- Email: [email protected]
Integral parameters for thermal fracture J′, θ and J∗
0013-7944190 $3.00 + 0.00 0 1990 Pcrgamon Press pk.
Engineering Fracture Mechanics Vol. 37, No. 3, pp. 697-700, 1990 Printed in Great Britain.
INTEGRAL
PARAMETERS FOR THERMAL J’, Je and J*
FRACTURE
JAROSLAV JOCH National Research Institute. for Machine Design, 190 11 Praha 9, Bechovice, Czechoslovakia Abstract-This paper discusses the difference between parameters J’, Je and J* in the case of linear elastic bodies under heat loading conditions. It shows for non-singular temperature at the crack tip that (i) J’ z Je; (ii) for temperature-dependent elastic parameters, J’ and/or Jo must be evaluated in an incremental way; (iii) for constant elastic parameters J’ and/or Jo are equal to the values obtained from the current state and they arc equal to J*. The physical meaning of the above motions parameters is discussed.
1. INTRODUCTION WHILETHET*-integral has been found to be be convenient for the elastic-plastic fracture d~ption in the case of a homogeneous temperature fiefal], the integral fracture parameters for thermal-mechanical loading (e.g. thermal shock) are still subject to investigation (see e.g. ref.[2]).Recently, Brust et a/.[21have produced a numerical study of six integral fracture parameters for heat loading. The study was made for a linear elastic material taking into account that some properties of integral parameters can be mom easily observed within the framework of linear elasticity as a special case of el~ti~t~l~ti~ty. Resides, the f’-integral, being a special case of the P-integral within elasticity (see e.g. ref.[3& will serve as a means to investigate the P-integral at heat loading. Results of that numerical study reveal, among other things, two facts: (i) parameters J’, Je and J* behave for the non-singular temperature at the crack tip ‘oughly in the same way when the material parameters are not tcmper&ture dependent, and (ii) the values of J’ and J’, when respecting and not respecting the temperature dependence of elastic parameters, respectively, ditherby more than four times in the case of the 304 Stainless Steel and thermal shock up to 288°C. In particular, the latter result seems to be of extraordinary importance and made the author undertake the minor analytic exercise presented in this paper. 2. HERON
OF REGAL
PARADER
UNDER
The parameters in question have been introduced by the following relationships: au, W’n, - uyn, ax,
J’ =
dl>
s s ‘I
uUd6, (Atluri[3])
w’s
0
Je =
we=
4
w%’
0 ~7 = Q - c&S, (Ainswotth et aI.[4n
Here, r, is a contour surrounding the crack tip of a very small radius 6 (sac e.g. ref.[2]);n, are components of the unit normal vector to r,; uU , Q and u, am stresses, strains and displacements, respectively; a and 0 are coe.fficientsof thermal expansion and temperature increment to the original state, respectively, and 6, is the Kronecker symbol. Let us introduce another intearel JD= >
It holds when
697
698
Technical note
where E, _
E for plane stress W) -I E/(1 - v’) for plane strain. K,is the stress intensity factor and E and v are the Young’s modulus and the Poisson’s ratio, respectively. Further, there is the parameter G, which is defined by energy balance as follows
Gd”
dwex dW dT
dt = dt
dt
dt ’
where a is the crack length, r is the time-like linear parameter, W, is the work of external forces, W is the work of internal forces (i.e. stresses) and T is the kinetic energy contained in the body. Henceforth, we shall follow the quasistatic case where T e 0,and W and IV, in a preselected time are understood as the functions of the crack length a only; in other words, relationship (5) can be rewritten as
&!!!&g.
(7)
Verbally, G defined by (6) can be expressed as “the mechanical energy which a body with a crack of length a receives ‘extra’ in addition to the ( W, - W’)balance and which results from the change of the body into another one, differing from each other merely by the infinitesimal increment of the crack length de, normalized by da”. Generally speaking, this mechanical “extra” energy can be dissipated remote from the crack tip, which makes such values of G uninteresting for the description of what occurs in the vicinity of the crack tip. If, however, eq. (6) is related to the near vicinity of the crack tip, G becomes physically a more appealing parameter. Obviously, under certain circumstances both of these cases will coincide (e.g. ref.[3]). Concerning the physical interpretation of eq. (5), related to the small vicinity of the crack tip, there are principially two different options. Always two infinitesimally different bodies are compared, though in one case the mechanical energy imbalance can be “drowned”
(ga)
in the course of the whole loading process, while in the other case it can be done so only within the current time. Another option in the physical interpretation jr” bu d(a@, )
(gb)
of eq. (5) is to decide whether the work of the internal forces has any share
@a)
does not have any share
(gb)
or
in the tearing process. It was demonstrated elsewhere (e.g. ref.[6]) how to transform parameters like G from eq. (6), characterized as above, into integrals like eqns (l)-(4). It is clear that the following parameters J and G introduced in the above will coincide: J’ [es. (l&G [eqs @aI, @all. 4 [eq. M-G hs @a),(gb)], J* Es. (3)l-G [es. (gbll, JD [eq. (4)l-G [eq. @W. Relationships (l)-(4) make it clear that J’, Je, J* and JD differ only in W-functions. The subsequent paragraph will analyse the J-integrals discussed above while examining the W-functions in a linear continuum.
