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Generalization of notch analysis and its extension to cyclic loading
Engineering Fracture Mechanics Vol. 32, No. 5, pp. 819-826, Printed in Great Britain.
1989
0013-7944/89 $3.00 + 0.00 0 1989 Perpmon Press pk.
GENERALIZATION OF NOTCH ANALYSIS AND ITS EXTENSION TO CYCLIC LOADING F. ELLYIN
and
D. KUJAWSKIt
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8 Abstract-A method is presented whereby the maximum stress/strain at notch roots can be determined for monotonic as well as cyclic loading. It is based on an averaged similarity measure of the stress and stain product within the elastic-plastic domain around the notch of two solutions, namely the theoretical elastic and actual elastoplastic, when the small scale yielding condition exists. The predicted values at the notch root are in good agreement with the experimental data. The method developed here is rather general, and can be used for the multiaxial states of stress. Two methods previously proposed to predict the stress and strain at the notch root are shown to be particular cases of the one developed herein.
INTRODUCTION MOST COMPONENTSof engineering structures contain geometric discontinuities such as notches, holes, etc. which causes stress/strain concentrations. The product of the nominal stress (strain) and theoretical stress concentration factor, KT,is generally used to estimate the local notch stress (strain) when the material at the notch root remains elastic. However, for most engineering materials, yielding often takes place in a small region around the notch root even at relatively low nominal elastic stresses. It is generally agreed that these local inelastic strains and stresses primarily determine the crack initiation life, and to a certain extent, the fatigue resistance and total life of the structure as a whole. Therefore, the first step to predict the crack initiation life of notched components is to estimate the local maximum stress and strain at the notch root. A number of attempts have been made to describe the nonlinear stress/strain behaviour of notches[ l-91. The most popular and frequently used formulae in notch analysis are: Neuber’s rule [l]; the modified version by Hardrath and Ohman[2] and, Stowell’s approximate formula[3]. It is known that for cyclic loading, Neuber’s, and Hardrath and Ohman’s formulae tend to over-estimate notch root strains[lO-1 11, and may not be accurate enough for predicting the crack initiation life. Based on an energy consideration, Molski and Glinka[4], Glinka[9] and present authors[5,6] recently have developed methods for the elastic-plastic notch analysis. For cyclic loading these relationships give more accurate predictions than those of Neuber’s or Hardrath and Ohman’s. A more general approach to notch analysis is presented in this paper. It is applicable for monotonic and cyclic loading in the case of uniaxial as well as multiaxial states of stress.
NOTCHED BODY ANALYSIS Consider a notched body of linear- or nonlinear-elastic to a slowly increasing surface force applied on the portion maximum applied force on the boundary S, is denoted corresponding displacements by Ui(as shown in Fig. 1). The within the body, i.e. aiij =
material, free of body forces, subjected S, of the boundary. Suppose that the by TiT the stress field by Q+ and the stress components cii are in equilibrium
0; aij = aji, (i, j = 1,2,3)
and satisfy boundary conditions on S, and on the notch surface, thus a,pj =
Ti on S,,
tPresent address: Institute of Machine Design Fundamentals, Poland. 819
Warsaw Technical University, Warsaw, Narbutta 84,
820
F. ELLYIN and D. KUJAWSKI
Fig. 1. A specimen with smooth-ended
op, = 0
notch.
on notch surface,
(31
where nj is the unit vector normal to the surface boundary directed towards the exterior of the body, and a comma denotes partial differentiation with respect to x,. The infinitesimal strain tensor is defined by, Eii = ;
fzfi,j+ u,,).
W
Using eqs (1) and (4) we can write,
Applying Gauss’ theorem of integration, i.e. passing from the volume to the surface integral, using eqs (2) and (3), we have JVcljs,dY=lY,
*iu,dS.
and
(6)
In deriving eq. (6) we have not used any constitutive law, only equilibrium equations, strain displacement relations and boundary conditions. Consider now a specimen of an elastic-plastic material with a notch, loaded by remotely applied load P,which causes yielding of the material in a zone defined by R, near the notch root (Fig. 2a) It is assumed that the plastic zone R, is small in size compared to notch length and other geometric dimensions of the specimen (small scale yielding). In the case of small scale yielding, the
for R,>>f$
a) Fig. 2. A smooth-ended
.Re
b) notch in a Rat specimen in the case of small scale yielding.
