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Energy dissipation and temperature rise associated with crack extension in a woven glass-epoxy laminate at low temperatures
PII: S0011-2275(97)00170-7
Cryogenics 38 (1998) 381–386 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$19.00
Energy dissipation and temperature rise associated with crack extension in a woven glass-epoxy laminate at low temperatures S. Ueda* and Y. Shindo Department of Materials Processing, Graduate School of Engineering, Tohoku University, Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
Received 26 June 1997 Failures, fracture (cracking) and debonding of filler materials used in the winding of a high-performance superconducting magnet generate heat. When combined with the high thermal response of the materials at low temperatures, the small heat input may result in premature quenching of the magnet. An analytical procedure, using a finite element method, was developed to calculate the dissipative energy and temperature rise associated with crack extension in a woven glass-epoxy laminate(G-10) at low temperatures. The amount of energy dissipated during partial fracture of the test specimen is calculated as a function of crack speed using a dynamic strain energy release rate. The dissipative energy is compared with the heat output determined experimentally at 77 K, and the conversion rate of dissipative energy into heat is obtained. From the average value for the conversion rate, the heat outputs at 77 K for total fracture and at 4 K for partial fracture are predicted. Temperature elevations at the crack tip are also calculated. 1998 Elsevier Science Ltd. All rights reserved Keywords: composites (A); mechanical properties (C); thermal conductivity (C); superconducting magnets (F)
Adiabatic, high-current-density superconducting magnets used in applications such as nuclear magnetic resonance and high energy physics are extremely susceptible to premature quenches triggered by small heat inputs generated within the magnet winding1. Because energy dissipated during crack formation or fracture of materials can affect the stability of the high performance superconducting magnets, quantification of such dissipative energy is important for superconducting magnet technology in particular and for the understanding of mechanical behavior of materials at low temperature in general2. Voccio et al.3 designed a experimental apparatus to measure the energy released during fracture in woven glass-epoxy laminates at 77 K. Fuchino and Iwasa2 also developed a new cryomechanics measurement technique and demonstrated that the technique was able to measure pulses of heating energy as small as 10 nJ at 4 K. A dynamic strain energy release rate was applied by Do¨ll4 to the fast fracture process of PMMA (polymethylmethacrylate) for which the amount of dissipated energy converted into heat was known from his previous work5. As a result of his calculations, it was found that in PMMA at the onset of rapid crack propagation
*To whom correspondence should be addressed.
approximately 60 percent of the dynamic strain energy release rate was turned into heat. The objective of the present study is to develop an analytical procedure to calculate the dissipative energy during crack extension and to relate it to the experimentally determined heat output3. A two dimensional finite element analysis is used to implement the procedure. The material is a woven glass-epoxy laminate (G-10) commonly used for cryogenic applications. A dynamic strain energy release rate based on Freund’s solution6 is employed to compute the dissipative energy. The dissipative energy is compared with the experimentally determined heat output at 77 K, and the conversion rate of dissipative energy into heat is obtained. Using the average value for the conversion rate, the heat output at 77 K for total fracture and at 4 K for partial fractures are predicted. Three dimensional finite element calculations are also carried out to obtain local temperature elevations accompanying the crack extension.
Statement of the problem We consider the notched woven glass-epoxy laminate (G10) tensile specimen3 shown in Figure 1. Figure 2(a) and (b) illustrate finite element models of the notched woven glass-epoxy laminate tensile specimen. Since the problem
Cryogenics 1998 Volume 38, Number 4 381
Energy dissipation and temperature rise in a woven glass-epoxy laminate: S. Ueda and Y. Shindo
∂ux(x,y) ∂x
∂uy(x,y) ∂y
∂u (x,y)
Figure 1 Notched glass-epoxy laminate(G-10) tensile specimen (dimensions in mm)
x
∂y
+
∂uy(x,y) ∂x
where xx(x,y), yy(x,y) and xy(x,y) are the stress tensors, E is the Young’s modulus and is the Poisson’s ratio. The effect of the plain weave fabrics of the woven laminate is separated from the material properties through the use of geometric efficiency factor  = 0.27. The load-displacement curve for a typical partial fracture event is presented in Figure 3. These partial fractures are generally characterized by a load drop and a corresponding sample extension. Point A corresponds to the critical condition just before the crack growth; the initial crack length is aA( = a0 ) ⬇ 0, the initial load is PA and the initial ydisplacement at loading point is uyA. When crack extends by aB-aA, the load drops to PB and the displacement increases linearly to uyB. The initial displacement uyA, the final displacement uyB, the initial load PA and the final load PB are the measured values 3. Using the values provided above, the Young’s modulus E of the specimen and the final crack length aB are determined. The crack extension problem is divided into K + 1 steps. If the crack of length ak (aAˆ = a0 ⬍ a1 ⬍ $ ⬍ ak ⬍ ak + 1 ⬍ $ ⬍ aK ⬍ aK + 1 = aB ) extends by an element size ak + 1 - ak, the load Pk + 1 (PA ⬎ P1 ⬎ $ ⬎ Pk ⬎ Pk + 1 ⬎ $ ⬎ PK ⬎ PK + 1 = PB )displacement uyk + 1 (uyA = uy0 ⬍ uy1 ⬍ $ ⬍ uyk ⬍ uyk + 1 ⬍ $ ⬍ uyK ⬍ uyK + 1 = uyB ) record would follow the linear path AB.
