Fracture Initiation Under Gross Yielding_Strain Enegy Density Criterion.pdf

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Enginesting Facture Mechan ics Vol. 18. No. 3, pp. W-677, 1983Printed in Great Britain.

00s7944/83/090667-1l$433.00/0Pergamon Press Ltd.

FRACTURE INITIATION UNDER GROSS YIELDING:STRAIN ENERGY DENSITY CRITERION

G. C. SIH and E. MADENCIInstitute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, PA 18015,U.S.A.

Ahstraet-The damage of metal alloys at the continuum level can be described by a combined process of yielding andfracture. Yielding is generally associated with excessive distortion and fracture with excessive dilatation. With theincreased use of low strength alloys, gross yielding can often precede the initiation of macrocracking. This paperapplies the strain energy density concept for examining failure by crack growth in a fully plastic material in order toshow that the criterion applies to linear elastic as well as nonlinear plastic materials. The incremental theory ofplasticity is employed in conjunction with the hnite element method for determining the stresses in a tensile specimencontaining a center crack. Net section yield is developed. Calculated are the different levels of intensity of yieldingaround the crack. As in the elastic case, crack growth in a fuhy plastic material also tends to coincide with the directionalong which the strain energy density function d V/d V attains a relative minimum. The onset of crack growth isassumed to occur when (d B$dV),, reaches some critical value (d W/dV): which must now be obtained from the truestress and strain curve of the material undergoing gross yielding.

INTRODUCTIONTHE PHILOSOPHYhat the structures behavior could be predicted from laboratory tests on small specimenswith different geometry and loading rates requires a viable theory that can consistently assess thedifferent modes of failure characterized by rapid crack growth in the elastic material to slow crackgrowth in the fully plastic material. This task is difficult because failure modes are governed by thecombined effect of geometry and component size, loading rate and material properties such as strengthand toughness. Unless these variables are treated simultaneously, no predictive capability could beestablished.

The theory of linear elastic fracture mechanics (LEFM) deals exclusively with the event of rapidcrack propagation occurring at nominal stresses that are well below the uniaxial yield stress of thematerial. Tests on precracked specimens undergoing very little plastic deformation are carried out tomeasure the so-called fracture toughness value of the material that is usually referred to as the Kl,parameter. In order to avoid plastic flow, the specimens must be sufficiently thick so that a large amountof energy is available for release at the onset of crack instability. Such a criterion limited to brittlefracture obviously cannot be applied to structures such as weldments, pressure vessels operating aboveroom temperature, pipelines, etc., that undergo plastic deformation. Although many competing failurecriteria[l] have attempted to explain crack growth behavior in the presence of plasticity, none of themhave reached the stage of application. The basic drawback is that the majority of the criteria cannotconsistently address the fracture process of crack initiation, slow growth and final termination. Theconditions to explain these three different stages of crack growth are often disconnected and lead toinconsistent results. Too much emphases have been placed on testing and collecting data without aknowledge of the underlying principles. The field of crack growth accompanied by plastic flow is still verymuch in flux and the present state of knowledge requires extensive scrutiny.

The aim of this communication is to extend the strain energy density criterion into the realm of fullyplastic material behavior. The basic idea is consistent with crack behavior in the elastic[2] andelastic-plastic[3] regime. Attention is focused on the fluctuation of the strain energy density functiondW/d V within the material. Regardless of whether the material undergoes plastic flow or not, thereprevails a location of (d W/d V),,, and (d W/d V)min.The former can be identified with distortion of thematerial and the latter with dilatation. This physical interpretation is shown to be valid for a crack in afully yielded material even though d W/dV can no longer be separated into the sum of dilatational anddistortional components for the nonlinear material. Hence, the d W/d V-criterion remains valid for theentire range of failure modes, loosely characterized in terms of material behavior as elastic, elastic-plastic and fully plastic.

667


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668 G. C. SIH and E. MADENCICRACK GROWTH IN FULLY PLASTIC MATERIAL

Depending on the combination of component size, loading rate, temperature and material. failure bycrack growth can occur in several different ways. This can be best described [4,5] by the behavior of acenter cracked specimen loaded in tension as illustrated schematically in Figs. l(a)-(d). When yielding isconfined to the immediate vicinity of the crack such that the energy released during crack growth ispredominantly elastic, the specimen is said to behave elasticallyi as shown in Fig. l(a). As yielding ofthe material near the crack increases, plastic enclaves are developed and the specimen behaves in anelastic-plastic manner, Fig. l(b). The medium strength alloys exhibit this characteristic. Crack growthtakes place in the elastic portion of the elastic-plastic material. The creation of plastic enclaves requiredadditional energy which is beneficial in that it reduces the available energy to drive the crack. Thisincreases the loading capacity of the system if the same amount of input energy were transferred to thespecimen in Fig. l(a), say, made of a brittle material and Fig. l(b), say, made of a ductile material. Aslong as crack growth at instability releases elastic energy, even though material away from the crackpath is yielded, the fracture toughness value is still governed by K,, or S,, i.e.

