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Contents lists available at ScienceDirect

**Theoretical and Applied Fracture Mechanics
**

journal homepage: www.elsevier.com/locate/tafmec

**Damage modeling in Small Punch Test specimens
**

E. Martínez-Pañeda a,⇑, I.I. Cuesta b, I. Peñuelas c, A. Díaz b, J.M. Alegre b

a

**Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
**

Structural Integrity Group, University of Burgos, Escuela Politécnica Superior, Avenida Cantabria s/n, 09006 Burgos, Spain

c

Department of Construction and Manufacturing Engineering, University of Oviedo, Gijón 33203, Spain

b

a r t i c l e

i n f o

Article history:

Received 23 June 2016

Revised 1 September 2016

Accepted 4 September 2016

Available online xxxx

Keywords:

Small Punch Test

Fracture toughness

Ductile damage

Finite element method

a b s t r a c t

Ductile damage modeling within the Small Punch Test (SPT) is extensively investigated. The capabilities of

the SPT to reliably estimate fracture and damage properties are thoroughly discussed and emphasis is

placed on the use of notched specimens. First, different notch profiles are analyzed and constraint conditions quantified. The role of the notch shape is comprehensively examined from both triaxiality and notch

fabrication perspectives. Afterwards, a methodology is presented to extract the micromechanical-based

ductile damage parameters from the load-displacement curve of notched SPT samples. Furthermore,

Gurson-Tvergaard-Needleman model predictions from a top-down approach are employed to gain insight

into the mechanisms governing crack initiation and subsequent propagation in small punch experiments.

An accurate assessment of micromechanical toughness parameters from the SPT is of tremendous

relevance when little material is available.

Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Many engineering applications require a mechanical characterization of industrial components from a limited amount of material. Under such circumstances, it is often not possible to obtain

specimens of the dimensions demanded by standard testing

methodologies. With the aim of overcoming this hurdle, a miniature non-standard experimental device was developed in the early

80s [1]. The aforementioned testing methodology, commonly

known as Small Punch Test (SPT), employs very small specimens

(generally, 8 mm diameter and 0.5 mm thickness) and may be considered as a non-destructive experiment. The SPT has consistently

proven to be a reliable tool for estimating the mechanical [2,3] and

creep [4,5] properties of metallic materials and its promising capabilities in fracture and damage characterization have attracted

great interest in recent years (see, e.g., [6–18]).

Although brittle fracture has been observed in certain materials

at low temperatures [10,16,17], the stress state inherent to the SPT

favors ductile damage. It therefore comes as no surprise that efforts

to characterize the initiation and subsequent propagation of cracks

in SPT specimens have mostly employed models that account for

the nucleation, growth and coalescence of microvoids (see, e.g.,

[8–12,18] and references therein). The model by Gurson [19], later

extended by Tvergaard and Needleman [20], is by far the most

⇑ Corresponding author.

**frequent choice, but other models - such as the one by Rousselier
**

[21] - have also been employed [9]. These models are able to quantitatively capture the experimental results by fitting several parameters that account for the ductile damage mechanisms taking place.

A variety of inverse techniques - including the use of evolutionary

genetic algorithms [11–13] and neural networks [8] - have been

proposed to compute the Gurson-Tvergaard-Needleman (GTN)

[19,20] parameters from the load-displacement curve of unnotched

SPT specimens. Void-based models have been particularly helpful

in the development of new methodologies to estimate fracture

toughness from SPT specimens [18]. However, some relevant

aspects remain to be addressed. The substantially different constraint conditions attained in the SPT, relative to conventional testing procedures, constitute the most important problem to

overcome. As depicted in Fig. 1, the high triaxiality levels (defined

as the ratio of the hydrostatic stress to the von Mises equivalent

stress) of standardized fracture toughness experiments - such as

compact tension or three point bending tests - translate into conservative estimations of the fracture resistance. This is not the case

of the SPT, hindering a direct comparison and leading to predictions

that may significantly differ from the plane strain fracture toughness. Hence, current research efforts are mainly devoted to the

development of notched or cracked SPT samples with the aim of

increasing the attained triaxiality level [7,18].