Notch analysis and cyclic loading
82I
stress state at a distance R, % RP is not perturbed much by the stress relaxation within Rp, and the traction and displacement vectors on & are essentially given by the linear-elastic solution. The actual configuration in Fig. 2(a) can be replaced by the region R, with boundary condition on R, given by linear-elastic solution of the problem (Fig. 2b). Let us now denote by o$, E;, us the actual values of the stress, strain and displacement fields, and by G;, E;, UTthose values obtained from the linear-elastic solution of the problem within R, of the body subjected to remotely applied load P. If the condition of l+, 4 & is satisfied, then
Using eq. (6) we can write
where VR, is the volume enclosed by R,. The relation (8) can be interpreted as an averaged linear-elastic, and elastic-plastic stress and strain produced notch. It is also a statement of equality of the total strain complemental strain energy) within the enclosed body for yielding. It can be shown that ~vq,zgdV
= IISd(~O+$]dV
similarity measure of the theoretical in the elastoplastic domain around the energy (sum of the strain energy and two solutions in the case of small scale
=~v[Ss,dL,+SEUd,,ldV.dV.
(9)
The first term in the brackets on the right hand side is the strain energy density, cij de,,
W(E) =
(10)
s and the second one is the complemental
strain energy density, i.e.
Therefore, eq. (9) can be written as jj,E,IV For a proportional from,
= jv [W(E) + W(o)] dV.
or nearly proportional
w(E)
where C& = 3 S&/2, Sii vs E,~curve. Similarly,
= f3# -
O&/3,
loading, strain energy density W(E) can be calculated
&.&SW’ Ifn’
=
(12)
E& =
1
gE4Ets
(13)
2.++/3, and n’ is the strain hardening exponent of ocs
W(0) = n/W(c).
(14)
Therefore, using (12), (14), (8) can be written in the following form: [qwa(&) - w’(E)] dV = 0
s VRc
(1%
where q = (1 + n ‘)/2 and is bounded by l/2 I 4 I I. The principle of minimum strain energy states that of all sets of admissible stress components, c~, the actual one (a”,) would make W&) a
822
F. ELLYIN and D. KUJAWSKl
minimum 1121.Thus, for any other stress state we will have We(s) 2: qW(.s). From condition (7) we have We(s) = q V(E) at the boundary, and also that the integral over the volume, eq. (15), is the same over R,.Thus, it follows that q w+(E) - W’(E) = 0,
(16)
or 5$&t = LT;&;.
(17)
A geometric proof of the condition (16) can be obtained by constructing a W(E) vs x, distribution. For example, when sectioned with the plane x2 = xJ = 0, integral (15) will denote the area under the curve W(E) vs x,. The two curves at the boundary coincide (see eq. 7) and the minimum condition of W(E) requires that We(s) be on or above W’(E). But condition (15) imposes that the two areas be the same, therefore, the integrand in (15) must be zero, i.e. (16) must hold true. The validity of (16) can further be established by noting that according to Irwin’s plastic zone correction factor1 131,for an elastic-perfectly-plastic material the increase in the near-tip stress field is ,,,/?, thus, the increase in the strain energy is by a factor of 2. For an elastic-perfectly-plastic material n’ = 0 and q = l/2, and (16) then holds true. It is interesting to note that when the path independent j-integral of Rice[l4] is used for the path around the smooth notch (Fig. l), then instead of (15) we get,
r
[W’(E) - W(E)] dy = 0
J notch
(18)
Furthermore, Hutchinson[l5] has shown that for a crack under mode I loading, the strain energy density for a fully-plastic regime is the same as that of purely elastic solution, for a material with a piecewise linear stress-strain relationship, i.e. W”(E) - We(&)= 0.
(19)
Comparing (19) with (16) we note that the first term on the left-hand side differs by a factor of q = (1 + n’)/2. However, it is believed that (16) would yield more accurate results for stress-concentration factors. SPECIAL
CASES
In particular case of a uniaxial stress field, eq. (17) reduces to: tF,&”=
CTaEa,
(20)
and it follows that, t&x&ax = ~~&,X.
(21)
Using the notations (T’& = &IT, and &Lax= &sn where o, and E, are nominal linear-elastic stress and strain, respectively, eq. (21) can be written in the following form, ;u: = K&.
(22)
This equation is the well-known Neuber’s formula where K, = [T;,,/(T, and I(, = E;,,/E, are stress and strain concentration factors, respectively. Let us now examine the differences between nonlinear-elastic and elastic-plastic material behaviour subjected to monotonic and cyclic loading. In the case of an elastic material, the unloading path is identical to that of loading (see Fig. 3), whereas for the inelastic material, the unloading path is different from that of loading. For example, the loading path of OAB of Fig. 3, is followed by the unloading path BDO- for a uniaxial stress state. Generally, the branches OAB and BDO have a similar shape, if the origin of the coordinate system for the unloading case is placed at B. The general equality condition (12) also applies for the unloading case as long as the proportionality of strain components hold throughout the unloading process.