s c = (1+ v)(l - 2v) 221rE Icwhere Y is the Poissons ratio and E the Youngs modulus. Any attempts made to introduce or define adifferent fracture toughness for the situation in Fig. l(b) would be fundamentally incorrect.For a very ductile material and/or very slow loading rates, yielding can spread to the specimenboundary prior to crack growth. This situation is shown in Fig. l(c) and can be described as grossyielding. Failure can occur by plastic collapse without macrocracking, Fig. l(c), or by macrocracking,Fig. l(d). The presence of a crack causes the material near the tips to further intensify the yielding. This

CT o-

(a) Elosti~Sehavior (b) Elostic~Plastic Behavior

o- o-(c) Net Section Yield (d) Fully Plastic BehaviorFig. 1. Brittle and ductile fracture behavior.

tElasticity does not imply linearity. The global behavior of a specimen can be nonlinear even if the material remains within theelastic range. Nonlinearity could be attributed by nonuniform and/or non self-similar crack growth[3].


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Fracture initiation under gross yielding 669is referred to as recuts degree yielding in Fig. l(d). Since crack growth now takes place in the yieldedmaterial, the energy required to create the free crack surfaces would be different from that required ifthe material were elastic.

According to the strain energy density criterion[7], the fracture toughness characterized by thecritical strain energy density function (d W/d& can be determined from the area under the truestress-strain curve 181:

where ii and cij are the stress and strain components. The relationship

(3)applies to all the cases shown in Figs. 1. Once (d W/d V), and S, are known from tests, the criticaldistance parameter r, follows from eqn (3). Referring to the situations in Figs. l(a and b), an element ofmaterial ahead of the crack experiences only elastic deformation while stressed to fracture at o, = a,and (d Wld V), can be calculated as aiiii/2. In Fig. l(d), the element ahead of the crack is already yieldedbefore fracturing at wf = u:. This corresponds to d W/dV as obtained from a nonlinear stress analysisand calculated from eqn (2) reaching a critical value (d W/d V): that is the energy density associated withfracturing A fully plastic element. The values of (d W/d V)$ for a variety of engineering materials can befound in [S] and the references therein.

THE CENTER CRACKED SPECIMEN: NONLINEAR STRESS ANALYSISIn order to demonstrate the validity of the strain energy density criterion applied to the fracture of

fully plastic materials, a center cracked specimen with dimensions 20 in. (50.8 cm) x 10 in. (25.4 cm)containing a crack of length 2a = 2 in. (5.08 cm) is solved. Refer to Fig. 3, the problem possesses onequarter symmetry and hence a finite element grid pattern need to be developed only for the shaded area,Fig. 4. The fixed boundaries correspond to lines of symmetry.

M ater i al propert ies and hdi ngs. A true stress and strain curve is given in Fig. 5 for a material withthe following properties:

V= 0.3E = 30 x lo6 psi (2.068x Id MPa) (41

gyys 7.5 x lo4 psi (5.171 x 10 MPa)(d %$dV): = 7.0 x lo3 in.-lb/in? (4.826 x 10 MJ/m3)

where a,, is the yield strength. The nonlinear uniaxial strain and stress response is approximated byemploying a multilinear model:

Oe Qc Qf Qct 1 1- -q &I - -!@a /$j&_# -7- # #

(a) Purely Elastic (b) Fully PlasticFii. 2. Schematic of elastic and fullyplastic material elements loaded to fracture.

(5)


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670in which

G. C. SIH and E. MADENCI

ff, =EAc,, Au,,A%is chosen to correspond with the current value of the effective stress[c)]. The specimen is loadedelastically up to 5.509 ksi and from there on, twenty-four (24) load increments are taken until a maximumstress of cr,, = 18.375x lo4 psi is reached, Table 1. At this load, the specimen has stretched and neckedconsiderably. Figure 6 shows the difference between the initial and final configuration of the crackedspecimen.

A stress analysis is performed using the modified version of PAPST[9, lo] that accounts for materialand geometric nonlinearities. The analysis incorporates the incremental theory of plasticity and the vonMises yield criterion. The state of affairs near the crack tip are treated by placing the two side nodes atone ninth (l/9) and four ninth (419) distance of the sides with reference to the common nodell I].Numerical results for the strain energy density function dW/dV at the nodes can vary from 10 to 20%depending on the interpolation technique used. This uncertainty can be avoided by displaying the valuesof d W/dV at the quadrature points of each element in the form of contour plots.