In this work, the influence of the shape of the notch on the SPT

response is extensively investigated, considering both the constraint conditions and the fabrication process. Crack initiation

**E-mail address: mail@empaneda.com (E. Martínez-Pañeda).
**

http://dx.doi.org/10.1016/j.tafmec.2016.09.002

0167-8442/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: E. Martínez-Pañeda et al., Damage modeling in Small Punch Test specimens, Theor. Appl. Fract. Mech. (2016), http://dx.

doi.org/10.1016/j.tafmec.2016.09.002

2

E. Martínez-Pañeda et al. / Theoretical and Applied Fracture Mechanics xxx (2016) xxx–xxx

Fig. 1. Influence of the specimen configuration on fracture toughness.

**and subsequent propagation is computed by means of the GTN
**

model for various geometries of notched SPT specimens and results

are compared to experimental data. Different methodologies to

extract the micromechanical-based ductile damage parameters

are proposed and the past, present and future capabilities of the

SPT to characterize fracture and damage are thoroughly discussed.

2. Experimental methodology

The SPT employs a miniature specimen whose entire contour is

firmly pressed between two dies with the load being applied at the

center by means of a 2.5 mm hemispherical diameter punch. The

special device outlined in Fig. 2 is coupled to a universal testing

machine. A free-standing extensometer is attached to the experimental device to accurately measure the punch displacement.

The experiments are performed at room temperature with a punch

**speed of v = 0.2 mm/min. Lubrication is employed to minimize the
**

effects of friction.

The mechanical response of the SPT specimen is therefore characterized by means of the measured applied load versus punch displacement curve. Fig. 3 shows the different stages that can be

identified in the characteristic SPT curve of a material behaving

in a ductile manner. Different criteria have been proposed to estimate mechanical and damage material parameters from the curve

[2,15].

3. Gurson-Tvergaard-Needleman model

The influence of nucleation, growth and coalescence of microvoids is modeled by means of the well-known Gurson-TvergaardNeedleman (GTN) [19,20] ductile damage model. Within the aforementioned framework, the yield function is defined by,

Fig. 2. Device and schematic description of the Small Punch Test.

Please cite this article in press as: E. Martínez-Pañeda et al., Damage modeling in Small Punch Test specimens, Theor. Appl. Fract. Mech. (2016), http://dx.

doi.org/10.1016/j.tafmec.2016.09.002

3

E. Martínez-Pañeda et al. / Theoretical and Applied Fracture Mechanics xxx (2016) xxx–xxx

Fig. 3. Regions of the load - punch displacement curve in a Small Punch Test.

Uðre ; rh ; ry ; f Þ ¼

re

ry

2

þ 2q1 f cosh

3q2 rh

2

1 þ q3 f

¼0

2ry

ð1Þ

**where f is the microvoid volume fraction, rh is the hydrostatic
**

stress, re is the conventional Von Mises equivalent stress, ry is

the yield stress of the matrix material and q1 ; q2 and q3 are fitting

parameters as defined by Tvergaard [22]. The modified void volume

**fraction f was introduced by Tvergaard and Needleman [20] to
**

model the decrease in load carrying capacity that accompanies void

coalescence, such that,

(

f ¼

for f 6 f c

f

f c

f c þ ff uf

ðf f c Þ for f > f c

c

4. Results

A numerical model of the SPT is developed by means of the

finite element software Abaqus/Standard. Attending to the specimen geometry and test setup, quasi-static conditions are assumed

ð2Þ

f

with f c being the critical void volume fraction, f f the void volume

**fraction at final fracture and f u ¼ 1=q1 the ultimate void volume
**

fraction. The current void volume fraction f_ evolves as a function

of the growth rate of existing microvoids and the nucleation rate

of new microvoids

f_ ¼ f_ growth þ f_ nucleation

**Fig. 4. Different notched SPT specimens examined. In all cases the notch radius
**

equals e=2 ¼ 100 lm.

