Analysis of load oscillations in instrumented impact testing

ANALYSIS OF LOAD OSCILLATIONS IN INSTRUMENTED

IMPACT TESTING

S. SAHRAOUI{

Laboratoire d'Acoustique de l'UniversiteÂ du Maine, U.M.R.-C.N.R.S. 6613, 72085 Le Mans Cedex 9,France

and

J. L. LATAILLADE

Laboratoire MateÂ riaux Endommagement FiabiliteÂ , ENSAM, Esplanade des Arts et MeÂ tiers, 33405Talence, France

AbstractÐThis work deals with the dynamic fracture tests of the 3-point bending notched specimen.The apparent frequency of the load oscillations is investigated for di�erent materials (steel, aluminium,PMMA and Al2O3). It is shown that the contact sti�ness between the striker and the specimen plays apredominant role in the vibrating aspect of the impact. The 2 degrees of freedom (dof), intensively usedin the literature, can predict the oscillations qualitatively, but cannot realistically model all the detailsof the vibrations of the specimen. A numerical model with several dof is considered by means of amodal analysis and analyzed through the experimental program. # 1998 Elsevier Science Ltd. Allrights reserved

KeywordsÐimpact testing, load oscillations, numerical analysis..

1. INTRODUCTION

THE INSTRUMENTED impact test is generally used to measure the toughness of materials. It is

general practice to measure and record the force which is acting on the specimen during the

impact. The simplicity of the 3-point bending impact test is sometimes taken as a justi®cation

for its use, where a more fundamental form of test would be di�cult or expensive to carry out.

Nevertheless, the dynamic e�ects resulting from the transverse impact are the principal inconve-

nience of these tests. Many investigations have been focused on the analysis of inertia e�ects for

evaluating the dynamic response of materials [1±4]. In order to overcome these di�culties, some

authors have proposed various solutions such as: the inverted pendulum [5], a soft material

between the specimen and the hammer [6], the impact response curve and time-to-fracture

measurements [7], and numerical modelling [8, 9]. To understand the dynamic e�ects of the 3-

point bending impact, some authors have proposed simple numerical models where the specimen

is represented by a spring-mass system[10±12].

The approach adopted for instrumented impact testing [13] assumes that a quasi-static equi-

librium prevails in the specimen. One of the ASTM requirements is based on the Server empiri-

cal relation [14] giving the period of the apparent specimen oscillations. Initially developed for

metallic materials, this standard is sometimes applied to polymers [15], ceramics [16] and

concrete [17].

The numerical and experimental work described below was initiated with two aims. Firstly,

to study the vibration response of the full mechanical system (specimen and loading device) to

the impact by means of an adapted model. Secondly, to examine the Server relation in various

experimental conditions.

Engineering Fracture Mechanics Vol. 60, No. 4, pp. 437±446, 1998# 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0013-7944/98/$ - see front matterPII: S0013-7944(98)00024-1

{Author to whom all correspondence should be addressed.

437

2. EXPERIMENTATION

2.1. Specimen details

Four materials were investigated in this study as indicated by Table 1. The concrete dataare derived from the literature[17].

A particular problem with polymeric materials is that the value of Young's modulus E isoften dependent upon the loading rate. The values presented above are obtained at 300 ÿ1 strainrate by means of a split Hopkinson tensile bar apparatus[18].

In all experiments described here, the geometry was that of the Charpy test employing 3-point bending, as shown in Fig. 1.

2.2. Experimental set-up

The tests were conducted using two di�erent devices.

1. A commercial Charpy pendulum (Wolpert, Germany). The tup of the hammer (7.5 J) hasbeen instrumented with strain gauges to sense the compression loading of the tup while it isin contact with the test specimen. The velocity with which the striker impacted against thespecimen could be varied by changing the angle from which the striker was released. Theimpact velocity can reach 4 m/s. This Charpy pendulum is able to test brittle ceramic speci-mens which need a low fracture energy and it has been used for the rebound tests describedbelow.

2. A horizontal line nose aluminium projectile guided by means of two V-supports. The frictionis minimized using te¯on bearings at the contact surfaces. This projectile is launched at adesired velocity (0.5±2 m/s) by a gas gun. Strain gauges are ®xed on the projectile at a chosenposition in order to avoid the superposition between incident and re¯ected waves at thebeginning of the impact. The striker length is increased so as to reduce its fundamental fre-quency (around 4.5 kHz) and then to minimize the load signal perturbation. Two gauge-bridges were ®xed at two diametrically opposite points on the striker surface. The geometricdetails are indicated in Fig. 2.

