C. Navarro

J. Fernandez-Saez

J. Loya

English

Crossref

IV France 110 (2003) 
Sciences, Les Ulis 
1/jp4:20030711 
simulation of dynamic TPB fracture test in a modified 
n bar 
J. Fernandez-Saez and C. Navarro 
305 
of ~ont~nuum Mechanics and Structural Analysis, Car/os Ill University of Madrid, 
de la Umvers1dad 30, 28911 Leganes, Madrid, Spain 
Abst ract. Dynamic three point bending fracture tests performed in a modified Hopkinson Bar 
are used to obtain material fracture properties (such as the fracture-initiation toughness) at high 
strain rates. This work presents a three-dimensional munerical analysis of the aforementioned 
tests, performed by the Finite Element Method, implemented in the commercial code ABAQUS. 
The relationship between the Stres.-; Intensity Factor and the Crack Mouth Opening Displacement 
was examined and compared with the bidimensional one. The results indicate that the use of 2D 
plane strain solutions can tmderestimate the fracture properties. 
1. INTRODU CTION 
Integrity of mechanical and structural components subjected to dynamic loading requires to know the 
material fracture behaviour at high strain rates. Considering mode I, several dynamic fracture pa-
rameters may be defined in relation to the crack propagation regime: the dynamic fracture- initiation 
toughness, K 1d , represents the value of the Stress Intensity Factor, SIF, at which a crack starts to 
propagate. Other dynamic fracture parameters are KID (dynamic fracture propagation toughness) 
and K 1A (crack arrest toughness) . These three material fracture parameters are used in design but 
the first is of special significance because it rates the effective propagation of a crack within the 
structural element subjected to impulsive load. 
In contrast to the determination of static fracture toughness, K1c, the methodology for dynamic 
fracture initiation toughness, Kid, is not yet standardized. The instrumented Charpy test has been 
widely used to evaluate the dynamic fracture properties of materials, but the maximum loading rate 
(Stress Intensity Factor loading rate, K1 ) achieved during the test is about K1 = 105 M Pa,jmf s. 
Descriptions have been published [1 , 2, 3, 4] of special arrangements of the Split Hopkinson Pressure 
Bar (SHPB) for dynamic bending tests at higher strain rates. 
This Hopkinson tests permit higher strain rates than those reached by instrumented Charpy 
bnpact tests. The system consists of a striker bar (called projectile or impactor), an inpu t pressure 
bar with a modified shape edge, a supporting device and the recording equipment. The cracked 
specimen is placed between the input bar and the supporting device and is loaded to fracture by 
means of a concentrated transverse force applied at its midspan. The projectile, moving at velocity 
Vo, strikes the input bar, generating a longitudinal strain compressive pulse, c:;(t) , that propagates 
along the bar. This pulse can be recorded by strain gages on its outer surface. Once the pulse 
reaches the right edge of the bar, part of its energy is directly transmitted to the specimen and to 
the supporting device, while t he remaining energy is reflected back to the input bar, but now, as a 
tensile pulse, C:r(t). The reflected pulse is recorded by the strain gages. Assuming the theory of one-
dimensional elastic wave propagation, the load exerted on the specimen, P;(t) , and t he displacement 
of the edge of the bar initially in contact with the specimen, 1~(t) , can be calculated [1 , 2, 3, 4]. 
To obtain Kid experimentally, both the time history of the Stress Intensity Factor, K1(t), during 
the specimen loading process, and the instant, t 1, at which the crack in the material begins to grow 
must be determined. Then Kid can be defined as: 
(1) 
To evaluate t he SI F throughout dynamic tests, different optical [5, 6, 7] and photoelastic [8] 
306 JOURNAL DE PHYSIQUE IV 
techniques have been proposed, but in general they require complex equipment. 
Numerical approaches [2], [9] based on the Finite Element Method (FEM) have been used to study 
the problem. Others [10, 11] derived simple formulas for the dynamic SIF of three-point bending 
specimens using the Euler-Bernouilli [10] or Timoshenko [11] beam theories, modelling the specimen 
as a simply supported beam with a crack at its midspan. These last models would be strictly 
