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Investigation of loading rate and plate thickness effects on dynamic fracture toughness of PMMA
Engineering Fracture Mechanics Vol. 54. No. 6. pp. 805-81 I, 1996 Copyright 0 1996 Elsevier Science Ltd.
Pergamon
0013-7944(95)00244-8
INVESTIGATION OF LOADING RATE THICKNESS EFFECTS ON DYNAMIC TOUGHNESS OF PMMA
Printed in Great Britain. All rights reserved 0013-7944/96
$15.00 + 0.00
AND PLATE FRACTURE
H. WADA and M. SEIKA Department of Mechanical Engineering, Daido Institute of Technology, 2-21 Daido-cho Minami-ku, Nagoya 457, Japan T. C. KENNEDY and C. A. CALDER Department of Mechanical Engineering, Oregon State University, Corvallis, OR 97331, U.S.A. K. MURASE Department of Transport Machine Engineering, Meijo University, Tenpaku-ku, Nagoya 468, Japan Abstract-To investigate the effects of loading rate and plate thickness on the fracture toughness of PMMA (polymethyl methacrylate) under impact loading, two methods, A method and B method, are applied as follows. In the A method, a dynamic finite element method and a strain gage method are applied to measure the dynamic fracture toughness in the fracture test using an air gun. In the B method, a single axis strain gage method is applied to measure the critical dynamic stress intensity factor, namely dynamic fracture toughness, in the fracture test using a weight dropping type apparatus. The dimensions of the PMMA specimen are L = 140 mm length and W = 30 mm width. Three values of the plate thickness E, 15.0 mm, 10.0 mm and 5.0 mm, are selected to investigate the plate thickness effect in the fracture test. Both results by the A and B methods precisely indicated the minimum value and the loading rate effect on the dynamic fracture toughness. Copyright 0 1996 Elsevier Science Ltd.
INTRODUCTION SINCE PMMA (polymethyl methacrylate) has many excellent properties, it is employed as an industrial material in general. Hence, investigating the fracture behavior for this material is of importance in the strength evaluation of structure. Particularly, dynamic fracture toughness Krd depends on loading rate x,(t), plate thickness B, etc. It has also a minimum value at some loading rate. Accordingly, investigating their effects on PMMA is reasonable to deepen universal understanding of the fracture toughness of several materials. Especially, it is known generally that static fracture toughness decreases as the plate thickness increases. We have an interest in this effect on the dynamic fracture toughness of PMMA under impact loading. The purpose of the present study is to investigate the effects of loading rate and plate thickness on the fracture toughness of PMMA under impact loading. The dynamic fracture toughness is measured for the specimen with three values of the plate thickness over a wide range of loading rates by the following two methods, A method [I] and B method. In the A method, a dynamic finite element method and a strain gage method are applied to measure the dynamic fracture toughness. The dynamic fracture toughness is determined by comparing the crack initiation time tf from the strain gage signal to the simulation curve of the dynamic stress intensity factor (SIF) K,(t) computed by a dynamic finite element method [2]. An air gun is used for the impact fracture test under single-point impact bending. In this method the dynamic fracture toughness is obtained over a range of relatively higher loading rates. In the B method, a single axis strain gage method [3] is applied to measure the dynamic stress intensity factor. The dynamic fracture toughness is determined from the fracture initiation point obtained from the strain gage signal. An impact loading apparatus with free fall of a striker is used for the impact fracture test under three-point bending. 805
806
H. WADA Table Young’s
1. Mechanical
modulus
E @Pa)
4.97
et al.
