ELASTOPLASTIC FRACTUREPresented By Dr. H.V.LakshminarayanaProfessor Post Graduate Engineering ProgrammesM.S Raimaih School of Advanced StudiesCenter For Engineering Design and Manufacturing ResearchNew BEL Road, Bangalore 560054

Session Title :ELASTOPLASTIC FRACTURE TopicsIntroduction.Non-linear elastic behavior.Characterizing elastoplastic behavior.J-integral in EPFM.ASTM standard test method to measure JIC a measure of fracture toughness.J-integral evaluation using NLFEA. SOURSE for Information and illustration : R.J. Sanford,Principles of fracture Mechanics, Prentice Hall.

Elastoplastic Fracture : IntroductionThe subject of LEFM is an applied elasticity study of bodies cantaining cracks.Limited plasticity at the crack tip was accounted for through plasticity correction. The approach is more than sufficient for a wide variety of problems, notably Fatigue Crack Growth(FCG) prediction for which the maximum stress due to applied loads are less than 30 percent of the yield stress.The LEFM approach also applies to fracture of materials which are naturally elastic upto failure : glass, ceramics, plastics, and high strength metals have limited amount of ductility prior to FRACTURE.The LFEM is not applicable when the combination of component geometry, crack type, shape, orientation, location and loading is such that the plastic zone is not confined.Also we should not expect the LEFM to extend to the case of large scale plastic behavior with extensive deformation prier to fracture. Including in this class of problems are some critical application that demand thorough analysis. Foe example, in the design of nuclear pressure vessels.

-material toughness is of paramount importance due to long term degradation of the material caused by neutron bombardment over the life time (> 20 years) of the pressure vessel. Despite the high initial toughness, subcritical flaws do developed, mainly in WELD METAL due to fatigue and stress corrosion cracking. As a results, FRACTURE (the new failure mode) must be considered in the event of accident. But the extend of plastic deformation at the event of the crack tip precludes the application and LEFM to determine crack growth Residual strength and service life. As illustrated in figure 11.1, there is a region between LEFM and plastic collapse that need to be characterized. it is this region to which the development of a theory of elastoplastic fracture mechanics (EPFM) is directed.Unfortunately, a direct approach incorporating a rigorous Theory of plasticity has proven elusive and alternative approaches that expand and generalize already firmly grounded LEFM approach have been adopted. These are presented in the next section.

Elasto plastic Fracture: Nonlinear Elastic BehaviorWe can extend the energy release rat approach to the class of nonlinear, but still elastic, materials by again considering the energy available for crack extension in terms of area in the load- load point displacement diagram.In this case the load-load point displacement diagram has the appearance Shown in figure 11.2. The area under the curve is the strain energU, and the area above the curve called complementary strain energy U*, is related to the potential energy p of the system by the relation 11.1For a two dimensional body with crack area, A, with surface tractions, Ti prescribed over a portion of the bounding surface, , the potential Energy of the body is given by 11.2

Where the strain-energy density W is and{For nonlinear elastic behavior}By analogy with linear Elastic case, we can define an energy release rate for nonlinear elastic bodies, denoted by J, as the area on the load- displacement diagram between crack area A and , as shown in figure 11.3, for (a) constant displacement or (b) constant load divided by the change of crack area :11.3In Eq, (11.3), the absolute value has been used to ensure that J>0, For the special case of a linear elastic behavior ,and

11.4Alternatively we can obtain a formal definition of J by differentiating Eq (11.2)11.5After some manipulation, this equation can be written in the form 11.6The integrand in Eq (11.6) is the energy momentum tensor developed by Eshelby (1910). This integral has the property that J=0 on any closed contour not encircling a singularity within an elastic solid body (a conservative system) then

Rice (1968) was the first to adapt the principle of conservation of energy momentum to a two dimensional body containing a singularity.Rice considered the contour . shown in fig 11.4 where the segment 1and 3 are chosen to be marginally on the material side of the traction free crack interface.We can write the condition for conservation of energy momentumwhereNote that , since dy =0 and on these segment.Therefore11.811.7

