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Effect of strain rate on crack growth in aluminum alloy 1100-0
Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
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Eect of strain rate on crack growth in aluminum alloy 1100-0 G.E. Papakaliatakis School of Engineering, Demokritus University of Thrace, GR-67100 Xanthi, Greece
Abstract Stress and damage analysis are performed to analyze the Mode I crack growth behavior of a central crack panel made of aluminum alloy 1100-0. On account of the highly nonhomogeneous stress state, each material element would experience a dierent strain rate depending on the location and loading rate. A data bank of uniaxial stress and strain curves is provided to cover the range local strain rates depending on the load time history. Such a approach is referred to as the strain rate dependent model in contrast to plasticity that utilizes a single constitutive relation. The strain energy density criterion is applied to determine the onset of crack initiation, stable crack growth and ®nal termination. A unique feature of the approach is that the same criterion could describe the foregoing three distinct events of fracture behavior. Results are obtained for applied loads with dierent strain rates and compared with those obtained from the classical theory of plasticity, which is unconservative. Ó 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Uniaxial tensile tests of metal alloys have shown that the mechanical response depends on the rate and history of the applied load. Changes in the post-yield behavior are more pronounced and can be signi®cant, particularly when the resulting stress and strain states are highly nonhomogeneous. The prevailing dierence between the local strain rate near a singular point such as the crack tip can dier from those at distances away by orders of magnitude. Hence, the use of a single and the same constitutive relation for all material elements as it is the case in the classical theory of plasticity is not adequate. A theory that accounts for nonhomogeneity arising from the local stress and strain states has been developed in Refs. [1,2]. Advocated is the notion that surface and volume energy density are no longer independent but they would interact via the rate change of volume with surface area which
has been traditionally assumed to vanish in the limit as the material element diminishes in size. Such a hypothetical assumption is removed in Refs. [1,2] and explains for inaccuracies and inconsistencies that would otherwise be present as in classical continuum mechanics theories. As the material elements are ®nite in size, their dimensions are determined from the conditions that the energy transmitted is the same in the three mutually perpendicular directions. This would provide a one-to-one correspondence between the uniaxial and multiaxial stress states. These elements have been referred to as the isoenergy density elements [2]. The result for stresses near a sharp crack can dier signi®cantly from the theory of plasticity [3]. An attempt to correct for the load strain rates was ®rst made in Ref. [4] without taking into account the full eect of nonhomogeneity due to stress and strain in addition to the material microstructure. This simpli®es the analysis and could still reveal the importance to extend the range of
0167-8442/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 9 9 ) 0 0 0 0 7 - 5
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G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
strain rates in the treatment of elastic±plastic crack problems. A modi®ed PAPST [5] (Plastic Axisymmetric/Planar Structure) computer program were used and crack problem similar to that in Ref. [4] was solved [6]. This work is a further application of the works [4,6] to analyze the fracture behavior of the aluminum alloy 1100-0. 2. Basic approach As mentioned earlier and in Refs. [4,6], a data bank of uniaxial curves with dierent strain rates is provided such that the strains (and/or stresses) for each load step can be determined at each location. The strains for two successive load or time steps would yield the strain rate and hence a particular stress±strain curve. The loci of these curve segments would give a complete stress±strain curve that is dierent for each material element. The PAPST [5] ®nite element computer program is applied to perform the elastic±plastic stress analysis by increments. Twelve-node isoparametric elements are used with a special singular crack tip element where the positions of two of the side nodes are shifted by distances of 1/9 and 4/9 from the crack tip corner node. The Von±Mises yield criterion is applied as the load is increased in a step-wise fashion. The computational procedure can be obtained as follows: · All elements are assumed to follow the same uniaxial stress±strain curve for the ®rst increment of loading. · The principal strain ej (j 1, 2, 3) are calculated at all Gaussian points of each element and hence the eective strain eeff can be obtained: i1=2 1 h 2 2 2 eeff p e1 ÿ e2 e2 ÿ e3 e3 ÿ e1 ; 2 1
ment. A knowledge of the element strain rate determines the particular stress±strain curve in the data bank. · Dierent segments of stress±strain curves with dierent strain rates for each load step are connected to yield a complete curve for a given material element. The above is repeated as the crack specimen is loaded step-by-step until the stable crack growth process is completed at which point rapid crack propagation is predicted to begin and the computation is terminated. 3. Specimen geometry and material Fig. 1 depicts a rectangular body in plane strain. It has a width of 2b 15 cm, height of
the corresponding stress being i1=2 1 h reff p r1 ÿ r2 2 r2 ÿ r3 2 r3 ÿ r1 2 : 2 2 · The strain rate of each element corresponds to the relative strains of two successive load increments divided by the corresponding time incre-
Fig. 1. Geometry of specimen.
