaluation of a broad material space with stiffness to 
proportional limit ratio varying 100 ≤ E / σo  ≤ 1000 and strain hardening exponent varying 3 ≤ n 
≤ 20 with Poisson’s ratio, ν = 0.3. The calculation of MK, uses the engineering yield strength 
defined at 0.2% plastic strain. This material space covers practically all engineering alloys. 
ASTM E399 allows a variety of specimens for determining linear-elastic fracture toughness. 
This study focuses on the C(T) specimen with the understanding that specimens of different 
geometry will need to be evaluated to quantify the effects of specimen geometry on plasticity-
induced compliance change. The slender ligament aspect ratio, W/B = 4, is used to represent the 
E399 lower limit on specimen thickness and is the worst-case for allowing plasticity to influence 
the non-linearity in the force-displacement record.  
The KIsi offset secant 
For this work, the change in compliance (C) is defined as a percent increase in compliance (or 
percent decrease in slope) of the force-displacement record with respect to the initial linear 
portion, as defined by the first, fully-elastic analysis load step. See Figure 2. Consistent with 
Wallin [2], the offset compliance where fracture toughness is evaluated, C, is proposed to 
follow the convention yielding C in percent:  
 C = A/(W-a), (1) 
with A = 135 for the C(T) specimen for dimensions in mm, which corresponds to crack extension 
of 0.5mm.  For common E399 C(T) specimen sizes with a/W = 0.5: 
W = 25.4 mm (1 inch) C = 10.6%  W = 50.8 mm (2 inch) C = 5.3% 
W = 63.5 mm (2.5 inch) C = 4.25%  W = 101.6 mm (4 inch) C = 2.7% 
 
For a KIc assessment per the E399 standard, the offset secant is fixed at C = 5% for all 
specimens.  Thus, to target 0.5mm crack extension, this corresponds to a C(T) with W = 54.0 
mm. At this specimen size, the KIc and KIsi methods are equivalent. Note that the coefficient A 
listed here is different from that in Wallin [2] because in that case, the coefficient was developed 
for the ASTM E1820 [6] specimen that measures displacement at the load line, whereas in the 
current paper the coefficient is developed for specimens that measure displacement at the 
specimen front face. 
In experiments for KIsi, the change in compliance throughout a test is estimated as a summation 
of crack extension, plasticity, and experimental error: 
C = Ccrack ext + Cplasticity + Cexperimental error (2) 
 
3 
 
Currently, experimental methods for KIsi assume Cplasticity and Cexperimental error are sufficiently 
small relative to Ccrack ext such that the offset secant will properly identify crack extension of 
0.5mm. The magnitude of Cplasticity is easily estimated in the finite element study where the 
crack length is fixed at a/W = 0.5, such that Ccrack ext ≡ 0. This study does not provide insight 
into the magnitude of Cexperimental error though this could be of importance as specimen size 
increases and the offset secant for KIsi gets smaller. 
Though not shown herein for brevity, the authors demonstrate that all calculations related to 
compliance change and deformation are fully scalable with geometry, thus one model provides 
results for all size specimens of the same proportion. 
RESULTS 
The main result of the analytical work is represented in a plot of change in compliance, C, 
versus specimen deformation, MK = bo*ys2 /K2 ≈ 0.15bo/rp.  Figure 3a illustrates this plot and its 
interpretation. The abscissa axis, MK, is plotted log scale and reversed, left to right—decreasing 
values of MK correspond to increasing force and deformation. The green box indicates the 
compliance and deformation requirements for KIc fixed at C = 5% and MK = 2.5, respectively.  
The solid line marked with open circles represents a typical result from the analysis, in this case 
representative of aluminum alloy 2219-T8.  Recall the analysis reflects only the C contribution 
from plasticity—compliance increases non-linearly with plastic zone size. In an experiment, the 
measured C includes all sources as considered in Eq 2, though for the current discussion, we 
will consider experimental error contributions negligible. The dashed line in Figure 3a represents 
a typical experimental result, in this case, one that is valid according to the deformation limit for 
KIc per E399, i.e., MK ≥ 2.5. Prior to the onset of crack extension, Cplasticity is the only source of 
compliance change. Once crack extension begins, Ccrack ext contributes strongly to the total C.  
In this example, the KIc test is valid for deformation because C = 5% is reached while MK ≥ 2.5.  
The pertinent detail is that, in this example, nearly 2/5ths of C is due to plasticity, not crack 
extension.  
 