Notch analysis and
cyclic loading
823
Ao
non-linear elastic
Aa’
_
Fig. 3. Cyclic Au-A& curves.
The cyclic variation in stresses, Aa, and strains, AE,depend only on the range of remotely applied load fluctuation. Therefore, for a nonlinear-elastic material upon cyclic loading, eq. (21) has to be written in terms of the representative stress and strain ranges, i.e. Aa” A.se= Aa” AEa,
(23)
where Aa” and A.9 are the notch root stress and strain ranges for the nonlinear-elastic material. For a fully reversed cycle of loading eq. (23), is simply the sum of strain energies for loading path OAB, and the reversed path BAO with origin placed at B. For an elastic-plastic material, one can use the plastic superposition method advocated by Hult and McClintock[l6] and Rice[l4] to calculate stress and strain ranges. In this approach the unloading stress-strain relationship is the same as the loading case, if the latter is referred to a set of reversed axes (point B in Fig. 3). Using the strain energy interpretation given above, eq. (23) can be written as: Aa”Aee=AWL+AWRL=AaaA~a+AWp,
(24)
where A W, and A W,, are the energies for loading path OAB, and the reversed path BAO with the origin placed at B, and A Wp is the absorbed strain energy per cycle (area of the hysteresis loop OABDO).? The above relationship holds for any point in the notch vicinity and thus can be written as,
(25) or Kt(iAa,As,)
= iAa&As,!,,,, + fA Wk,.
(26)
In eq. (26), l/2 Aa,As, is the nominal strain energy density at the far field, A W,, and the right-hand side of (26) is the cyclic strain energy density represented by the area under the loading curve OAB in Fig. 3. The relation between stress and strain ranges is given by [17] As AP Asp __=-+-_=_+ 2 2 2
Aa 2E
(27)
where E is the elastic modulus, K’ and n’ are cyclic strength and hardening exponent of the material. tFor a nonlinear-elastic material A Wp = 0.
824
F. ELLYIN
and D. DUJAWSKI
Using relationship (27), eq. (26) can be written as
G-3) where Acr and AE~ are the maximum notch root stress and plastic strain ranges, respectively.
DISCUSSION Relation (28) was proposed by Molski and Glinka[4], and present authors[5,6] to estimate the maximum stress and strain at the notch root during cycle loading when the applied nominal stress (or strain) are below material yield condition. On the other hand, Neuber’s formula (eq. 22), is applicable for the notch root stress and strain calculation in the case of monotonic loading. Note that both eqs (22) and (28) are derived here as particular cases of eq. (17). The proposed theory is thus more general and can also be used for the multiaxial states of stress. To compare the predictions of the proposed theory with the experimental results, two types of loading, i.e. monotonic and cyclic, are considered. Plates containing circular and elliptical central holes were tested under monotonic loading in ref. [l 11. Figure 4 shows the maximum monotonic notch strain E;,,, against the theoretical notch strain, &E,. Prediction of the present method (i.e. eq. 21), lies between the experimental data. A similar plate with a circular hole was tested under cyclic stress controlled conditions[lO]. Prediction of the present method (eq. 26), as well as that of Neuber’s rule are depicted in Fig. 5. The results, with the exception of a single point, appear to lie closer to that predicted by the present theory. It is seen in Fig. 5 that Neuber’s rule tends to over estimate the strains at the notch root. Attempts have been made to modify Neuber’s relation for the cyclic loading by replacing KT in (22) by a fatigue notch reduction factor, Kf, based on experimental observation[l8]. Note that while KT depends on the notch geometry, Kf on the other hand may vary with number of cycles and type of material[l9,20]. For example, values of & may vary from close to KT at low strain levels to a lower value at high strain levels. From Fig. 5, it is evident that there is no need to define an empirical factor such as Kc and the proposed relationship (28) tends to predict the experimental observations in a consistent manner. Further comparison with the experimental data can be found in refs [5, 61.
-
Center of Notch
IO3
Monotonic g--e Curve (2024-T351 Al Alloy) ____-----_ -----(Scale of Stress)-
Cl Ki4.60
Elliptical Notch
-
max Max.Monotonic Notch Strain Ea Fig. 4. Theoretical
notch
strain
vs maximum
monotonic
notch
strain
for centrally
notched
plates(ll].
825
Notch analysis and cyclic loading
8008 z
_
P
700 -
%
3 %
600 -
2 sg
~
P
500 -
Cycilc 0-s curve (245T3 Al. alloy) K,=2.56
1 F *, 100
b
t/ 0
I
0
0.002
I
I
I
I
I
I
0004
0.006
0.008
0.010
01312
0.014
E max,
00 16
Max. Notch Strain Amplitude
Fig. 5. Theoretical notch stress amplitude vs maximum notch strain amplitude for a circular notched plate under stress-controlled condition[lO].