ð3Þ

1

**where, according to Chu and Needleman [23], the latter is assumed
**

to follow a Gaussian distribution, given by,

ð4Þ

with e_ p being the equivalent plastic strain rate, and,

2 !

f

1 ep en

A ¼ pnﬃﬃﬃﬃﬃﬃﬃ exp

2

Sn

Sn 2 p

ð5Þ

**Here, en is the mean strain, Sn is the standard deviation and f n is the
**

void volume fraction of nucleating particles.

Different methodologies have been proposed to fit model

parameters from a variety of experimental tests (see, e.g.,

[8,11,18]). A common procedure in the literature is to assume constant values of the parameters q1 and q2 (with q3 ¼ q21 ) based on

the micromechanical cell studies by Tvergaard [22,24], but more

complex models have also been proposed [25].

0.8

Triaxiality ξ

f_ nucleation ¼ Ae_ p

0.9

0.7

0.6

0.5

L

0.4

L+T

0.3

0.2

0.15

C

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Notch depth a/t

Fig. 5. Triaxiality levels in the direction of fracture at r ry =J ¼ 1 for several notch

types, different notch depths and t ¼ 1 mm.

Please cite this article in press as: E. Martínez-Pañeda et al., Damage modeling in Small Punch Test specimens, Theor. Appl. Fract. Mech. (2016), http://dx.

doi.org/10.1016/j.tafmec.2016.09.002

4

E. Martínez-Pañeda et al. / Theoretical and Applied Fracture Mechanics xxx (2016) xxx–xxx

**and a 3-D approach is adopted, taking advantage of symmetry
**

when possible. As described elsewhere [18,26], 8-node linear brick

elements are employed, with the mesh gradually being refined

towards the notch, where the characteristic element length is

determined from a sensitivity study. The lower matrix, the fixer

and the punch are modeled as rigid bodies and their degrees of

freedom are restricted except for the vertical displacement of the

punch. The friction coefficient was set to l ¼ 0:1, which is a common value for steel-to-steel contact under partial lubrication. Ductile damage is captured by means of the GTN model, which is

implemented in ABAQUS by means of a UMAT subroutine, where

the consistent tangent moduli is computed through the Euler backward algorithm, as proposed by Zhang [27].

1

0.9

Triaxiality ξ

0.8

**4.1. The role of the notch geometry
**

The influence of the notch geometry on the stress triaxiality is

thoroughly examined. Thus, as depicted in Fig. 4, three different

notch classes have been considered; 10 10 mm2 square specimens with (i) a longitudinal notch (L), (ii) a longitudinal and transverse notch (L + T), and (iii) a circular notch of 3 mm diameter (C).

Furthermore, for each geometry calculations are performed for two

thicknesses (t ¼ 0:5 mm and t ¼ 1 mm) and four notch depths

(a=t ¼ 0:2; a=t ¼ 0:3; a=t ¼ 0:4 and a=t ¼ 0:5). Hence, a total of

24 different configurations have been examined.

First, the stress triaxiality n, defined as,

n¼

0.7

0.6

0.5

L

0.4

L+T

0.3

0.2

0.15

**As discussed before, focus is placed in notched SPT specimens,
**

as introducing a defect in the sample paves the way to establishing

a direct correlation with standardized tests and allows for fracture

resistance predictions applicable to a wide range of stress states.

Hence, different geometries are modeled as a function of the various types of notches considered.

C

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Notch depth a/t

Fig. 6. Triaxiality levels in the direction of fracture at r ry =J ¼ 1 for several notch

types, different notch depths and t ¼ 0:5 mm.

rh

re

ð6Þ

**is computed in the direction of fracture at a normalized distance
**

from the notch tip of r ry =J ¼ 1. With J denoting the J-integral, which

is computed by means of the domain integral method. Results

obtained at the precise instant in which cracking initiates (i.e.,

f ¼ f c in all the integration points of an element) are shown in

Fig. 5 for the three notch classes considered, different notch depths

and a specimen thickness of t ¼ 1 mm.