Energy dissipation and heat output associated crack extension An energy method8 is used to calculate the static stress intensity factor at the crack tip. For plane stress conditions
Figure 2 Finite element models for G-10 tensile specimen
is symmetric, only the top half of the specimen was modeled. The specimen is assumed to be under plane stress conditions. The coordinates x and y coincide with the principal axes of the laminate. By representing the x and y components of displacements as ux(x,y) and uy(x,y), respectively, the constitutive equations for the woven laminate can be written as E 1 − 2
E 1 − 2
0
E yy(x,y) = 1 − 2 xy(x,y) 0
E 1 − 2
0
冤 冥 xx(x,y)
382
0
E 2(1 + )
Cryogenics 1998 Volume 38, Number 4
(1) Figure 3 Load-displacement curve for a typical partial fracture event and crack extension
Energy dissipation and temperature rise in a woven glass-epoxy laminate: S. Ueda and Y. Shindo the stress intensity factor KSk (k = 0 苲 K) for the opening mode is given by KSk =
再 冎 E⌬Uk ⌬ak
1/2
(k = 0 苲 K)
(2)
where ⌬Uk and ⌬ak indicate the slightly different strain energy and crack size. In Equation (2) it is assumed that the external loading is kept constant during differentiation. A dynamic stress intensity factor Kk(v) and a dynamic strain energy release rate Gk(v) are given by6 Kk(v) = Gk(v) =
1 − (v/vR ) K (k = 0 苲 K) 兵1 − q(v/vR )其1/2 Sk
r2(v/vR )2兵1 + q2(v/vR )2其1/2Kk(v)2 [4兵1 − q2(v/vR )2其1/2兵1 − r2(v/vR )2其1/2 − 兵2 − r2(v/vR )2其2 ]E
(3)
(4)
(k = 0 苲 K)
where q = r兵(1- )/2(1-2 )其1/2, r = vR/c2, v is the crack velocity, vR is the Rayleigh surface wave velocity and c2 is the shear wave velocity. The dissipative energy f(v) and the conversion rate ␣(v) of dissipative energy into heat become:
冘 G (v)(a
∂ ∂x +
∂ ∂z
再
x( )
再
z( )
冎 再 冎
∂(x,y,z,t) ∂ + ∂x ∂y
y( )
冎
∂(x,y,z,t) ∂y
(7)
∂(x,y,z,t) ∂(x,y,z,t) = cp( ) ∂z ∂t
where t is the time, (x,y,z,t) is the temperature, x( ), y( ) and z( ) are the thermal conductivities, cp( ) is the specific heat and is the mass density. The initial condition is
(x,y,z,t) = ⌽A (t ⬍ tA )
(8)
where ⌽A is an initial temperature. Figure 4 shows the thermal boundary conditions. Reflective symmetry about the crack plane (y = 0.0) and the longitudinal midplane (z = 0.0) enable the use of one-quarter model as indicated. Three types of boundary conditions are considered: ∂(x,y,z,t) = q0(t) ∂y
(tA ⱕ t ⱕ tB )
∂(x,y,z,t) = h( )兵(x,y,z,t) − ⌽A其 (tB ⬍ t) ∂y
冧
on ⌫1
(9)
K
f(v) = 2b
k
k+1
− ak )
(5)
k=0
␣(v) =
exp f(v)
(6)
where 2b is the specimen thickness and exp is the experimenally determined heat output3. For each event the conversion rate for the partial fracture at 77 K is obtained as a function of the crack velocity. The average value for the rate ␣(v) is used to determine the heat output h(v) for a total fracture event at 77 K. Superconducting magnets are maintained at 4 K and it is important to estimate the heat output during crack extension at 4 K. Cooling G-10CR woven glass-epoxy laminate from 77 K to 4 K increases the value of parameters: Young’s modulus about 7% and tensile strength about 5%9. G-10CR is the cryogenic grade material of G-10 and meets federal and military specifications for G-10 product. Under the assumption that the values of PA and PB at 4 K are 7% higher than those at 77 K, we can also calculate the dissipated energy as a function of the crack velocity at 4 K using Equation (5). With this dissipated energy and the average value for the conversion rate at 77 K, the heat output at 4 K can be determined.