properties
of specimen
Poisson’s
ratio
0.388
Y
material
Density
p (kg/m3)
1.18 x 10’
EXPERIMENTS AND DETERMINATIONS OF & Material and specimen The material chosen for testing is commercially available PMMA. The material properties are given in Table 1. Young’s modulus E in the table is a dynamic value measured from the velocity of the stress wave propagation. The dimensions of the PMMA specimen are L = 140 mm length and W = 30 mm width as shown in Fig. 1. Three values of the plate thickness B, 15.0, 10.0 and 5.0 mm, are selected to investigate the plate thickness effect in the fracture test. A strain gage is mounted at a position Y+ 1.5 mm away from the crack tip to measure the crack initiation time tf or the dynamic SIF K,(t) in the fracture test with the A method, as shown in Fig. 1. The distance r is measured by precisely magnifying it 20 times with a universal projector. A method A dynamic finite element method and strain gage method are applied to measure the dynamic fracture toughness. The dynamic fracture toughness is determined by comparing the crack initiation time from the strain gage signal to the simulation curve of the dynamic stress intensity factor computed by a dynamic finite element method. An air gun is used for the impact fracture test under single-point bending. In this method the dynamic fracture toughness is obtained over a range of relatively higher loading rates. Figure 2 illustrates the impact test apparatus using the air gun produced by the authors and the measuring system. The apparatus consists of an incident bar, a stand of specimen, a striker, a barrel and a compressor. The striker made of PMMA is 21.8 mm in diameter and 200 mm in length. The PMMA-striker bar is projected through the barrel made of a steel pipe (22.7 mm in inside diameter, 45 mm in outside diameter and 1000 mm in length) by releasing compressed air from an air tank with a solenoid valve. Then the projected striker bar strikes the incident bar of PMMA (2000 mm in length and 20 mm in diameter) contacting the specimen, resulting in the impact load applied to the specimen. Since the specimen, mounted on two smooth thin rods, has no supports in the direction of the impact load as shown in Fig. 3, it is assumed that the specimen fractures only with its own inertia. The velocity of the striker can be changed freely by adjusting the pressure of air. A time history of the impact load applied to the specimen is measured with the strain gage mounted at the center of the incident bar, while the crack initiation time tf is determined with the strain gage mounted in the neighborhood of a crack tip. The output signal from each strain gage is amplified with an amplifier (Kyowa Dengyo Co., CDV/CDA-230C, frequency response 200 kHz) through a gage bridge, recorded in a transient converter (Rikendensi Co., TCL-005DG, 6 cha, 4096 word/cha, 10 bit) and finally processed with a personal computer (Nippon Electric Co., PC-980M). A part of the strain pulse occurring in the incident bar due to the collision of the striker reflects to the incident bar and another part of the pulse transmits in 140
B=( 5,10 ,15)
(Dimensions Fig. 1. Specimen
configuration
in
mm)
and gage position.
Investigation
of loading
rate and plate thickness
effects
807
(Dimensions
in
mm)
Oscilloscope
Fig. 2. Impact
test apparatus
and block diagram
of instrument
system
the specimen. A time history of the applied load to the specimen is obtained from the signal of the strain gage mounted at the center of the incident bar [2]. In the dynamic finite element simulation, assuming a plate element of homogeneous and isotropic material, an equation of motion can be expressed in matrix form as follows:
[Ml(~) + [KlCd> = P’) 3
(1)
where [K] and [M] are the assembled stiffness and mass matrices, respectively, (P} the incremental vector of an external force and {d} the incremental vector of nodal displacement. The superposed dot (.) implies differentiation with respect to time. Equation (1) is solved by the so-called j3 method of Newmark [ 4, 51, using the consistent mass matrix for [Ml, obtained from the quadratic isoparametric plane elements [6] with eight nodes for the stiffness matrix [K] and the values measured on the experiment for {P}. The quarter point element is used around the crack tip as shown in Fig. 4 [7]. The dynamic SIF is calculated by K,(t) = 2uG(27c/e)“*/(k-+ 1) ,
(2)
where G is the shear modulus of specimen material, v is the nodal displacement near the crack tip as shown in Fig. 4, e is the distance from the crack tip to the nodal point near the crack tip and K = 3 - 4v for the plane strain condition or K = (3 - v)/(l + v) for the plane stress condition where v is Poisson’s ratio of the specimen material. Figure 5 is a typical example of the finite element mesh based on the symmetry of the specimen used for two-dimensional analysis. The crack initiation time tf is measured from the signal of a strain gage mounted at the position e + 1.5 mm
Specimen
Strain *
gage
Cra
Fig. 3. Loading
configuration.
Fig. 4. Crack
tip elements.
