CA-3 43/76) GEORGIA INSTITUTE OF TECHNOLOGY OFFICE OF CONTRACT ADMINISTRATION SPONSORED PROJECT INITIATION Project Title: Date: March 10, 1980 Finite Strain Creep and Creep Crack Growth Analyses Project No: E-20-607 Project Diiector: Dr. S. N. Atluri Sponsor: NASA, Lewis Research Center Until Agreement Period: From 6/1/80 Type Agreement: Grant NAG 3-38 Amount: $49,992 NASA 7,315 E-20-379 (Cost-Sharing) $57,307 Total Reports Required: Semi-Annual Status, Final Technical Sponsor Contact Person (s): a.(Technical Mattet; sitqlf1/24)Ley Gordon '1'7-Smith Lewis Research Center 21000 Brookpark Road Mail Stop 49-3 Cleveland, OH 44135 (216) 433-4000, X6671 Contractual Matters (thru OCA) Grants Specialist 10516 /4 , g/we- su Lewis Research Center 21000 Brookpark Road Cleveland, OH 44135 ( 216) 433-4000, X55 4 1 Defense Priority Rating: n/a Assigned to: Civil Engineering COPIES TO: Project Director Division Chief (EES) School/Laboratory Director Deen/Director—EES Accounting Office Procurement Office Security Coordinator (OCA) /Reports Coordinator (OCA) OCA Research Property Coordinator (School/nig%) Ubrary. Technical Reports Section EES Information Office EES Reports & Procedures Project File (OCA) Project Code (GTRI) Other 0 GEORGIA INSTITUTE OF TECHNOLOGY OFFICE OF CONTRACT ADMINISTRATION SPONSORED PROJECT TERMINATION SHEET Date 4/15/83 Project Title: Finite Strain Creep and Creep Crack Growth Analysis Project No: E-20-607 Project Director: Dr. S. N. Atluri Sponsor: NASA, Lewis Research Center Effective Termination Date: 10/31/82 Clearance of Accounting Charges: 10/31/8 2 Grant/Contract Closeout Actions Remaining: Final Invoice and Closing Documents Final Fiscal Report Final Report of Inventions Govt. Property Inventory & Related Certificate ❑ Classified Material Certificate Other Assigned to: CE (School/1 COPIES TO: Administrative Coordinator Research Property Management Accounting Procurement/EES Supply Services Reseatch_Security Services Coordinator (OC Legal Services OCA Library EES Public Relations (2) Computer Input Project File Other Atluri Creep Crack-Growth: A New Path-Independent Integral (T c), and Computational Studies t R.B. Stonesifer* S.N. Atluri** Center for the Advancement of Computational Mechanics School of Civil Engineering Georgia Institute of Technology, Atlanta, GA 30332 tReport cr! Research under NASA Grant NAG 3-38 NASA - Lewis Research Center, Cleveland, Ohio * Doctoral Candidate ** Regents' Professor of Mechanics Preface This report is based on research performed at Georgia Tech. under NASA grant NAG 3-38 from the NASA - Lewis Research Center, Cleveland, Ohio, with S.N. Atluri as the Principal Investigator at Georgia Tech. The authors thank Drs. M. Hirschberg, B. Gross, and J. Srawley of NASA - Lewis Research Center for their continuous encouragement and timely advice during the course of this research. The authors express their appreciation to Ms. M. Eiteman for her assistance in the preparation of this report. This report also forms a part of a Doctoral thesis submitted by R.B. Stonesifer to Georgia Institute of Technology, in December 1981. Table of Contents Page Abstract I. Introduction II. Derivation of the (AT) c , J, and C* Integrals 10 III. Derivation of Finite Element Equations 27 IV. Elements for Singular Crack-tip Bel'.avior 34 V. Creep Crack Growth Computations 47 VI. Conclusions 89 APPENDICES A. Existence of Limits for Contour 93 Integral Definitions B. Numerical Difference Between (T1css 96 C. Numerical Methods for Evaluation of 97 Contour Integrals D. Simulation of Crack Extension 99 E. Analytical Evaluation of Cic for the Strip Problem 103 REFERENCES 106 ABSTRACT The development of valid creep fra.:Iture criteria is a major topic in cur-rent engineering research. Two path-independent integral parameters which show some degree of promise as fracture criteria are the C* and (AT)c integrals. The first portion of the present work reviews the mathematical aspects of these parameters. This is accomplished by deriving generalized vector forms of the parameters using conservation laws which are valid for arbitrary,three-dimensional, cracked bodies with crack Surf ace tractions (or applied displace-ments), body forces, inertial effects and large deformations. Two principal conclusions are that (AT) is a valid crack-tip parameter during nonsteady as — c well as steady-state creep and that (ST) has an energy rate interpretation c whereas C* does not. The development and application of fracture criteria often involves the solution of boundary/initial value problems associated with deformation and stresses in either laboratory specimens or actual components. Due to the power of the finite element method in treating complex geometries and non-linearities, it has often been used for this purpose. In the present work, an efficient, small displacement, infinitesimal strail, displacement based finite element model is developed for general elastic/plastic material behavior. For the present numerical studies, this model is specialized to two dimensional plane stress and plane strain and to power law creep constitutive relations. A mesh shifting/remeshing procedure is used for simulating crack growth: The model is implemented with the quarter-point node technique and also with specially developed, conforming, crack-tip singularity elements which provide for the1.n/(1+n) strain singularity associated with the HRR crack-tip field. Comparisons are made with a variety of analytical solutions and alternate numerical solutions for a number of problems. SECTION I INTRODUCTION Fracture Problems and Fracture Criteria: A Review Characterizing the displacement, stress and strain fields associated with stationary and propagating cracks in solids characterized by various idealized constitutive relations is one of the most important areas of study in fracture mechanics. The importance of these studies is not only that we can predict the stress or displacement fields in a cracked body, but also that knowing the nature of such fields we can possibly correlate observed fracture behavior with some aspect of these fields and thus arrive at valid fracture criteria. Criteria for Crack Growth Initiation The two macroscopic aspects of fracture for which correlations are commonly sought are the initiation of crack growth and the rate of crack growth. The most notable initiation correlations are with the elastic stress intensity factor, KI' for the elastic (and/or small scale yielding) case [1,2] and with the J1-integral for cases in which plasticity may not be limited to the crack-tip [3,4]. The conditions under which these correlations are independent of-geometry are discussed in the cited references. The critical values of K 1 and Ji for a given material are denoted KIc and Jlc, respectively. It is implied by the use of the subscripts "I" and "1" that these criteria are for the crack - 1 - opening mode (i.e., mode I). While similar criteria might be expected for the sliding and tearing modes (i.e., modes II and III), the experimental data for such studies is lacking. Both K and J1 have been shown to be crack-tip field parameters and both can be evaluated experimentally through energy considerations. The linear elas-tic, mode I crack-tip field determined in [5] shows that the asymptotic crack-tip fields are proportional to K1. Similarly, the asymptotic, mode I, crack-tip fields for power-law deformation theory plasticity have been shown [6.7] to depend upon the single parameters J 1 .1 In the case of small scale yielding, K is easily related to the energy release rate, GI [8], which is a measure of the potential energy decrease due to an increase in crack length. The quantity J 1 has a similar potential energy interpretation in the case of deformation theory plasticity, and becomes identical to GI for small scale yielding of a stationary crack. In the foregoing discussion, the time dependence of the material's response and of the applied loading is assumed to be negligible. For creep crack growth these assumptions are no longer valid. We now consider crack growth initiation in materials which exhibit creep behavior. While a significant number of creep fracture experiments have been reported in the literature, it appears that the primary interest has been to find a creep crack growth rate criterion as opposed to an initiation criterion. As a result of this emphasis, many investigators use notched specimens rather than precracked specimens and many do not report data which could be useful in addressing the question of initiation. At present there seems to be some indication [9,10] that when precracked specimens are used, the time required for creep crack growth initiation is negligible when 1 The deformation theory of plasticity precludes elastic unloading from an elastic- plastic state and thus is mathematically equivalent to nonlinear elasticity. The crack-tip fields associated with power-law deformation plasticity are commonly referred to as HRR fields after the au,:hors of references [6] and [7]. compared to the life of the specimen. It should be understood that this is not a universally acknowledged conclusion [11] and that further study is indicated. As noted previously, the second macroscopic aspect of fracture for which correlations with crack-tip fields are commonly sought is crack growth or propa-gation. The following summarizes the development of criteria relating to this aspect. Criteria for Crack Propagation Slow crack growth occurring under constant load implies that the material response is time dependent and is generally