applicable if pure three-point bending test conditions were maintained throughout the experiment, 
that is , with no contact loss between the specimen and the support device, nor between the specimen 
and the input bar. However, recent two-dimensional numerical analysis of the dynamic three point 
bending test [12], accounting for the possible loss of contact between the interacting bodies, showed 
that the dynamic SI F can be evaluated with high precision from the measurement of Crack Mouth 
Opening Displacement CMOD throughout the test, assuming that the same relationships between 
SIF and CMOD used in the static cases apply to the dynamic ones. In this last work [12] a high 
speed photography system was used to measure the CMOD directly on the specimen. Popelar et 
al. [13] proposed a dynamic test method in which the CMOD was measured by means of a gage. 
The aforementioned analyses are based on 2-D solutions and less information about 3-D analy-
ses is found in the literature, but several studies have considered three-dimensional effects in the 
determination of fracture properties [14, 15]. 
Here we present a three-dimensional numerical analysis of the dynamic three point bending frac-
ture tests performed in a modified Hopkinson Bar. The numerical analysis is by Finite Element 
Method, implemented in the commercial code ABAQUS. The relationship between the SI F and the 
CMOD is examined and compared with the bidimensional one. 
2. NUMERlCAL SIMULATION 
A three-dimensional numerical simulation of the impact bending fracture test was performed by the 
Finite Element Method using the commercial code ABAQUS [16]. The finite element model includes 
the projectile, the input bar, the specimen and its supporting device. The simulated projectile and 
input bar were cylindrical, 22 mm in diameter and 330 mm and 1 m length, respectively. One of the 
input bar edges, precisely that in contact with the specimen, is sketched in fip;ure 1. 
The geometry and dimensions of the specimen are given in figure 2. Its dimensions were: width 
W = 20 mm, span S =80 mm, and total length, L = 100 mm. The specimen had a fatigue crack of 
length a. Three dimensional effects were studied varying the thickness, B , (5 mm and 10 mm) and 
the crack length (a/W =0.3,0.5 and 0.7) . 
If-----------'. ~ b~I m 
Figure 1: Input bar scheme. 
Given the symmetry in geometry and boundary conditions, only one quarter of the input bar, 
projectile and specimen, and only half of one roller support were modelled. The input bar mesh was 
of 3020 nodes and 1988 elements, and that of the projectile had 870 nodes and 550 elements. 
For the mesh of the specimen, eight layers were used through the thickness. To take into account 
the stress and strain concentrations at the crack tip, the mesh was refined in this zone, the smallest 
element side being of 2.8 J.Lm. The mesh was also refined at the contact zone between the specimen 
and the input bar, and between the specimen and the supporting device. 
The number of nodes and elements used in the mesh of the specimen depends on the case analyzed, 
and varies between 5982 and 6174 nodes, and 7303 and 7744 elements. 
The supporting device was modelled by a half-roller 8 mm in diameter. To simulate the rest of 
the support, 60 special elements called " infinity elements" (CIN3D8) provided by ABAQUS finite 
element package [16] were added. This kind of element is often used in boundary value problems 
EURODYMAT 2003 
L ~ lOO mm 22mm 
t 
B 
30rnm 
Figure 2: Bending specimen scheme. 
defined in unbounded domains, or for problems in which the region of interest is small as compared 
to the surrounding medium. These last elements can be used in conjunction with conventional 
finite elements [16]. In this dynamic analysis, the infinity elements were chosen for their ability to 
transmit energy outside the finite element mesh, without trapping or reflecting it. This transmission 
is optimized when the boundary between finite and infinite elements is orthogonal to the direction 
from which the waves impinge on this boundary [16]. Figure 3 shows the meshes for the specimen, 
the end of the input bar and the support used in the analysis for the case of B = 10 mm and ajW 
= 0.5. 
~ 
~ 
59 
I 
~ ~ 