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
2h 20 cm and a through-the-thickness crack of length 2a 4 cm. Equal and opposite uniform stress of magnitude r is increased monotonically. The problem thus possesses one quarter symmetry. Mechanical properties of the cracked specimen corresponds to those for the commercial grade aluminum alloy 1100-0. They are listed in Table 1. This metal alloy 1100-0 is rate sensitive as exhibited by the true stress and true strain curves [7] in Fig. 2 for strain rates of e_ 3 10ÿ3 , 1.8, 0:220 103 and 1:100 103 sÿ1 . The terminal points of these curves were stipulated on the basis that the failure strains decrease with increasing strain rates. Henceforth, these four strain rates will be referred to as Cases I, II, III and IV as in Table 2.
Table 1 Mechanical properties of aluminum alloy 1100-0 Young's modulus E (MPa)
Poisson's ratio m
Yield strength rys (MPa)
3310.6
0.33
44.8
133
4. Preliminary considerations 4.1. Plasticity solution The rectangular body with a central crack is discretized by ®nite elements. Fig. 3 shows the ®nite element grid pattern with one-quarter symmetry. Twenty four (24) elements and 151 nodes were used to compute for the stress and displacements as the Case I, II, III and IV are solved incrementally using the PAPST [5] computer program. Some typical results are displayed in Figs. 4 and 5. Plotted in Fig. 4 are the normal stress ry and eective reff as a function of the distance r from the crack tip for e_ 3 ´ 10±3 sÿ1 and 1.100 ´ 103 sÿ1 . The decay in stress magnitude becomes much more pronounced as the strain rate increases. Similar results are given in Fig. 5 for the eective strain eeff and strain energy density function dW/dV which is given by
Fig. 2. True stress±true strain curves of aluminum alloy 1100-0 for dierent strain rates.
Table 2 Applied strain rates I
II
III
IV
3 ´ 10ÿ3
1.8
0.220 ´ 103
1.100 ´ 103
Fig. 3. Finite element grid pattern with one quarter symmetry.
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G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
Fig. 4. Variation of stress component ry and eective stress with distance from crack tip for applied stress 38.25 MPa and strain rate 0.003 and 1100 sÿ1 using plasticity theory.
where, particularly near the crack tip region where dW/dV would attain its largest value. Let the size of this region be denoted by ro taken as the radius of a circle around the crack tip. In continuum mechanics, ro would be macroscopic in size, say 0.01 cm. Crack initiation is assumed to occur when dW/dV in one of the elements on the periphery of the circle reaches a critical value, say (dW/dV)cr [3]. For Mode I loading where symmetry prevails across the crack plane, the position of this critical element corresponds to the location where the change of volume dV is the largest or dW/dV is the least, i.e., dW/dV minimum or (dW/dV)min . Since this work consider only symmetrical loading, the subscript min will be dropped. The critical value of dW/dV can be computed from the area under the uniaxial true stress and true strain curve:
dW dV
Zef cr
r de:
4
0
This area decreases with increasing strain rate e_ as in Fig. 2. Hence, (dW/dV)cr is also expected to decrease with fracture strain ef . A graphical display of this trend is given by the dotted curve in Fig. 6. The corresponding applied stress ri to initiate crack growth is shown by the solid curve in Fig. 6; it also decreases with increasing strain rate which is to be expected.
Fig. 5. Variations of eective strain and strain energy density function with distance from crack tip for applied stress 38.25 MPa and strain rate 0.003 and 1100 sÿ1 using plasticity theory.
dW dV
Zeij rij deij :
3
0
These curves decay much more sharper than those in Fig. 4 for the stresses. 4.2. Crack growth initiation As the applied load increases by steps, the energy density will also increase accordingly every-
Fig. 6. Variations of critical applied stress for onset of crack growth and critical strain energy density with applied strain rate.
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
5. Strain rate dependent model
Table 3 Constant strain rate contours for r 23.25 MPa
5.1. Stress analysis
Contour no.
Strain rate (´ 10ÿ5 sÿ1 )
1 2 3 4 5 6 7 8
3.00 4.69 6.38 8.07 9.76 11.46 13.15 14.84
For a given strain rate, the load for the ®rst step is increased to a level when most of crack tip nodes experience plastic deformation. For a constant incremental increase of 5 MPa, the corresponding time increment is 0.533 s. The strain rate is 3 ´ 10ÿ3 sÿ1 . To illustrate how the local strain rate increases as the crack tip is approached, constant contours of strain rate are plotted as shown in Fig. 7. Note that each contour is labelled 1; 2; . . . ; 8 they correspond to increasing local strain rates from 3 ´ 10ÿ5 to 14.84 ´ 10ÿ5 sÿ1 and greater as given in Table 3 for the ®fth increment of loading with r 23.25 MPa. For the region very near the crack tip which is shaded, the strain rate is greater than 18.22 ´ 10ÿ5 sÿ1 . The near tip grid pattern is shown at the left hand side corner of Fig. 7. Displayed in Fig. 8 is a plot of the eective stress versus eective strain curve for element no. 1 located next to the crack tip, Fig. 3. It is compared with the base curve for an applied strain rate of 3 ´ 10ÿ3 sÿ1 that would have been used in the classical theory of plasticity. The solid curve represents the stress±strain relation that is developed by segments with increasing strain rates as the load is increases. The deviation increases with increasing strain rates and becomes more signi®cant for high applied loading rates. This can be seen from
135
Fig. 8. Equivalent uniaxial stress±strain response of element 1 and base material for strain rate 0.003 sÿ1 .