Fig. 5 reveals higher stress triaxiality levels in the configurations

with a circular notch (C), with the longitudinal notch configuration

(L) showing the lowest triaxiality and the longitudinal and

transversal notch (L + T) case falling in between. Besides, a high

sensitivity to the notch depth is observed in the (L) geometry,

Fig. 7. Experimental observations and numerical predictions of crack initiation and growth.

doi.org/10.1016/j.tafmec.2016.09.002

E. Martínez-Pañeda et al. / Theoretical and Applied Fracture Mechanics xxx (2016) xxx–xxx

**while the opposite is shown for the (C) and (L + T) cases. Results are
**

however substantially different when a smaller specimen thickness is assumed (h ¼ 0:5 mm) as depicted in Fig. 6.

As shown in Fig. 6, the constraint conditions are now highly

dependent on the notch depth, with the longitudinal notch configuration (L) attaining the maximum levels when a=t ¼ 0:5. An

2

1.8

Triaxiality ξ

1.6

1.4

L

L+T

1.2

C

1

0.8

0.6

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Notch depth a/t

Fig. 8. Triaxiality levels in the direction of maximum n at r ry =J ¼ 1 for several

notch types, different notch depths and t ¼ 1 mm.

2

1.8

L

L+T

Triaxiality ξ

1.6

C

1.4

1.2

1

0.8

0.6

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Notch depth a/t

Fig. 9. Triaxiality levels in the direction of maximum n at r ry =J ¼ 1 for several

notch types, different notch depths and t ¼ 0:5 mm.

5

**increase in n is observed for both (L) and (C) cases when the defect
**

size increases while the opposite trend is shown for the (L + T) configuration. The high sensitivity of the results to the notch depth is

explained by the different location of the onset of damage. Thus, in

the circular notch configuration, large defect sizes lead to crack initiation sites located at the notch tip, while this is not the case for

ratios of a=t lower than 0.4. In all cases the initiation and subsequent propagation of damage trends computed in the numerical

model agree with the experimental observations, as depicted in

Fig. 7.

As the location for the onset of damage is highly dependent on

the notch to thickness ratio, it may be more appropriate to estimate the triaxiality level in the direction of maximum n. Fig. 8

shows the results obtained according to this criterion for a thickness of t ¼ 1 mm and the aforementioned configurations. As in

Figs. 5 and 6, the stress triaxiality is computed at a normalized distance r rY =J ¼ 1 as a function of the ratio between the notch length

and the sample thickness.

Fig. 8 reveals that the triaxiality levels attained with the longitudinal notch configuration (L) are significantly higher than those

relevant to the circular (C) and longitudinal and transversal (L

+ T) notch configurations. A similar trend is observed for a smaller

sample thickness, as shown in Fig. 9.

Differences between configurations are however smaller when

the sample thickness decreases, and the triaxiality levels attained

with the longitudinal notch specimen are significantly lower than

those shown for t ¼ 1 mm. Moreover, reducing the thickness of

the sample beyond 0.5 mm could have further implications, as size

effects may influence the mechanical response [28,29]. Highest triaxiality levels seem therefore to be attained with a longitudinal

notch for a specimen thickness of 1 mm.

One further aspect to take into consideration is the fabrication

process [14]. Two techniques are mainly being used: (i) highprecision micromachining and (ii) laser-induced micromachining,

which will be respectively referred to as micromachining and laser.

Each manufacturing procedure leads to a different notch geometry,

as shown in Fig. 10. Thus, laser procedures lead to sharper notches

with smaller depths than micromachining. Substantial differences

are observed in the notch radius as well, with laser-induced techniques leading to values one order of magnitude lower

(e=2 ¼ 10 lm).

The constraint conditions in the direction of maximum triaxiality are examined for notch geometries resembling the outcome of

micromachining and laser fabrication approaches and the results

are shown in Fig. 11 for the (L) configuration. As shown in the figure, higher triaxialities are obtained with the laser technique, particularly for larger notch depths.