CONCLUSIONS A method has been presented whereby the maximum stress and strain at a notch root can be determined for elastic-plastic or nonlinear-elastic materials from the knowledge of the theoretical stress concentration factor, &, which is a notch geometry dependent factor. The method applies to both monotonic and cyclic loading and can be used in the case of multiaxial states of stress. The predicted values at the notch roots are in good agreement with the experimental data. Two previously proposed methods to estimate the maximum stress/strain at notches, can be derived as particular cases of the present method. ~cknowlege~enrs-The results reported herein are part of a general investigation on the behaviour of materials and reliability of components under various stress states and environments. The research is supported, in part, by the Natural Science and Engineering Research Council of Canada (NSERC Grant No. A-3808). Thanks are also due to Izaak Killam Memorial Foundation for awarding a Post-Doctoral Fellowship to D. Kujawski.
REFERENCES [I] H. Neuber, Theory of stress concentration for shear strained prismatical bodies with arbitrary non-linear stress-strain law. J. appi. Me&. 28, 544-550 (1961). [2] H. F. Hardrath and L. Ohman, A study of elastic and plastic stress concentration factors due to notches and fillets in Bat plates. NACA Report 117 (1953). [3] E. Z. Stowell, Stress and strain concentration at a circular hole in an infinite plate. NACA Technical Note, No. 2073 (1950). [4] K. Molski and G. Glinka, A method of elastic-pfastic stress and strain calculation of a notch root. Mater. Sci. Engng SO, 93-100 (1981). [S] D. Kujawski and F. Ellyin, An energy-based method for stress and strain calculation of notches. Proc. 8th 1st. Conf. on Structural Me&zanies in Reactor Technology, Brussels, paper L 413, pp. 173-178 (August 1985). [6] F. Eilyin and D. Kujawski, Notch root stress/strain prediction for elastic-plastic loading. RES Me&mica 20, 177-190 (1987). [7] J. D. Morrow, R. M. Wetzel and T. H. Topper, Laboratory simulation of structural fatigue behaviour, in EfSecrs of Environment and Complex Load History on Fatigue Life, ASTM STP 462, 79-91 (1970). [8] M. Hoffmann and T. Seeger, A generalized method for estimating elastic-plastic notch stresses and strains. J. Engng Mater. Technol. 107, 250-260 (1985). [9] G. Glinka, Calculation of inelastic notch-tip strain-stress histories under cyclic loading. Engng Fracture Mech. 22, 836854 (1985).
826
F. ELLYIN and D. KUJAWSKI
[lo] B. N. Leis and N. D. Frey, Cyclic-inelastic deformation and fatigue resistance of notched-thin aluminum plates. Proc. of the Society for Experimental Stress Analysis, Vol. 39, pp. 287-295 (1982). [1I] B. N. Leis, C. V. B. Gowda and T. H. Topper, Some studies of the influence of localized and gross plasticity on the monotonic and cyclic concentration factors. J. Test. Eual. 1, 341-348 (July 1973). [12] J. Washizu, Variational Methods in Elasticity and Plasticity. Pergamon Press, Oxford (1968). [13] G. R. Irwin, Linear fracture mechanics, fracture transition and fracture control. Engng Fracture Mech 1, 241-257 (1968). [14] J. R. Rice, Mechanics of crack tip deformation and extension by fatigue. Fafigue Crack Propagation. ASTM STP 415, 267-309,
(1967).
[15] J. W. Hutchinson, Singular behaviour at the end of a tensile crack in a hardening material J. Mech. Phys. Sol. 16. 13-31 (1968). [16] J. A. H. Hult and F. A. McClintock, Elastic-plastic stress and strain distributions around sharp notches under repeated shear. Proc. 9th Int. Congr. of Applied Mechanics, Vol. 8, Brussels, pp. 51-58 (1956). [17] F. Ellyin and D. Kujawski, Plastic strain energy in fatigue failure. J. Press. Vess. Technol. 106, 342-347 (1984). [18] T. H. Topper, R. M. Wetzel and J. D. Morrow, Neuber’s rule applied to fatigue of notched specimen. J. Mater. 4. 2W209 (1969). [19] T. Udoguchi and T. Wada, Notch effect on low-cycle fatigue strength of metals. Proc. 1st Int. Cor$ on Pressure Vessel Technology, Delft, Paper 11-92, pp. 1191-1202 (1969). [20] T. V. Duggan and M. W. Proctor, Prediction of crack formation life in notched specimens. Proc 5th Int. Con!. on Fracture, Cannes, France (Edited by D. Fracois), Vol. 2, pp. 589-596 (1981). (Received 14 March 1988)
1989
0013-7944/89 $3.00 + 0.00 0 1989 Perpmon Press pk.