Efective stress contours. For material elements in a multiaxial stress state, it is convenient todescribe yielding in terms of an equivalent or effective stress? Q. Figure 7 displays the contours of oCfffor one quarter of the specimen 5 in. (12.70 cm) x 10 in. (25.4 cm). The material surrounding the crack isstressed entirely beyond the yield stress 75 ksi, the intensity of local yielding is seen to increase as thecrack tip is approached. The contour labelled 3 resembles the enclaves in Fig. l(d) that undergo seconddegree of yielding. Within the dark region, the effective stress reaches 290 to 920 ksi. More details of thegeff contours are displayed in Figs. 8 to 10 as the region near the crack tip is enlarged starting from theouter region elements No. 7 to 10 in Fig. 4, No. 3 to 6 and finally No. 1 and 2 at the crack tip. Thecontour No. 11 shows that the local cr,, is more than twelve (12) times that of the yield stress cYswhilethe material near the specimen is stressed only more than twice (2) above yield.

20

Fig. 3. Center cracked panel for

-One Qucdront

-Inonlinear analysis.

tThe effective stress aeff = V/(34) is also often referred to as the &flow criterion.


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Fracture initiation under gross yielding

GridPot ternFatSerkSD I

-

in.

t t t t t 1 t a t

Fig. 4. Finite element grid pattern with ane quarter symmetry.

STRAINFig. 5. Uniaxial stress-strain curve for material used in nonlinear analysis.


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672 G. C. SIH and E. MADENCIurn 183.750 si

tttttf

. __ m m

Fully PlasticMaterial

0.182

Deformed/Configurationt_Q.l15/ fo.292cm).

I fnitialW+Configurotiofi

-4462 cm!Fig. 6. Distorted configuration of specimen at maximum load.

Contour Na I - 130 ksl2 - 1703 - 2104 - 25f)

Dark Region - 290 to 920

LitX?Crock

&TO cm

EiUn

-

Fig.7. Effective stress contours for a = 2.54cm at maximum load for one quarter specimen.


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Fracture initiation under gross yielding 613Table 1. Load increments applied to the center cracked specimen

I ncrment stress 1IKre6n?ntNo. (ksi) No.1 5.5092 15.ooo3 16.1024 19.2975 37.5006 55.5007 50.0978 6O.ooO9 61.075

10 65.62511 71.25012 75.octo

Contour No. I - 130 ksi2 - 1703 - 2104 - 250

13 84.37514 93.75015 103.12516 112.50017 121.87518 131.25019 140.62520 15o.ocm21 159.37522 168.75023 178.12524 183.750

No. 5 - 290 ksi6 -3307 - 380

Dork Region 420 to 920----_-_----

I- 1.016 cm 1.016 cmFig. 8. Effective stress contours at maximum load for details in elements T-10.

Contour No. I - 150 ksi No. 5 310 ksi2 - 190 6 - 3503 - 230 7 - 3904 - 270 8 -430

Dork Region 470 to 920

Fig. 9. Effective stress contours at maximum load for details in elements 34


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674 G. C SIW and E. MADENCIThe continuum mechanics analysis, however, must stop short at a limiting distance r. from the cracktip within which the details will be dominated by the material microst~cture. Thus, the focal element

under consideration is assumed to lie outside of a core region having a radius, say r,, Fig. 11. Therelation in eqn (3) then applies for r > ro. This leads to the discussion of failure criterion for crackgrowth in a yielded material.

STRAIN ENERGY DENSITY CRITERIONBecause of the versatility of eqn (2) such that d W/dV can be computed for any materials, the strain

energy density concept applies equally well to cracks in fully plastic materials.Eiypotheses. The basic assumptions concerning the criterion can be stated as follows:Hypothesis (1): The location of maximum yielding is assumed to coincide with maximum strain energy

density (d W/d V),,, and of fracture initiation with minimum strain energy density (d Wjrd&in.Hypothesis (2): Failure by yielding and fracture are assumed to occur when (dud V),,, and

(d W/d I)min each their respective critical values.

Contour No. I - 190 No. 7 - 4302 - 230 8 - 4703 - 270 9 - 5104- 310 IO - 5505 - 350 II - 9206 - 390

Fig. 10. Effective stress contours at maximum load for details in elements l-2.

Fully Plastic Case

L ro= Radius of Core RegionFig. Il. Core region near a crack tip or site of possible failure.


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Fracture initiation under gross yielding 675Hypothesis (3): The amount of incremental growth rl, r2,. . . , rj,. . . , r, of the yield zone or crack is

governed by a relation of the type

( 1W =s,=s,= =z- SCm c r, r2 ,. . . , ri -, . . . , =y =const. (7)of the process of yielding or crack growth leads to global instability?. Then

r, ri >,. . . , > rQ.SThe strain energy density criterion has been used in a quasi-linear analysisLl21 to account for both crack growth and plasticdeformation.PThe strain energy density criterion when used in conjunction with an elastic stress analysis predicts maximum yielding at0 = cos-(1 - 2~). For Y= 0.3, the angle is 66.4which is much less than the present case of full plasticity.