However, micromachining leads to a better control of the

notching process, which translates in a uniform notch along the

Fig. 10. Schematic view of (a) high-precision micromachining and (b) laser-induced micromachining notch fabrication approaches.

doi.org/10.1016/j.tafmec.2016.09.002

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E. Martínez-Pañeda et al. / Theoretical and Applied Fracture Mechanics xxx (2016) xxx–xxx

than those necessary to introduce defects by means of laser or femtolaser techniques. Consequently, the use of high-precision micromachining is generally recommended.

Triaxiality ξ

2

1.8

Micromachining profile

1.6

Laser profile

4.2. GTN parameters identification through the SPT curve in edge

notched specimens

1.4

1.2

1

0.8

0.6

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Notch depth a/t

Fig. 11. Triaxiality levels in the direction of maximum n at r ry =J ¼ 1 for a

longitudinal notch (L) resembling laser and micromachining fabrication techniques,

different notch depths and t ¼ 0:5 mm.

specimen length. As shown in Fig. 12, this is not the case in laserbased techniques, where less uniformity is observed in the surface

finish, with the shape of the notch varying significantly along the

specimen length as the depth increases.

The aforementioned drawbacks may be alleviated by the use of

femtolaser, which allows for a good surface finish and a greater

depth accuracy (see Fig. 13). However, the notch losses uniformity

far from the center region. Moreover, the manufacturing costs of

notched specimens by micromachining are substantially lower

**Fig. 12. Modified SEM image showing the lesser notch uniformity attained with
**

laser-induced micromachining.

**A novel methodology to extract the parameters that govern the
**

nucleation, growth and coalescence of microvoids in GursonTvergaard-Needleman model is presented. The proposed procedure is employed with SPT specimens partially precracked

throughout the thickness and numerical predictions are compared

with experimental data for a precipitation hardened martensitic

stainless steel of Young’s modulus E ¼ 192 GPa, ultimate strength

ru ¼ 1200 MPa, yield stress ry ¼ 1100 and strain hardening coefficient n ¼ 40.

The proposed methodology, outlined in Fig. 14, aims to assess

the critical void volume fraction at the onset of coalescence f c for

given values of the remaining GTN parameters. Thus, following

[24], q1 ; q2 and q3 are considered to be respectively equal to 1.5,

1 and 2.25. While, for illustration purposes, it is assumed that

**Fig. 14. Outline of the proposed methodology to identify the GTN parameters from
**

a notched SPT specimen.

Fig. 13. Cross section of the notch obtained from (a) laser-induced micromachining and (b) femtolaser-induced micromachining.

doi.org/10.1016/j.tafmec.2016.09.002

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E. Martínez-Pañeda et al. / Theoretical and Applied Fracture Mechanics xxx (2016) xxx–xxx

**en ¼ 0:1; Sn ¼ 0:1 and f f ¼ 0:15. The initial void volume fraction f 0
**

is assumed to be equivalent to the volume fraction of intermetallic

particles and it is therefore considered to be equal to 0. By having

previously fixed the value of f n , which equals 0.01 in the aforementioned case study, the critical void volume fraction f c can be

obtained by means of a number of steps:

– Firstly, the nucleation and growth of micro-voids in the SPT is

modeled without considering coalescence. In that way, the value

of f n can be easily obtained by fitting the experimental curve.

– Afterwards, the punch displacement corresponding to the 90%

of the maximum load in the experimental curve D1 is measured.

This quantity is identified as the punch displacement at the

onset of failure, as observed in interrupted tests.

– The first estimation of the critical void volume fraction f c1 is

then obtained from the void volume fraction versus punch displacement curve, as it corresponds to the punch displacement

at the onset of failure D1 . For this purpose, the void volume fraction variation with punch displacement considered corresponds

to the node with higher porosity at the precise instant when the

experimental and numerical predictions deviate.

– A coalescence-enriched simulation is then performed with the

previously extracted value of f c . Afterwards, the difference

between the numerical and experimental predictions of the

punch displacement at the maximum load level is computed

d ¼ DPmax;sim DPmax;expt .