3. BEHAVIOUR OF W’, We, W” AND WD The behaviour of these parameters will be studied on an uniaxial example. Let us suppose that a bar is heated and loaded by specific force u over a long time interval described by linear parameter t within (0; 1) as follows
e=e,ta 0 = u, ty.
VW W)
Here, p, y > 0 and u, and 8, correspond in time to t = 1. The coefficients /I and y characterize the development in time of heating and mechanical loading; e.g. p-rcc and y -rO mean that the bar was first loaded by specific force u, and then in time near t = 1 heated by 0,. The expressions for W-parameters according to (lb), (2b), (3b) and (4b) within linear elasticity and at the earlier mentioned loading am as follows
(114
(lib) (114
T&mica1 note
699
Here E, and a, are E and u at time t = 1. Loading (10) can be considered as sufficiently representative, and using (I 1) we shall try to formulate some findings. (i) Since uj b u, near the crack tip, we can write J’ * Je % J * = JD under the condition that the temperature at the crack tip is not singuIar and the elastic material properties do not depend on temperature. (ii) If the crack tip temperature is singular and the elastic material is constant then J* s J’ under the linear temperature and mtchanicaf Io&~J de~lo~t. (iii) J* ssJD for constant elastic material regardless of temperature behaviour at the crack tip. In (i) the symbol “ R ” should be understood as follows: if the radius6 of the contour tends to zero, the symbol “ % ” should tend to “ = “. The contour radius is, however, for reasons discussed e.g. in ref.[3], small but finite. The relationships for the R-parameters and loading (10) will also be developed for material with E and a -dent on temperature. The material dependence can be considered, e.g. in accordance with ref.[2] for the 304 Stainless St& for which the following linear approximation can be applied within 0400°C E=E,+k&-6)=E,+kEB,(1-fB)
(12a)
a = a, - k.(ff t - 6) = a, - k,B, (1 - te),
(I2b)
where k, and k, are positive numbers. Then, the W-functions can be expressed as
(134
Using (13), we shall infer the properties of J-integrals for temperaturedependmt materials. With non-singular temperature at the crack tip it holds that (i) J’ % Je and (ii) J’ and/or Je are affected only by the temperature dependence of E, unlike a; (iii) while with temperature-independent E the values of J’ and/or Je do not depend on the history of loading, it is not the case with thermally dependent E and the calculation must be performed incrementally even within linear elasticity. How significant the difference between incremental and non-incremental computation with &ermally depeadent E [reqecting the difference between (1 la) and (13b)] can be illustrated, for the earlier rn~tio~ material and thermal shock within 0-2%VC, by the computation quoted in the introduction and d ocumented in ref. [2]. (iv) For a non-singular temperature at the crack tip, J* s Ja. (v) J’ and J.. differ more markedly only when the crack tip temperatureis singular. (vi) Jo and JD are generally different only with temperature-dependent material properties.
4. DISCUSSION The previous paragraphs show that unless the crack tip ~m~rature is singular, it is very mu& the same whether we consider (9a) or (9b) while interpreting G from (5). It has to be specified, though, when @a) and (Sb) are to be used in this interpretation. It will be understood that cracks with small length increments throughout the loading process will have to be interpreted in terms of (8a), and consequently, the parameters describing the crack growth will be J’ and/or T* with the corresponding R curves. If the crack does not move at all, we must interpret G in terms of (Sb) and the goveming parameters for the non-stability of the crack will be JD and Jlc. From a practical standpoint, however, only the &st of the two cases ems to be frequent enough, especially if the ductile tearing description is considered. In this discussion we must also cope with the fact that the same stresses and strains near the crack tip can cornspond with different J’s under thermal loading if material properties depend on temperature. Thus, J’ and/or T* are not “for&’ parameters as it may appear from comparison with J within HRR theory, but essentially “energetic” ones. As indicated elsewhere (e.g. ref. [3]), an infinite number of integral parameters can describe one stress-strain field near the craak tip, but only J’ and/or T* have the meauing of crack driving force. It is a matter of experimental verification to show whether T* under thermal-mechanical loading is able to describe the crack growth (using R curves) in the same accomplished way as in the case of mechanical loading only[l].