the curves in Figs. 9 and 10 for applied strain rates of 1.8 and 220 sÿ1 , respectively. Table 4 gives a comparison of the critical strain energy density function (dW/dV)cr for the three solid curves in Figs. 8±10 as compared with the base curve values. 5.2. Strain rate eect on crack growth initiation
Fig. 7. Near ®eld contours of strain rate for applied stress 23.25 MPa.
The critical load to initiate crack growth would dier if the strain rate dependent model is used instead of the classical plasticity theory. Consider a step load increase of 5 MPa occurring in a time interval of 88.8 ´ 10ÿ5 , 0.76923 ´ 10ÿ5 and 0.1538 ´ 10ÿ5 s, this would correspond to the three applied strain rates of 1.8, 220 and 1100 sÿ1 . Application of the strain energy density criterion [1,3,4] gives rise to the critical applied stress ri
136
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139 Table 4 Critical strain energy density function Strain rate (sÿ1 )
0.003 1.8 220.0 1100.0
(dW/dV)cr (MJ/m3 ) Plasticity (base curve)
Strain rate dependent model
21.97 10.19 8.25 6.57
11.66 9.09 6.33 4.99
Fig. 9. Equivalent uniaxial stress±strain response of material element 1 and base material for strain rate 1.8 sÿ1 .
Fig. 11. Variation of critical stress and critical value of the strain energy density function calculated by plasticity theory and by the strain rate dependent model.
Fig. 10. Equivalent uniaxial stress±strain response of element 1 and base material for strain rate 220 sÿ1 .
which are represented by the solid curves in Fig. 11. Note that the plasticity theory solution is not conservative. The same applies to the critical strain energy density functions. That is the plasticity theory would predict a higher load for crack initiation. Fig. 12 gives a comparison of the two theories for crack growth with 2b 15 cm and 2h 20 cm. It can be seen that for a given critical stress, more crack growth would be predicted by the plasticity theory. By the same token, a much lower critical strain energy density function corresponds to the strain rate dependent model being more conservative.
Fig. 12. Variation of critical stress and critical value of the strain energy density function with half crack length calculated by plasticity theory and from the strain dependent model.
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
5.3. Eect of specimen height Plotted in Fig. 13 are the critical stress and strain energy density functions against the half specimen height h that varies from 3 to 18 cm. The other dimensions of the specimen are 2b 15 cm and 2a 4 cm. Load is increased in a step of 5 MPa within 0.5333 s. This corresponds to a constant strain rate of 0.003 (sÿ1 ). Numerical values of the eective strain and strain rates in element no. 1 for a 1, 3 and 5 cm and dierent applied stresses can be found in Table 5. The critical stress tends to increase with h up to approximately 10.5 cm after which it remains nearly constant. Again, the
Fig. 13. Variation of critical stress and critical value of the strain energy density function with half specimen height calculated by plasticity theory and from the strain dependent model.