The notation, -4 denotes a line integral taken in the direction opposite to Since and are any two arbitrary line integrals encircling the crack tip both taken in the same direction Eq. (11.8) demonstrate that J is path independent.In recognition of his contribution, the quantity J is often described as Rices J-integral or simple J-integral.Path independence of the J-integral offers a verity of opportunities for defining its value and with the aid of Eq. (11.4) the corresponding stress intensity factor K for the Linear Elastic case.The J-integral evaluation approach for determining the SIF (K) is often used in computational fracture mechanics (i.e. by Finite Element Analysis).Since J-integral can taken at a distance somewhat removed from the crack tip, this approach lessens the need for a highly refined Finite Element Mesh at the crack tip or the use of singular element around the crack tip.However care must still be taken to ensure that the increased stiffness of the numerical model does not lead to an under estimate of the facture parameters (J or K).

Judicious choice for the integration path can often simplify the calculation of J.The J- integral evaluation approach has a application in ANALYTICAL METHODS OF ANALYSIS OF CRACKED BODIES.With the introduction of J- integral, the LEFM has been extended to nonlinear elastic bodies. The theory is completely rigorous.Unfortunately, there are few engineering application for the generalization, except as an alternative method to extraction of SIF (K).The primary interest in the J-integral lies in its application to an approximate theory for elasto plastic fracture, called EPFM.

Elasto plastic Fracture: Characterizing Elastoplastic Material Behavior.The extension of the theory of the nonlinear fracture mechanics to include elasto plastic material behavior in which the size of the plastic zone exceeds the small scale yielding approximation is based on the DEFORMATION THEORY OF PLASTICITY.The argument for the choice of the deformation theory of plasticity is that, provided no unloading is permitted to occur, the prediction of an elasto plastic material model and a nonlinear elastic material model is indistinguishable to an outside observer. By this we means that a load- load point displacement diagram, such as figure 11.2 would look the same for either material model. The mechanisms going on inside the two materials are markedly different, but out wordily there is no difference.For this argument to be valid, the stress must be non decreasing every where and the stress component must remain in fixed proportion as the deformation proceeds.

The latter condition is not strictly satisfied due to plasticity induced stress redistribution but outside the plastic zone the stresses are assumed to be nearly proportional for simple loading.Finite Element Analysis using THE INCREMENTAL THEORY OF PLASTICITY by Mc Mee King (1977) support this assumption.With these restriction the J-integral has been proposed as an Elasto plastic Fracture Mechanics parameter.Despite the similarities between nonlinear elastic and elasto plastic behavior, there are important difference even if we do not permit any unloading to occur (i.e. monotonically increasing load only).Let us reexamine (fig 11.3) for the elasto plastic case. Since we can not permit any unloading the change in the load- load point displacement curve cause by a change in crack length cannot be in principle, be measured with a single specimen.There fore we will consider two specimen identical in all respect except for their crack lengths (a) and (a+a) respectively each loaded to the same total displacement .

From the deformation theory of plasticity arguments, the change in potential energy is given by a shaded area in figure 11.3a.However unlike the nonlinear elastic case, not all of their energy is available to form NEW SURFACE. In the elasto plastic case, much of the energy is non recoverable and we cannot interpret J as an energy available for crack extension.Working independently Hulchinson (1968) and Rice and Rosengren (1968) proposed an alternative interpretation of J as a measure of the stress state under some condition.They all assume that the elasto plastic behavior (Stress Vs Strain) can be modeled fitting the Ramberge Osgood flow rule.

Where, o is the flow stresses o is the flow strain ( = )

11.15

Measured in true stress- strain units n- strain hardening exponentTypical Ramberg-Osgood stress-strain curve for several values of exponent n are shown in fig 11.6.Since J is path independent, we are free to evaluate it along a circular path surrounding the crack tip that is r = constant as shown in the figure 11.7 for this path,

And Eq (11.6) can be written as Note that the terms in the integrand of Eq, (11.16) are the product of stress and strain and J is independent of choice of r, Eq, (11.16b) can be satisfied ifor11.16a11.16b