GENERALIZATION OF NOTCH ANALYSIS AND ITS EXTENSION TO CYCLIC LOADING F. ELLYIN
and
D. KUJAWSKIt
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8 Abstract-A method is presented whereby the maximum stress/strain at notch roots can be determined for monotonic as well as cyclic loading. It is based on an averaged similarity measure of the stress and stain product within the elastic-plastic domain around the notch of two solutions, namely the theoretical elastic and actual elastoplastic, when the small scale yielding condition exists. The predicted values at the notch root are in good agreement with the experimental data. The method developed here is rather general, and can be used for the multiaxial states of stress. Two methods previously proposed to predict the stress and strain at the notch root are shown to be particular cases of the one developed herein.
INTRODUCTION MOST COMPONENTSof engineering structures contain geometric discontinuities such as notches, holes, etc. which causes stress/strain concentrations. The product of the nominal stress (strain) and theoretical stress concentration factor, KT,is generally used to estimate the local notch stress (strain) when the material at the notch root remains elastic. However, for most engineering materials, yielding often takes place in a small region around the notch root even at relatively low nominal elastic stresses. It is generally agreed that these local inelastic strains and stresses primarily determine the crack initiation life, and to a certain extent, the fatigue resistance and total life of the structure as a whole. Therefore, the first step to predict the crack initiation life of notched components is to estimate the local maximum stress and strain at the notch root. A number of attempts have been made to describe the nonlinear stress/strain behaviour of notches[ l-91. The most popular and frequently used formulae in notch analysis are: Neuber’s rule [l]; the modified version by Hardrath and Ohman[2] and, Stowell’s approximate formula[3]. It is known that for cyclic loading, Neuber’s, and Hardrath and Ohman’s formulae tend to over-estimate notch root strains[lO-1 11, and may not be accurate enough for predicting the crack initiation life. Based on an energy consideration, Molski and Glinka[4], Glinka[9] and present authors[5,6] recently have developed methods for the elastic-plastic notch analysis. For cyclic loading these relationships give more accurate predictions than those of Neuber’s or Hardrath and Ohman’s. A more general approach to notch analysis is presented in this paper. It is applicable for monotonic and cyclic loading in the case of uniaxial as well as multiaxial states of stress.
NOTCHED BODY ANALYSIS Consider a notched body of linear- or nonlinear-elastic to a slowly increasing surface force applied on the portion maximum applied force on the boundary S, is denoted corresponding displacements by Ui(as shown in Fig. 1). The within the body, i.e. aiij =
material, free of body forces, subjected S, of the boundary. Suppose that the by TiT the stress field by Q+ and the stress components cii are in equilibrium
0; aij = aji, (i, j = 1,2,3)
and satisfy boundary conditions on S, and on the notch surface, thus a,pj =
Ti on S,,
tPresent address: Institute of Machine Design Fundamentals, Poland. 819
Warsaw Technical University, Warsaw, Narbutta 84,
820
F. ELLYIN and D. KUJAWSKI
Fig. 1. A specimen with smooth-ended
op, = 0
notch.
on notch surface,
(31
where nj is the unit vector normal to the surface boundary directed towards the exterior of the body, and a comma denotes partial differentiation with respect to x,. The infinitesimal strain tensor is defined by, Eii = ;
fzfi,j+ u,,).
W
Using eqs (1) and (4) we can write,
Applying Gauss’ theorem of integration, i.e. passing from the volume to the surface integral, using eqs (2) and (3), we have JVcljs,dY=lY,
*iu,dS.
and
(6)
In deriving eq. (6) we have not used any constitutive law, only equilibrium equations, strain displacement relations and boundary conditions. Consider now a specimen of an elastic-plastic material with a notch, loaded by remotely applied load P,which causes yielding of the material in a zone defined by R, near the notch root (Fig. 2a) It is assumed that the plastic zone R, is small in size compared to notch length and other geometric dimensions of the specimen (small scale yielding). In the case of small scale yielding, the
for R,>>f$
a) Fig. 2. A smooth-ended
.Re
b) notch in a Rat specimen in the case of small scale yielding.