137
solution from the classical plasticity theory is not conservative. The same applies to the critical strain energy density which is not sensitive to changes in h. Tabulated in Table 6 are values of the eective strain and strain rate for 2b 15 cm, 2a 4 cm and h 3, 10 and 20 cm. 5.4. Stable crack growth The strain energy density theory also predicts stable crack growth, the details of which can be found in Refs. [1,3,4] and will not be repeated here. Increments of stable crack growth are determined for specimen with 2b 15 cm, 2h 20 cm and an initial crack of 2a 2 cm. The load is increased in steps of 5 MPa and at a strain rate of 0.003 (sÿ1 ). Fig. 14 gives a plot of applied stress as a function of crack growth a0 ÿ a. The dotted and solid curve corresponds, respectively, to the strain rate dependent model and plasticity theory. While both curves have the same trend, less crack growth is predicted by the plasticity theory. This result is the same as that found in [4]. Table 7 gives the numerical values of a0 ÿ a and ri which is the half crack length increment as the applied stress is increased from 38.25 to 68.25 MPa. 6. Concluding remarks A strain rate dependent model is applied to analyze the stable growth in an aluminum alloy
Table 5 Eective strain and strain rate in element 1 for 2h 20 cm, 2b 15 cm and a 1, 3 and 5 cm Applied stress r (MPa)
a 1 cm
a 3 cm
a 5 cm
eeff (´10ÿ2 )
e_ (sÿ1 )
eeff (´10ÿ2 )
e_ (sÿ1 )
eeff (´10ÿ2 )
e_ (sÿ1 )
8.25 13.25 18.25 23.25 28.25 33.25 38.25 43.25 48.25
0.842 1.390 2.060 3.880 5.108 6.587 8.580 11.604 15.678
0.0062 0.0093 0.0114 0.0138 0.0171 0.0209 0.0252 0.0339 0.0515
1.516 2.805 4.499 6.700 9.544 13.732 15.633 ÿ ÿ
0.0163 0.0196 0.0258 0.0335 0.0433 0.0638 0.0971 ÿ ÿ
2.832 5.743 10.349 15.458 ÿ ÿ ÿ ÿ ÿ
0.0322 0.0391 0.0619 0.1284 ÿ ÿ ÿ ÿ ÿ
138
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
Table 6 Eective strain and strain rate in element 1 for 2b 15 cm, 2a 4 cm and h 3, 10 and 20 cm Aplied stress r (MPa)
h 3 cm eeff (´10 )
e_ (s )
eeff (´10 )
e_ (s )
eeff (´10ÿ2 )
e_ (sÿ1 )
8.25 13.25 18.25 23.25 28.25 33.25 38.25 43.25 48.25
1.544 2.878 4.651 6.975 9.920 14.303 15.692 ÿ ÿ
0.0147 0.0177 0.0235 0.0309 0.0391 0.0582 0.0883 ÿ ÿ
1.162 2.027 3.159 4.567 6.306 8.607 11.519 15.696 ÿ
0.0133 0.0162 0.0212 0.0263 0.0326 0.0431 0.0545 0.0877 ÿ
1.142 1.983 3.080 4.445 6.132 8.288 11.023 15.211 15.406
0.0129 0.0157 0.0205 0.0255 0.0316 0.0404 0.0512 0.0785 0.1393
ÿ2
h 10 cm ÿ1
ÿ2
Fig. 14. Variation of applied stress with half crack growth, calculated by the plasticity theory (dashed line) and from the strain rate dependent model (rigid line).
1100-0. In contrast to the classic theory of plasticity, stress±strain curves with dierent strain
h 20 cm ÿ1
rate are used such that the response for each material elements is determined according to the local strain state for each load segment. The resulting equivalent stress±strain relation for each element would thus be dierent, especially for those near the crack tip where the stress and strain ®eld is highly nonhomogeneous. The difference diminishes with distance away from the crack tip. The strain energy density theory is applied to determine crack initiation, stable growth and onset of rapid crack propagation. It is shown that the results obtained from the plasticity theory is not conservative. More speci®cally, it yields higher critical applied stress and less crack growth. These conclusions are consistent with those found in Ref. [4] and suggest the need to correct for the high strain rate eect near tip.
Table 7 Comparison of strain rate dependent model with plasticity for crack increment and crack growth with 2a 2 cm, 2b 15 cm and 2h 20 cm Applied stress r (MPa) 38.25 43.25 48.25 53.25 58.25 63.25 68.25
Strain rate dependent model
Plasticity theory
ri (cm)
a0 ÿ a (cm)
ri (cm)
a0 ÿ a (cm)
0.0129234 0.0201153 0.0300927 0.0439344 0.0553654 0.0651848 0.0987003
0.0129234 0.0330387 0.0631354 0.1070698 0.1624352 0.2276200 0.3263203
ÿ 0.0169715 0.0271205 0.0391945 0.0500820 0.0611205 0.0915350
ÿ 0.0169715 0.0440920 0.0832865 0.1333685 0.1944890 0.2860240
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
References [1] G.C. Sih, Mechanics and physics of energy density theory, Theoret. Appl. Fracture Mech. 4 (1985) 157±173. [2] G.C. Sih, Thermomechanics of solids: Nonequilibrium and irreversibility, Theoret. Appl. Fracture Mech. 9 (1988) 175± 198. [3] G.C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer Academic Publishers, Dordrecht, 1991, pp. 383±393. [4] G.C. Sih, D.Y. Tzou, Plasticity deformation and crack growth behavior, in: G.C. Sih, A.J. Ishlinshy, S.T. Mileiko
139
(Eds.), Plasticity and Failure Behavior of Solids, Kluwer Academic Publishers, Dordrecht, 1990, pp. 91±114. [5] PAPST (Plastic Axisymmetric/Planar Structures), David W. Taylor Naval Ship Research and Development Center, Bethesda, Maryland, 1981. [6] G.E. Papakaliatakis, E.E. Gdoutos, E. Tzanaki, A strain rate dependent model for crack growth, in: G.C. Sih, E.E. Gdoutos (Eds.), Mechanics and Physics of Energy Density, Kluwer Academic Publishers, Dordrecht, 1992, pp. 109± 119. [7] U.S. Lindholm, L.M. Yeakley, High strain-rate testing: Tension and compression, Exp. Mech. 8 (1968) 1±9.