∂(x,y,z,t) = 0(tA ⱕ t) on ⌫2 ∂y
(10)
(x,y,z,t) = ⌽A (tA ⱕ t) on ⌫3
(11)
where h( ) is the coefficient of convective heat transfer, ⌫1 苲 ⌫3 represent the boundaries shown in Figure 4. To calculate the temperature distribution around the crack tip by the finite element procedure, we use the heat production q0(t) as follows: q0(t) = Q[1 − exp兵 − (t − tA )/TR其]
(12)
Temperature rise near the crack tip At time tA the crack rapidly advances causing a sudden drop in the load until the crack arrests at time tB = tA + aB/v. We analyze the transient temperature state near the crack tip and include the effect of temperature-dependent material properties. A three dimensional finite element analysis is used to implement the procedure. The differential equation governing the unsteady heat conduction problem is10
Figure 4 Thermal boundary conditions
Cryogenics 1998 Volume 38, Number 4 383
Energy dissipation and temperature rise in a woven glass-epoxy laminate: S. Ueda and Y. Shindo Table 1 Young’s modulus E and crack length aB
Event No. Partial fracture Partial fracture Partial fracture Partial fracture Total fracture
Q=
1 2 3 4
Numerical results E (Gpa) aB (mm)
PA (N)
PB (N)
8.86 8.18 8.67 9.20 8.81
260 533 233 450 1120
224 518 188 441 0
0.931 0.226 0.674 0.272 –
h(v)
冕
(13)
tB
[1 − exp兵 − (t − tA )/TR其]dt
tA
where TR denotes the relaxation time. As the time dimension is of an infinite extent, we shall deal with finite time domains tm − 1 ⬍ t ⱕ tm (m = 1,2,$) and repeat the calculation for subsequent domains, where tA = t0 ⬍ t1 ⬍ t2 ⬍ $ ⬍ tm − 1 ⬍ tm ⬍ $. The process will thus lead to a step-by-step calculation and we use the procedure of time-stepping derived by Crank-Nicholson formula. In the time domains tm − 1 ⬍ t ⱕ tm (m = 1,2,$), the thermal properties of the nth element are approximated by the properties at the mean temperature n(tm − 1 ) (n = 1,2,$,N, m = 1,2,$) of the nth element, where N is the number of the total elements. The mean temperature n(tm − 1 ) is given by
n(tm − 1 ) =
1 ⍀n
冕冕冕
(x,y,z,tm − 1 )dxdydz
(14)
⍀n
(n = 1,2,$,N, m = 1,2$) where ⍀n (n = 1,2,$,N) is the region of the nth element.
Experimental results 3 uyA (mm) uyB (mm) 0.36 0.80 0.33 0.60 1.42
exp (mJ)
0.47 0.83 0.35 0.64 –
11.3 6.0 5.8 5.6 900.0
Numerical results and discussion The accepted value for the Poisson’s ratio at 77 K is 0.29. For each event the predicted Young’s modulus E and crack length aB are given in Table 1, along with the initial load PA, the final load PB, the initial displacement uyA, the final displacement uyB and the measured heat output exp3. Normalized dynamic stress intensity factor Kk(v)/0a1/2 B and the dynamic strain energy release rate Gk(v) (J/m2 ) for partial fracture 1 are presented in Table 2. 0 is the average stress
0 =
PA S
(15)
where S is the ligament cross-section. The dynamic stress intensity factor and the dynamic strain energy release rate decrease with increasing the crack velocity. The conversion rates ␣(v) of dissipative energy f(v) into heat exp as a function of the crack velocity for event no. 1 苲 4 are presented in Figure 5 as continuous lines. The average value, which is shown as a dashed line, increases from about 0.52 (v/vR = 0.0) to about 0.91 (v/vR = 0.3). To make the significance of this point a little clearer, the actual crack velocity should be measured. The dissipated energy f(v) and the heat output h(v) as a function of the crack velocity for the total fracture event are given in Figure 6. The dissipative energy decreases with increasing the crack
Table 2 Normalized dynamic stress intensity factor Kk(v)/0aB1/2 and dynamic strain energy release rate Gk(v) (J/m2 ) for partial fracture 1
k
1
2
3
4
5
6
ak/aB
0.154
0.308
0.462
0.616
0.770
0.924
8.71 8.64 8.37 8.05 7.33 6.61
9.00 8.94 8.66 8.32 7.59 6.83
26518 26121 24545 22595 18825 15265
28362 27938 26252 24166 20134 16326
Kk(v)/0aB1/2
v/vR 0.0 0.01 0.05 0.1 0.2 0.3
6.77 6.72 6.51 6.26 5.70 5.14
7.15 7.10 6.88 6.61 6.02 5.43
8.32 8.26 8.00 7.69 7.01 6.32
Gk(v) (J/m2 )
v/vR 0.0 0.01 0.05 0.1 0.2 0.3
384
8.62 8.55 8.29 7.96 7.29 6.54
16050 15810 14856 13676 11394 9239
17895 17627 16563 15247 12703 10300
Cryogenics 1998 Volume 38, Number 4
24225 23863 22423 20641 17197 13945
25970 25581 24037 22128 18435 14949
Energy dissipation and temperature rise in a woven glass-epoxy laminate: S. Ueda and Y. Shindo
Figure 5 Conversion rates ␣( v ) as a function of the crack velocity for partial fractures at 77 K
Figure 7 Dissipated energy f( v ) and heat output h( v ) as a function of the crack velocity for the partial fracture 1 at 77 K and 4 K
where qb(⌬ ) is the heat flux and ⌬ = -⌽A. The transient temperature distributions were evaluated by taking TR = 1/100, 1/500 and 1/1000 sec in Equation (12). These results were in substantial agreement. Thus it may be said that the result for TR = 1/500 sec is, from a practical view point, quite satisfactory. Figure 8 shows the variation of the temperature with time at the crack tip for the partial fracture 1 and v/vR = 0.1 at ⌽A = 77 K. The solid line and the dashed line denote the results at z/b = 0 and 0.5, respectively. The temperatures first increase, go through maximums, and then decrease as t increases. The peak value of the temperature at z/b = 0.0, which occurs at t = tB, is 476 K. Figure 9 shows the temperature at the crack tip (z/b = 0.0) as a function of time for the partial fracture 1 and v/vR = 0.1 at ⌽A = 77 K, 4 K. The continuous line and the dashed line denote the results at ⌽A = 4 K and 77 K, respectively. As a result of our calculations, it is found that the temperature rise at ⌽A Figure 6 Dissipated energy f( v ) and heat output h( v ) as a function of the crack velocity for the total fracture at 77 K