Notch analysis and cyclic loading
82I
stress state at a distance R, % RP is not perturbed much by the stress relaxation within Rp, and the traction and displacement vectors on & are essentially given by the linear-elastic solution. The actual configuration in Fig. 2(a) can be replaced by the region R, with boundary condition on R, given by linear-elastic solution of the problem (Fig. 2b). Let us now denote by o$, E;, us the actual values of the stress, strain and displacement fields, and by G;, E;, UTthose values obtained from the linear-elastic solution of the problem within R, of the body subjected to remotely applied load P. If the condition of l+, 4 & is satisfied, then
Using eq. (6) we can write
where VR, is the volume enclosed by R,. The relation (8) can be interpreted as an averaged linear-elastic, and elastic-plastic stress and strain produced notch. It is also a statement of equality of the total strain complemental strain energy) within the enclosed body for yielding. It can be shown that ~vq,zgdV
= IISd(~O+$]dV
similarity measure of the theoretical in the elastoplastic domain around the energy (sum of the strain energy and two solutions in the case of small scale
=~v[Ss,dL,+SEUd,,ldV.dV.
(9)
The first term in the brackets on the right hand side is the strain energy density, cij de,,
W(E) =
(10)
s and the second one is the complemental
strain energy density, i.e.
Therefore, eq. (9) can be written as jj,E,IV For a proportional from,
= jv [W(E) + W(o)] dV.
or nearly proportional
w(E)
where C& = 3 S&/2, Sii vs E,~curve. Similarly,
= f3# -
O&/3,
loading, strain energy density W(E) can be calculated
&.&SW’ Ifn’
=
(12)
E& =
1
gE4Ets
(13)
2.++/3, and n’ is the strain hardening exponent of ocs
W(0) = n/W(c).
(14)
Therefore, using (12), (14), (8) can be written in the following form: [qwa(&) - w’(E)] dV = 0
s VRc
(1%
where q = (1 + n ‘)/2 and is bounded by l/2 I 4 I I. The principle of minimum strain energy states that of all sets of admissible stress components, c~, the actual one (a”,) would make W&) a
822
F. ELLYIN and D. KUJAWSKl
minimum 1121.Thus, for any other stress state we will have We(s) 2: qW(.s). From condition (7) we have We(s) = q V(E) at the boundary, and also that the integral over the volume, eq. (15), is the same over R,.Thus, it follows that q w+(E) - W’(E) = 0,
(16)
or 5$&t = LT;&;.
(17)
A geometric proof of the condition (16) can be obtained by constructing a W(E) vs x, distribution. For example, when sectioned with the plane x2 = xJ = 0, integral (15) will denote the area under the curve W(E) vs x,. The two curves at the boundary coincide (see eq. 7) and the minimum condition of W(E) requires that We(s) be on or above W’(E). But condition (15) imposes that the two areas be the same, therefore, the integrand in (15) must be zero, i.e. (16) must hold true. The validity of (16) can further be established by noting that according to Irwin’s plastic zone correction factor1 131,for an elastic-perfectly-plastic material the increase in the near-tip stress field is ,,,/?, thus, the increase in the strain energy is by a factor of 2. For an elastic-perfectly-plastic material n’ = 0 and q = l/2, and (16) then holds true. It is interesting to note that when the path independent j-integral of Rice[l4] is used for the path around the smooth notch (Fig. l), then instead of (15) we get,
r
[W’(E) - W(E)] dy = 0
J notch
(18)
Furthermore, Hutchinson[l5] has shown that for a crack under mode I loading, the strain energy density for a fully-plastic regime is the same as that of purely elastic solution, for a material with a piecewise linear stress-strain relationship, i.e. W”(E) - We(&)= 0.
(19)
Comparing (19) with (16) we note that the first term on the left-hand side differs by a factor of q = (1 + n’)/2. However, it is believed that (16) would yield more accurate results for stress-concentration factors. SPECIAL
CASES
In particular case of a uniaxial stress field, eq. (17) reduces to: tF,&”=
CTaEa,
(20)
and it follows that, t&x&ax = ~~&,X.
(21)
Using the notations (T’& = &IT, and &Lax= &sn where o, and E, are nominal linear-elastic stress and strain, respectively, eq. (21) can be written in the following form, ;u: = K&.
(22)
This equation is the well-known Neuber’s formula where K, = [T;,,/(T, and I(, = E;,,/E, are stress and strain concentration factors, respectively. Let us now examine the differences between nonlinear-elastic and elastic-plastic material behaviour subjected to monotonic and cyclic loading. In the case of an elastic material, the unloading path is identical to that of loading (see Fig. 3), whereas for the inelastic material, the unloading path is different from that of loading. For example, the loading path of OAB of Fig. 3, is followed by the unloading path BDO- for a uniaxial stress state. Generally, the branches OAB and BDO have a similar shape, if the origin of the coordinate system for the unloading case is placed at B. The general equality condition (12) also applies for the unloading case as long as the proportionality of strain components hold throughout the unloading process.
Notch analysis and
cyclic loading
823
Ao
non-linear elastic
Aa’
_
Fig. 3. Cyclic Au-A& curves.