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Eect of strain rate on crack growth in aluminum alloy 1100-0 G.E. Papakaliatakis School of Engineering, Demokritus University of Thrace, GR-67100 Xanthi, Greece
Abstract Stress and damage analysis are performed to analyze the Mode I crack growth behavior of a central crack panel made of aluminum alloy 1100-0. On account of the highly nonhomogeneous stress state, each material element would experience a dierent strain rate depending on the location and loading rate. A data bank of uniaxial stress and strain curves is provided to cover the range local strain rates depending on the load time history. Such a approach is referred to as the strain rate dependent model in contrast to plasticity that utilizes a single constitutive relation. The strain energy density criterion is applied to determine the onset of crack initiation, stable crack growth and ®nal termination. A unique feature of the approach is that the same criterion could describe the foregoing three distinct events of fracture behavior. Results are obtained for applied loads with dierent strain rates and compared with those obtained from the classical theory of plasticity, which is unconservative. Ó 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Uniaxial tensile tests of metal alloys have shown that the mechanical response depends on the rate and history of the applied load. Changes in the post-yield behavior are more pronounced and can be signi®cant, particularly when the resulting stress and strain states are highly nonhomogeneous. The prevailing dierence between the local strain rate near a singular point such as the crack tip can dier from those at distances away by orders of magnitude. Hence, the use of a single and the same constitutive relation for all material elements as it is the case in the classical theory of plasticity is not adequate. A theory that accounts for nonhomogeneity arising from the local stress and strain states has been developed in Refs. [1,2]. Advocated is the notion that surface and volume energy density are no longer independent but they would interact via the rate change of volume with surface area which
has been traditionally assumed to vanish in the limit as the material element diminishes in size. Such a hypothetical assumption is removed in Refs. [1,2] and explains for inaccuracies and inconsistencies that would otherwise be present as in classical continuum mechanics theories. As the material elements are ®nite in size, their dimensions are determined from the conditions that the energy transmitted is the same in the three mutually perpendicular directions. This would provide a one-to-one correspondence between the uniaxial and multiaxial stress states. These elements have been referred to as the isoenergy density elements [2]. The result for stresses near a sharp crack can dier signi®cantly from the theory of plasticity [3]. An attempt to correct for the load strain rates was ®rst made in Ref. [4] without taking into account the full eect of nonhomogeneity due to stress and strain in addition to the material microstructure. This simpli®es the analysis and could still reveal the importance to extend the range of
0167-8442/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 9 9 ) 0 0 0 0 7 - 5
132
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
strain rates in the treatment of elastic±plastic crack problems. A modi®ed PAPST [5] (Plastic Axisymmetric/Planar Structure) computer program were used and crack problem similar to that in Ref. [4] was solved [6]. This work is a further application of the works [4,6] to analyze the fracture behavior of the aluminum alloy 1100-0. 2. Basic approach As mentioned earlier and in Refs. [4,6], a data bank of uniaxial curves with dierent strain rates is provided such that the strains (and/or stresses) for each load step can be determined at each location. The strains for two successive load or time steps would yield the strain rate and hence a particular stress±strain curve. The loci of these curve segments would give a complete stress±strain curve that is dierent for each material element. The PAPST [5] ®nite element computer program is applied to perform the elastic±plastic stress analysis by increments. Twelve-node isoparametric elements are used with a special singular crack tip element where the positions of two of the side nodes are shifted by distances of 1/9 and 4/9 from the crack tip corner node. The Von±Mises yield criterion is applied as the load is increased in a step-wise fashion. The computational procedure can be obtained as follows: · All elements are assumed to follow the same uniaxial stress±strain curve for the ®rst increment of loading. · The principal strain ej (j 1, 2, 3) are calculated at all Gaussian points of each element and hence the eective strain eeff can be obtained: i1=2 1 h 2 2 2 eeff p e1 ÿ e2 e2 ÿ e3 e3 ÿ e1 ; 2 1
ment. A knowledge of the element strain rate determines the particular stress±strain curve in the data bank. · Dierent segments of stress±strain curves with dierent strain rates for each load step are connected to yield a complete curve for a given material element. The above is repeated as the crack specimen is loaded step-by-step until the stable crack growth process is completed at which point rapid crack propagation is predicted to begin and the computation is terminated. 3. Specimen geometry and material Fig. 1 depicts a rectangular body in plane strain. It has a width of 2b 15 cm, height of
the corresponding stress being i1=2 1 h reff p r1 ÿ r2 2 r2 ÿ r3 2 r3 ÿ r1 2 : 2 2 · The strain rate of each element corresponds to the relative strains of two successive load increments divided by the corresponding time incre-
Fig. 1. Geometry of specimen.