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676 G. C. SIH and E. MADENCI(dW/dV),a, at 8 = 80.

II 1 I I I I 10 30 60 so* 120. 150. iwANGLE, 6

Fig. 13. Variations of strain energy density with angle 0 at maximum load for r = 0.079 n. (0.200 m).

0 60 1200 180ANGLE, 8

Fig. 14. Strain energy density as a function of 8 at load increment No. 24 for I = 0.050 n. (0.127 m).

Table 2. Angular distribution of strain energy density for I = O.OSOin.0.127cm)Angl e dWdVe (ksi )00 155.700

15. 450 166.08030. 28" 185.73745. 400 205.40959. 250 213.19774. 940 216.15889. 930 221.890

Anglee dW/dV(ksl)91. 540 259. 115104.750 232. 210120.13" 209. 130134.810 l ag. 934149.501 170. 688164.069 i 51.ao5177.186 141. 115


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Fracture initiation under gross yielding 677the path of crack growth. Hypothesis (2) states simply that there will be a critical value (d W/d V):outside the core region corresponding to crack initiation as the load is increased. In essence, the samecriterion that were used for crack extension in elastic materials has been shown to hold for cracks infully plastic materials.

CONCLUSIONSAside from determining the details of pfasticity near the crack tip, the objective of this work is toextend the strain energy density criterion to crack extension in the f&y plastic materia1. This is

significant because many structural members retain their load carrying capacity even when gross yieldinghas occurred. Although the critical value (d W/d V): to initiate fracture in the plastic material is differentfrom that of (d W/d V), for the elastic material, the basic hypotheses of the d wd V-criterion governingcrack growth remain unchanged regardless of material behavior. Hence, a consistent procedure foranalyzing the combined effect of structural component geometry, material and loading can now bedeveloped to account for failure modes ranging from the very brittle behavior to the fully plasticbehavior. It is also essential to note that the nonuniform rate of yielding and racck growth dependingsensitively on load path must be treated simultaneously. This concerns with the application ofHypothesis (3) a topic that will be discussed in the future.

REFERENCES[I] Post-Yield Fracture Mechanics, (Edited by D. G. H. Latzko). Applied Science, London (1979).[21 G. C. Sih, A special theory of crack propagation, In Mechanics of Fracture, Vol. I: Methods of Analysis and Solutions of CrackProblems, (Edited by G. C. Sih), pp. 21-45. Noordhoff International Publishing, The Netherlands (1973).131G. C. Sih and B. V. Kiefer, Nonlinear response of solids due to crack growth and plastic deformation, Tn Nonlinear and DynamicFracture ~echaaics, (Edited by N. Penone and S. N. Athui), Vol. 35, pp. 136156. ASME Applied Mechanics Division (I~9).141G. C. Sih, Mechanics of Ductile Fracture, In Pruceedings of~nte~ationa~ Conference in Fracture Mechanics Technology,(Editedby G. C. Sih and C. L. Chow), pp. 767-784. Sijthoff and Noordhoff International Publishers, The Netherlands (1977).151G. C. Sih, Phenomena of Instability: Fracture mechanics and flow separation. Naoal Research Reuiews 32(3), 30-42 (1980).[61 G. C. Sih, Fracture toughness concept. ASTM Special Technical Publication STP 605,3-15 (1976).[71 G. C. Sih, Experimental fracture mechanics: Strain energy density criterion, In Mechanics of Fracture, Vol. VII: ExperimentalEualuation of Stress Concentration and Intensity Factors, (Edited by G. C. Sib), pp. XVIII-LVI. Martinus Nijhoff Publishers, The

Netherlands (1981).181L. F. Gillemot, Criterion of crack initiation and spreading. f. Engng Fracture Mech. 8,239-253 (1976).191L. N. Gifford and P. D. Hilton, P~fimin~y ~cumentation of PAPST-Non-linear fracture and stress analysis by finite elements.NSRDC-Preliminary Documentation (January 1981).(101 L. N. Gifford and P. D. Hilton, DTNSRDC-79/052,Naval Ship Research Development Center Report, Bethesda, Maryland (1979).[Ill S. L. Pu, M. A. Hussain and W. E. Lorensen, The collapsed cubic isoparametric element as a singular element for crackproblems. Int. I. Numerical Methods Engng 12, 1727-1742 1978).[I21 G. C. Sih and P. Matic, A pseudo-linear analysis of yielding and crack growth: Strain energy density criterion, In Proceedings ofDefer& Fracture and Fatigue,(Edited by G. C. Sih and J. W. Provan). Martinus Nijhoff Publishers, The Netherlands, pp. 223-232119833.(Received 25 May 1982; received for publication 20 July 1982)

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