– Finally, f c will be estimated from the f versus displacement

curve by considering the void volume fraction that corresponds

to a punch displacement of D1 d.

The final estimation of f c allows to accurately capture the

experimental trends by means of the GTN model, as shown in

Fig. 15. Two experimental curves are shown (SPT I and SPT II) to

give an indication of the experimental scatter.

**4.3. GTN parameters identification through a top-down approach
**

While the capabilities of the SPT to accurately estimate

mechanical and creep properties are widely known, several uncertainties hinder its use in fracture toughness predictions. Useful

insight can be gained by means of micromechanical-based ductile

damage models, paving the way for the development of a combined experimental-numerical methodology that will allow to conduct structural integrity evaluations from a very limited amount of

material. With this aim, the nucleation and propagation of damage

in notched SPT specimens is examined by means of the GTN model.

The structural integrity of a CrMoV steel welding joint is assessed

by examining the base metal before (CrMoV) and after an intermediate heat treatment of 4 h at 350 °C (CrMoV IHT). The mechanical

properties relevant to both materials are shown in Table 1, as

extracted from the uniaxial tensile tests. Here, the hardening

behavior is fitted with a Hollomon type power law, with k being

the strength coefficient and n the strain hardening exponent.

Following the conclusions extracted from Section 4.1, SPT specimens with a longitudinal notch are employed. The GTN parameters

are obtained by fitting through a top-down approach [18] the loaddisplacement curve of uniaxial tests in notched round bars. Different specimen geometries are employed in the two material cases

considered, being the inner radius of 2.63 mm (CrMoV) and 2 mm

(CrMoV IHT). The vertical displacement is accurately measured by

means of digital image correlation (DIC), as depicted by the center

image of Fig. 16; the samples geometry and the mesh employed

are also shown in the figure. Taking advantage of the double symmetry, only one quarter of the specimens is modeled, employing

8-node quadrilateral axisymmetric elements.

GTN

parameters

are

obtained

by

first

assuming

q1 ¼ 1:5; q2 ¼ 1:0; q3 ¼ 2:25 [24] and en ¼ 0:3; Sn ¼ 0:1 [23];

while f 0 ; f n ; f c and f f are identified by calibrating with experiments

through a top-down approach. As in the previous section, a zero

Fig. 15. Numerical and experimental correlation for SPT experiments with an edge notch: (a) Load-displacement curve and (b) crack growth predictions. The crack length

equals 5 mm.

Table 1

Mechanical properties.

CrMoV

CrMoV IHT

E (GPa)

ry (MPa)

ru (MPa)

k (MPa)

n

200

210

595

762

711

822

1019

1072

0.107

0.071

doi.org/10.1016/j.tafmec.2016.09.002

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E. Martínez-Pañeda et al. / Theoretical and Applied Fracture Mechanics xxx (2016) xxx–xxx

**Fig. 16. Mesh and geometry of the notched uniaxial tensile specimens employed
**

for (a) CrMoV and (b) CrMoV IHT; a representative image of the DIC characterization is also shown. All dimensions are given in mm.

**initial void volume fraction f 0 ¼ 0 is adopted, as it is assumed to
**

correspond to the volume fraction of intermetallic particles. The

remaining parameters (f n ; f c and f f ) are identified from the

×104

3

2.5

2.5

2

2

Force (N)

Force (N)

3

experimental load-displacement curve of the notched uniaxial samples, as outlined in Fig. 17. First, the void volume fraction of nucleating particles f n is obtained by correlating the experimental data with

the numerical results obtained without considering void coalescence. Afterwards (Fig. 17b and c), the critical void volume fraction

f c is identified by assuming that it corresponds with the rapid loss

in strength characteristic of void coalescence. And lastly, the slope

of the experimental curve after the onset of failure determines the

value of f f (Fig. 17d).

Damage parameters obtained for the base metal before and

after the intermediate heat treatment are displayed in Table 2.

By employing uniaxial tensile tests on notched specimens for the

GTN parameter identification it is possible to clearly establish the

location of the onset of damage and accurately measure the displacement through the DIC technique.