The load signals derived from each apparatus are digitally stored in a transient recorder andthen analyzed in a computer station (Fig. 3).

Table 1. Specimen's characteristics

Steel Aluminium PMMA Alumina Concrete

Dimensions 40�4�1040�7.8�10

L�B�W 40�10� 10 40�10�10 40�5�10 32� 4�4 229� 25� 76(mm) 40�10�10

Notch 2 1.5 0.5length (mm) 5 5 2 1

5 2E(GPa) 200 80 5 270r (Kgdmÿ3) 7.8 2.7 1.2 3.8 1.1

Fig. 1. Specimen geometry

S. SAHRAOUI and J. L. LATAILLADE438

2.3. Contact sti�ness determination

In a simple spring and mass model of dynamic e�ects in impact tests, the shape of the

force-time response depends on the ratio of the contact sti�ness between the specimen and the

striker, and the bending sti�ness of the specimen [19]. The techniques for determining the com-

pliance (Cm) of a Charpy impact machine are described by Ireland in ref. [2]. We performed

rebound tests on a known compliance specimen at low impact velocity for measuring the con-

tact time (tc) as shown on the typical load-time record (Fig. 4). Then:

tc � p��mCt

p;

where m and Ct are, respectively, the striker mass (the specimen mass is neglected) and the total

compliance. We obtain:

Cm � Ct ÿ Cs;

where Cs is the specimen compliance. We assume a rigid contact at the two supporting points

Fig. 2. Line nose striker.

Fig. 3. Instrumentation.

Analysis of load oscillations in instrumented impact testing 439

and then Cm is considered as the contact compliance C (C = 1/kc) between the striker and thespecimen.

Table 2 gives the contact sti�ness values for di�erent experimental conditions.

3. NUMERICAL APPROACH

3.1. Beam vibration theory

The linear vibrations of beams are governed by the well-known partial di�erentialequation:

EI@4

@x4y�x; t� � rA

@2

@t2y�x; t� � 0; �1�

where E, I, r and A represent, respectively, Young's modulus, the second moment of area, thevolumic mass and the cross-sectional area. The de¯ection y(x,t) is illustrated in Fig. 5.

Each vibration mode has an angular frequency on given by:

on � l2n

��EI

rAL4

s; �2�

where ln is determined by the boundary conditions. In the case of a simply supported specimen,we have:

sin ln � 0

then:

ln � np; n � 1; 2; . . .

In the case of free specimen, the boundary conditions lead to:

Fig. 4. Typical load-time record in a rebound test.

Table 2. kc (MNmÿ1) values for various specimens

PMMA Steel Aluminium Alumina

Charpy 16 34 22 ÐpendulumAluminium Ð Ð Ð 13striker


cos ln cosh ln � ÿ1;hence:

l1 � 3:011p2;

ln�2 � �2n� 1� p2:

If we introduce the specimen compliance (in 3-point bending con®guration):

C � L3

48EI; �3�

the angular frequency takes the form:

on � l2n4��3p �rCLBW�ÿ1=2 �4�

where the thickness B and the depth W are indicated in Fig. 1.

3.2. Numerical models

Many numerical investigations are focused on the 3-point bending impact. In the ®nite el-ement approach, the contact interaction between the specimen and the striker is not taken intoaccount. The specimen loading is treated as an imposed load problem (®nal-peak-sawtooth-pulse loading) [8] or as an imposed displacement problem (i.e. constant velocity at the impactpoint). In order to introduce the contact sti�ness, simpler models (spring-mass) have been pro-posed: Williams and Birch [10] consider successive collisions between the specimen and the stri-ker with a given restitution coe�cient; Suaris and Shah [11] and Williams [12] study a 2 degreeof freedom (dof) mechanical system where the striker is replaced by another spring-mass model(this spring represents the contact sti�ness); and Mills and Zhang [6] introduce the damping anda unilateral contact at the impact point.

A more elaborated discretization is performed to include the ¯exural oscillations and thecontact interaction between the striker and the specimen [20]. The simply supported beam isreplaced by a chain of n elements connected by hinges, supporting concentrated masses withsprings giving a restoring torque. The beam-tup mechanical system is represented by n + 1 dof(Fig. 6), where the linear sti�ness kc represents the specimen-tup contact interaction. The impactis treated as an initial conditions problem with n + 1 dof and solved through a modal analysis.Further details of this approach may be found in previous papers [20, 21] from which we cansummarize the following results:

. only the symetric modes are present in the specimen oscillations (Oi is the angular frequencyof the ith symetric mode);

. Figure 7 illustrates (for n = 9) the variation of O2/o1 through Cs/C (o1 is the fundamentalangular frequency of the simply-supported specimen).