808
H. WADA
Fig. 5. Finite element
et al.
mesh for specimen.
away from the crack tip. Since the signal indicates a sudden increase at the crack initiation instant, the dynamic fracture toughness Kid [2] is determined by comparing the tr to the simulation curve of the dynamic SIF computed by FEM, as shown in Fig. 6. B method A single axis strain gage method [3] is applied to measure the dynamic stress intensity factor. The dynamic fracture toughness is determined from the fracture initiation point obtained from the strain gage signal. An impact loading apparatus with free fall of a striker is used for the impact fracture test under three-point bending. In order to perform dynamic three point bending tests, the weight dropping type apparatus was developed as shown in Fig. 7. Impact velocities are controlled by manual lifting to the desired height and the mass of striker is 15 kg. The figure indicates the instrument system for the fracture test. The dynamic SIF K,(t) and crack initiation time tr are determined from the signal of the strain gage mounted in the neighborhood of the crack tip in the specimen. The K,(t) [3] is calculated from eq. (3) with the single axis strain gage. K,(t) = EEy(27rr)“*/(1- v) )
(3)
where E is Young’s modulus, E, the strain measured with the strain gage near crack tip and r the distance from the crack tip to the gage position. The output signal from the strain gage is amplified with an amplifier (Kyowa Dengyo Co., CM/GA-230C, frequency response 200 kHz) through a gage bridge, recorded in a digital oscilloscope (Philip, D14-125GR/177) and finally processed with a personal computer (Nippon Electric Co., PC-9801Vm), as shown in Fig. 7. Figure 8 illustrates a typical example of the time history of the dynamic SIF calculated using eq. (3) from the strain gage signal mounted near the crack tip. We can see the fracture initiation point clearly in the figure.
Timet(,Us) Fig. 6. Determination
of dynamic
fracture
toughness
(A method).
Investigation
of loading
rate and plate thickness
effects
809
Striker
c Strain gage bridge
(Dimensions Fig. 7. Instrument
system of weight
dropping
in
mm)
type apparatus
RESULTS AND DISCUSSION
To obtain results of dynamic fracture toughness over a wide range of loading rates, two dynamic fracture tests are conducted using the air gun (A method) and the weight dropping type apparatus (B method). Figure 9 illustrates the relationship between the dynamic fracture toughness Z& and the loading rate &l(t) for each plate thickness of PMMA. The figure includes results of the dynamic fracture toughness determined by A and B methods, together with the result by Kobayashi et al. [8] over a range of relatively lower loading rates with the instrumented Charpy impact test. We find that the dynamic fracture toughness becomes larger as the plate thickness decreases, from 15 to 5 mm for the results by the A method in the figure. We suppose that this results from a relaxation of the deformation around the crack tip in the direction of the plate thickness with decreasing plate thickness. However, plots of the results by the B method lie scattered over a range of relatively lower loading rates compared with those by the A method. The scatter in the results arises due to errors involving the conditions at the crack tip, the position of the strain gage etc. These results suggest a need to conduct further investigation. However, results by both A and B methods precisely indicate the minimum value and the loading rate effect on the dynamic fracture toughness.
TImet( US) $1)=1_04 I 104(Mpa J m/s) Fig. 8. Determination
of dynamic
fracture
toughness
(A method).
810
H. WADA
et al.
x Kobayashi et al.
i
4 Fig. 9. Relationship
’
4 Lo&(t)
between
dynamic
(MPaJm/s) fracture
toughness
and loading
rate.
Since the fracture initiation time is determined from the strain gage signal on the surface of the specimen and we assume that the fracture initiates from inside the specimen thickness, a problem arises in determination of dynamic fracture toughness. Although we can get the dynamic SIF inside the specimen thickness by three-dimensional finite element analysis, it is very difficult to measure the fracture initiation time inside the plate thickness. In the A method the dynamic fracture toughness is determined by comparing the fracture initiation time to the simulation curve of the dynamic SIF by a two dimensional finite element analysis assuming plane strain condition for the thick specimen. Further we think that the definition of loading rate has to be considered. We will inquire further into these problems concerning the determination of dynamic fracture toughness and the definition of the loading rate. CONCLUSIONS Consequently,
the following conclusions may be drawn:
1. The single axis strain gage method can be applied to measure the dynamic stress intensity factor in a practical manner. 2. The impact loading apparatus with free fall of a striker is a possible means to conduct a fracture test over a range of loading rates giving a minimum fracture toughness for PMMA. 3. The dynamic fracture toughness of PMMA becomes larger as the plate thickness decreases, from 15 to 5 mm. 4. The dynamic fracture toughness K ,d of PMMA indicates a minimum value at ca k,(r) = 1.3 x lo4 (MPa,,f-/m s) in the loading rate. The minimum value is K,,&O.3 (MPa,,&) and is about 20 percent value of the static fracture toughness. 5. The dynamic fracture toughness of PMMA increases rapidly as the in the range between k,(t) = 1.0 x lo4 (MPafi/s) and k,(t) = 6.7 x lo4 (MPa thickness. Acknowledgements-The authors would like to thank Daido All of the present calculations were carried out using HITAC Technology.