Where hij (n,) are suitable functions describing the angular variations in the stresses and strains.In the very near field of the crack tip the linear term in Equation (11.15) is generally small compared with the power hardening term and can be neglected with this approximation Eq, (11.15) and (11.17) can be combined to yield Stress-field: Strain-field:11.1711.18a11.18b

Where fij and gij describe the angular variation of the stress and strain in the near field of the crack tip and In is an integration constant that depends on n and the state of stress.Extensive tables of the function fij and gij have been generated by Shih(1993)In has been approximated in series for m, see Saxena (1998) asIn= 6.568-0.4744n+0.040n2-0.001262n3 for plane stress stateIn= 4.546-0.2827n+0.0175n2-0.451610-4n3 for plane strain stateFrom Eq, (11.18) we can observe that J plays the same role in EPFM that K plays in LEFM- namely it describes STRENGTH OF THE SINGULARITY, that is the CRACK TIP STRESS and STRAIN FIELDS scale with J.The order of the singularity is determined by the strain hardening exponent. Note that when n=1, these equations reduce to their corresponding forms for linear elastic material models ,(i.e.LEFM)The stress and strain field described by Eq,(11.18) are called HRR fields. And play an important role in the extension of nonlinear elastic fracture mechanics into the elasto plastic domain i.e (EPFM)

In 1972, Begley and Landes proposed the use of the J-integral as a plane-strain elasto plastic fracture criterion .i.e J=JcThe combination of the HRR crack tip stress field interpretation of the J-integral along with the ability to measure its value experimentally from potential energy argument, led them to suggest that by analogy, there must be a parameter, J1C for elasto plastic material that is comparable with K1C for linear elastic materials.Begley and landes proposed that a series of test specimens of increasing crack lengths be used to construct a set of MASTER CURVS for each combination of material and specimen.In the their experiments the area under the load Vs load displacement curve up to a fixed total displacement was determined with a polar planimeter with modern digital instrumentation the area would be computed on the fly, using a suitable integration rule for each specimen.If we were to follow the definition of J, from Eq.(11.3) literally, we would integrate the area above the load- load point displacement curve, however, since Begley and Landes were concerned only with the difference between the curves, the area under the curve was more convenient to measure.The computed potential energy was then plotted as a function of the crack area and numerically differentiated to determine J as

The critical value, JIC was the value of J corresponding to the displacement at the onset of crack extension i.e fracture.To be a valid material property the measured value of J1C should be independent of the specimen used to measure. In a companion paper Landes and Begley (1972-1076) determine J1c for a rotor steel from two specimen geometries having significantly different plastic slip-line fields. Their results shows that to within experimental error, J1C was independent of specimen geometry over the full range from elastic to fully plastic More resent studies have shown that measurement of the value of the J integral under less then plane strain state may depend on the specimens geometry.The currently adopted procedure is describe in ASTM Standard E1820-99. Standard Test Method For Measurement of Fracture Toughness (1999). This standard replaces ASTM standards E813 standard Test method for J1c, a measure of fracture toughness (discontinued 1997). E1152 standard test method for determining J-R curves (discontinued 1996) and E1737 standard test method for J-integral characterization of fracture toughness (discontinued 1998).11.19

In many respect ASTM E813 is similar to ASTM standard E399, except that, at the expence of additional INSTRUMENTATION, it permits the determination of Jc value when the stringent requirement for a valid K1C are not met.ASTM E813 requires continues measurement of both Load- Load-Line displacement (needed for Jc) and the load- crack mouth opening displacement (used to determine K1C).The procedure for determining the germane fracture parameter is the same for all three type of certified specimen geometries. The single-edge notch bend specimen(SEN(B)], the compact Tension specimen [C(T)] and the disk-shaped compact tension specimen DC (T).Since ASTM E813 has one of its options, the determination of valid K1C, the pre-cracking and fixturing requirement are the same as described in E 399.After the test is completed KQ is computed, and the relevant validation checks are perform to verify if KQ KIC. If this check FAILS, Jc is computed from Note that K is calculated by a=ao(i.e. the fatigue pre-cracked crack length/and11.20