classified as creep crack growth. If the material's time dependent nature is negligible under the subject con-ditions, then it is assumed that crack growth requires an increase in applied load. This latter case is typical of situations in which small scale yield conditions are not met and for which J1 has been found to correlate with crack growth initiation. The primary interest in this quasi-static mode of crack growth is that for some materials and geometries, the increase in load carrying capacity of the structure during quasi-static crack growth is significant. This implies that design procedures can be developed to take advantage of this added margin of safety. To justify such a procedure, however, there muast be some dependable means of predicting the crack growth versus load behavior as well as predicting at what load the crack becomes unstable (i.e., is no longer quasi-static). As noted, J 1 is generally accepted as a valid initiation criterion for this problem. For prediction of the subsequent growth, however, there are at least two proposed criteria which appear to provide reasonable correlations with experimental data. The first growth criterion can be stated as J 1=J1R (Aa), where J 111 (0) = Jlc and j1R(Aa) is assumed to be a material property which depends on the amount of crack growth, a [12,13]. The subscript "R" denotes that this quantity characterizes the material's resistance to cracking. While strong theoretical - 3 - arguments can be given as to why this criterion should not be valid (except possibly for very limited amounts of crack growth [14]), it has been demon-strated that reasonable predictions can result from the use of this criterion for at least some classes of problems [12,15]. Based on the theoretical objections to the use of J ilt except for limited crack growth, a second criterion which is based on the crack-tip deformation has been proposed [16,17]. This criterion results from finite element simu-lations of quasi-static crack growth experiments which indicate that the crack-tip opening angle, CTOA (defined by the first finite element behind the crack-tip) becomes constant during crack growth. Whereas the CTOA, so defined, is clearly a mesh dependent quantity, the concept of crack-tip deformations becoming constant with crack growth is physically meaningful. The procedure for applying this criterion in finite element based predictions of crack growth behavior is to use Jlc for initiation and J1R for crack growth prediction until the computed CTOA has become constant with crack growth. Continued growth is then governed by this constant value of CTOA. Alternatively, a predetermined CTOA resistance curve can be used throughout growth. Crack growth instability is assumed to oc-cur (for either CTOA or J1R as the criterion) when further increase in crack length results in the criterion for growth being exceeded without further increase in applied loading. The J1R and CTOA criteria appear to provide reasonable correlation of ductile slow crack growth behavior for a variety of materials, geometries and load conditions [15,18,19]. Creep crack growth generally becomes a concern when components are operated at elevated temperatures. Whereas quasL-static crack growth can be on the order of mm/sec, typical creep crack growth rates are on the order of pm/sec. Compared to elastic-plastic quasi-static crack growth, the problem of creep crack initiation and growth is a relatively new area of study. - 4 - Numerous experimental studies have been undertaken with the purpose of finding a parameter which correlates with creep crack propagation rate. (See, for example, the review article [20] and [21-24].) Most of these investigations consider as candidate parameters, KI' some form of net section (or reference) stress, and in more recent studies C*. The C* parameter is the steady-state creep analogue of JI (in the sense of [25]) in that the definition of C* is the same as that for J1 except displacements and strains are replaced by their res-pective rates [26]. It is illustrated in Fig. 1.1 that the above three parameters can be expected to correlate three distinctly different creep crack growth situations. In Fig. 1.1a, a crack and its associated ligament are shown for a material and geometry which results in negligible creep strains everywhere except in the vicinity of the crack-tip. This condition is analogous to that of small scale yielding in elastic-plastic fracture. Fig. 1.1b represents a situation in which C* might be considered an appropriate parameter. This situation is characterized (i) by the body being essentially at steady-state creep conditions (which implies very slow crack propagation) and (ii) by the creep-damage process-zone being local to, and therefore controlled by, the crack-tip field. Fig. 1.1c illustrates the type of situation for which net section stress might be expected to control crack growth. In this case, the main feature is the widespread creep damage zone. It is seen from Fig. 1.1 that intermediate situations can occur. For example, suppose a particular material and geometry results in a crack propagation rate such that elastic strain rates are not negligible compared to creep strain rates (i.e., nonsteady creep) and at the same time, creep strains are no longer localized to the crack-tip region. While neither KI or C* could be valid parameters for this case, it appears reasonable to expect that crack growth - 5 - a. K r ,(AT)c controlled behavior • b. C ,((T)c controlled behavior c. crnet controlled behavior region with creep damage Fig. 1.1 Conditions for which creep crack growth parameters are expected to be valid - 6 - rate is still determined by the local crack-tip field since the creep damage process zone is still assumed to be local to the crack-tip. A parameter which is apparently capable of spanning the gap between KI controlled growth and C* controlled growth has been introduced [27]. This parameter is referred to as (AT)c and is defined by a path-independent vector integral. A detailed discussion of a generalized C* (i.e., C*) and (AT) c is given in Section II of this work with a principal result being that the energy relationship commonly used for experimental measurement of C* does not apply to C* but rather applies to the (AT)c parameter. This means that the experimental results are actually showing a correlation with (AT) rather — c than with C. Based on the theoretical validity of (AT) c as a crack-tip field parameter for nonsteady as well as steady-state creep and based on the mounting experimental evidence that crack propagation rate correlates well with (AT) c , it seems the creep crack growth rate problem is close to having a solution. Motivation for the Present Work In the following, we review previous studies to the extent required to place the present study in perspective and briefly introduce the present work. The nonlinear nature of creep constitutive relationa precludes analytical solu-tions for either stationary or propagating cracks in a creeping material: For stationary cracks in a power-law creep material, however, it is known that the HRR fields are present in the vicinity of the crack-tip [26]. (Since the singularity in creep strain rates is greater than that in the stresses, and thus elastic strains, it follows that the HRR field exists at the crack-tip during nonsteady as well as steady-state creep.) For propagating cracks, it appears that the HRR fields no longer exist at the crack-tip, but that analytical tools exist to determine the fields which do exist [28,]9]. Whi:_e knowledge of the crack-tip field is valuable, the solution of boundary value problems must depend on numerical methods. The finite element method, in particular, shows promise for solving creep crack growth problems. Only a few studies on finite element modeling of creep crack growth have been reported. The earliest is apparently that of Ohtani and Nakamura [30]. This study simulated crack growth with a node-release technique and assumed a critical crack-tip plastic strain criterion for creep crack growth. The rate constitutive law contained an elastic term and a creep term based on the gen-eralization of the uniaxial Norton power law. Hinnerchs [31] uses the Bodner-Partom constitutive law [32] and a node-release technique for modeling crack growth. In this work, several candidate criteria are examined by simulating crack growth experiments. Due to the ap-parently limited crack growth (<0.5 mm), the short test durations (one hour) and the lack of crack growth measurement data (which requires the development of a so-called hybrid experimental-numerical procedure to estimate the crack growth history), it seems the general applicability of the conclusions from this study are questionable. It also seems likely that the methods for evalu-ating C* in this study are incorrect1 and thus the conclusions concerning C* should be regarded accordingly. Ehlers and Riedel [33] have conducted a finite element analysis of a stationary crack in a compact specimen. The primary emphasis in this study is on the nature of the crack-tip field during the transition from the initial elastic field to the steady-state creep field. 