5. CONCLUDING
REMARKS
This paper has analysed the integral fracture parameters J’, Je and J* in terms of linear elasticity under thermal-mechanical loading. The main facts arising from the analysis can be summarized as follows. (i) J’ is to be confronted with Je only if the temperature at the crack tip is singular. (ii} For non-unbar crack tip tem~ratu~ the J*-integral describes the situation in the same way as the stress ~tc~i~ factor within linear elasticity. (iii) In terms of linear elasticity the incremental computation of J’ is needed only when material properties aTe temperature dependent. Otherwise the description of fracture using J’ is identical to that using the stress intensity factor. Further investigations in this Bold should bring progress in the description of the crack growth caused by repeated thermal shock.
700
Technical note
REFERENCES [l] F. W. Brust, T. Nishioka, S. N. Atluri and M. Nakagaki, Further studies on elastioplastic stable fracture utilising the T*-integral. Engng Fracture Mech. 22, 10794103 (1985). [2] F. W. Brust. M. Nakagaki and C. Springtield, Integral parameters for thermal fracture. Engng Fracture Mech. (in press). [3] S. N. Atluri, Energetic approaches and path-independent integrals in fracture mechanics, in Computational Methoak in the Mechanics of Fracture (Edited bv S. N. Atluri). North-Holland Press, Amsterdam (1986). [4] R. A. Ainsworth, B. K. Neal; and R.-H. Price, Fracture behaviour in the presence of thermal strains. I. Mech. E. Conference on Defects in Pressure Vessels (1978). [S] W. S. Blackburn, Path independent integrals to predict onset of crack instability in an elastioplastic material. Int. J. Fracture Mech. 8, 343-346 (1972). [6] J. R. Rice, Mathematical analysis in the mechanics of fracture. Fracture, Vol. II. Mathematical Fundamentals (Edited by H. Liebowitz). Academic Press, New York (1968). (Received 18 May 1989)
Engineering Fracture Mechanics Vol. 37, No. 3, pp. 697-700, 1990 Printed in Great Britain.
INTEGRAL
PARAMETERS FOR THERMAL J’, Je and J*
FRACTURE
JAROSLAV JOCH National Research Institute. for Machine Design, 190 11 Praha 9, Bechovice, Czechoslovakia Abstract-This paper discusses the difference between parameters J’, Je and J* in the case of linear elastic bodies under heat loading conditions. It shows for non-singular temperature at the crack tip that (i) J’ z Je; (ii) for temperature-dependent elastic parameters, J’ and/or Jo must be evaluated in an incremental way; (iii) for constant elastic parameters J’ and/or Jo are equal to the values obtained from the current state and they arc equal to J*. The physical meaning of the above motions parameters is discussed.
1. INTRODUCTION WHILETHET*-integral has been found to be be convenient for the elastic-plastic fracture d~ption in the case of a homogeneous temperature fiefal], the integral fracture parameters for thermal-mechanical loading (e.g. thermal shock) are still subject to investigation (see e.g. ref.[2]).Recently, Brust et a/.[21have produced a numerical study of six integral fracture parameters for heat loading. The study was made for a linear elastic material taking into account that some properties of integral parameters can be mom easily observed within the framework of linear elasticity as a special case of el~ti~t~l~ti~ty. Resides, the f’-integral, being a special case of the P-integral within elasticity (see e.g. ref.[3& will serve as a means to investigate the P-integral at heat loading. Results of that numerical study reveal, among other things, two facts: (i) parameters J’, Je and J* behave for the non-singular temperature at the crack tip ‘oughly in the same way when the material parameters are not tcmper&ture dependent, and (ii) the values of J’ and J’, when respecting and not respecting the temperature dependence of elastic parameters, respectively, ditherby more than four times in the case of the 304 Stainless Steel and thermal shock up to 288°C. In particular, the latter result seems to be of extraordinary importance and made the author undertake the minor analytic exercise presented in this paper. 2. HERON
OF REGAL
PARADER
UNDER
The parameters in question have been introduced by the following relationships: au, W’n, - uyn, ax,
J’ =
dl>
s s ‘I
uUd6, (Atluri[3])
w’s
0
Je =
we=
4
w%’
0 ~7 = Q - c&S, (Ainswotth et aI.[4n
Here, r, is a contour surrounding the crack tip of a very small radius 6 (sac e.g. ref.[2]);n, are components of the unit normal vector to r,; uU , Q and u, am stresses, strains and displacements, respectively; a and 0 are coe.fficientsof thermal expansion and temperature increment to the original state, respectively, and 6, is the Kronecker symbol. Let us introduce another intearel JD= >
It holds when
697
698
Technical note
where E, _
E for plane stress W) -I E/(1 - v’) for plane strain. K,is the stress intensity factor and E and v are the Young’s modulus and the Poisson’s ratio, respectively. Further, there is the parameter G, which is defined by energy balance as follows
Gd”
dwex dW dT
dt = dt
dt
dt ’
where a is the crack length, r is the time-like linear parameter, W, is the work of external forces, W is the work of internal forces (i.e. stresses) and T is the kinetic energy contained in the body. Henceforth, we shall follow the quasistatic case where T e 0,and W and IV, in a preselected time are understood as the functions of the crack length a only; in other words, relationship (5) can be rewritten as
&!!!&g.