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
2h 20 cm and a through-the-thickness crack of length 2a 4 cm. Equal and opposite uniform stress of magnitude r is increased monotonically. The problem thus possesses one quarter symmetry. Mechanical properties of the cracked specimen corresponds to those for the commercial grade aluminum alloy 1100-0. They are listed in Table 1. This metal alloy 1100-0 is rate sensitive as exhibited by the true stress and true strain curves [7] in Fig. 2 for strain rates of e_ 3 10ÿ3 , 1.8, 0:220 103 and 1:100 103 sÿ1 . The terminal points of these curves were stipulated on the basis that the failure strains decrease with increasing strain rates. Henceforth, these four strain rates will be referred to as Cases I, II, III and IV as in Table 2.
Table 1 Mechanical properties of aluminum alloy 1100-0 Young's modulus E (MPa)
Poisson's ratio m
Yield strength rys (MPa)
3310.6
0.33
44.8
133
4. Preliminary considerations 4.1. Plasticity solution The rectangular body with a central crack is discretized by ®nite elements. Fig. 3 shows the ®nite element grid pattern with one-quarter symmetry. Twenty four (24) elements and 151 nodes were used to compute for the stress and displacements as the Case I, II, III and IV are solved incrementally using the PAPST [5] computer program. Some typical results are displayed in Figs. 4 and 5. Plotted in Fig. 4 are the normal stress ry and eective reff as a function of the distance r from the crack tip for e_ 3 ´ 10±3 sÿ1 and 1.100 ´ 103 sÿ1 . The decay in stress magnitude becomes much more pronounced as the strain rate increases. Similar results are given in Fig. 5 for the eective strain eeff and strain energy density function dW/dV which is given by
Fig. 2. True stress±true strain curves of aluminum alloy 1100-0 for dierent strain rates.
Table 2 Applied strain rates I
II
III
IV
3 ´ 10ÿ3
1.8
0.220 ´ 103
1.100 ´ 103
Fig. 3. Finite element grid pattern with one quarter symmetry.
134
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
Fig. 4. Variation of stress component ry and eective stress with distance from crack tip for applied stress 38.25 MPa and strain rate 0.003 and 1100 sÿ1 using plasticity theory.
where, particularly near the crack tip region where dW/dV would attain its largest value. Let the size of this region be denoted by ro taken as the radius of a circle around the crack tip. In continuum mechanics, ro would be macroscopic in size, say 0.01 cm. Crack initiation is assumed to occur when dW/dV in one of the elements on the periphery of the circle reaches a critical value, say (dW/dV)cr [3]. For Mode I loading where symmetry prevails across the crack plane, the position of this critical element corresponds to the location where the change of volume dV is the largest or dW/dV is the least, i.e., dW/dV minimum or (dW/dV)min . Since this work consider only symmetrical loading, the subscript min will be dropped. The critical value of dW/dV can be computed from the area under the uniaxial true stress and true strain curve:
dW dV
Zef cr
r de:
4
0
This area decreases with increasing strain rate e_ as in Fig. 2. Hence, (dW/dV)cr is also expected to decrease with fracture strain ef . A graphical display of this trend is given by the dotted curve in Fig. 6. The corresponding applied stress ri to initiate crack growth is shown by the solid curve in Fig. 6; it also decreases with increasing strain rate which is to be expected.
Fig. 5. Variations of eective strain and strain energy density function with distance from crack tip for applied stress 38.25 MPa and strain rate 0.003 and 1100 sÿ1 using plasticity theory.
dW dV
Zeij rij deij :
3
0
These curves decay much more sharper than those in Fig. 4 for the stresses. 4.2. Crack growth initiation As the applied load increases by steps, the energy density will also increase accordingly every-
Fig. 6. Variations of critical applied stress for onset of crack growth and critical strain energy density with applied strain rate.
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
5. Strain rate dependent model
Table 3 Constant strain rate contours for r 23.25 MPa
5.1. Stress analysis
Contour no.