1 While the details of the numerical procedures for evaluating C* are not given in [31], it appears that the W* term of C* (see Section II) is incorrectly in-terpreted as a history dependent quantity as opposed to a quantity dependent solely on the steady-state stress and strain rate. - 8- 1 The strength of the HRR field during the transition period is determined through fitting the near tip equivalent stress field. The calculations use eight-noded isoparametric elements with quarter-point elements being used at the crack-tip so as to have an r-1/2 strain singularity. Creep crack growth and creep crack growth criteria are not considered in this study. The finite element equations fortFe creep crack growth model being used in the present study are derived from the principle of virtual work in Section III. Section V presents the results of several analyses involving both stationary cracks and propagating cracks. The creep crack growth simulation is via a mesh shifting/ remeshing procedure. Calculations are made using the quarter-point element tech-nique as well as with a specially developed (SectionIV) compatible element which incorporates the HRR, r-n/(1), strain singularity. An important aspect of the current work is the study of the (AI)c parameter. In particular, the meaning of (AT)c, its relationship to Ct, and its calcula-tion within the context of finite element analysis are explored in depth. A series of crack propagation calculations are combined with analytical and experimental results in Section V to show that creep crack growth in 304 stain-less steel at 650°C occurs under essentially steady-state creep conditions. This implies that the crack growth rate for a given crack length and load can be determined from a steady-state creep solution which does not depend on the previous load and crack growth histories. This observation implies that simple crack growth prediction methodologies may be developed. SECTION II DERIVATION OF THE (AT) c , J AND C* INTEGRALS — — Preliminaries We shall consider problems which exhibit the following constitutive be-havior: ij ceij + cij c = LijkZ kZ + (3/2)7(a eq) n cy id (2.1) We denote the cartesian coordinates of the undeformed body as x.. Defining as the rate of displacement (or velocity) of a material particle from the cur-rent configuration, thenis the symmetric part of the rate of displacement aa.ii _ . gradient eij E (V—tT = — a - Eijij. The gradient operator V is with res- . 373 pect to the current coordinates yi where it is understood that yi = xi + i. Lijk2 is the tensor of instantaneous elastic moduli. We let a the corotational rate (or "Zaremba-Jaumann rate") of the Kirchhoff stress a.. aym wherea..is related to the Cauchy stress T.. by aij = JT ij (J=det[ ax ]). The equivalent Kirchhoff stress Geq is related to the deviatoric Kirchhoff stress aij (= a " - 1/3 akk6 ij ) by a eq = (3/2)(al.aij )1/2. The parameters Y and n are those of the familiar Norton's law eq I = -(a )n eq were eq = [(2/3)..E..] 1/2 13 13 - 10 - We will use the notation: (_) denotes a second order tensor; (_) implies a vector; a = B . c implies a. = B..c.; A = B . C implies A.. = k B. .C..; .. - 1 13 j 513 J ik 3 A:B = A. B . Also note that V B:C implies C and V . B implies Y1 - - ij ij -t- - 3jk —il t -t - 3B.. 3y i A Conservation Law for Finite Elastic and Nonsteady Creep Material Behavior The discovery of conservation laws and the possibility of deriving path-independent integrals from these laws are not particularly recent occurrences as discussed in [34]. However, the literature in this area has been rather piece-meal and therefore difficult to assimilate. The recent work of Atluri [27] has done much to unify and generalize this subject and is the basis for the following presentation. We will consider a very general conservation law which has been given by Atluri, but will limit our discussion of this law to materials characterized by (2.1). We will use cartesian coordinates exclusively. Note that by special selection of material constants (i.e., y=0), (2.1) can be specialized to elasticity. Alternatively, by assuming that the stresses are invariant with time, (2.1) can be specialized to steady-state creep behavior. In the following presentation, the current configuration (i.e., the config-uration at time t) is the reference configuration. There may be initial stresses existing for this reference configuration. If stresses do exist, then they are assumed to satisfy the linear and angular momentum balance condition (i.e., equilibrium) V t . T p t (f-t - a-t ) = 0; T = TT - where p t , t f and at are the current mass density, ..x)dy force vector and accel- eration vector. - 11 A conservation integral relation given by Atluri [27] for a closed volume Vt (at the current time, t), which is free from singularities and any other defects (which would preclude the application of the divergence theorem), is: 0 = f {V AW - (V-t -t T):Ae - V . [(T+At) . Ae] Vt t (2.2) - p t (f-a) . Ae}dV + f [nt . (T+At) - E] . AedS S + f n . (T+AO . (Ae-Ae)dS -t - Se In (2.2), At is the incremental first-Piola-Kirchhoff (nonsymmetric) stress (At = [Aa - Ae . a]/J), where Ao is the material increment of Kirchhoff stress. The current mass density is denoted p t , and f and a are the body force and acceleration vectors at time t+At, respectively. St and Se are the portions of the boundary of Vt upon which prescribed tractions, t, are acting and at which prescribed displacement gradients, Ae, exist, respectively. The current outward normal to St or Se is nt . The quantity AW, discussed in detail in [27], is the incremental stress-working density in time At, and is given by: AW = T:Ae + 1T :Ae E T:Ae + AU (2.3) where 1 AU = 2-At :Ae (2.4) The validity of (2.2) is readily verified through the two identities [27]: SAW -t AW = V (T:Ae) + EAU = T:e -t -t- + Vt Ae:T + V Ae:AtT - -t (2.5) and t . [(T+At) . Ae] = [V . (T+At)] . Ae + V Ae:(T+At)T (2.6) - 12 - t the satisfaction of linear momentum balance in Vt: -t V . [T+At] + p t(f-a) = 0 (2.7) and the satisfaction of the boundary conditions: 1 nt . [T+At] = E on St (2.8) Ae = to on Se (2. 9) Note that identity (2.5) assumes that T ( the initial stress for the incre-ment) is an explicit function of its position in V t . The existence of AU is shown and discussed in the work of Atluri [35]. Having the relation (2.2) it is now possible to specialize this relation to finite elastic behavior or to steady-state creep behavior. However, since we are primarily interested in the path-independent integrals which can be ob-tained from (2.2) we will postpone the specialization till after we have de-rived the general path-independent integral (AT) c . Path-Independent Integrals for Fracture Analysis The conservation integral (2.2) is used [27] to obtain a path-independent integral which is applicable to the analysis of cracks by considering a volume Vt - V such as illustrated in Fig. 2.1. (Note that a two-dimensional case is illustrated for simplicity). The use of the divergence theorem for the region depicted in Fig. 2.1 results in (2.2) being rewritten fr 234 [nt AW - at . (T+At) . Ae]dS (2.10) [(-V : e -p (f-a) . Ae]dV Vt-V E 1The validity of (2.2) does not require S t+S e=3Vt where 3V denotes the surface bounding V t . Therefore, ali t need not coincide with the botundary of the body under consideration. - 13 - Lt - c c4-0 f -(-1t -t (AT) = [ AW - n . (T+At) . Ae]dS r (2.11) + Jrr. n AWdS + f n AWdS - Jr . AedS -t -t 12 45 S t - n . (T+At) . AedS -t - S e = [nt AW - n . (T+At) . Ae]dS E (AT) 6 (2.10) In writing (2.10) it has been assumed that Se+St=F12+r45' which implies that r234 does not coincide with any exterior boundaries. This has been assumed purely for convenience of notation. We have also used the notation r = F 165' Noting that (2.10) contains two equalities, it can be verified by inspection that (AT) E depends on E (or more generally r ) but that it does not depend - c E i on the selection of r234 . In this sense (AI) c s path-independent independent of the selected far-field path). Following the reasoning of Atluri [27], we define (AT)c as the limit of (AT)E as E goes to zero.1 [nt AW - n--t . (T+At) . Ae]dS - 44° L t -t € t-V [(-V T):Ae - p t (f-a) . Ae]dV + Jrn AWdS + Jr n AWdS - Jr t. AedS -t -t r 12 r45 St -Jr n . (T+At) . Aed4 -t -Se ) 1The existence of the limit is shown in Appendix A. - 14 - fr 234 r23 4 Fig. 2.1 Contours for applying the conservation law to a two-dimensional, cracked body By defining (AT)c as the limit of (AT)c as E goes to zero, it is seen that (AT) c is entirely determined by the asymptotic near-tip fields. It will be shown later that the converse to this statement is also valid when the near-tip fields are the HRR fields. That is, it will be shown that (AT)c determines the asymptotic near-tip fields. Often it happens that only the first component of the vector quantity (AT) c is of interest. We will write the first component of (AT)c as (AT1)c. Also, the quantities (i) c and (T 1 ) c will often be used in place of (AT) c and (AT i ) c . These quantities are related by (AT) c (.k) c = Lt — At-)-0 At (2.12) However, in the presentation of numerical evaluations of (AT i ) c we use (T1)c as a convenient, approximate notation for (AT1 ) c /At. We now consider two special cases of (2.11). For symmetrical deformation about the x1 axis and cracks oriented along the x1 axis with traction free crack surfaces, no body forces and negligible inertial effects, the first component of (AT) c is (AT1 ) = Jr [n1AW - n.(r..+At. i )Ae. )(IS c E-)- Lt 0 j 31 j FE (2.13) = J [n1J AW - n.(r..+Atji)Ae )dS r234 J(at.. V 37 1_ jidV t Note that the limit of the volume integral has been written in its explicit- form as a result of the arguments for the existence of this limit, given in Appendix A. If, in addition to the above conditions, the strains are in- finitesimal and the deformations small, then there is no need to differentiate - 16 - between xi and yi, At becomes identical to ATij and we have (recalling Aeij E MuMy.): 1 j (AT1)c = Lt E+(31 [n 1 AW - n.