velocity. Using the average value for the conversion rate at 77 K and the dissipated energy f(v), the heat output h(v) becomes about 600 (mJ). The measured heat output exp shown in Table 1 is 900 (mJ). This difference may arise mainly from the neglect of the kinetic energy due to the sudden motion of the bellows sealing the system3. The dissipative energy and the heat output for partial fracture 1 at 4 K are shown in Figure 7 as continuous lines. The results for partial fracture 1 at 77 K are also shown as dashed lines. The dissipative energy and the heat output at 4 K are about 5% higher than those at 77 K. The specific heat cp( ), the thermal conductivity ( ) and the mass density for G-10CR have been used9,11. The heat transfer coefficient h( ) is calculated from the boiling curve of the surrounding medium as follows12: h( ) =
qb(⌬ ) ⌬
(16)
Figure 8 Temperature rise at the crack tip ( z/b = 0.0, 0.5) for the partial fracture 1 at 77 K
Cryogenics 1998 Volume 38, Number 4 385
Energy dissipation and temperature rise in a woven glass-epoxy laminate: S. Ueda and Y. Shindo 3. The heat output for the total fracture event is estimated, and the heat output in the fracture process at liquid helium temperature is also predicted. 4. Because of the extremely low heat capacity of the materials at 4 K, the peak value of the temperature at 4 K is more than ten times greater than that at 77 K.
References
Figure 9 Temperature rise at the crack tip ( z/b = 0.0) for the partial fracture 1 at 77 K and 4 K
= 4 K is much higher than the rise at ⌽A = 77 K and the peak value of the temperature at ⌽A = 4 K is 5128 K.
Concluding remarks A procedure is developed to predict the energy dissipation and temperature rise during crack extension in woven glassepoxy laminates at low temperatures. The procedure is implemented in two and three dimensional finite element programs. Based on this study the following conclusions are made: 1. 2.
386
The dissipative energy associated with crack extension at 77 K is calculated as a function of the crack velocity using a dynamic strain energy release rate. In order to determine the conversion rate of dissipateve energy into heat, the calculated dissipative energy is compared with the experimentally determined heat output at 77 K. The conversion rates for the partial fractures are obtained as a function of the crack velocity.
Cryogenics 1998 Volume 38, Number 4
1. Iwasa, Y., Experimental and theoretical investigation of mechanical disturbances in epoxy-impregnated superconducting coils. 1. General introduction. Cryogenics, 1985, 25, 304–306. 2. Fuchino, S. and Iwasa, Y., A cryomechanics technique to measure dissipated energies of 10 nJ. Experimental Mechanics, 1990, 30(6), 356–359. 3. Voccio, J.P., Iwasa, Y., Maksimov, I.L. and Bovrov, E.S., Experimental and theoretical study of fracture energy and crack dynamics in G-10 at 77 K. Advances in Cryogenic Engineering, 1988, 34, 99–106. 4. Do¨ll, W., Application of an energy balance and an energy method to dynamic crack propagation. International Journal of Fracture, 1976, 12(4), 595–605. 5. Do¨ll, W., An experimental study of the heat generated in the plastic region of a running crack in different polymertic materials. Engineering Fracture Mechanics, 1973, 5, 259–268. 6. Freund, L.B., Crack propagation in an elastic solid subjected to general loading-I. Constant rate of extension. Journal of the Mechanics and Physics of Solids, 1972, 20, 129–140. 7. Zako, M., Mori, T., and Miyoshi, T., Study on fracture toughness for composite materials (Fracture toughness evaluation by equivalent stress intensity factor Ke ) Transactions of the Japan Society of Mechanical Engineers 1979, 45, 726–733 (in Japanese). 8. Aamodt, B. and Klem, F. Application of numerical techniques in practical fracture mechanics, in: Fracture Mechanics in Engineering Practice (Ed. P.Stanley), Applied Science Publishers Ltd. (1976) 33–40 9. Kasen, M.B., MacDonald, G.R., Beekman, D.H. and Schramm, R.E., Mechanical, electrical, and thermal characterization of G-10CR and G-11CR glass-cloth/epoxy laminates between room temperature and 4 K. Advances in Cryogenic Engineering, 1980, 26, 235–244. 10. Carslaw, H.S., and Jeager, J.C., Conduction of Heat in Solids, Clarendon Press (1959) 1–49 11. Shindo, Y. and Ueda, S., Transient thermal-mechanical behavior of cracked glass-cloth-reinforced epoxy laminates at low temperatures. Advances in Cryogenic Engineering, 1996, 42, 137–146. 12. Cryogenic Association of Japan (Ed.), Cryogenic Engineering Handbook, Uchida Rokakuho Publishing Co., Ltd. 1983, 207–237 (in Japanese)
Cryogenics 38 (1998) 381–386 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$19.00
Energy dissipation and temperature rise associated with crack extension in a woven glass-epoxy laminate at low temperatures S. Ueda* and Y. Shindo Department of Materials Processing, Graduate School of Engineering, Tohoku University, Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
