A number of papers have been presented on the evaluation of energy release rate for thermoelasticity and corresponding J integral. Two main approaches were developed to treat energy release rate in elasticity. The first is based on direct calculation of the potential energy rate with respect to crack length. The second makes use of Lagrangian formalism. The translational and expansional energy release rates in thermoelasticity are studied by employing the formalism of irreversible thermodynamics and the Crack Layer Approach

Chudnovsky, A.

Gommerstadt, B.

English

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https://ntrs.nasa.gov/search.jsp?R=19850013375 2020-03-20T18:56:41+00:00Z
NASA Contractor Report 174872
Translational and Extensional Energy Release Rates (the J- and M-Integrals)
for a Crack Layer in Thermelasticity
A. Chudnovsky and B. Gomerstadt
Case Western Reserve University
Cleveland. Ohio
•
I
(NASA-C8-174872) TBINSLITICNIL AND
EITENSIONIL ENEEGI RELEASE BITES (THE J- IND
B-INTEGRALS) FOE A CRICK LITIB IN
THUROB LASTICIT2 Final Beport (Case Western
Fesei-re Univ.) 9 p HC 102,iB! A01
	 CSCL 209 G3/39
V85-21685
41^
Unclas
14425
March 1985
Prepared for
•	 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Lewis Research Center
Under Contract NAG 3-223
TRANSLATIONAL AND EXTENSIONAL ENERGY RKLEASK RATES
(THE J- AND M-INTEGRALS) FOR A CRACK LAYER IN THERHOKLASTICITY
A. Chudnovsky and B. Gonaerstadt
Department of Civil Engineering
Case Western Reserve University
Cleveland, Ohio 44106
I.
Since 1979 there has been a number of papers on evaluation of energy
release rate for thermoelasticity and corresponding J-integral. Two main
approaches are developed to treat energy release rate in elasticity. The
first is based on direct calculation of potential energy rate with respect to
clack length (1). The second makes use of Lagrangian formalise (Mother's
theorem) (2). The direct method was extended to the problems of
themoelasticity by Wilson and Yu (3), K. Kishimoto, S. Aoki, and M. Sakata
(41 and Gurtin (S). A useful discussion on the extension of the approach to 	
s^
thermoelasticity was presented by McCartney (6].
The second approach was elaborated by Atkinson and Smelser (7) and
	 :f
discussed in 181. In elasticity the energy release rate is expressed as a
path independent integral. The path invariancy of the similar integrals for
thermoelasticity is discussed in aforementioned papers.
In 13, 41 the J-integral consist of two parts: first is a common for
elastic problems integral along a path and the aecond includes a nonvanishing
integral over the volumr. ( area for two dimensional applications).
Path independent integral introduced in (5) has limitations in path
invariancy as was noticed in (7): the integration paths should start and end
at certain points. It is not discussed in (S) whether the integral represents
any energy release rate.
Another path independent integral expressed in terms of energy-momentum
tensor has been derived in (7). The derivation follows the Lagrangian
formalism of (2) and essentially depends on Lagrangian chosen. A comparison
of this integral and that obtained in present paper (see also (61) indicates
that an entropy term is omitted.
In this paper we consider translational and expansional energy release
rates employing the formalism of irreversible thermodynamics and Crack Layer
Approach. A damage zone usually precedes and surrounds a propagating crack.
A system of a crack and the associated damage is treated as a crack layer (CL)
(9). The damage parameter for simplicity is defined as an areu of
discontinuity surfaces per unit volume. A part of the CL within which the
rate of damage accumulation is positive, (p > o) is defined as an active
zone. The CL theory thus models fracture propagation as the active zone
movements: translation and rotation as a rigid, isotropic expansion and
distortion for a small active zone. There are potential energy release rates
corresponding to each of these elementary movements. Energy release rates
corresponding only to the active zone translation, the .) -integral, and
expansion, the N-integral are considered here. The rate of the damage density
resulting from these two movements of active zone is ;10,111.
	
P =-v,ap- ^ xK app	 (1)
where v  and a stands for the rates of translation in direction tangent to
the crack path and expansion, respectively, and Z is a position vector with
	