Steel Company in Nagoya for partially funding this work. M-840/20 at the Computation Center in Daido Institute of
REFERENCES [I] H. Wada, Determination of dynamic fracture toughness for PMMA. Engng Fracture Mech. 41, 821-831 (1992). A novel impact three-point bend test method for determining dynamic [2] T. Yokoyama and K. Kishida, fracture-initiation toughness. Exp. Mech. 29, 188-194 (1989). [3] H. Kurosaki, H. Nozaki and S. Hukuda, Mode I stress-intensity factor measurements by the electrical strain gage. Trans. of the JSME (in Japanese) 56, 875-882 (1990). [4] N. M. A. Newmark, A method of computation for structural dynamics. J. Engng Mech. Div., ASCE 85,67-94 (1959). [S] H. Wada, T. Takagi and T. Nishimura, A method obtaining stress intensity factor by FEM and its application to dynamic problem. BUN. of the JSME 25, l-8 (1982).
investigation of loading rate and plate thickness effects
811
3rd edn. Wiley [6] R. D. Cook, D. S. Malkus and M. E. Plesha, Conceprs and Applications of’Finite Element Analk, (1988). [7] R. S. Barsoum. On the use of isoparametric finite elements in linear fracture mechanics. ht. J. numer. Mrth. Engng IO, 25-37 (1978). [8] T. Kobayashi. H. Miyata, K. Kikukawa and T. Higashihara, Instrumented impact test and evaluation of fracture toughness in PMMA. Trans. qf the JSME (in Japanese) 55, 24342438 (1989). (Received 24 Fehruar), 1995)
Pergamon
0013-7944(95)00244-8
INVESTIGATION OF LOADING RATE THICKNESS EFFECTS ON DYNAMIC TOUGHNESS OF PMMA
Printed in Great Britain. All rights reserved 0013-7944/96
$15.00 + 0.00
AND PLATE FRACTURE
H. WADA and M. SEIKA Department of Mechanical Engineering, Daido Institute of Technology, 2-21 Daido-cho Minami-ku, Nagoya 457, Japan T. C. KENNEDY and C. A. CALDER Department of Mechanical Engineering, Oregon State University, Corvallis, OR 97331, U.S.A. K. MURASE Department of Transport Machine Engineering, Meijo University, Tenpaku-ku, Nagoya 468, Japan Abstract-To investigate the effects of loading rate and plate thickness on the fracture toughness of PMMA (polymethyl methacrylate) under impact loading, two methods, A method and B method, are applied as follows. In the A method, a dynamic finite element method and a strain gage method are applied to measure the dynamic fracture toughness in the fracture test using an air gun. In the B method, a single axis strain gage method is applied to measure the critical dynamic stress intensity factor, namely dynamic fracture toughness, in the fracture test using a weight dropping type apparatus. The dimensions of the PMMA specimen are L = 140 mm length and W = 30 mm width. Three values of the plate thickness E, 15.0 mm, 10.0 mm and 5.0 mm, are selected to investigate the plate thickness effect in the fracture test. Both results by the A and B methods precisely indicated the minimum value and the loading rate effect on the dynamic fracture toughness. Copyright 0 1996 Elsevier Science Ltd.