The cyclic variation in stresses, Aa, and strains, AE,depend only on the range of remotely applied load fluctuation. Therefore, for a nonlinear-elastic material upon cyclic loading, eq. (21) has to be written in terms of the representative stress and strain ranges, i.e. Aa” A.se= Aa” AEa,
(23)
where Aa” and A.9 are the notch root stress and strain ranges for the nonlinear-elastic material. For a fully reversed cycle of loading eq. (23), is simply the sum of strain energies for loading path OAB, and the reversed path BAO with origin placed at B. For an elastic-plastic material, one can use the plastic superposition method advocated by Hult and McClintock[l6] and Rice[l4] to calculate stress and strain ranges. In this approach the unloading stress-strain relationship is the same as the loading case, if the latter is referred to a set of reversed axes (point B in Fig. 3). Using the strain energy interpretation given above, eq. (23) can be written as: Aa”Aee=AWL+AWRL=AaaA~a+AWp,
(24)
where A W, and A W,, are the energies for loading path OAB, and the reversed path BAO with the origin placed at B, and A Wp is the absorbed strain energy per cycle (area of the hysteresis loop OABDO).? The above relationship holds for any point in the notch vicinity and thus can be written as,
(25) or Kt(iAa,As,)
= iAa&As,!,,,, + fA Wk,.
(26)
In eq. (26), l/2 Aa,As, is the nominal strain energy density at the far field, A W,, and the right-hand side of (26) is the cyclic strain energy density represented by the area under the loading curve OAB in Fig. 3. The relation between stress and strain ranges is given by [17] As AP Asp __=-+-_=_+ 2 2 2
Aa 2E
(27)
where E is the elastic modulus, K’ and n’ are cyclic strength and hardening exponent of the material. tFor a nonlinear-elastic material A Wp = 0.
824
F. ELLYIN
and D. DUJAWSKI
Using relationship (27), eq. (26) can be written as
G-3) where Acr and AE~ are the maximum notch root stress and plastic strain ranges, respectively.
DISCUSSION Relation (28) was proposed by Molski and Glinka[4], and present authors[5,6] to estimate the maximum stress and strain at the notch root during cycle loading when the applied nominal stress (or strain) are below material yield condition. On the other hand, Neuber’s formula (eq. 22), is applicable for the notch root stress and strain calculation in the case of monotonic loading. Note that both eqs (22) and (28) are derived here as particular cases of eq. (17). The proposed theory is thus more general and can also be used for the multiaxial states of stress. To compare the predictions of the proposed theory with the experimental results, two types of loading, i.e. monotonic and cyclic, are considered. Plates containing circular and elliptical central holes were tested under monotonic loading in ref. [l 11. Figure 4 shows the maximum monotonic notch strain E;,,, against the theoretical notch strain, &E,. Prediction of the present method (i.e. eq. 21), lies between the experimental data. A similar plate with a circular hole was tested under cyclic stress controlled conditions[lO]. Prediction of the present method (eq. 26), as well as that of Neuber’s rule are depicted in Fig. 5. The results, with the exception of a single point, appear to lie closer to that predicted by the present theory. It is seen in Fig. 5 that Neuber’s rule tends to over estimate the strains at the notch root. Attempts have been made to modify Neuber’s relation for the cyclic loading by replacing KT in (22) by a fatigue notch reduction factor, Kf, based on experimental observation[l8]. Note that while KT depends on the notch geometry, Kf on the other hand may vary with number of cycles and type of material[l9,20]. For example, values of & may vary from close to KT at low strain levels to a lower value at high strain levels. From Fig. 5, it is evident that there is no need to define an empirical factor such as Kc and the proposed relationship (28) tends to predict the experimental observations in a consistent manner. Further comparison with the experimental data can be found in refs [5, 61.
-
Center of Notch
IO3
Monotonic g--e Curve (2024-T351 Al Alloy) ____-----_ -----(Scale of Stress)-
Cl Ki4.60
Elliptical Notch
-
max Max.Monotonic Notch Strain Ea Fig. 4. Theoretical
notch
strain
vs maximum
monotonic
notch
strain
for centrally
notched
plates(ll].
825
Notch analysis and cyclic loading
8008 z
_
P
700 -
%
3 %
600 -
2 sg
~
P
500 -
Cycilc 0-s curve (245T3 Al. alloy) K,=2.56
1 F *, 100
b
t/ 0
I
0
0.002
I
I
I
I
I
I
0004
0.006
0.008
0.010
01312
0.014
E max,
00 16
Max. Notch Strain Amplitude
Fig. 5. Theoretical notch stress amplitude vs maximum notch strain amplitude for a circular notched plate under stress-controlled condition[lO].