(7)
Verbally, G defined by (6) can be expressed as “the mechanical energy which a body with a crack of length a receives ‘extra’ in addition to the ( W, - W’)balance and which results from the change of the body into another one, differing from each other merely by the infinitesimal increment of the crack length de, normalized by da”. Generally speaking, this mechanical “extra” energy can be dissipated remote from the crack tip, which makes such values of G uninteresting for the description of what occurs in the vicinity of the crack tip. If, however, eq. (6) is related to the near vicinity of the crack tip, G becomes physically a more appealing parameter. Obviously, under certain circumstances both of these cases will coincide (e.g. ref.[3]). Concerning the physical interpretation of eq. (5), related to the small vicinity of the crack tip, there are principially two different options. Always two infinitesimally different bodies are compared, though in one case the mechanical energy imbalance can be “drowned”
(ga)
in the course of the whole loading process, while in the other case it can be done so only within the current time. Another option in the physical interpretation jr” bu d(a@, )
(gb)
of eq. (5) is to decide whether the work of the internal forces has any share
@a)
does not have any share
(gb)
or
in the tearing process. It was demonstrated elsewhere (e.g. ref.[6]) how to transform parameters like G from eq. (6), characterized as above, into integrals like eqns (l)-(4). It is clear that the following parameters J and G introduced in the above will coincide: J’ [es. (l&G [eqs @aI, @all. 4 [eq. M-G hs @a),(gb)], J* Es. (3)l-G [es. (gbll, JD [eq. (4)l-G [eq. @W. Relationships (l)-(4) make it clear that J’, Je, J* and JD differ only in W-functions. The subsequent paragraph will analyse the J-integrals discussed above while examining the W-functions in a linear continuum.
3. BEHAVIOUR OF W’, We, W” AND WD The behaviour of these parameters will be studied on an uniaxial example. Let us suppose that a bar is heated and loaded by specific force u over a long time interval described by linear parameter t within (0; 1) as follows
e=e,ta 0 = u, ty.