Strain rate (´ 10ÿ5 sÿ1 )
1 2 3 4 5 6 7 8
3.00 4.69 6.38 8.07 9.76 11.46 13.15 14.84
For a given strain rate, the load for the ®rst step is increased to a level when most of crack tip nodes experience plastic deformation. For a constant incremental increase of 5 MPa, the corresponding time increment is 0.533 s. The strain rate is 3 ´ 10ÿ3 sÿ1 . To illustrate how the local strain rate increases as the crack tip is approached, constant contours of strain rate are plotted as shown in Fig. 7. Note that each contour is labelled 1; 2; . . . ; 8 they correspond to increasing local strain rates from 3 ´ 10ÿ5 to 14.84 ´ 10ÿ5 sÿ1 and greater as given in Table 3 for the ®fth increment of loading with r 23.25 MPa. For the region very near the crack tip which is shaded, the strain rate is greater than 18.22 ´ 10ÿ5 sÿ1 . The near tip grid pattern is shown at the left hand side corner of Fig. 7. Displayed in Fig. 8 is a plot of the eective stress versus eective strain curve for element no. 1 located next to the crack tip, Fig. 3. It is compared with the base curve for an applied strain rate of 3 ´ 10ÿ3 sÿ1 that would have been used in the classical theory of plasticity. The solid curve represents the stress±strain relation that is developed by segments with increasing strain rates as the load is increases. The deviation increases with increasing strain rates and becomes more signi®cant for high applied loading rates. This can be seen from
135
Fig. 8. Equivalent uniaxial stress±strain response of element 1 and base material for strain rate 0.003 sÿ1 .
the curves in Figs. 9 and 10 for applied strain rates of 1.8 and 220 sÿ1 , respectively. Table 4 gives a comparison of the critical strain energy density function (dW/dV)cr for the three solid curves in Figs. 8±10 as compared with the base curve values. 5.2. Strain rate eect on crack growth initiation
Fig. 7. Near ®eld contours of strain rate for applied stress 23.25 MPa.
The critical load to initiate crack growth would dier if the strain rate dependent model is used instead of the classical plasticity theory. Consider a step load increase of 5 MPa occurring in a time interval of 88.8 ´ 10ÿ5 , 0.76923 ´ 10ÿ5 and 0.1538 ´ 10ÿ5 s, this would correspond to the three applied strain rates of 1.8, 220 and 1100 sÿ1 . Application of the strain energy density criterion [1,3,4] gives rise to the critical applied stress ri
136
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139 Table 4 Critical strain energy density function Strain rate (sÿ1 )
0.003 1.8 220.0 1100.0
(dW/dV)cr (MJ/m3 ) Plasticity (base curve)
Strain rate dependent model
21.97 10.19 8.25 6.57
11.66 9.09 6.33 4.99
Fig. 9. Equivalent uniaxial stress±strain response of material element 1 and base material for strain rate 1.8 sÿ1 .
Fig. 11. Variation of critical stress and critical value of the strain energy density function calculated by plasticity theory and by the strain rate dependent model.
Fig. 10. Equivalent uniaxial stress±strain response of element 1 and base material for strain rate 220 sÿ1 .
which are represented by the solid curves in Fig. 11. Note that the plasticity theory solution is not conservative. The same applies to the critical strain energy density functions. That is the plasticity theory would predict a higher load for crack initiation. Fig. 12 gives a comparison of the two theories for crack growth with 2b 15 cm and 2h 20 cm. It can be seen that for a given critical stress, more crack growth would be predicted by the plasticity theory. By the same token, a much lower critical strain energy density function corresponds to the strain rate dependent model being more conservative.
Fig. 12. Variation of critical stress and critical value of the strain energy density function with half crack length calculated by plasticity theory and from the strain dependent model.
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
5.3. Eect of specimen height Plotted in Fig. 13 are the critical stress and strain energy density functions against the half specimen height h that varies from 3 to 18 cm. The other dimensions of the specimen are 2b 15 cm and 2a 4 cm. Load is increased in a step of 5 MPa within 0.5333 s. This corresponds to a constant strain rate of 0.003 (sÿ1 ). Numerical values of the eective strain and strain rates in element no. 1 for a 1, 3 and 5 cm and dierent applied stresses can be found in Table 5. The critical stress tends to increase with h up to approximately 10.5 cm after which it remains nearly constant. Again, the
Fig. 13. Variation of critical stress and critical value of the strain energy density function with half specimen height calculated by plasticity theory and from the strain dependent model.