The GTN model parameters shown in Table 2 are subsequently

employed to model nucleation, growth and coalescence in the SPT.

The experimental and numerical results obtained for both materials are shown in Figs. 18 and 19. Fig. 18 shows the damageenhanced numerical predictions along with the experimental data

and the conventional elasto-plastic simulations; GTN results precisely follow the experimental curve in both cases, showing the

good performance of the top-down methodology employed.

fn = 0.015, 0.01, 0.005, 0

1.5

1

×104

fn = 0.01

fn = 0.004

1.5

1

CrMoV

CrMoV IHT

0.5

CrMoV

CrMoV IHT

0.5

fn = 0.007, 0.004, 0.001, 0

0

0

0

0.2

0.4

0.6

0.8

1

0

(b)

Force (N)

Void volume fraction, f

fc = 0.02

fc = 0.012

0.03

2.5

2

1.5

0.02

1

0.01

0.5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

**Experimental data CrMoV
**

GTN model fn = 0.01; fc = 0.02; ff = 0.22

Experimental data CrMoV IHT

GTN model fn = 0.004; fc = 0.012; ff = 0.15

3

0.05

0.8

×104

3.5

CrMoV IHT

0.04

0.6

(a)

CrMoV

0.06

0.4

Displacement (mm)

4

0.07

0.2

Displacement (mm)

0

0

0.2

0.4

0.6

Displacement (mm)

Displacement (mm)

(c)

(d)

0.8

1

Fig. 17. Outline of the top-down approach: (a) experimental data and numerical predictions for different values of f n , (b) identification of the sudden load drop associated

with void coalescence, (c) void volume fraction in the center of the specimen versus displacement for the chosen value of f n , (d) numerical damage simulation.

doi.org/10.1016/j.tafmec.2016.09.002

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**E. Martínez-Pañeda et al. / Theoretical and Applied Fracture Mechanics xxx (2016) xxx–xxx
**

Table 2

Ductile damage modeling parameters (GTN model) obtained from a notched tensile test through a top-down approach.

CrMoV

CrMoV IHT

q1

q2

q3

f0

en

Sn

fn

fc

ff

1.5

1.5

1.0

1.0

2.25

2.25

0

0

0.3

0.3

0.1

0.1

0.01

0.004

0.02

0.012

0.22

0.15

1800

Experimental data

Damage-free numerical results

GTN model results

1600

1400

1200

1500

Force (N)

Force (N)

Experimental data

Damage-free numerical results

GTN model results

2000

1000

800

1000

600

400

500

CrMoV

CrMoV IHT

200

0

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

0.2

0.4

0.6

0.8

1

Displacement (mm)

Displacement (mm)

(a)

(b)

1.2

1.4

1.6

Fig. 18. SPT experimental and numerical (with and without damage) load-displacement curves for (a) CrMoV and (b) CrMoV IHT.

Fig. 19. Different notched SPT specimens examined.

**In Fig. 19 one can easily observe that the onset of damage and
**

subsequent propagation is accurately captured by the numerical

model. This is particularly useful for the development of new

methodologies for fracture toughness assessment within the SPT,

as it allows to identify crack propagation patterns and measure

the crack tip opening displacement [18].

top-down approach is employed to gain insight into the mechanisms of crack growth in the SPT, with the ultimate goal of developing an standardized procedure to accurately assess fracture

toughness from small scale experiments.

5. Conclusions

**The authors gratefully acknowledge financial support from the
**

Ministry of Economy and Competitiveness of Spain through grant

MAT2014-58738-C3.

**Ductile damage modeling within notched SPT specimens has
**

been thoroughly examined. The different perspectives adopted

have been reviewed and the choice of an appropriate notch geometry has been extensively studied, from both triaxiality and manufacturing considerations.

Particular emphasis is placed on the identification of the GTN

model parameters. On the one hand, a novel methodology is proposed with the aim of enabling ductile damage modeling from

the load versus punch displacement curve. On the other hand, a

Acknowledgments

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