(Tj ..+AT.j .) Dx, i]dS 1 i (2.14) BAu [n13 31 1 AW - n.(T.. 3x1 +A3T..) ildS 234 jr ST.' AEijdV V Bx t 1 The replacement AE.. in the volume integral is made possible by 13 ij the symmetry of T i . and thus 8T../6X1. j lj Physical Interpretation of (AT) c It has been shown by Atluri [27], that the vector (AT) c has the following physical meaning. Let two bodies with non-propagating1 cracks be identical except for the second body having an additional, arbitrarily directed, in-finitesimal increment in crack length characterized by the vector dc. It is assumed that both bodies experience identical load histories. Define total potential energy increments corresponding to the time increment At as 2 .LE1 =1 + AO1 + AK1 AE2 = Alp2 + AQ2 + AK2 for the first and second bodies, respectively. In (2.15), is the incre- mental work of external forces, AO is the incremental stress-work and AK is the increment in the kinetic energy. (It should be noted that AQ includes the inelastically dissipated energy.) Then 1Atluri [27] has shown that the l/r singularity in kinetic energy, which is as-sociated with dynamically propagating cracks, changes the interpretation. 2Note that sign convention for AE 1 and AE2 is opposite to [27] so as to conform - 17 - _ to conventional usage. (AT.) c bdci = -(AE2 - AE1 ) 1 (2.16) where b is the length of the crack front. If one is only interested in self-similar crack extension in the x1-direction, then dc2 = dc3 = 0 and AE2 - AE1 (AT1 ) c = bdcl (2.17) Therefore, (AT) is related to the incremental potential energy difference — c between two bodies which are identical except for an incremental crack length difference dc. Finite Elasticity and J As noted previously, the constitutive law (2.1) can be specialized to elastic behavior by choosing y to be zero. Therefore, (AT) as defined by — c (2.11) is a valid crack-tip characterizing parameter for general nonlinear elasticity with finite strains, large deformation, body forces and inertial effects. Howerver, the basic premise of elastic behavior is that the con-stitutive relations are independent of the histories of deformation and stress. This means that the constitutive relations can be derived from a potential. For instance, a potential, U, exists for t, the first Piola-Kirchhoff stress, such that T au t (2.18) In the following, we consider the reference configuration to be the stress-free, undeformed configuration at t=0, and therefore drop the subscript t for con-venience. As a result of the existence of the relation (2.18), it is possible to state two identities which are analogous to those of (2.5) and (2.6) for the case tEO. 3e DeDU 3x1 mn = = Ve:tT - inn - 18 - (2.19) and V [t . el = V . t.e + Ve:t (2.20) Similarly, the linear momentum balance ;i.e., equilibrium) condition is now V . t + p(f - a) = 0 (2.21) and the boundary conditions n . t = t - on St (2.22) e = e- on Se (2.23) Noting the similarity of equations (2.18) through (2.23) with (2.4) through (2.9) it is easy to arrive at the following conservation law J {VU - V . [t . T] - p(f - a) . e}dV (2.24) V + ir [n . t - t] . edS + f n . t . (e - e)dS St Se Following the procedure used'in deriving (AT) c from (2.2) we apply the diver- gence theorem to (2.24) for the volume V - V and take the limit as E goes to zero. The path-independent vector quantity resulting from this procedure will be called J. J = Lt [nU - . . e]dS c-)-0 (2.25) L c4-0 t / jr1.7_v =234 [nU - n . t . e]dS + - f) . edV — — + J . + Jr nUdS - jr . edS - Jr n . . edS ! r12 F45 St S e In writing (2.22) through (2.25) it is understood that o is the mass density in the reference configuration, n is the unit normal in the reference configu-ration, f are arbitrary body forces per unit mass, a is the absolute material - 19 - acceleration, and t are prescribed tractions per unit undeformed area St. We now consider several special cases of (2.25). If the problem being considered involves a crack oriented along the x1 coordinate direction and is loaded so that only mode I crack-tip behavior occurs, then J I is of primary interest and we have: [n U - n.t..e. ]dS + Jr jl = Jr 1 ij Jl E40) (ai - fi)edV r234 V-V T.1e. dS - f n.t..e. dS 11 1 1J J1 t Se (2.26) If in addition the problem involves infinitesimal strains, small displacements and traction-free crack surfaces, we have jl = Jrr 234 au. -I dS + Jr1 p(a. V - f.) 1 91.1 4 (2.27) Ln U - n.T - 1 ij DX1 dV Dx1 where use has been made of the existence arguments of Appenix A in taking the limit of the volume integral. For elastic behavior and non-propagating cracks, Atluri [27] shows that J has the meaning of energy release rate to a process zone V in the sense that DE Jkbdck = - dt Dt where b is the length of the crack front, DE D4,_ DS2E DKE ---E- _ Dt Dt Dt Dt and [DE dt = E (t + dt) - E Dt (2.28) For an elasto-dynamically propagating crack (i.e., singular kinetic energy) Atluri [27] concludes that - 20 - DL Jkbdck = [-- Dt dt (2.29) where L is the Lagrangian (i.e., L E K - ib -Q) such that DL Dtp e DS-2 DK — — Dt Dt - Dt ▪ Dt Therefore, Jk has the meaning of "rate of change of Lagrangian per unit crack growth". We now consider the special case of steady-state creep behavior. Steady-State Creep and C* It has been shown that (AT) characterizes the crack-tip field for mat-- c erials which exhibit creep behavior such as in (2.1). It is known that under certain conditions of applied loading, the constitutive relation (2.1) can (after long times) result in a steady-state. This steady-state is primarily charcter-ized by the time independence of the stresses (i.e., AU = At.. = 0). Specializing J 1 (2.13) to steady-state conditions, we define the steady-state value of (AT ) : -1 c Lt (AT ) = [n .Ae. - n.T Ae 1 css E4-0 1 ij ij j ji ]dS (2.30) jr aT.. [n1ij j T Ae - n.T..Ae. ]dS - Aeij .dV • r234 V Dy t Because (2.1) results in a power-law relation at steady-state, which is analogous to the power law deformation-theory plasticity (or essentially non-linear elasticity), Goldman and Hutchinson [26] have suggested a path-independent C*1 integral, where f au C * = J [11 w* - n.r.. I ] dS ij 3x 1 W* = j( 13 rijdij 0 (2.31) (2.32) - 21 - The question of how C*1 and (AT1)css are related, is a natural one. Before obtaining an equation relating C* to (AT1)css' however, the conservation integral (2.2) will be used to derive a generalized vector integral C*. In specializing (2.2) to steady-state we note that now stress is a function of the strain rate, and that stress increments are zero. Thus, AwEt:Ae. Also we may write: [V AW - (V T):Ae]dV = T:V AedV -t -t- - -t /t iVt Thus, at steady-state, we may write (2.2) as: 0 = f [T:(V-t Ae) - V . (T.Ae) - p t (f - a) . te]dV Vt -t [nt . T - t] . AedS + nt (fie - ie) dS - +t irse or equivalently, in 0 = t + [nt - Using the symmetry JO [T: T:V e = T:V -t- -t As a result of the incompressibility T:Vt- — E = T T :V and W* = -1.1 T T i rate form, (V 6) - Vt • ( I-0 - P t (f - a) -t - - n . . - . edS + fs t of T we note that: . . eT 1 (e + )] = t:V 2 - -t- condition G E • dE ij . ii &NV - = e)dS we have (2.33) (2.34) (2.35) (2.36) which leads to _ 71 _ Lt -->.0 Jr j 31 11 C1 E [n 114* - n ... .T idS = (C1 )E r (2.40) 3W* = T. 3.* ij Combining the results of the above manipulations we have: 3W* 31,1* D mn V w* - T / : V E = T:V -t ay i a mn ay. -t- -t- 1 (2.37) (2.38) Using (2.38) and the divergence theorem while applying (2.33) to V t - VE , we define the vector quantity (C*) E : .11 [nt W* - n-t . T .e]dS - f t (E - a) . edV (2.39) V- 234 t-V E + Jr n W*dS + jr n W*dS - jr . edS - jr n . T. TAB -t -t -t r 12 r 45 S t s e = Jr [n W* - n . I . e]dS E (C*) If we define the limit of (C*) E as E->.-0 to be C*, we have a quantity which characterizes the crack-tip field and is independent of the selection of r234' Restricting our attention to problems involving symmetric deformations about the x1 axis and cracks oriented along the x1 axis, with traction-free crack faces, no body forces and negligible inertia effects, we find that -t -t - = f [n1W* - n. .e. IdS Y234 31 11 In computing W* it is convenient to invert (2.1), substitute the re-suit into (2.36) and use the following identity to complete the integration: l+n 1-n (2.41) d( eq) n 2 l+n = 3- (-17-17) Ueq) n EiJ - 23 - The result of this manipulation is: 1 l+n W* - l+n (1)n( eq) n (2.42) or W' - 1 Y(6eq )1n Relationship of (T1)css and C* for Steady-State Creep (2.43) Now we will relate C* of (2.40) to the steady-state value of (AT1)c. First we rewrite (2.30) in rate form as: (i 1 ) css Lt E4•0 [n 1 . . - n. T ](IS 1 ij ij j j il (2.44) [n . - n.T..e. ldS - 1. B e dV 1 13 ij 3 ji 11 t Y1 Using the notation W=T..e. we have 1J 1j (i1 ) css = if (n13 11 W - n.T..e.1 )dS 234 DT.. - ay ij dv Noting that: . W = T •• = Ti1 • (e • e..) = T ' • • lj lj 2 i3 j • • (2.45) (2.46) it is seen that W is the rate of stress-working density, while W* is just a mathematical potential for T!.. As a result of incompressibility we can write iJ 1 l+n W = aeqeq = Y(aeq )l+n = (1) n ) y eq (2.47) as contrasted to W* of (2.42) and (2.43). Comparing the left equalities of 1This result is only valid for steady-state creep and is obtained through the substitution of the steady-state specialization of (2.1) into (2.46). - 24 - (2.40) and (2.44), it is seen that (i1) css and C* are related by: ( ii) c* Lt css 1 ▪ e4-0F 1 n (W W) dS (2.48a) = c* Lt Jr n,(a n+1dS e )q 1 n+1 E-÷c) r (2.48b) E Appendix B gives several numerical examples of relation (2.48) for two rather extreme values of n. It is now clear that C* and (AT)c are-not equivalent quantities under any condition despite their being derivable from the same conservation law1. The quantity (AI)c follows more directly from the conservation law and is the more gen-eral quantity not only in that it is applicable to nonsteady as well as steady- state creep but also in that it is applicable to constitutive laws which are more gen-eral than (2.1). The quantity C* relies on the special property of (2.1) which allows the existence of a potential W* for the stresses (T'). Furthermore, since W* does not have any physical meaning, whereas W has the meaning of stress-working density, it is understandable that (AT) has an energy interpretation whereas C* does not. c It is for this reason that it seems more appropriate to refer to experimental measure-. dE ments of - (1a as measurements of (T1)c as opposed to measurements of C or J . 