Received 26 June 1997 Failures, fracture (cracking) and debonding of filler materials used in the winding of a high-performance superconducting magnet generate heat. When combined with the high thermal response of the materials at low temperatures, the small heat input may result in premature quenching of the magnet. An analytical procedure, using a finite element method, was developed to calculate the dissipative energy and temperature rise associated with crack extension in a woven glass-epoxy laminate(G-10) at low temperatures. The amount of energy dissipated during partial fracture of the test specimen is calculated as a function of crack speed using a dynamic strain energy release rate. The dissipative energy is compared with the heat output determined experimentally at 77 K, and the conversion rate of dissipative energy into heat is obtained. From the average value for the conversion rate, the heat outputs at 77 K for total fracture and at 4 K for partial fracture are predicted. Temperature elevations at the crack tip are also calculated. 1998 Elsevier Science Ltd. All rights reserved Keywords: composites (A); mechanical properties (C); thermal conductivity (C); superconducting magnets (F)
Adiabatic, high-current-density superconducting magnets used in applications such as nuclear magnetic resonance and high energy physics are extremely susceptible to premature quenches triggered by small heat inputs generated within the magnet winding1. Because energy dissipated during crack formation or fracture of materials can affect the stability of the high performance superconducting magnets, quantification of such dissipative energy is important for superconducting magnet technology in particular and for the understanding of mechanical behavior of materials at low temperature in general2. Voccio et al.3 designed a experimental apparatus to measure the energy released during fracture in woven glass-epoxy laminates at 77 K. Fuchino and Iwasa2 also developed a new cryomechanics measurement technique and demonstrated that the technique was able to measure pulses of heating energy as small as 10 nJ at 4 K. A dynamic strain energy release rate was applied by Do¨ll4 to the fast fracture process of PMMA (polymethylmethacrylate) for which the amount of dissipated energy converted into heat was known from his previous work5. As a result of his calculations, it was found that in PMMA at the onset of rapid crack propagation
*To whom correspondence should be addressed.
approximately 60 percent of the dynamic strain energy release rate was turned into heat. The objective of the present study is to develop an analytical procedure to calculate the dissipative energy during crack extension and to relate it to the experimentally determined heat output3. A two dimensional finite element analysis is used to implement the procedure. The material is a woven glass-epoxy laminate (G-10) commonly used for cryogenic applications. A dynamic strain energy release rate based on Freund’s solution6 is employed to compute the dissipative energy. The dissipative energy is compared with the experimentally determined heat output at 77 K, and the conversion rate of dissipative energy into heat is obtained. Using the average value for the conversion rate, the heat output at 77 K for total fracture and at 4 K for partial fractures are predicted. Three dimensional finite element calculations are also carried out to obtain local temperature elevations accompanying the crack extension.
Statement of the problem We consider the notched woven glass-epoxy laminate (G10) tensile specimen3 shown in Figure 1. Figure 2(a) and (b) illustrate finite element models of the notched woven glass-epoxy laminate tensile specimen. Since the problem
Cryogenics 1998 Volume 38, Number 4 381
Energy dissipation and temperature rise in a woven glass-epoxy laminate: S. Ueda and Y. Shindo
∂ux(x,y) ∂x
∂uy(x,y) ∂y
∂u (x,y)
Figure 1 Notched glass-epoxy laminate(G-10) tensile specimen (dimensions in mm)
x
∂y
+
∂uy(x,y) ∂x
where xx(x,y), yy(x,y) and xy(x,y) are the stress tensors, E is the Young’s modulus and is the Poisson’s ratio. The effect of the plain weave fabrics of the woven laminate is separated from the material properties through the use of geometric efficiency factor  = 0.27. The load-displacement curve for a typical partial fracture event is presented in Figure 3. These partial fractures are generally characterized by a load drop and a corresponding sample extension. Point A corresponds to the critical condition just before the crack growth; the initial crack length is aA( = a0 ) ⬇ 0, the initial load is PA and the initial ydisplacement at loading point is uyA. When crack extends by aB-aA, the load drops to PB and the displacement increases linearly to uyB. The initial displacement uyA, the final displacement uyB, the initial load PA and the final load PB are the measured values 3. Using the values provided above, the Young’s modulus E of the specimen and the final crack length aB are determined. The crack extension problem is divided into K + 1 steps. If the crack of length ak (aAˆ = a0 ⬍ a1 ⬍ $ ⬍ ak ⬍ ak + 1 ⬍ $ ⬍ aK ⬍ aK + 1 = aB ) extends by an element size ak + 1 - ak, the load Pk + 1 (PA ⬎ P1 ⬎ $ ⬎ Pk ⬎ Pk + 1 ⬎ $ ⬎ PK ⬎ PK + 1 = PB )displacement uyk + 1 (uyA = uy0 ⬍ uy1 ⬍ $ ⬍ uyk ⬍ uyk + 1 ⬍ $ ⬍ uyK ⬍ uyK + 1 = uyB ) record would follow the linear path AB.