137
solution from the classical plasticity theory is not conservative. The same applies to the critical strain energy density which is not sensitive to changes in h. Tabulated in Table 6 are values of the eective strain and strain rate for 2b 15 cm, 2a 4 cm and h 3, 10 and 20 cm. 5.4. Stable crack growth The strain energy density theory also predicts stable crack growth, the details of which can be found in Refs. [1,3,4] and will not be repeated here. Increments of stable crack growth are determined for specimen with 2b 15 cm, 2h 20 cm and an initial crack of 2a 2 cm. The load is increased in steps of 5 MPa and at a strain rate of 0.003 (sÿ1 ). Fig. 14 gives a plot of applied stress as a function of crack growth a0 ÿ a. The dotted and solid curve corresponds, respectively, to the strain rate dependent model and plasticity theory. While both curves have the same trend, less crack growth is predicted by the plasticity theory. This result is the same as that found in [4]. Table 7 gives the numerical values of a0 ÿ a and ri which is the half crack length increment as the applied stress is increased from 38.25 to 68.25 MPa. 6. Concluding remarks A strain rate dependent model is applied to analyze the stable growth in an aluminum alloy
Table 5 Eective strain and strain rate in element 1 for 2h 20 cm, 2b 15 cm and a 1, 3 and 5 cm Applied stress r (MPa)
a 1 cm
a 3 cm
a 5 cm
eeff (´10ÿ2 )
e_ (sÿ1 )
eeff (´10ÿ2 )
e_ (sÿ1 )
eeff (´10ÿ2 )
e_ (sÿ1 )
8.25 13.25 18.25 23.25 28.25 33.25 38.25 43.25 48.25
0.842 1.390 2.060 3.880 5.108 6.587 8.580 11.604 15.678
0.0062 0.0093 0.0114 0.0138 0.0171 0.0209 0.0252 0.0339 0.0515
1.516 2.805 4.499 6.700 9.544 13.732 15.633 ÿ ÿ
0.0163 0.0196 0.0258 0.0335 0.0433 0.0638 0.0971 ÿ ÿ
2.832 5.743 10.349 15.458 ÿ ÿ ÿ ÿ ÿ
0.0322 0.0391 0.0619 0.1284 ÿ ÿ ÿ ÿ ÿ
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G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
Table 6 Eective strain and strain rate in element 1 for 2b 15 cm, 2a 4 cm and h 3, 10 and 20 cm Aplied stress r (MPa)
h 3 cm eeff (´10 )
e_ (s )
eeff (´10 )
e_ (s )
eeff (´10ÿ2 )
e_ (sÿ1 )
8.25 13.25 18.25 23.25 28.25 33.25 38.25 43.25 48.25
1.544 2.878 4.651 6.975 9.920 14.303 15.692 ÿ ÿ
0.0147 0.0177 0.0235 0.0309 0.0391 0.0582 0.0883 ÿ ÿ
1.162 2.027 3.159 4.567 6.306 8.607 11.519 15.696 ÿ
0.0133 0.0162 0.0212 0.0263 0.0326 0.0431 0.0545 0.0877 ÿ
1.142 1.983 3.080 4.445 6.132 8.288 11.023 15.211 15.406
0.0129 0.0157 0.0205 0.0255 0.0316 0.0404 0.0512 0.0785 0.1393
ÿ2
h 10 cm ÿ1
ÿ2
Fig. 14. Variation of applied stress with half crack growth, calculated by the plasticity theory (dashed line) and from the strain rate dependent model (rigid line).
1100-0. In contrast to the classic theory of plasticity, stress±strain curves with dierent strain
h 20 cm ÿ1
rate are used such that the response for each material elements is determined according to the local strain state for each load segment. The resulting equivalent stress±strain relation for each element would thus be dierent, especially for those near the crack tip where the stress and strain ®eld is highly nonhomogeneous. The difference diminishes with distance away from the crack tip. The strain energy density theory is applied to determine crack initiation, stable growth and onset of rapid crack propagation. It is shown that the results obtained from the plasticity theory is not conservative. More speci®cally, it yields higher critical applied stress and less crack growth. These conclusions are consistent with those found in Ref. [4] and suggest the need to correct for the high strain rate eect near tip.
Table 7 Comparison of strain rate dependent model with plasticity for crack increment and crack growth with 2a 2 cm, 2b 15 cm and 2h 20 cm Applied stress r (MPa) 38.25 43.25 48.25 53.25 58.25 63.25 68.25
Strain rate dependent model
Plasticity theory
ri (cm)
a0 ÿ a (cm)
ri (cm)
a0 ÿ a (cm)
0.0129234 0.0201153 0.0300927 0.0439344 0.0553654 0.0651848 0.0987003
0.0129234 0.0330387 0.0631354 0.1070698 0.1624352 0.2276200 0.3263203
ÿ 0.0169715 0.0271205 0.0391945 0.0500820 0.0611205 0.0915350
ÿ 0.0169715 0.0440920 0.0832865 0.1333685 0.1944890 0.2860240
G.E. Papakaliatakis / Theoretical and Applied Fracture Mechanics 31 (1999) 131±139
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(Eds.), Plasticity and Failure Behavior of Solids, Kluwer Academic Publishers, Dordrecht, 1990, pp. 91±114. [5] PAPST (Plastic Axisymmetric/Planar Structures), David W. Taylor Naval Ship Research and Development Center, Bethesda, Maryland, 1981. [6] G.E. Papakaliatakis, E.E. Gdoutos, E. Tzanaki, A strain rate dependent model for crack growth, in: G.C. Sih, E.E. Gdoutos (Eds.), Mechanics and Physics of Energy Density, Kluwer Academic Publishers, Dordrecht, 1992, pp. 109± 119. [7] U.S. Lindholm, L.M. Yeakley, High strain-rate testing: Tension and compression, Exp. Mech. 8 (1968) 1±9.