VW W)
Here, p, y > 0 and u, and 8, correspond in time to t = 1. The coefficients /I and y characterize the development in time of heating and mechanical loading; e.g. p-rcc and y -rO mean that the bar was first loaded by specific force u, and then in time near t = 1 heated by 0,. The expressions for W-parameters according to (lb), (2b), (3b) and (4b) within linear elasticity and at the earlier mentioned loading am as follows
(114
(lib) (114
T&mica1 note
699
Here E, and a, are E and u at time t = 1. Loading (10) can be considered as sufficiently representative, and using (I 1) we shall try to formulate some findings. (i) Since uj b u, near the crack tip, we can write J’ * Je % J * = JD under the condition that the temperature at the crack tip is not singuIar and the elastic material properties do not depend on temperature. (ii) If the crack tip temperature is singular and the elastic material is constant then J* s J’ under the linear temperature and mtchanicaf Io&~J de~lo~t. (iii) J* ssJD for constant elastic material regardless of temperature behaviour at the crack tip. In (i) the symbol “ R ” should be understood as follows: if the radius6 of the contour tends to zero, the symbol “ % ” should tend to “ = “. The contour radius is, however, for reasons discussed e.g. in ref.[3], small but finite. The relationships for the R-parameters and loading (10) will also be developed for material with E and a -dent on temperature. The material dependence can be considered, e.g. in accordance with ref.[2] for the 304 Stainless St& for which the following linear approximation can be applied within 0400°C E=E,+k&-6)=E,+kEB,(1-fB)
(12a)
a = a, - k.(ff t - 6) = a, - k,B, (1 - te),
(I2b)
where k, and k, are positive numbers. Then, the W-functions can be expressed as
(134
Using (13), we shall infer the properties of J-integrals for temperaturedependmt materials. With non-singular temperature at the crack tip it holds that (i) J’ % Je and (ii) J’ and/or Je are affected only by the temperature dependence of E, unlike a; (iii) while with temperature-independent E the values of J’ and/or Je do not depend on the history of loading, it is not the case with thermally dependent E and the calculation must be performed incrementally even within linear elasticity. How significant the difference between incremental and non-incremental computation with &ermally depeadent E [reqecting the difference between (1 la) and (13b)] can be illustrated, for the earlier rn~tio~ material and thermal shock within 0-2%VC, by the computation quoted in the introduction and d ocumented in ref. [2]. (iv) For a non-singular temperature at the crack tip, J* s Ja. (v) J’ and J.. differ more markedly only when the crack tip temperatureis singular. (vi) Jo and JD are generally different only with temperature-dependent material properties.
4. DISCUSSION The previous paragraphs show that unless the crack tip ~m~rature is singular, it is very mu& the same whether we consider (9a) or (9b) while interpreting G from (5). It has to be specified, though, when @a) and (Sb) are to be used in this interpretation. It will be understood that cracks with small length increments throughout the loading process will have to be interpreted in terms of (8a), and consequently, the parameters describing the crack growth will be J’ and/or T* with the corresponding R curves. If the crack does not move at all, we must interpret G in terms of (Sb) and the goveming parameters for the non-stability of the crack will be JD and Jlc. From a practical standpoint, however, only the &st of the two cases ems to be frequent enough, especially if the ductile tearing description is considered. In this discussion we must also cope with the fact that the same stresses and strains near the crack tip can cornspond with different J’s under thermal loading if material properties depend on temperature. Thus, J’ and/or T* are not “for&’ parameters as it may appear from comparison with J within HRR theory, but essentially “energetic” ones. As indicated elsewhere (e.g. ref. [3]), an infinite number of integral parameters can describe one stress-strain field near the craak tip, but only J’ and/or T* have the meauing of crack driving force. It is a matter of experimental verification to show whether T* under thermal-mechanical loading is able to describe the crack growth (using R curves) in the same accomplished way as in the case of mechanical loading only[l].
5. CONCLUDING
REMARKS
This paper has analysed the integral fracture parameters J’, Je and J* in terms of linear elasticity under thermal-mechanical loading. The main facts arising from the analysis can be summarized as follows. (i) J’ is to be confronted with Je only if the temperature at the crack tip is singular. (ii} For non-unbar crack tip tem~ratu~ the J*-integral describes the situation in the same way as the stress ~tc~i~ factor within linear elasticity. (iii) In terms of linear elasticity the incremental computation of J’ is needed only when material properties aTe temperature dependent. Otherwise the description of fracture using J’ is identical to that using the stress intensity factor. Further investigations in this Bold should bring progress in the description of the crack growth caused by repeated thermal shock.
700
Technical note
REFERENCES [l] F. W. Brust, T. Nishioka, S. N. Atluri and M. Nakagaki, Further studies on elastioplastic stable fracture utilising the T*-integral. Engng Fracture Mech. 22, 10794103 (1985). [2] F. W. Brust. M. Nakagaki and C. Springtield, Integral parameters for thermal fracture. Engng Fracture Mech. (in press). [3] S. N. Atluri, Energetic approaches and path-independent integrals in fracture mechanics, in Computational Methoak in the Mechanics of Fracture (Edited bv S. N. Atluri). North-Holland Press, Amsterdam (1986). [4] R. A. Ainsworth, B. K. Neal; and R.-H. Price, Fracture behaviour in the presence of thermal strains. I. Mech. E. Conference on Defects in Pressure Vessels (1978). [S] W. S. Blackburn, Path independent integrals to predict onset of crack instability in an elastioplastic material. Int. J. Fracture Mech. 8, 343-346 (1972). [6] J. R. Rice, Mathematical analysis in the mechanics of fracture. Fracture, Vol. II. Mathematical Fundamentals (Edited by H. Liebowitz). Academic Press, New York (1968). (Received 18 May 1989)