Energy dissipation and heat output associated crack extension An energy method8 is used to calculate the static stress intensity factor at the crack tip. For plane stress conditions
Figure 2 Finite element models for G-10 tensile specimen
is symmetric, only the top half of the specimen was modeled. The specimen is assumed to be under plane stress conditions. The coordinates x and y coincide with the principal axes of the laminate. By representing the x and y components of displacements as ux(x,y) and uy(x,y), respectively, the constitutive equations for the woven laminate can be written as E 1 − 2
E 1 − 2
0
E yy(x,y) = 1 − 2 xy(x,y) 0
E 1 − 2
0
冤 冥 xx(x,y)
382
0
E 2(1 + )
Cryogenics 1998 Volume 38, Number 4
(1) Figure 3 Load-displacement curve for a typical partial fracture event and crack extension
Energy dissipation and temperature rise in a woven glass-epoxy laminate: S. Ueda and Y. Shindo the stress intensity factor KSk (k = 0 苲 K) for the opening mode is given by KSk =
再 冎 E⌬Uk ⌬ak
1/2
(k = 0 苲 K)
(2)
where ⌬Uk and ⌬ak indicate the slightly different strain energy and crack size. In Equation (2) it is assumed that the external loading is kept constant during differentiation. A dynamic stress intensity factor Kk(v) and a dynamic strain energy release rate Gk(v) are given by6 Kk(v) = Gk(v) =
1 − (v/vR ) K (k = 0 苲 K) 兵1 − q(v/vR )其1/2 Sk
r2(v/vR )2兵1 + q2(v/vR )2其1/2Kk(v)2 [4兵1 − q2(v/vR )2其1/2兵1 − r2(v/vR )2其1/2 − 兵2 − r2(v/vR )2其2 ]E
(3)
(4)
(k = 0 苲 K)
where q = r兵(1- )/2(1-2 )其1/2, r = vR/c2, v is the crack velocity, vR is the Rayleigh surface wave velocity and c2 is the shear wave velocity. The dissipative energy f(v) and the conversion rate ␣(v) of dissipative energy into heat become:
冘 G (v)(a
∂ ∂x +
∂ ∂z
再
x( )
再
z( )
冎 再 冎
∂(x,y,z,t) ∂ + ∂x ∂y
y( )
冎
∂(x,y,z,t) ∂y
(7)
∂(x,y,z,t) ∂(x,y,z,t) = cp( ) ∂z ∂t
where t is the time, (x,y,z,t) is the temperature, x( ), y( ) and z( ) are the thermal conductivities, cp( ) is the specific heat and is the mass density. The initial condition is
(x,y,z,t) = ⌽A (t ⬍ tA )
(8)
where ⌽A is an initial temperature. Figure 4 shows the thermal boundary conditions. Reflective symmetry about the crack plane (y = 0.0) and the longitudinal midplane (z = 0.0) enable the use of one-quarter model as indicated. Three types of boundary conditions are considered: ∂(x,y,z,t) = q0(t) ∂y
(tA ⱕ t ⱕ tB )
∂(x,y,z,t) = h( )兵(x,y,z,t) − ⌽A其 (tB ⬍ t) ∂y
冧
on ⌫1
(9)
K
f(v) = 2b
k
k+1
− ak )
(5)
k=0
␣(v) =
exp f(v)
(6)
where 2b is the specimen thickness and exp is the experimenally determined heat output3. For each event the conversion rate for the partial fracture at 77 K is obtained as a function of the crack velocity. The average value for the rate ␣(v) is used to determine the heat output h(v) for a total fracture event at 77 K. Superconducting magnets are maintained at 4 K and it is important to estimate the heat output during crack extension at 4 K. Cooling G-10CR woven glass-epoxy laminate from 77 K to 4 K increases the value of parameters: Young’s modulus about 7% and tensile strength about 5%9. G-10CR is the cryogenic grade material of G-10 and meets federal and military specifications for G-10 product. Under the assumption that the values of PA and PB at 4 K are 7% higher than those at 77 K, we can also calculate the dissipated energy as a function of the crack velocity at 4 K using Equation (5). With this dissipated energy and the average value for the conversion rate at 77 K, the heat output at 4 K can be determined.