Where is a factor that depends on Test specimen geometry and Apl is the Plastic component of the area under the load- load point displacement curve, shown in figure 11.8.The J calculated at the final point of instability is denoted JQ. If JQ meets certain requirements spelled out in the ASTM standard E 1820-99, , otherwise, the notation Ju is applied to signify that the determined value of the J does not meet the necessary criteria to be considered in-plane size independent , Note that Jc may be dependent on thickness.ASTM standard E 1820 99 goes beyond the measurement of Resistance to unstable crack extension (i.e. Jc) to characterize stable crack extension, referred to as stable tearing.The stable crack extension behavior of a material is characterized by the fracture toughness (J) Vs crack extension (a) CURVE, such as the one, illustrated in fig, 11.9, the resulting J-R curve is similar to the K-R curve in LEFM. 11.21

The standard procedure for determining the J-R curve is based on the elastic compliance method, where multiple point are determined from a single test specimen.The standard procedure for higher degree of signal resolution than that required for determining Jc or K1c within the standard.The difficulties with standing, J-R curves do characterizes fracture behavior in high toughness material.In the United State of America the use of the J- integral to characterize elasto plastic fracture is almost universal, however, for historical reason, an alternative measure, the crack tip opening displacement (CTOD), is more widely used in the United Kingdom and on the European continent.In principle CTOD can be measured directly at the crack tip. In practice its value is inferred from CMOD measurement and ASTM standard E1820-99 includes procedure for reporting. However the CTOD and J are not independent quantities.Although various definitions of the CTOD, denoted by t, have been proposed, we recommend the one proposal by Tracy (1979), illustrated in fig 11.10, as the intercept of two 45 lines originating at the deform crack tip and intersecting the crack profile.

Where kii represents the angular variation in ui.Note that point on the crack profile in fig 11.10 are displaced in both the x and y direction relative to there unreformed positions (point A) letting

Note that is not known apriori, and solving Eqs. (11.22 and 11.23) together to eliminate results in an expression for t ofter form Integrating Eq. (11.18b) and evaluating it for = to determine the opening profile of the crack yield an expression of the11.2411.2311.22

Shih (1981) evaluated dn using finite element analysis for a wide ranges of values of and (o/E). The results for plane stress state are shown in figure 11.11. For plane strain state, the values are about 20 percent lower.For plane stress state, the values are about 20 percent lower.As a results the CTOD approach for characterizing elasto plastic fracture and the J- integral are alternative representations of the same measure.For perfectly plastic behavior , and Eqs, (11.24) for plane stress state has the simpler form

Despite the lack of as firm mathematical foundation as enjoyed by LEFM, EPFM plays an important role in the analysis and prevention of FRACTURE within its domain of influence.Crack instability in the elasto plastic domain is only one of several competing mechanisms of failure. Net section yielding for example. 11.25

J-Integral as Elasto Plastic Fracture Mechanics ParameterThe use of J-integral for characterizing elasto plastic fracture is based on several assumptions. It is worthwhile to review them here.In an elasto plastic material, energy momentum is not necessarily conserved on an arbitrary closed contour not containing a singularity. As a result, we can not use J=0 to prove path independence, and the magnitude of J may depend on the integration path.The change in potential energy between two crack lengths, represented by the shaded areas in figure 11.3, is not equal to the energy required to create new surface. Some if not most, of the loss in potential energy goes to irreversible losses, such as heat, and there is no known algorithm for allocating this area among the various energy absorbing mechanisms.The use of deformation theory of plasticity to model the behavior of a component or structure undergoing elasto plastic deformation is appropriate only as long as there is no UNLOADING any where in the component or structure. Fracture (failure due to crack extension), by its nature, violates this assumption- that is , the crack growth involves the

Separation of two planes into new stress- free surfaces. Hence the stress at the crack tip must be redistributed to the remaining portion of the body.The interpretation of J as a parameter to measure the crack tip stress state depends on the applicability of the Ramberg-Osgood constitutive model to the material. It is tacitly assumed that ductile metals can be modeled by this flow rule without testing the models validity in each case. Not all materials can be modeled by a smoothly varying transition from the linear elastic to nonlinear elastic regime one obvious example is mild steel with clearly defined upper and lower yield point.The development of the HRR crack tip stress field concept ignores the linear portion of the Ramberg-Osgood constitutive model. This is under assumption that in the very near tip region, the strain are large enough that the contribution of the linear part is small compared with the total deformation, but small enough that the small strain theory of elasticity is still applicable. In principle, but not in practice, this assumption would rule out the use of the J- integral as an alternative characterizing parameter for materials that just barely miss satisfying the requirement for valid K1C as described in ASTM stand E1820-99.