INTRODUCTION SINCE PMMA (polymethyl methacrylate) has many excellent properties, it is employed as an industrial material in general. Hence, investigating the fracture behavior for this material is of importance in the strength evaluation of structure. Particularly, dynamic fracture toughness Krd depends on loading rate x,(t), plate thickness B, etc. It has also a minimum value at some loading rate. Accordingly, investigating their effects on PMMA is reasonable to deepen universal understanding of the fracture toughness of several materials. Especially, it is known generally that static fracture toughness decreases as the plate thickness increases. We have an interest in this effect on the dynamic fracture toughness of PMMA under impact loading. The purpose of the present study is to investigate the effects of loading rate and plate thickness on the fracture toughness of PMMA under impact loading. The dynamic fracture toughness is measured for the specimen with three values of the plate thickness over a wide range of loading rates by the following two methods, A method [I] and B method. In the A method, a dynamic finite element method and a strain gage method are applied to measure the dynamic fracture toughness. The dynamic fracture toughness is determined by comparing the crack initiation time tf from the strain gage signal to the simulation curve of the dynamic stress intensity factor (SIF) K,(t) computed by a dynamic finite element method [2]. An air gun is used for the impact fracture test under single-point impact bending. In this method the dynamic fracture toughness is obtained over a range of relatively higher loading rates. In the B method, a single axis strain gage method [3] is applied to measure the dynamic stress intensity factor. The dynamic fracture toughness is determined from the fracture initiation point obtained from the strain gage signal. An impact loading apparatus with free fall of a striker is used for the impact fracture test under three-point bending. 805
806
H. WADA Table Young’s
1. Mechanical
modulus
E @Pa)
4.97
et al.
properties
of specimen
Poisson’s
ratio
0.388
Y
material
Density
p (kg/m3)
1.18 x 10’
EXPERIMENTS AND DETERMINATIONS OF & Material and specimen The material chosen for testing is commercially available PMMA. The material properties are given in Table 1. Young’s modulus E in the table is a dynamic value measured from the velocity of the stress wave propagation. The dimensions of the PMMA specimen are L = 140 mm length and W = 30 mm width as shown in Fig. 1. Three values of the plate thickness B, 15.0, 10.0 and 5.0 mm, are selected to investigate the plate thickness effect in the fracture test. A strain gage is mounted at a position Y+ 1.5 mm away from the crack tip to measure the crack initiation time tf or the dynamic SIF K,(t) in the fracture test with the A method, as shown in Fig. 1. The distance r is measured by precisely magnifying it 20 times with a universal projector. A method A dynamic finite element method and strain gage method are applied to measure the dynamic fracture toughness. The dynamic fracture toughness is determined by comparing the crack initiation time from the strain gage signal to the simulation curve of the dynamic stress intensity factor computed by a dynamic finite element method. An air gun is used for the impact fracture test under single-point bending. In this method the dynamic fracture toughness is obtained over a range of relatively higher loading rates. Figure 2 illustrates the impact test apparatus using the air gun produced by the authors and the measuring system. The apparatus consists of an incident bar, a stand of specimen, a striker, a barrel and a compressor. The striker made of PMMA is 21.8 mm in diameter and 200 mm in length. The PMMA-striker bar is projected through the barrel made of a steel pipe (22.7 mm in inside diameter, 45 mm in outside diameter and 1000 mm in length) by releasing compressed air from an air tank with a solenoid valve. Then the projected striker bar strikes the incident bar of PMMA (2000 mm in length and 20 mm in diameter) contacting the specimen, resulting in the impact load applied to the specimen. Since the specimen, mounted on two smooth thin rods, has no supports in the direction of the impact load as shown in Fig. 3, it is assumed that the specimen fractures only with its own inertia. The velocity of the striker can be changed freely by adjusting the pressure of air. A time history of the impact load applied to the specimen is measured with the strain gage mounted at the center of the incident bar, while the crack initiation time tf is determined with the strain gage mounted in the neighborhood of a crack tip. The output signal from each strain gage is amplified with an amplifier (Kyowa Dengyo Co., CDV/CDA-230C, frequency response 200 kHz) through a gage bridge, recorded in a transient converter (Rikendensi Co., TCL-005DG, 6 cha, 4096 word/cha, 10 bit) and finally processed with a personal computer (Nippon Electric Co., PC-980M). A part of the strain pulse occurring in the incident bar due to the collision of the striker reflects to the incident bar and another part of the pulse transmits in 140
B=( 5,10 ,15)
(Dimensions Fig. 1. Specimen
configuration
in
mm)
and gage position.