∂(x,y,z,t) = 0(tA ⱕ t) on ⌫2 ∂y
(10)
(x,y,z,t) = ⌽A (tA ⱕ t) on ⌫3
(11)
where h( ) is the coefficient of convective heat transfer, ⌫1 苲 ⌫3 represent the boundaries shown in Figure 4. To calculate the temperature distribution around the crack tip by the finite element procedure, we use the heat production q0(t) as follows: q0(t) = Q[1 − exp兵 − (t − tA )/TR其]
(12)
Temperature rise near the crack tip At time tA the crack rapidly advances causing a sudden drop in the load until the crack arrests at time tB = tA + aB/v. We analyze the transient temperature state near the crack tip and include the effect of temperature-dependent material properties. A three dimensional finite element analysis is used to implement the procedure. The differential equation governing the unsteady heat conduction problem is10
Figure 4 Thermal boundary conditions
Cryogenics 1998 Volume 38, Number 4 383
Energy dissipation and temperature rise in a woven glass-epoxy laminate: S. Ueda and Y. Shindo Table 1 Young’s modulus E and crack length aB
Event No. Partial fracture Partial fracture Partial fracture Partial fracture Total fracture
Q=
1 2 3 4
Numerical results E (Gpa) aB (mm)
PA (N)
PB (N)
8.86 8.18 8.67 9.20 8.81
260 533 233 450 1120
224 518 188 441 0
0.931 0.226 0.674 0.272 –
h(v)
冕
(13)
tB
[1 − exp兵 − (t − tA )/TR其]dt
tA
where TR denotes the relaxation time. As the time dimension is of an infinite extent, we shall deal with finite time domains tm − 1 ⬍ t ⱕ tm (m = 1,2,$) and repeat the calculation for subsequent domains, where tA = t0 ⬍ t1 ⬍ t2 ⬍ $ ⬍ tm − 1 ⬍ tm ⬍ $. The process will thus lead to a step-by-step calculation and we use the procedure of time-stepping derived by Crank-Nicholson formula. In the time domains tm − 1 ⬍ t ⱕ tm (m = 1,2,$), the thermal properties of the nth element are approximated by the properties at the mean temperature n(tm − 1 ) (n = 1,2,$,N, m = 1,2,$) of the nth element, where N is the number of the total elements. The mean temperature n(tm − 1 ) is given by
n(tm − 1 ) =
1 ⍀n
冕冕冕
(x,y,z,tm − 1 )dxdydz
(14)
⍀n
(n = 1,2,$,N, m = 1,2$) where ⍀n (n = 1,2,$,N) is the region of the nth element.
Experimental results 3 uyA (mm) uyB (mm) 0.36 0.80 0.33 0.60 1.42
exp (mJ)
0.47 0.83 0.35 0.64 –
11.3 6.0 5.8 5.6 900.0
Numerical results and discussion The accepted value for the Poisson’s ratio at 77 K is 0.29. For each event the predicted Young’s modulus E and crack length aB are given in Table 1, along with the initial load PA, the final load PB, the initial displacement uyA, the final displacement uyB and the measured heat output exp3. Normalized dynamic stress intensity factor Kk(v)/0a1/2 B and the dynamic strain energy release rate Gk(v) (J/m2 ) for partial fracture 1 are presented in Table 2. 0 is the average stress
0 =
PA S
(15)
where S is the ligament cross-section. The dynamic stress intensity factor and the dynamic strain energy release rate decrease with increasing the crack velocity. The conversion rates ␣(v) of dissipative energy f(v) into heat exp as a function of the crack velocity for event no. 1 苲 4 are presented in Figure 5 as continuous lines. The average value, which is shown as a dashed line, increases from about 0.52 (v/vR = 0.0) to about 0.91 (v/vR = 0.3). To make the significance of this point a little clearer, the actual crack velocity should be measured. The dissipated energy f(v) and the heat output h(v) as a function of the crack velocity for the total fracture event are given in Figure 6. The dissipative energy decreases with increasing the crack
Table 2 Normalized dynamic stress intensity factor Kk(v)/0aB1/2 and dynamic strain energy release rate Gk(v) (J/m2 ) for partial fracture 1
k
1
2
3
4
5
6
ak/aB
0.154
0.308
0.462
0.616
0.770
0.924
8.71 8.64 8.37 8.05 7.33 6.61
9.00 8.94 8.66 8.32 7.59 6.83
26518 26121 24545 22595 18825 15265
28362 27938 26252 24166 20134 16326
Kk(v)/0aB1/2
v/vR 0.0 0.01 0.05 0.1 0.2 0.3
6.77 6.72 6.51 6.26 5.70 5.14
7.15 7.10 6.88 6.61 6.02 5.43
8.32 8.26 8.00 7.69 7.01 6.32
Gk(v) (J/m2 )
v/vR 0.0 0.01 0.05 0.1 0.2 0.3
384
8.62 8.55 8.29 7.96 7.29 6.54
16050 15810 14856 13676 11394 9239
17895 17627 16563 15247 12703 10300
Cryogenics 1998 Volume 38, Number 4
24225 23863 22423 20641 17197 13945
25970 25581 24037 22128 18435 14949
Energy dissipation and temperature rise in a woven glass-epoxy laminate: S. Ueda and Y. Shindo
Figure 5 Conversion rates ␣( v ) as a function of the crack velocity for partial fractures at 77 K
Figure 7 Dissipated energy f( v ) and heat output h( v ) as a function of the crack velocity for the partial fracture 1 at 77 K and 4 K
where qb(⌬ ) is the heat flux and ⌬ = -⌽A. The transient temperature distributions were evaluated by taking TR = 1/100, 1/500 and 1/1000 sec in Equation (12). These results were in substantial agreement. Thus it may be said that the result for TR = 1/500 sec is, from a practical view point, quite satisfactory. Figure 8 shows the variation of the temperature with time at the crack tip for the partial fracture 1 and v/vR = 0.1 at ⌽A = 77 K. The solid line and the dashed line denote the results at z/b = 0 and 0.5, respectively. The temperatures first increase, go through maximums, and then decrease as t increases. The peak value of the temperature at z/b = 0.0, which occurs at t = tB, is 476 K. Figure 9 shows the temperature at the crack tip (z/b = 0.0) as a function of time for the partial fracture 1 and v/vR = 0.1 at ⌽A = 77 K, 4 K. The continuous line and the dashed line denote the results at ⌽A = 4 K and 77 K, respectively. As a result of our calculations, it is found that the temperature rise at ⌽A Figure 6 Dissipated energy f( v ) and heat output h( v ) as a function of the crack velocity for the total fracture at 77 K