From this model, the load oscillations are considered as a combination of a ``quasi-static'' forcecorresponding to the bending specimen (1st mode), and a dynamic force which is related to thevibration of the specimen around its equilibrium con®guration (Fig. 8).

Fig. 5. Beam coordinate.


4. RESULTS AND DISCUSSION

The typical load-time records in instrumented impact testing are represented in Fig. 9.These experimental records are used for determining the apparent frequency.

We observe that the records correspond qualitatively to a superposition of two modes: theso-called quasi-static mode and the second mode as described in the above numerical approachsection. The frequency of the oscillations is not the fundamental frequency of the specimen. Itwas interesting to note that the relation in eq. (4) does not give an approximate value of thisobserved frequency in any boundary conditions. After some transformations, eq. (4) can be writ-ten as:

Tn � 2p=on � anW

c0�EBCs�1=2; �5�

where c0 is the sound velocity in the material and an is a coe�cient listed in Table 3.The apparent period t of the specimen oscillations is given by the empirical relation

(known as SERVER relation):

t � 3:36W

c0�EBCs�1=2: �6�

This relation, obtained for metallic specimens at a speci®c shape (L/W = 4), is the basis of theASTM standard for Charpy impact testing [14]. The empirical value 3.36 in eq. (6) can be com-pared to the an values in free±free con®guration (Table 3). As suggested by Server [14], the oscil-lations during the impact appear to be a combination of ®rst and second symetric specimenmodes with mode 1 dominating. In fact, this combination is qualitatively comparable with thenumerical shape given in Fig. 8 and corresponding to this ®rst and second symetric modes ofthe specimen-tup system (Fig. 6).

Fig. 6. Modelization of the specimen-tup system.

Fig. 7. O2/o1 vs kc/ks.


According to the above described numerical model, the experimental values O2/o1 (normal-ized apparent oscillations frequency) are determined and compared with numerical results(Fig. 10). The contact sti�ness kc has been obtained from each material using the rebound testdescribed above. This sti�ness is assumed independent of the specimen dimensions and theimpact velocity. The value 2.6 (dashed line in Fig. 10), computed from the relations in eqs (5)and (6), gives the normalized apparent frequency by the fundamental frequency of the simply-supported specimen (L/W = 4).

The 2 dof model curve (Fig. 7) is obtained by simple calculations[17]:

O2

o1� 1� kc

ks

� �1=2

: �7�

We observe a good agreement between the present numerical model, the 2 dof model and theexperimental values for low kc/ks ratio.

It was interesting to examine the concrete data derived from the literature [17] and indicatedin Fig. 10. The authors have pointed out the analogous values between the apparent half periodcomputed by using their 2 dof model and the Server relation (eq. (6). They have confused the

Fig. 8. (a) Load oscillations, (b) corresponding specimen shape.

Table 3. an values (eq. (5)

Simply supported Free±free

First mode Third mode First mode Third mode

L/W= 4 8.83 0.98 3.89 0.72L/W= 8 12.48 1.39 5.50 1.02


apparent period relation with the half period one. Obviously, their results correspond to a lowervalue of the frequency and are corroborated by our model as mentioned in Fig. 10.

For high kc/ks ratio the 3 dof model is too far from experimental results which are enclosedby Server relation values and the 10 dof model.

The measured contact sti�ness kc through the rebound test is lower than its real value. Thisfact is following the in®nite rigidity hypothesis at the two specimen supports and can explainthe gap between experimental values and numerical curve at low kc/ks.

Globally, the Server relation is an interesting tool for dynamic fracture toughness testing ofmetallic materials, as well as some non metallic ones. It should be useful to investigate more sys-

Fig. 9a and bÐCaption opposite


temically the dynamic response of such materials with the objective to check the apparent fre-quency relation for other specimen shapes (L/W = 6, 8, . . .).

5. CONCLUSION

The present study has pointed out the importance of the second mode of vibrations of thefull mechanical system (specimen and tup or striker) which is dependent on the contact sti�ness.The contact sti�ness hypothesis, for characterizing the interaction between the tup and the speci-men, seems to be an appropriate method to evaluate the load oscillations frequency.

The model presented in this work gives a more realistic result than the 2 dof model whichis less accurate for high relative contact sti�ness.

Fig. 9. Fracture test load-time record: (a) steel, (b) aluminium, (c) PMMA and (d) alumina.


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(Received 10 June 1997)

Fig. 10. Experimental apparent frequencies vs contact sti�ness (see Fig. 7).


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Analysis of load oscillations in instrumented impact testing