Investigation
of loading
rate and plate thickness
effects
807
(Dimensions
in
mm)
Oscilloscope
Fig. 2. Impact
test apparatus
and block diagram
of instrument
system
the specimen. A time history of the applied load to the specimen is obtained from the signal of the strain gage mounted at the center of the incident bar [2]. In the dynamic finite element simulation, assuming a plate element of homogeneous and isotropic material, an equation of motion can be expressed in matrix form as follows:
[Ml(~) + [KlCd> = P’) 3
(1)
where [K] and [M] are the assembled stiffness and mass matrices, respectively, (P} the incremental vector of an external force and {d} the incremental vector of nodal displacement. The superposed dot (.) implies differentiation with respect to time. Equation (1) is solved by the so-called j3 method of Newmark [ 4, 51, using the consistent mass matrix for [Ml, obtained from the quadratic isoparametric plane elements [6] with eight nodes for the stiffness matrix [K] and the values measured on the experiment for {P}. The quarter point element is used around the crack tip as shown in Fig. 4 [7]. The dynamic SIF is calculated by K,(t) = 2uG(27c/e)“*/(k-+ 1) ,
(2)
where G is the shear modulus of specimen material, v is the nodal displacement near the crack tip as shown in Fig. 4, e is the distance from the crack tip to the nodal point near the crack tip and K = 3 - 4v for the plane strain condition or K = (3 - v)/(l + v) for the plane stress condition where v is Poisson’s ratio of the specimen material. Figure 5 is a typical example of the finite element mesh based on the symmetry of the specimen used for two-dimensional analysis. The crack initiation time tf is measured from the signal of a strain gage mounted at the position e + 1.5 mm
Specimen
Strain *
gage
Cra
Fig. 3. Loading
configuration.
Fig. 4. Crack
tip elements.
808
H. WADA
Fig. 5. Finite element
et al.
mesh for specimen.
away from the crack tip. Since the signal indicates a sudden increase at the crack initiation instant, the dynamic fracture toughness Kid [2] is determined by comparing the tr to the simulation curve of the dynamic SIF computed by FEM, as shown in Fig. 6. B method A single axis strain gage method [3] is applied to measure the dynamic stress intensity factor. The dynamic fracture toughness is determined from the fracture initiation point obtained from the strain gage signal. An impact loading apparatus with free fall of a striker is used for the impact fracture test under three-point bending. In order to perform dynamic three point bending tests, the weight dropping type apparatus was developed as shown in Fig. 7. Impact velocities are controlled by manual lifting to the desired height and the mass of striker is 15 kg. The figure indicates the instrument system for the fracture test. The dynamic SIF K,(t) and crack initiation time tr are determined from the signal of the strain gage mounted in the neighborhood of the crack tip in the specimen. The K,(t) [3] is calculated from eq. (3) with the single axis strain gage. K,(t) = EEy(27rr)“*/(1- v) )
(3)
where E is Young’s modulus, E, the strain measured with the strain gage near crack tip and r the distance from the crack tip to the gage position. The output signal from the strain gage is amplified with an amplifier (Kyowa Dengyo Co., CM/GA-230C, frequency response 200 kHz) through a gage bridge, recorded in a digital oscilloscope (Philip, D14-125GR/177) and finally processed with a personal computer (Nippon Electric Co., PC-9801Vm), as shown in Fig. 7. Figure 8 illustrates a typical example of the time history of the dynamic SIF calculated using eq. (3) from the strain gage signal mounted near the crack tip. We can see the fracture initiation point clearly in the figure.
Timet(,Us) Fig. 6. Determination
of dynamic
fracture
toughness
(A method).
Investigation
of loading
rate and plate thickness
effects
809
Striker
c Strain gage bridge
(Dimensions Fig. 7. Instrument
system of weight
dropping
in
mm)
type apparatus
RESULTS AND DISCUSSION
To obtain results of dynamic fracture toughness over a wide range of loading rates, two dynamic fracture tests are conducted using the air gun (A method) and the weight dropping type apparatus (B method). Figure 9 illustrates the relationship between the dynamic fracture toughness Z& and the loading rate &l(t) for each plate thickness of PMMA. The figure includes results of the dynamic fracture toughness determined by A and B methods, together with the result by Kobayashi et al. [8] over a range of relatively lower loading rates with the instrumented Charpy impact test. We find that the dynamic fracture toughness becomes larger as the plate thickness decreases, from 15 to 5 mm for the results by the A method in the figure. We suppose that this results from a relaxation of the deformation around the crack tip in the direction of the plate thickness with decreasing plate thickness. However, plots of the results by the B method lie scattered over a range of relatively lower loading rates compared with those by the A method. The scatter in the results arises due to errors involving the conditions at the crack tip, the position of the strain gage etc. These results suggest a need to conduct further investigation. However, results by both A and B methods precisely indicate the minimum value and the loading rate effect on the dynamic fracture toughness.