1 The HRR Field We now give the HRR field in terms of (AT1)c. Whereas similar relations have been written in terms of C* for steady-state creep [36], the relations in terms of (,6,T1)c will be valid for nonsteady creep as well as steady-state creep. The HRR field as given in [37] but modified for creep by replacing E and ui by 13 and a. respectively, is: 13 1Note that these equations are derived on the assumption that (i.e., creep steady-state). Therefore, in order to have a well defined creep constitutive law we must have y#0 and n finite. - 25 - 1 1* = I + 1711-1 jr [a eg ( 0 )] 11+1 cosede -Tf TT (2.51) -1 [T..,a eq ] = K arn+1 rs..(0),Ffreq(en lj -n = yK rn+1 E.(0) 13 E 13 1 n+1 1:1 = yK r u.(0) 1 (2.49a) (2.49b) (2.49c) whereeq(0) has been normalized to have a maximum value of unity and K a and Kc are amplitude factors which are related by K = (Ka) n (2.49d) It can therefore be seen that the asymptotic crack-tip fields are entirely determined when Ka (or K ) is known or specified. Combining (2.49a,b,c) with the first equality of (2.14), using (2.49d) and rearranging, gives: 1 1 ((ATI.) c) n+1 ( (il)c n+1 Ka _1 yi*At - yI* (2.50) where I* is analogous to I defined by Eq. (24) of [6] except for the factor n/(n+1) multiplying the energy density term. To be explicit, It is therefore seen that knowing the value of (T1)c is equivalent to knowing Ka and thus is sufficient for defining all aspects of the asymptotic crack-tip field during nonsteady creep as well as under steady-state creep conditions. SECTION III DERIVATION OF FINITE ELEMENT EQUATIONS Before stating the variational principle and deriving the finite element equations, it is beneficial to illustrate the constitutive behavior to which the finite element model is addressed. Elastic/Viscoplastic Constitutive Relations A rather general rate constitutive law proposed by Perzyna [38], can be written in incremental form as: AE. = AT? 4. 1- 211 2v AT kk 6. y where 0 for F < 0 0 The relative magnitudes for the incremental viscoplastic strain components are seen to depend on the factor of/T...1 This implies that AE.v., when con-sidered as a vector in nine-dimensional stress space, is always directed along the normal to the surface f(Tij,Evp k ). By choosing FE f=a [aeq = (3/2(T' 1 .)1/2] such that af/3Tij = (3/2) eq lJ iJ (Ti./a ), and choosing (D(F) = (F) n we find that (3.1) becomes ij eq AE.. ij = 1 AT , 4. 1-2v &r 6 4. (3/2)y(a : 211 ij 3E kk ij eq 13 At This represents the special case of creep behavior which is considered exclusively in this study.2 It should be understood, however, that the finite element model which is described below is applicable to the more general behavior rep-resented by (3.1). Derivation of Finite Element Equations The finite element model is derived from the principle of virtual work T..(Se..dV - jr t idu idS = 0 (3.4) Sa rft the present finite element analysis, we assume only infinitesimal defor-mations and strains; hence there is no need to differentiate between the deformed and undeformed configurations. Inwriting(3.4)itshouldbenotedthatT..are the stresses existing at time t + At (where t is the current time), ti are the prescribed tractions on S at t + At, and Su.[SE.. = (1/2)(ou. .+6u. .)] are arbitrary Q 1 13 1,3 3,1 compatible virtual displacements. Following customary procedures we introduce the element displacement shape functions which relate element displacement u to element nodal iIt is common to choose fEF in which case we have what is called an associative law. 2While (3.3) is known to deviate from real material behavior (especially for primary creep) it is a widely used constitutive law and therefore has been adopted in the present study. - 28 - (3.3) displacements { q } u = {u} = [N]{q}; du = {Su} = [N]{6q} (3.5) 1 We again use the customary notation wherein strain (and stress) components are placed in one-dimensional arrays {c} = [B]{q}; {6E} = [B]fSql (3.6) Substituting (3.5) and (3.6) into (3.4) and applying conventional procedures for assembling element matrices into global matrices we have Eele [ {T}T[B]dV e - Jr {i}T[N]dli ele{601 E {F}T {6Q} = 0 ae Since {6q} are arbitrary virtual nodal displacements, it follows that r teleL f {T} T [B]dV - f a1T [N]dSA = {F} T = {0}T S (3.7) We now express the stresses {t} at t + At in terms of the current stresses, {T } I , and the incremental stresses corresponding to the time in-crement At: {T} = fT1 I+1 = {T} + {AT} I+1 (3. 8) In (3.8) and in the following, the I and I-I-1 subscripts designate the incremental solution with which the quantity is associated. Application of the incremental elastic constitutive law results in {T} I+1 = {T} + [E]{AEel}I+1 (3.9) = {T} I + [E][{AE} I+1 - f A 'vp l i+11 where {AEvp}I+1 are the incremental viscoplastic strains and [E] is the matrix _ 2 0 _ of elastic constants. Substituting (3.9) into (3.7), taking the transpose, and placing the known terms on the right hand side we have the final form of the finite element equations: where • [1(]141 I+1 = 1T1 I+1 IS - {R} I vp 1+1 [K] ele [B]T[E][B]dV e = {T} I+1 ele J [N]T{t} 1+1dS o'e fS vp 1I+1 ele = [B]T[E]fAcvp1I+1dV E e {R}I ele = Jr [B] T fT) dV V e (3.10) (3.11) (3.12) (3.13) (3.14) The above volume integrals are evaluated in the current work by 2x2 Gauss quadrature. The array fT} is input directly in terms of node point forces. Solution Procedures It should be noted that [K] of (3.10) is just the elastic stiffness and therefore only needs to be formed and decomposed1 once. This results in significant savings in the number of computations per time step as compared to methods using stiffness matrices which mast be reformed at each step (i.e., tangent stiffness methods). It should also be noted that the term {Svp} I+1 is computed from incremental viscoplastic strains {4Evp}1+1 which are esti-mated using {T} I and the material constitutive law (3.1). Only for the 1The equations (3.10) are solved in the current work by the decomposition [K]= [L][D][L]T where [D] is a diagonal matrix (the only nonzero entries are those on its diagonal) and [L] is a lower triangular matrix (the only nonzero entries are those below its diagonal); see for example [39]. - 30 - special situation when the stresses do not change with time will this estimate be exact. Having obtained the incremental nodal displacements {AOI+1 by solving (3.10), one can easily find the total incremental strains {AE)I+1 via the incremental analogue of (3.6). We now describe two procedures for ob- taining 1711+1. The first and simpler method to cbtain {T}I+1 is to substitute the esti- matedfAEvp1 I+1 used in solving for {AO I+1 into (3.9). If one does this, then it happens that {R}I+1 = {T}I+1 and therefore (3.10) becomes for the next step1 [K]{AQ} I+2 = f 1 -T- 1+2 ISvpI+2 {T}I+1 . {AT}1+2 + {Svp}1+2 (3.15) (3.16) This method was compared to the following method and was found to require smaller time steps to achieve similar results. Rather than using the estimated values of {1cvp}1+1 and (3.19) to determine {T}I+1' the constitutive relation (2.1) is integrated over the cur-rent time step at each Gaussian quadrature point with the condition that total strain {E} varies linearly with respect to time from {E}I to {E}1+1" (The present study uses an Eulerian scheme with each time step being divided into five subincrements.) The result of this procedure is better adherence to the postulated constitutive law at the expense of introducing a somewhat unequili-brated stress state. The amount of disequilibrium depends on the accuracy of the original estimate for the incremental viscoplastic strains and thus on the time step size. At this point one has two alternatives. The first is to use the 1This procedure results in the current model reducing to that of Zienkiewicz and Cormeau [40]. - 31 - viscoplastic strain increments obtained through the time integration procedure as an improved estimate and to re-solve (3.10) for the current time step. This procedure would, after several iterations, result in a stress state which is equilibrated to within some small user specified tolerance. With this type of procedure the time steps could be as large as those used with tangent stiffness methods. Further, it is reasonable to expect the solution to be at least as accurate as if a tangent stiffness method were used.1 The second alternative is to go immediately to the next time step with the understanding that the term till, in (3.10) results in the disequilibrium from the Ith step being corrected in the I+1 step. This feature is the result of the virtual work statement (3.4) being written in terms of total stress and tractions rather than incremental