The HRR stress field is not unique. There are an infinite number of ways to satisfy Eqs (11.17), corresponding to an infinite number of constitutive model for elasto plastic materials, by this, we mean that constitutive relations are not intrinsic to nature that is, they are not LAWS of nature in the same sense as for example, stress or strain transformation laws , but they are EMPERICAL relations chosen to model the observed behavior of the material they purport to represent.Despite the lack of as firm a mathematical foundation as enjoyed by LEFM, EPFM plays an important role in the analysis and prevention of fracture within its domain of influence. Just as there are restriction on the critical SIF (Kc) in the LEFM to ensure that the measured critical value is specimen independent, similar restrictions exits for J.Since the shape of the load- load point displacement curve for a cracked component or structure undergoing elasto plastic deformation ( around crack tips) depends on both structure (and its slip line field) and the constitutive model there is no reason to pre-suppose that the change in potential energy (i.e the shaded area in fig 11.13) will be structure independent.

In LEFM, we insisted that the plastic zone be fully contained within the singularity domination zone as a necessary ( but not sufficient ) condition to employ a one parameter (i.e stress intensity factor K) representation of the stress state.In EPFM, the corresponding requirement for crack-tip stress field is called J-dominance and the confined field is not the plastic zone, but the region of finite strain.Hutchinson (1983) has studied the size of the J- dominance zone in relation to the CTOD (t) are through equation (11.25) to the magnitude of J.Using the FEA results of Mc Meeking (1997), Hutchinson determined that one condition for a valid determination of J where R> 3t , t=J/o Where R as the radius of the J- dominates zone depicted schematically in figure (11.12)A second requirement is that the FRACTURE PROCESS ZONE be fully contained within the J-dominated zone.

Since the primary mechanism of ductile crack growth is that of void nucleation and coalescence, which are themselves finite strain events this requirement is also satisfied by Eqs (11.26)Based on nonlinear finite element analysis, Hutchinson estimated R to be on the order of 20 to 25 percent of the corresponding plastic zone diameter 2y computed from

For the specific case of compact tension and beam bend test specimens, the radius R of the J-dominated zone is equal to 7 percent often uncracked ligament.At the present time the primary application of Jc in EPFM is as a measure of fracture toughness when plastic deformation renders the Kc concept moot.The development of a fracture criteria Jappl=Jc, similar to K=Kc requires that Japplied be determined for each combination of component geometry and material being considered.11.27

Even if nonlinear elasticity concepts are employed rather than elasto plasticity theory, the task of evaluating Japplied is formidable and few solution exits. As a results, the FEM is usually employed to determine estimates of J1 where the benefit of path independence is used to simplify the computation of the line integral.Not to be forgotten in all this process is the fact that fracture (failure due to unstable crack growth/propagation) is only one of several competing MECHANISM OF FAILURE. As illustrated in fig 11.1 when the net section stress approaches approximately 80 percent of the yield stress, the difference between the EPFM estimate of failure load and that predicted by classical limit theory is not significantly different. In this event computing the maximum load- carrying capacity of limit theory and applying a generous factor of safety may be a reasonable alternative to the more costly and time consuming fracture mechanics calculation based on J.

EXPERIMENTAL MEASUREMENT OF JIC

SENB Test Specimen3 Point Bending LoadExperiment to determine J1C are designed such that a large amount of plastic deformation is allowed to occur near the crack tip.The Plastic zone size may be as large as the crack length.Formation of a plastic hinge and rotation of the beam occurs.In the rest of the beam, small elastic deformation prevails.