Figure 3b illustrates the key findings of the analysis effort needed to answer two questions 
regarding the consequence of allowing the deformation measure to extend to MK = 1.1 as first 
proposed for KIsi: 1) Does the increased deformation invalidate LEFM assumptions for the use of 
K; and 2) Can plasticity alone create sufficient C to reach the KIsi limit before crack extension 
begins? Either case is undesirable, leading to a non-relevant underestimate of linear-elastic 
toughness.  In Figure 3b, the analysis result of Cplasticity remains shown by the solid line with 
open circles. The contribution to KJ from plasticity (KJ-plastic, given in percent) is shown by the 
curve marked with solid circles and the proposed deformation limit of MK = 1.1 is shown by the 
vertical dotted line. Recall that analysis results plotted here are examples representative of Al 
2219-T8 material; however, the findings are fully supported by results from the larger material 
space. 
 
The KJ-plastic result answers question 1.  In this example, KJ-plastic = 4.4% at MK = 1.1.  In the full 
material space of the analysis set, at MK = 1.1, KJ-plastic ranges from 3% and 6.5%.  In the LEFM, 
E399 test method, the plastic contribution to fracture energy is not captured (KI computed from 
only force, not absorbed energy in the specimen). The LEFM toughness measurements that 
ignore the small plastic contribution are low, or conservative, by this percentage.  This range of 
4 
 
conservative error would generally not be grounds to consider the result invalid for LEFM 
assumptions; therefore, the use of MK = 1.1 appropriately maintains LEFM assumptions.   
 
The Cplasticity result answers question 2.  The C limits for KIsi are shown for various size 
specimens by horizontal dashed lines in Figure 3b.  It is clear that the Cplasticity result crosses 
many of these lines before reaching MK = 1.1.  In the example given by Figure 3b, only the 
smallest specimen of W = 25 mm avoids having Cplasticity cause the test record cross the KIsi 
offset secant prior to MK = 1.1. If non-linearity in the test record due to Cplasticity causes the test 
record to cross the KIsi offset secant prior to crack extension, the result is not a measurement of 
toughness, but merely a measure of yielding in the specimen.  Additional limits on specimen 
deformation are needed to control plasticity to ensure crack extension has occurred at the secant 
crossing.  The proposed remedy is to make MK a function of specimen size, for example, MK ≥ 
bo/12.5mm, which is equivalent to a fixed maximum size for rp.  Using MK ≈ 0.15bo/rp, this 
criterion equates approximately to rp ≤ 1.9 mm for all specimen sizes. 
 
CONCLUSIONS 
This body of work has provided a concise and convenient way to visualize and quantify the 
relationship between compliance change in the C(T) specimen due to plasticity and the 
deformation level in the specimen.  By using non-dimensional metrics, the conclusions from the 
single analysis are valid for all specimen sizes of identical proportion.  Though not detailed 
herein, the material space covered by the analysis allows the evaluation to extend to all practical 
engineering alloys.  Two key conclusions were developed: 
1. An increase in the allowable limit for deformation to MK = 1.1 would not invalidate 
LEFM assumptions, rendering only small conservative errors in the evaluation of K.  
2. An increase in the allowable limit for deformation to MK = 1.1 would create sufficient 
compliance change in the force-displacement record due to plasticity to result in potential 
misidentification of toughness prior to crack extension for specimens larger than 
approximately W = 25mm. The proposed remedy is to make MK a function of specimen 
size, e.g. MK ≥ bo / 12.5mm, which equates to a fixed maximum plastic zone size for all 
specimens.    
 
REFERENCES 
[1] ASTM E399-12e3 “Standard Test Method for Linear-Elastic Plane-Strain Fracture 
Toughness KIc of Metallic Materials,” ASTM International, 2016. 
[2] Wallin, K. R. W., “Critical Assessment of the Standard ASTM E 399,” Fatigue and 
Fracture Mechanics: 34th Volume, ASTM STP 1461, S. R. Daniewciz, J. C. Newman and 
K.-H. Schwalbe, Eds., ASTM International, West Conshohocken, PA, 2004. 
[3] Wang, Yangyi, “A Two-Parameter Characterization of Elastic-Plastic Crack Tip Fields and 
Applications to Cleavage Fracture,” PhD diss., Massachusetts Institute of Technology, 
1991. 
[4] http://www.questintegrity.com/software-products/feacrack 
[5] http://www.warp3d.net 
[6] ASTM E1820 “Standard Test Method for Measurement of Fracture Toughness,” ASTM 
International, 2016. 
 
 
5 
 
 
 
Figure 1. Finite element C(T) mesh exploiting 1/4 symmetry (side and fracture-plane views). 
 
 
 
Figure 2. The change in compliance (ΔC) is defined as a percent increase in compliance (or 
percent decrease in slope) of the force versus displacement trace with respect to the initial linear 
portion as defined by the first analysis load step. 
 
6 
 
 
Figure 3a. Percent change in compliance versus normalized loading level, illustrating sources of 
compliance change. 
 
Figure 3b. Percent change in compliance and percent plastic contribution to KJ versus 
normalized loading level, illustrating issues with Cplasticity at high deformation. 


A Review of the Proposed KIsi Offset-Secant Method for Size-Insensitive Linear-Elastic Fracture Toughness Evaluation

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