quantities. Owing to this corrective nature and to the diminishing returns one obtains from additional iterations, the second alternati7e is used in the present study. Regulation of Time Steps The creep calculations use a variable time step size which is auto-matically regulated by the finite element program based on two criteria. The first criterion is the maximum percent difference between the incremental equivalent estimated creep strain and the incremental equivalent integrated creep strain for all the Gauss points in the mesh: C1 = Max AEEST - AEINT (3.17) AEINT The second criterion is the maximum ratio of incremental equivalent integrated 1 This procedure could actually be more accurate if similar constitutive law integration procedures and equilibrium iterations are not performed with the tangent stiffness procedure. Also, it has been shown [41] that many element types become overly stiff when using the tangent stiffness method for modeling constitutive behavior approaching incompressibility. This problem is not en-countered with the current method. - 32 - creep strain to the equivalent elastic strain: C2 = Max [AEINT 5-] (3.18) The user specified, maximum permissible values for C1 and C2 are C1- and C2, respectively. The size of the next step is then obtained from [T AtI+1 = AtI . Min Cl , c 2 (3.19) Note that the initial time increment cannot be determined from (3.19) and must be specified by the user so as to satisfy the two step size criteria. In the present study, the values of C1- and C2- are 0.2 and 1.0, res-pectively. With these values, it has been found that the initial time steps are controlled by C1 while later time steps are controlled by C 2. The values of C1 and C2 are strongly affected by the mesh refinement since a finer mesh results in Gauss points being closer to the crack-tip and therefore having larger stresses and strain rates. To determine the sensitivity of the solution to the selection of C1 and C 2 , a compact specimen was analyzed with the above criteria and also with C1 and C being halved (i.e., C = 0.1 and C = 0.5). It was 2 1 2 found that the load point displacement differed by less than 0.5% for all time and that the steady-state solutions were essentially identical. It there-fore appears these values of C1 and C2 are small enough to ensure that the solutions to be discussed do not depend on these step size criteria. SECTION IV ELEMENTS FOR SINGULAR CRACK-TIP BEHAVIOR This section describes and compares several two-dimensional crack-tip singularity elements. Perhaps the primary motivation for introducing singu-lar crack-tip elements into finite element models is the significant savings in computational expense. It is shown in Section V, for example, that 57 element model with elastic, r-1/2, singular elements results in a more accurate solution than a non-singular 102 element model. The savings in CP time in this case is greater than 50%. Generally, one must consider that some additional effort is required to develop and implement a special crack-tip element and that this tends to offset the savings in CP time. It has been discovered, however, that the very commonly used , eight-noded, isoparametric element can be made to produce an r-1/2 strain singularity by merely shifting mid-side node locations via the node definition input data [42,43]. Therefore, a very convenient means for modeling linear elastic crack-tip behavior exists. It has also been shown [44] that a l/r type strain singularity can be obtained with this element type thus providing a suitable element for non-hardening plasticity problems. For more general singularity behavior, such as the r-n/(n+1) strain singularity associated with the HRR crack-tip field of power-law plasticity or creep, one must resort to sepcially formulated elements. In the following sections, we consider special elements for linear - 34 - elastic problem (r-1/2 strain singularity) as well as special elements for problems involving the HRR, r-n/(n+1) , strain singularity. In discussing these elements, an attempt is made to point out their advantages and dis-advantages. Elements for Linear Elastic Materials Although many special elements have been used for linear elastic frac-ture analysis (see Atluri [45] for review), we consider here only the eight-noded isoparametric element. There are two basic forms in which the eight-noded isoparametric element can be used as a linear elastic crack-tip element. In the first form, the two midside nodes adjacent to the corner node located at the crack-tip are shifted toward the corner node so that they occupy the quarter-point of their respective sides. This form is illustrated in Fig. 4.1 by the Type A crack-tip mesh. The second form in which the element can be used is illustrated in Fig. 4.1 by the Type B crack-tip mesh. In this form, the eight-noded element is degenerated to a triangular element by defining two corner nodes and their midside node to be the same node which is located at the crack tip. Then the two midside nodes adjacent to the crack-tip corner node are shifted to their quarter-points. It is important that only one node be used at the crack-tip, as opposed to three superposed nodes, since the latter case has been shown [44] to result in the 1/r type strain singularity. Barsoum [43,44] notes that numerical experimentation shows the degen-erate triangular form yields more accurate results than the nondegenerate element. He goes on to recommend that the four-sided configuration be abandoned based on the premise that the r-1/2 singularity exists only along the edges of the element and also that the strain energy for this element becomes unbounded if exact integration is used [44]. Ying [46], on the other hand, concludes that the r-1/2 singularity does exist within the four-sided element as well as along its edges and that the strain energy for the element is bounded (and thus -3 5 - \— crack tip Mesh type A Square 8- node isoparametric elements crack tip Mesh type B Degenerate B- node isoparametric elements / / plane of A symmetry a 0_,.1 region of special crack tip modeling Fig. 4.1 Several crack-tip mesh configurations the element stiffness is well defined). Apart from the above considerations, there are two aspects of the degenerate Type B element which inherent:y make it preferable to the nondegenerate element. The first is that the process of collapsing one side of the element to coincide with the crack-tip results in the element local coordinates being transformed into a form of polar coordinates. Since the element's shape functions are defined in the element local coordinates it is to be expected that angular bias will be much less apparent for this element type. The second feature of the degenerate triangular element which makes it preferable is that it is geometrically better suited for creating crack-tip finite element meshes of arbitrary refinement. Since the angular dependence of the near tip solution is significant, this flexibility for increasing the mesh refinement in the angular direction is important. Combining these two aspects with the numerical evidence cited by Barsoum, it seems the triangular, degenerate element is the better element for modeling the elastic crack-tip singularity. For this reason, all quarter-point element calculations in the present study use the degenerate triangular form. In particular,themesh configuration Type B of Fig. 4.1 has been used exclusively. Elements for Materials with HRR Crack-Tip Fields In the previous section, we discussed crack-tip elements for linear elastic material behavior. It has been seen that the standard eight-noded isoparametric element can be made to have the r-1/2 strain singularity and thus is useful for analysis of cracks in linear elastic materials. It can be shown (see for example, Atluri [451) that this is the only singularity which this eight-noded element can exhibit. However, it can also be shown that higher order elements of the isoparametric family can result in singularities of the type -n/(n+1) provided n is an integer. If we denote the order of the isoparametric -37 - interpolation as m, then singularities of the type r(1-0/t can be obtained for t an integer such that t13 Jr Ae dV Vt -V E 1 Now consider the limit of the V t -VE integral of (A.2). Inspection of this integral shows that it can be put into the form: E÷0 in f 2 Lt 1 C + — g(r,e)rdrde V t -VE r (A.3) R 7 0 where V' is a small circular region centered at the crack-tip and C is the integral over the region Vt - Vt . The function g(r,e) is a nonsingular function which becomes g(e) in the limit as r goes to zero, where g(e) is known in terms of the HRR fields. Upon a first inspection of (A.3) one is tempted to conclude that the limit does not exist since the integrand has a non-integrable singularity at r = 0. If, however, we look at the right equality of (A.2) it is seen that this conclusion results in a contradiction. Since we have shown that the limit of the integral on r E does exist (and therefore (A.2) requires that the limit of the integral over V t - VE must exist). A re-inspection of (A.3) shows that the only way for this apparent contraction to be resolved is if the g(r,e) of (A.3) has the following property: Tr g(r,e)de g(e)de = 0 (A.41 -7 If function g(e) is known explicitly for the linear elastic case and there-fore (A.4) can be directly verified. For the HRR field g(e) is not known explicitly and therefore (A.4) can only be verified numerically. - 94 - = C + Lt 1 (-)g(r,e)dedr f€1-7r For infinitesimal strain, nonlinear elasticity, the following