velocity. Using the average value for the conversion rate at 77 K and the dissipated energy f(v), the heat output h(v) becomes about 600 (mJ). The measured heat output exp shown in Table 1 is 900 (mJ). This difference may arise mainly from the neglect of the kinetic energy due to the sudden motion of the bellows sealing the system3. The dissipative energy and the heat output for partial fracture 1 at 4 K are shown in Figure 7 as continuous lines. The results for partial fracture 1 at 77 K are also shown as dashed lines. The dissipative energy and the heat output at 4 K are about 5% higher than those at 77 K. The specific heat cp( ), the thermal conductivity ( ) and the mass density for G-10CR have been used9,11. The heat transfer coefficient h( ) is calculated from the boiling curve of the surrounding medium as follows12: h( ) =
qb(⌬ ) ⌬
(16)
Figure 8 Temperature rise at the crack tip ( z/b = 0.0, 0.5) for the partial fracture 1 at 77 K
Cryogenics 1998 Volume 38, Number 4 385
Energy dissipation and temperature rise in a woven glass-epoxy laminate: S. Ueda and Y. Shindo 3. The heat output for the total fracture event is estimated, and the heat output in the fracture process at liquid helium temperature is also predicted. 4. Because of the extremely low heat capacity of the materials at 4 K, the peak value of the temperature at 4 K is more than ten times greater than that at 77 K.
References
Figure 9 Temperature rise at the crack tip ( z/b = 0.0) for the partial fracture 1 at 77 K and 4 K
= 4 K is much higher than the rise at ⌽A = 77 K and the peak value of the temperature at ⌽A = 4 K is 5128 K.
Concluding remarks A procedure is developed to predict the energy dissipation and temperature rise during crack extension in woven glassepoxy laminates at low temperatures. The procedure is implemented in two and three dimensional finite element programs. Based on this study the following conclusions are made: 1. 2.
386
The dissipative energy associated with crack extension at 77 K is calculated as a function of the crack velocity using a dynamic strain energy release rate. In order to determine the conversion rate of dissipateve energy into heat, the calculated dissipative energy is compared with the experimentally determined heat output at 77 K. The conversion rates for the partial fractures are obtained as a function of the crack velocity.
Cryogenics 1998 Volume 38, Number 4
1. Iwasa, Y., Experimental and theoretical investigation of mechanical disturbances in epoxy-impregnated superconducting coils. 1. General introduction. Cryogenics, 1985, 25, 304–306. 2. Fuchino, S. and Iwasa, Y., A cryomechanics technique to measure dissipated energies of 10 nJ. Experimental Mechanics, 1990, 30(6), 356–359. 3. Voccio, J.P., Iwasa, Y., Maksimov, I.L. and Bovrov, E.S., Experimental and theoretical study of fracture energy and crack dynamics in G-10 at 77 K. Advances in Cryogenic Engineering, 1988, 34, 99–106. 4. Do¨ll, W., Application of an energy balance and an energy method to dynamic crack propagation. International Journal of Fracture, 1976, 12(4), 595–605. 5. Do¨ll, W., An experimental study of the heat generated in the plastic region of a running crack in different polymertic materials. Engineering Fracture Mechanics, 1973, 5, 259–268. 6. Freund, L.B., Crack propagation in an elastic solid subjected to general loading-I. Constant rate of extension. Journal of the Mechanics and Physics of Solids, 1972, 20, 129–140. 7. Zako, M., Mori, T., and Miyoshi, T., Study on fracture toughness for composite materials (Fracture toughness evaluation by equivalent stress intensity factor Ke ) Transactions of the Japan Society of Mechanical Engineers 1979, 45, 726–733 (in Japanese). 8. Aamodt, B. and Klem, F. Application of numerical techniques in practical fracture mechanics, in: Fracture Mechanics in Engineering Practice (Ed. P.Stanley), Applied Science Publishers Ltd. (1976) 33–40 9. Kasen, M.B., MacDonald, G.R., Beekman, D.H. and Schramm, R.E., Mechanical, electrical, and thermal characterization of G-10CR and G-11CR glass-cloth/epoxy laminates between room temperature and 4 K. Advances in Cryogenic Engineering, 1980, 26, 235–244. 10. Carslaw, H.S., and Jeager, J.C., Conduction of Heat in Solids, Clarendon Press (1959) 1–49 11. Shindo, Y. and Ueda, S., Transient thermal-mechanical behavior of cracked glass-cloth-reinforced epoxy laminates at low temperatures. Advances in Cryogenic Engineering, 1996, 42, 137–146. 12. Cryogenic Association of Japan (Ed.), Cryogenic Engineering Handbook, Uchida Rokakuho Publishing Co., Ltd. 1983, 207–237 (in Japanese)