respect to the center of isotropic expansion.
	 ..
Potential energy release rate P associated with damage growth is
•	 anp =	 a	 (2)
2
111-A
' n
t
where V is a potential energy density and integral is taken over the active
zone. Upon substitution of (1) this relation takes the form
P = -,,, a^ a, v -	 Yk a dv	 (3)S P	 ^ ^
where k = 1,2
The first integral in the right -hand side of (3) is a translational energy
release rate (J-integral), the second one is an expansional energy release
rate W integral).
2. Translational Energy Release Rate in Thermoelasticity
The stress tensor jrand the absolute temperature T constitutes a
conventional set of parameters for a thermoelastic medium. This set, extended
by adding the damage parameter p is chosen as a system of thermodynamic
state parameters
	 J
TA
The corresponding thermodynamic potential is a Gibbs' free energy density
which is usually designated in continuum mechanics as the potential energy
density
Then, the following constitutive equations for thermoelasticity are held:
ar = -s
(5)aT o'.aNsr
3
f.
W
4-
I
where S is the entropy. and fX is the total strain tensor. According to Sq.
(3).
I
a^	
i
Jap	
(7)
V
Since (4)
a^  aKp = aK^'- ^"' aK o'^^- a T &T 	 (8)P	 aa^^	 a
Using local constitutive equations ( S) and (6) this expression can be
rewritten as
an aKp = akf -a;(0C ^ K^ s aKr << 	 (9)
^p
Hero, f is the strain energy density, or the specific Helmholtz free energy,
and u  the component of the displacement vector.
For d being a linear coefficient of thermal expansion the specific
entropy for theemoelasticity
8 ' CC Gkk
	 (10)
Substituting (9) and (10) into expression (7) for J 1 and using Gauss theorem
we finally arrive at the integral
^1
J1 
-	 ( f nl - Ti j nj u i.1 )dl +^d Okk 0, 1 dv	 (11)
r
	 V
eN
Here, n3 is J - th component of the unit normal to ar. integration path
I J 8 = T - To , where To is the absolute temperature for the initial
equilibrium state of the system when the strain is everywhere zero. The first
I^ term in (11), i.e., integral along the path r repeats that for elastic case
with the fields f, 9  and ?i corresponding to the thermoelastic equilibrium.
The second term, i.e., the area integral, cannot be in general converted into
a line integral. However, such a conversion can be easily done for linearly
varying temperature field.
3. Expansional Energy Release Rate
Expression for the K-integral follows from the second part of the
right-hand side of (3):
M '=
ap
r1 Xk ^Kp JV
V 
Using the same approach like in the previous section the expression for this
integral is obtained in the following form.
M = ^ XK ^'I K f ^; 40 Xk
	 P-1 (:(	 9 + X O( elm) Q V	 (13)
V
where i, J, k and J = 1,2.
The position vector Xk
 is taken with respect to the center of isotropic
expansion.
The K -integral consists of two parts similarly to the J-integral (11): one
is a line integral and another is an area integral.
(12)
J
4. Disctasion
Since p is not zero within the active zone V
A only the following integral
over an arbitrary area V Vg is reduced to
i
t
S
}
a P
JV	
aP
JV
V 3A P	 A
It leads to the following relations
^^ O, d v =	 a v^^ ^^	 (15)a P	 appV 'q	 VA
and
at xKaK dv = i xka^ dv	 (16)ap	 ^	 ap	 p
V A	 VA
These imply the invariancy of J- and H-integral with respect to the domain
of integration V VA' As it was shown above the area integral can be
decomposed into the linear and area integrals. The linear integral formally
coincides with that for the elastic case, however, for thermoelasticity it is
not path independent. Only the sum of the linear and area integrals in (11)
and (13) obeys the conservation law.
The expressions (11) and (13) agree with corresponding integrals of [12]
and differ from all the rest mentioned above expressions for Energy release
rate.
. a
A
0
6
Is
References
1. J. R. Rice, Trans., ASMP,. j. Appl. loch., 90, 379-386 (1968).
2. Eshelby J. D. in "Inelastic Behavior of Solids" (Ed. M. F. Kanninen at
al.) p. 77. McGraw-Hill, New York (1970),
3. Wilson, W. K., and Yu, I. W., Int. J. of Fracture, Vol. 15, No. 4, p.
377-387, (1979).
4. Kishimoto, K., Aoki, S., and Sakata, M., Engineering Fracture Mechanics,
Vol. 13, pp. 841-850 (1980).
5. Curtin, M. E., Int. J. of Fracture, Vol. 15, No. 5, pp. 8169-8170 (1979).
6. McCartney, L. M., Int. J. of Fracture, Vol. 15, No. 6, pp. 8217-8221
(1979).
7. Atkinson, L. and Smelser, R. E. Int. J. Solids and Structures, Vol. 18,
pp. 533-549 (1982).
8. Frankfort, G. and Golebiewska Herman,, A., Trans ASME, J. of Appl. Koch.,
Vol. 49, p. 710-714 (1987).
9. Chudnovsky, A., NASA Report, N174634, March, 1984.
10. V. Khandogin, A. Chudnovsky, In the book "Dynamics and Strength of
Aircraft Constructions", 4 p. 148-175, Novosibirsk, 1978 (in Russian).
11. Chudnovsky, A., Workshop on Damage Eds. Stouffer D. et al. Kentucky, May
4-7, p. 79-87.
12. Aoki, S., Kishimoto, K., Sakata, M., Trans. ASME, J. Appl. Hoch., Vol.
48, p. 825-829 (1981).
7
J


Translational and extensional energy release rates (the J- and M-integrals) for a crack layer in thermoelasticity

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