relation provides an alternative to verifying (A.4) direclty Da Du au fV-V axk ij J ij axl ij axk Ac dV = jr n.Aa.. dS - Jr n.Aa.. dS (A.5) r234 r E The relation (A.5) (which assumes zero crack surface tractions and no body forces) illustrates that this volume integral of type (A.3) can be expressed in terms of the contour integral of type (A.1). The relation (A.5) can be verified through the divergence theorem, the linear momentum balance condition and the following identities: \ a _ __a ( aw a a€ as aE 3xk €3xk aE. a .. arm 3Xk ) li ij MI1 = aE.. 3x mn m n 1.] k as mn = Aar E ij au aE mn axnamn axk mn axk L. Aa APPENDIX B Numerical Difference Between (T1)css and C* 1 The purpose of this appendix is to give some examples to illustrate • the numerical difference between (T1)css and C* as given by (2.48). Using (2.50), (2.49a) and (2.51), we have n+1 (i 1T* ) css - I * = 1 + (n+1)I aeq(0)cosed9 1 -Ti (B. 1) The values tabulated in Table B.1 were computed approximately from values of I and plots of 6 eq (8) given in [6] and should be viewed accordingly. Table B.1 Comparison of (T ) and C* 1 css 1 Plane Strain Plane Stress (T 1 ) css C* 1 n = 3 n = 13 n = 3 n = 13 0.98 1.00 1.11 1.14 It is seen that for the range of n commonly encountered, (i1)css and C*1 are numerically very similar for plane strain but differ significantly for plane stress. APPENDIX C Numerical Methods for Evaluation of Contour Integrals The numerical procedures for evaluating J 1 as defined by (2.27), (AT i ) c as defined by (2.14) and Ct as defined by (2.31) are described in the following. General Procedures In studying the contour integral paths indicated in the finite element meshes of Figs. 5.2,3,5 (dashed lines) it is seen that the paths always pass through the centers of elements as opposed to along their edges. This procedure has been adopted so as to be-lefit from the presumably more accurate solution within the elements. Each element contour is divided into two segments with the integration being accomplished by two point Gaussian quadrature. All of the integrations are performed in the element local coordinates. The J1-Integral for Linear Elastic Analyses The contour integral portion of (2.27) involves the stresses, T.., and thedisplacementderivatives,DuJ3xl . Both of these quantities can be evalu-ated at the required Gauss points through the element nodal displacements and simple manipulations with element matrices. In the current study, J 1 is only considered as a parameter for linear elastic material behavior and therefore U = (1/2)a .E... The C*-Integral 1 The C* integral of (2.31) consists only of a contour integral. The W* of (2.31) is evaluated using (2.43). The gradient rates are approximated by 31 1 DAu .., ax At Dx 1 1 and therefore are average rates for the increment as opposed to the rates at the end of the increment. The contour integration procedure for C* is as described above and uses two point quadrature for each element segment. Whereas stresses are easily computed at the required contour Gauss points in the elastic case, the stresses must be computed incrementally in creep analyses and there-fore stress information must be stored for each contour integration point unless nonstandard element interpolations are used. In the present study, the stresses at the contour Gauss points are interpolated from the 2x2 element Gauss points through bilinear Lagrangian interpolation (in local coordinates), thus eliminating the need for additional storage. The ( T1)c-Integral In the evaluation of (2.14) it is understood that T.. are the stresses 13 at the beginning of the time increment being considered. The procedures for evaluating the contour integral portion of (2.14) are the same as used in evalu-ating C*1. The incrmental stress-work density, WW, is computed from AW = (t.. + 1 — lj 2 ij The stress derivative appearing in the area integral of (2.14) is evaluated based on the 2x2 element Gauss point values and the assumption that the stresses are distributed bilinearly wi .:h respect to element: local coordinates. Elements which are entirely within Vt are integrated with the usual 2x2 Gauss quadrature. Elements which are only partially within Vt have each applicable quadrant in-tegrated by one point Gauss quadrature. APPENDIX D Simulation of Crack Extension Modeling the propagation of a crack using the finite element method requires some special procedure for representing the creation of new crack surface. A common procedure is to relax the nodal forces at the crack-tip node, thus in effect allowing the crack to extend to the next node along its path of propagation. This relaxation process car. be accomplished in one time step but usually is allowed to extend over several time steps due to the large change in nodal forces which is inherent in the process. The major attrac-tion of this node-release procedure is its simplicity. There are two draw- backs of this procedure which resulted in an alternate procedure being adopted in this study. The first is that the increment in crack growth is directly determined by the nodal spacing in the mesh, therefore restricting the flex-ibility one has in selecting a time step size, the mesh size and /or the number of nodal force relaxation steps. The second and perhaps more important drawback is that the method is not adaptable to models which use crack-tip singularity elements. A typical mesh in the vicinity of a crack-tip is shown in Fig. D.1. The region A represents the region being modeled by singular crack-tip elements which in the present case remain centered on the crack-tip. The Type B elements are eight-noded isoparametric elements which distort so that the region A can remain centered on the crack-tip. The sequence of element configu-rations in Fig. D.1 illustrates the shifting/remeshing procedure used in [53, 54] and adopted here. The region A is moved by shifting nodes without altering element connectivity until the Type B element ahead of the region A becomes - 99 - • • • • • • • • • • • • • • B • B•B •B • C• C• • • • • • • • • • • • • • • B • A •B • C• C• Fig. D.1 Example of mesh shifting/remeshing procedure for simulation of crack growth overly distored. At this point, the elements in the vicinity are redefined so that further shifting is possible. It can be seen that this procedure allows the increment in crack length to be arbitrarily small and does not involve release of nodes in the same sense as for the previously described node- release procedure. The added flexibility afforded b Ythis shifting procedure does require some additional work. For example, in the creep crack growth application, nodal displacements and element integration point stresses are interpolated. The method of interpolation which is employed in this procedure is discussed next. We consider that the solution at time t, has been obtained and we now must find the solution at time t 2 . During the interval (t 1 ,t 2) the crack has grown by an amount Aa. Since the crack growth simulation procedure requires that nodes be shifted, and since the solution at t1 must be represented in terms of nodal and Gauss point quantities for the shifted mesh, it is necessary to submit the affected nodes and Gauss point to an interpolation or fitting pro-cedure. The simplest interpolation procedure for nodal displacements and the one used in [53,54] as well as for calculations in the present study is one which directly uses the element shape functions. In this method, the nodal positions for the mesh at t 2 are located in the mesh at t l . Knowing which element of the mesh at t1 encompasses this new node position allows the immediate calculation of displacements by use of the element shape functions and the nodal quantities for the mesh at t l . While this is a consistent procedure for transferring the solution at t 1 to the mesh at t 2 , it should be understood that the transfer cannot be perfect. That this must be the case can be seen by considering that spatial derivatives of displacements, etc., are not continu-ous across element boundaries. Since the element boundaries change position during the shifting process, points which had continuous derivatives at tl will - 101 - have discontinous derivatives in the mesh at t2 and vice-versa. When the mesh is shifted, the element Gauss points are also shifted; this means the Gauss points represent different material points before and after the shift. In order that the new Gauss point stresses accurately represent the cur-rent stress state, it is necessary to interpolate stresses for the new Gauss point locations using the old Gauss point values and locations. The procedure for doing this is to assume the element stresses are distributed bilinearly with respect to the element local coordinates. Then it becomes possible to use bilinear Lagrangian interpolation polynomials and the 2x2 element Gauss point stresses to interpolate within each element. For all creep crack growth calculations in this study, the crack growth increment sizes were chosen small enough that the new Gauss point stresses for each shifted element were always the result of interpolation within that same element. APPENDIX E Analytical Evaluation of C* for the Strip Problem 1 This appendix briefly outlines the anlaytical evaluation of C* for two infinite strip problems and then summarizes the results in tabular form. The first strip problem is that which is illustrated in Fig. 5.3. We shall refer to this problem as Case A. The second problem, or Case B, is similar to Case A in every respect except the top and bottom edges of the