TImet( US) $1)=1_04 I 104(Mpa J m/s) Fig. 8. Determination
of dynamic
fracture
toughness
(A method).
810
H. WADA
et al.
x Kobayashi et al.
i
4 Fig. 9. Relationship
’
4 Lo&(t)
between
dynamic
(MPaJm/s) fracture
toughness
and loading
rate.
Since the fracture initiation time is determined from the strain gage signal on the surface of the specimen and we assume that the fracture initiates from inside the specimen thickness, a problem arises in determination of dynamic fracture toughness. Although we can get the dynamic SIF inside the specimen thickness by three-dimensional finite element analysis, it is very difficult to measure the fracture initiation time inside the plate thickness. In the A method the dynamic fracture toughness is determined by comparing the fracture initiation time to the simulation curve of the dynamic SIF by a two dimensional finite element analysis assuming plane strain condition for the thick specimen. Further we think that the definition of loading rate has to be considered. We will inquire further into these problems concerning the determination of dynamic fracture toughness and the definition of the loading rate. CONCLUSIONS Consequently,
the following conclusions may be drawn:
1. The single axis strain gage method can be applied to measure the dynamic stress intensity factor in a practical manner. 2. The impact loading apparatus with free fall of a striker is a possible means to conduct a fracture test over a range of loading rates giving a minimum fracture toughness for PMMA. 3. The dynamic fracture toughness of PMMA becomes larger as the plate thickness decreases, from 15 to 5 mm. 4. The dynamic fracture toughness K ,d of PMMA indicates a minimum value at ca k,(r) = 1.3 x lo4 (MPa,,f-/m s) in the loading rate. The minimum value is K,,&O.3 (MPa,,&) and is about 20 percent value of the static fracture toughness. 5. The dynamic fracture toughness of PMMA increases rapidly as the in the range between k,(t) = 1.0 x lo4 (MPafi/s) and k,(t) = 6.7 x lo4 (MPa thickness. Acknowledgements-The authors would like to thank Daido All of the present calculations were carried out using HITAC Technology.
Steel Company in Nagoya for partially funding this work. M-840/20 at the Computation Center in Daido Institute of
REFERENCES [I] H. Wada, Determination of dynamic fracture toughness for PMMA. Engng Fracture Mech. 41, 821-831 (1992). A novel impact three-point bend test method for determining dynamic [2] T. Yokoyama and K. Kishida, fracture-initiation toughness. Exp. Mech. 29, 188-194 (1989). [3] H. Kurosaki, H. Nozaki and S. Hukuda, Mode I stress-intensity factor measurements by the electrical strain gage. Trans. of the JSME (in Japanese) 56, 875-882 (1990). [4] N. M. A. Newmark, A method of computation for structural dynamics. J. Engng Mech. Div., ASCE 85,67-94 (1959). [S] H. Wada, T. Takagi and T. Nishimura, A method obtaining stress intensity factor by FEM and its application to dynamic problem. BUN. of the JSME 25, l-8 (1982).
investigation of loading rate and plate thickness effects
811
3rd edn. Wiley [6] R. D. Cook, D. S. Malkus and M. E. Plesha, Conceprs and Applications of’Finite Element Analk, (1988). [7] R. S. Barsoum. On the use of isoparametric finite elements in linear fracture mechanics. ht. J. numer. Mrth. Engng IO, 25-37 (1978). [8] T. Kobayashi. H. Miyata, K. Kikukawa and T. Higashihara, Instrumented impact test and evaluation of fracture toughness in PMMA. Trans. qf the JSME (in Japanese) 55, 24342438 (1989). (Received 24 Fehruar), 1995)