Figure 3: Finite element mesh, case B = 10 mm, ajw = 0.5. 
Linear elastic behaviour was assumed for the projectile, bar and support material with the fol-
lowing properties: Young modulus E = 200 GPa and Poisson coefficient v = 0.3. The mass density 
of this material wasp= 7850 kgjm3 . 
For the specimen material (7075-T651 aluminum alloy) an elastic-viscoplastic behaviour with 
Young modulus E = 72 GPa and Poisson ratio v = 0.3 was considered. The viscoplastic constitutive 
law used was: 
(2) 
where cpl is the equivalent plastic strain and ipl the equivalent plastic strain rate. The dependence 
of rJo on cpl was computed as a0 (cpt) = 300 · E:pl + 550 (a0 expressed in MPa). The values of the 
307 
308 JOURNAL DE PHYSIQUE IV 
parameters D and n were: D = 9164 and n = 0.93. The mass density of this material was p = 2800 
kgfm3 . The dynamic cases solved correspond to a projectile impact velocity of 10 m/s. 
The SI F can be obtained from the general path independent dynamic J integral, J', derived by 
Nishioka and Atluri [17](see also [18]). For a mode I stationary crack (as in our case), the SI F of 
the crack in the plane of symmetry, K 1(t)\.P, was calculated, as in this section purely plane strain 
conditions exist, as: 
(3) 
where the dynamic J integral was computed along integration paths contained in the symmetry 
plane. 
For comparison, static analysis was performed by applying an imposed displacement at nodes of 
the specimen in contact with the input bar, obtaining the SI F from the conventional J integral. 
3. RESULTS 
Figure 4 shows the evolution of K1\sp versus CMOD measured at the points on the surface of the 
specimen, CMOD\.,, for the case of a dynamic test on a specimen of B = 10 mm and a ratio a/W 
of 0.5. 
s; 
e 
~ 
~ 
~ 
50 
40 
30 
20 
10 
5.0xlo·5 t.Oxl0-4 J.5xl0,. 2.0xto"" 2.5xto"" 
CMOD(m) 
Figure 4: Relationship between K1\sp and CMOD\sf· 
A linear relation can be drawn between SIF and CMOD, i.e: 
KI(t)\sp = k3v · CMOD(t)\s! 
This result was obtained for all the cases analyzed. 
In bidimensional situations the SI F can be related to the CM 0 D as: 
where the constant k20 is [19]: 
K1(t) = k2v · CMOD(t) 
~D = EJW kp(a) 
4a vp(a) 
(4) 
(5) 
(6) 
In this expression, kp(a) and vp(a) are non-dimensional functions depending on a (a = a/w) 
and (3 ((3 = L/W) ratios, and their expressions are found in Guinea el al. [19]. For the case (3 = 4, 
they are: 
EURODYMAT 2003 
k13(a) = fi (1.9 + 0.41a + 0.51a2 - 0.17a3) 
(1- a)>(1 + 3a) 
2 3 0.66 
v13 (a) = 0.76- 2.28n + 3.87a - 2.04a + (1 _ a)2 
(7) 
(8) 
To determine three-dimensional effects in the SI F - CM 0 D relationship, the ratio between the 
slopes k30 and k20 were obtained and plotted in figure 5(a) (dynamic cases) and 5(b) (static cases). 
1.06 
1.04 ; 
" _,[' 1.02 
G-t> 8 = 5mm 
A--1:>. 8 = 10 mm 
0.98 '---'---~----'--~----'--~----'--~----'----' 
0.3 0.4 0.5 
a/W 
(a) Dynamic cases. 
0.6 0.7 
1.04 
" ~ 102 
.} 
0.3 0.4 0.5 
a/W 
(b) Static cases. 
Figure 5: Ratios of k3D and k2D slopes. 
0.6 0 .7 
These figures show that in all the cases considered, except for a/W = 0.3, the slope calculated 
from the 3D analysis is greater than the 2D plane strain slope, specially for thin specimens (B = 5 
mm). This indicates that the use of 2D plane strain solutions can slightly underestimate the values 
of dynamic fracture initiation toughness for specimens with ratio a/W above 0.4. 
4. CONCLUDING REMARKS 
A three-dimensional numerical simulation of the impact bending fracture test was performed by 
the Finite Element Method. The model includes the projectile, the input bar, the specimen and 
its supporting device. The relationship between the Stress Intensity Factor and the Crack Mouth 
Opening Displacement was examined and compared with that of the bidimensional one. It was 
concluded that the slope of the linear relationship between the SI F and CM 0 D calculated from the 
3D numerical analysis is greater than that with 2D plane strain solutions, normally used to interpret 
the fracture test results, so, the Kid determined from CMOD measurement and 2D plane strain 
relations may be slightly underestimated in specimens with ratio a/ W above 0.4. 
Acknowledgements 
This work was supported by the Comisi6n Interministerial de Ciencia y Tecnologia under Grant 
PB98-0027. 
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