Numerical Evaluation of J-Integral Stress-strain relations such as Ramberg-Osgood law describing the elastic- plastic material ( only for monotonically increasing loading ) are available.The task is to evaluate J-Integral knowing the constitutive equation, the geometry of the cracked component, and the loading condition.J-integral is generally evaluated using a numerous method J-integral is evaluated by integrating on an arbitrary path around the crack tip. We need not compute strain- stress field very accurately near the crack tip. In fact we can choose the integration path far away from the Crack tip.We can avoid use of SINGULAR elements around a crack tip.With increasing acceptance of FEM and availability of commercial FEA programs, numerical evaluation of J-integral is no longer difficult.However, the FE model and J evaluation procedure needs to be VALIDATED.

Exercise : Computational Determine the J-Integral for the semi-infinite strip of height h1 subjected to in-plane bending moment M acting on the free end in a linear elastic material.

A standard 1T aluminum C(T) specimen with W=2.01 inch is fatigue pre cracked to a/w=0.48 following ASTME 399 procedure. After completing of FRACTURE TEST, the load F- load line displacement (VM) was filled to the function. VM= (1.3P + 0.005P2) 10-6 inch and failure occurred at 7000 lb. Calculate J.Perform a FEA of a strip of final width in tension containing a central crack of length 2 a/w =0.6. Using a contour of your choice. Evaluate J-Integral, and compare your numerical value with target values.

Begley, J.A and Landes, J.D,(1979) Serendipity and the J.Integral Int. J. of Fracture, 12, pp. 764-766.Saxena, A, (1998) Nonlinear Fracture Mechanics for Engineers, CRC Press, Boca Ratan, Florida.ASTM Standard E 1820-99, 1999, Standard Teat Method for Measurement of Fracture Toughness, Annual Book of ASTM Standards Vol. 03.01, ASTM, West-Conshohocken, PA.Hutchin son, J.W.(1983) Fundamentals of the phinomenological Theory of Nonlinear Fracture Mechanics , J. App.Mech, Vol.50,pp,1042-1051.Mc Meeking, R.M (1977) Finite Deformation Analysis of DTOD in Elasto-Plastic material and implication for Fracture, J.Mech, Phys, Solid,Vol.25 ,pp 357-381.Rice, J.R.(1968 a) A path Independent Integral and the approximate analysis of strain concentration by Notch and Cracks, J.Applied Mech, Vol. 35, pn, 379-386.REFERENCE

Rice, J.R. and Rosen gren, GF(1968), plane strain deformation near the crack tip in a power law hardening material, J.Mech,phys, solid, m16,pp 1-12.Shih , CF (1981). Relationship between J-integral and the CTOD for stationary and extending cracks. T.Mech, phys, solid, Vol. 29, 305-326.Shih CF (1993) table of Hutchinson, Rice and Rosengren (HRR) Singular field Quantities, MRL-E-147 Report, Brown Univ, Providence, RI.Tracy, DM (1976). Finite Element Solution for Crack Tip Behaviour in small scale yielding, J. eng. Matlsa Technology, Vol. 98, pp, 146-151.Eshelby JD (1970). Energy Relations and the Energy Momentum tensor in continuum mechanics Inelastic behavior of solid, Mc Graw-hill, N y1. Pp 77-115.Rice JR (1968 a) A Path Independent Integral and the Approximate Analysis of strain concentration on by notches and Cracks, ASME, applied Mechanics, Vol.35, New York, pp. 191-311.References (contd.)

Rice JR ( 1968 d). Mathematical Analysis in the Mechanics of Fracture in Fracture: An Advanced treatise : Volume I I , mathematical fundamentals, Academic press, New York, pp. 191-311.Begley, J. A., and Landes. J. D1972, The J Integral as a Fracture Criterion, Fracture Toughness, Proceeding of the 1971 National Symposium on Fracture Mechanics, part 2, ASTM STP 514, American Soc. For Testing and Materials, Philadelphia, pp.1-20.Hutchinson, J. W., 1983, Singular Behavior and at the End of a Tensile Crack in a Hardening Material, J. Mech. Phys. Solids , Vol. 16, pp. 13-31.Kanninen, M. F., and Popelar, C. H., 1985, Advanced Fracture Mechanics, Oxford University Press, New York.