CONCLUSIONS A method has been presented whereby the maximum stress and strain at a notch root can be determined for elastic-plastic or nonlinear-elastic materials from the knowledge of the theoretical stress concentration factor, &, which is a notch geometry dependent factor. The method applies to both monotonic and cyclic loading and can be used in the case of multiaxial states of stress. The predicted values at the notch roots are in good agreement with the experimental data. Two previously proposed methods to estimate the maximum stress/strain at notches, can be derived as particular cases of the present method. ~cknowlege~enrs-The results reported herein are part of a general investigation on the behaviour of materials and reliability of components under various stress states and environments. The research is supported, in part, by the Natural Science and Engineering Research Council of Canada (NSERC Grant No. A-3808). Thanks are also due to Izaak Killam Memorial Foundation for awarding a Post-Doctoral Fellowship to D. Kujawski.
REFERENCES [I] H. Neuber, Theory of stress concentration for shear strained prismatical bodies with arbitrary non-linear stress-strain law. J. appi. Me&. 28, 544-550 (1961). [2] H. F. Hardrath and L. Ohman, A study of elastic and plastic stress concentration factors due to notches and fillets in Bat plates. NACA Report 117 (1953). [3] E. Z. Stowell, Stress and strain concentration at a circular hole in an infinite plate. NACA Technical Note, No. 2073 (1950). [4] K. Molski and G. Glinka, A method of elastic-pfastic stress and strain calculation of a notch root. Mater. Sci. Engng SO, 93-100 (1981). [S] D. Kujawski and F. Ellyin, An energy-based method for stress and strain calculation of notches. Proc. 8th 1st. Conf. on Structural Me&zanies in Reactor Technology, Brussels, paper L 413, pp. 173-178 (August 1985). [6] F. Eilyin and D. Kujawski, Notch root stress/strain prediction for elastic-plastic loading. RES Me&mica 20, 177-190 (1987). [7] J. D. Morrow, R. M. Wetzel and T. H. Topper, Laboratory simulation of structural fatigue behaviour, in EfSecrs of Environment and Complex Load History on Fatigue Life, ASTM STP 462, 79-91 (1970). [8] M. Hoffmann and T. Seeger, A generalized method for estimating elastic-plastic notch stresses and strains. J. Engng Mater. Technol. 107, 250-260 (1985). [9] G. Glinka, Calculation of inelastic notch-tip strain-stress histories under cyclic loading. Engng Fracture Mech. 22, 836854 (1985).
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F. ELLYIN and D. KUJAWSKI
[lo] B. N. Leis and N. D. Frey, Cyclic-inelastic deformation and fatigue resistance of notched-thin aluminum plates. Proc. of the Society for Experimental Stress Analysis, Vol. 39, pp. 287-295 (1982). [1I] B. N. Leis, C. V. B. Gowda and T. H. Topper, Some studies of the influence of localized and gross plasticity on the monotonic and cyclic concentration factors. J. Test. Eual. 1, 341-348 (July 1973). [12] J. Washizu, Variational Methods in Elasticity and Plasticity. Pergamon Press, Oxford (1968). [13] G. R. Irwin, Linear fracture mechanics, fracture transition and fracture control. Engng Fracture Mech 1, 241-257 (1968). [14] J. R. Rice, Mechanics of crack tip deformation and extension by fatigue. Fafigue Crack Propagation. ASTM STP 415, 267-309,
(1967).
[15] J. W. Hutchinson, Singular behaviour at the end of a tensile crack in a hardening material J. Mech. Phys. Sol. 16. 13-31 (1968). [16] J. A. H. Hult and F. A. McClintock, Elastic-plastic stress and strain distributions around sharp notches under repeated shear. Proc. 9th Int. Congr. of Applied Mechanics, Vol. 8, Brussels, pp. 51-58 (1956). [17] F. Ellyin and D. Kujawski, Plastic strain energy in fatigue failure. J. Press. Vess. Technol. 106, 342-347 (1984). [18] T. H. Topper, R. M. Wetzel and J. D. Morrow, Neuber’s rule applied to fatigue of notched specimen. J. Mater. 4. 2W209 (1969). [19] T. Udoguchi and T. Wada, Notch effect on low-cycle fatigue strength of metals. Proc. 1st Int. Cor$ on Pressure Vessel Technology, Delft, Paper 11-92, pp. 1191-1202 (1969). [20] T. V. Duggan and M. W. Proctor, Prediction of crack formation life in notched specimens. Proc 5th Int. Con!. on Fracture, Cannes, France (Edited by D. Fracois), Vol. 2, pp. 589-596 (1981). (Received 14 March 1988)