strip are "clamped" rather than "on rollers". These boundary conditions are summarized as follows: Case A: 6 (x,h) = -6 = 6 (E.1) T xy (x,h) = T xy(x,-h) = 0 Case B: (x,h) = -A (x,-h) = (E. 2) 171x(x,h) =x(x,-h) = 0 The crack surfaces are traction-free in both cases. We can select a C*-integral contour which allows C* to be evaluated 1 1 quite easily.1 Consider a contour of rectangular shape which coincides with the top and bottom edges of the strip, extends far enough ahead of the crack-tip so as to be in a steady-state stress field which is unaffected by the presence of the crack-tip, and extends far enough behind the crack-tip so as to be in stress-free material. We now evaluate C*1, as defined by (2.31), through the use of this contour. It can be seen that for both Case A and Case B, the horizontal portions of the contour at y = + h do not contribute to the 1This procedure parallels that used by Rice [50] for the evaluation of J1 in a similar elastic strip problem. - 103 - integral, nor does the portion in the stress-free material. At the veritcal portion of the contour ahead of the crack-tip, the only non-zero term is that involving W. Therefore, it is seen that for both Case A and Case B we have C* = 2W*h (E.3) where W* implies W* existing far ahead of the crack-tip. Using the boundary conditions (E.1) and (E.2) and the assumption of steady-state conditions, it is possible to evaluate the remote steady-state stresses, T... Using (2.43) re- l..) sults in W* and. thus C*. The results of this exercise are summarized in Table E. 1. The corresponding linear elastic strip problem which is obtained by re-placing the displacement rate boundary conditions by the corresponding dis-placement boundary onditions has been treated in a similar manner. These re-sults are also given in Table E.1. Table E .1 Analytical Solutions for the Infinite Strip problem Steady-State Creep 1 n B(1-1 Vrc° n+1 \ IVY/ \8) 1\ 2n1-1/- Linear Elasticity Case A plane stress plane strain 2)n+1 2 1 1 1-v 2 Case B plane stress 2)n+1 2 1 1-v 2 plane strain 1-v (l+v ) ( 1-2v ) * This case does not have a steady-state solution since the boundary conditions require a volumetric strain rate. REFERENCES [1] Srawley, J.E. and Brown, W.F., "Fracture Toughness Testing Methods", ASTM STP 381, 1965, pp. 133-145. [2] Brown, W.F. and Srawley, J.E., "Plane Strain Crack Toughness Testing of High Strength Metallic Materials", ASTM STP 410, 1966. [3] Begley, J.A. and Landes, J.D., "The J-Integral as a Fracture Criterion", ASTM STP 514, 1972, pp. 1-20. [4] Landes, J.D. and Begley, J.A., "Test Results from J-Integral Studies: An Attempt to Establish a I le. Testing Procedure", ASTM STP 560, 1974, pp. 170-186. [5] Westergaad, H.M., "Bearing Pressures and Cracks", J. Appl. Mech., 61, 1939, pp. A49-53. [6] Hutchinson, J.W., "Singular Behavior at the End of a Tensile Crack in a Hardening Material" J. Mech. Phys. Sol., 16, 1968, pp. 13-31. [7] Rice, J.R., Rosengren, G.F., "Plain Strain Deformation Near a Crack Tip in a Power-Law Hardening Material", J. Mech. Phys. Sol., 16, 1968, pp. 1-12. [8] Irwin, G.R., "Fracture Mechanics", Structural Mechanics, (Proceedings of First Naval Symposium), Pergamon Press, 1960, pp. 557-594. [9] Nikbin, K.M., Webster, G.A. and Turner, C.E., "Relevance of Nonlinear Fracture Mechanics to Creep Cracking", ASTM STP 601, 1976, pp. 47-62. [10] Harper, M.P. and Ellison, E.G., "The Use of the C* Parameter in Predic- ting Creep Crack Propagation Rates", J. of Strain Analysis, Vol. 12, Nov. 3, 1977, pp. 167-179. [11] Vitek, V., "A Theory of the Initiaion of Creep Crack Growth", Int. J. of Fracture, Vol. 13, No. 1, 1977, pp. 39-50. [12] Shih, C.F. and Kumar, V., "Estimation Technique for the Prediction of Elastic-Plastic Fracture of Structural Components of Nuclear Systems," Contract RP1237-1, First Seminannual Report for Electric Power Research Institute, General Electric Co., Report, 1979. [13] Zahoor, A. and Paris, P.C., "Ductile Tearing Instability of a Center- Cracked Panel of an Elastic-Plastic Strain Hardening Material", Engng. Materials and Technology, 103, 1981, pp. 46-54. [14] Hutchinson, J.W. and Paris, P.C., "Stability Analysis of J-Controlled Crack Growth", ASTM STP 668, 1979, pp. 37-64. [15] Aihara, S., Machida, S. and Kanazawa, T., "A Study on Unstable Ductile Fracture", Advances in Fracture Research, Vol. 5, Francois, D., (Ed.), Proceedings Fifth Intl. Conf. on Fracture, Cannes, 1981, pp. 2329-2336. [16] Kanninen, M.F., Hahn, G.T., Broek, D., Stonesifer, R.B., Marchall, C.W., Abou-Sayed, I.S. and Zahoor, A., "Development of a Plastic Fracture Methodology", Battelle olumbus Laboratory Report to ERPI, Contract 601-1, March 1981. [17] Kanninen, M.F., Rybicki, E.F., Stonesifer, R.B., Broek, D., Rosenfield, A.R., Marschall, C.W. and Hahn, G.T., "Elastic-Plastic Fracture Mechanics for Two Dimensional Stable Crack Growth and Instability Problems", ASTM STP 668, 1979, pp. 121-150. [18] Abou-Sayed, I.S., Marschall, C.W. and Kanninen, M.F., "An Assessment of the Fracture Toughness Associated with Flat and Slant Crack Growth in A533B Steel", Advances in Fracture Research, Vol. 1, Francois, D. (Ed.), Proceedings Fifth Intl. Conf. on Fracture, Cannes, 1981, pp. 227-234. [19] Abou-Sayed, I.S., Broek, D., Forte, T.P. and Stonesifer, R.B., "Plane Stress Fracture under Biaxial Loading", Advances in Fracture Research, Vol. 4, Francois, D. (Ed.), Proceedings Fifth Intl. Conf. on Fracture, Cannes, 1981, pp. 1707-1714. [20] Fu, L.S., "Creep Crack Growth in Technical Alloys at Elevated Temperature - A Review", Engng. Fracture Mech., Vol. 13, 1980, pp. 307-330. [21] Landes, J.D. and Begley, J.A., "A Fracture Mechanics Approach to Creep Crack Growth", ASTM STP 590, 1976, pp. 128-148. [22] Branco, C.M. and Radon, J.C., "Analysis of Creep Cracking by the J-Integ- ral Concept", Engng. Aspects of Creep, Proc. Conf. at Univ. Sheffield, I. Mech. E., 1980, pp. 43-48. [23] Ohji, K., Ogura, K. and Kubo, S., "The Application of Modified J-Integral to Creep Crack Growth in Austentic Stainless Steel and Cr-Mo-V Steel," Engng. Aspects of Creep. , Proc. Conf. at Univ. Sheffield, I. Mech. E., 1980, pp. 9-16. [24] Koterazawa, R. and Mori, T., "Applicability of Fracture Mechanics Parameters to Crack Propagation Under Creep Condition", J. Engng. Materials and Technology, Vol. 99, Series H, No. 4, 1977, pp. 298-305. [25] Hoff, N.J., "Approximate Analysis of Structures in the Presence of Moderately Large Creep Deformations", Quart. Appl. Math., 12, 1954, pp. 49-55. [26] Goldman, N.L. and Hutchinson, J.W., "Fully Plastic Crack Problems: The Center-Cracked Strip Under Plane Strain", Int. J. Solids Structures, 11, 1975, pp. 575-591. - 107 - [27] Atluri, S.N., "Path-Independent Integrals in Finite Elastiicty and Inelasticity, with Body Forces, Inertia, and Arbitrary Crack-Face Conditions", Report No. GIT-CACM-SNA-81-8, Georgia Institute of Technology, March, 1981, also Engng. Fracture Mech., (in press). [28] Riedel, H. and Wagner, W., "The Growth of Macroscopic Cracks in Creeping Materials", Advances in Fracture Research, Vol. 2, Francois, D. (Ed.), Proceedings Fifth Intl. Conf. on Fracture, Cannes, 1981, pp. 683-390. [29] Hui, C.Y. and Riedel, H., "The Asymptotic Stress and Strain Field Near the Tip of a Growing Crack under Creep Conditions", Brown University Engineering Report MRL E-117. [30] Ohtani, R. and Nakamura, S., "Crack Propagation in Creep (Finite Element Analysis)", Journal of the Society of Materials Science, Japan, Vol. 25, No. 275, 1976. [31] Hinnerichs, T.D., Viscoplastic and Creep Crack Growth Analysis by the Finite Element Method, doctoral dissertation, Air Force Institute of Technology, 1980. [32] Bodner, S.R. and Partom, Y., "Constitutive Equations for Elastic-Viscoplastic Strain Hardening Materials", J. Appl. Mech., 42, 1975, pp. 385-389. [33] Ehlers, R. and Riedel, H., "A Finite Element Analysis of Creep Deformation in a Specimen Containing a Macroscopic Crack", Advances in Fracture Research, Vol. 2, Fracture, Cannes, 1981, pp. 691-698. [34] Knowles, J.K. and Sternberg, E., "On a Class of Conservation Laws in Linearized and Finite Elastostatics", Archive for Rational Mechanics and Analysis, Vol. 44, No. 3, 1972, pp. 187-211. [35] Atluri, S.N., "On Some New General and Complementary Energy Theorems for the Rate Problems of Finite Strain, Classical Elastoplasticity", J. of Structural Mechanics, Vol. 8, No. 1, 1980, pp. 61-92. [36] Riedel, H. and Rice, J.R., "Tensile Cracks in Creeping Solids", Brown University Report E(11-1) 3084A64 to U.S. Dept. of Energy, 1979. [37] Hutchinson, J.W., "Plastic Stress and Strain Fields at a Crack Tip", J. Mech. Phys. Solids, Vol. 16, 1969, pp. 337-347. [38] Perzyna, P., "Fundamental Problems in Viscoplasticity", Advan. Appl. Mech., 9, 1966, pp. 243-377. [39] Bathe, K.J. and Wilson, E.L., Numerical Methods in Finite Element Analysis, Prentice-Hall, 1976. [40] Zienkiewicz, 0.C. and Cormeau, I.C., "Viscoplasticity - Plasticity and Creep in Elastic Solids - A Unified Numerical Solution Apprach", Int. J. for Num. Meth. Engng., 8, 1974, pp. 821-845. [41] Nagtegall, J.C., "arks, D.M. and Rice, J.R., "On Numerically Accurate Finite Element Solutions in the Fully Plastic Range", Comp. Meth. in Appl. Mech. and Engng., 4, 1974, pp. 153-177. [42] Henshell, R.D. and Shaw, K.G., "Crack Tip Finite Elements are Jnneces-sary", Int. J. Num. Meth. Engng., 9, 1975, pp. 495-509. [43] Barsoum, R.S., "On the Use of Isoparametric Finite Elements in Linear Fracture Mechanics", Int. J. Num. Meth. Engng . , 10, 1976, pp. 25-37. [44] Barsoum, R.S., "Triangular Quarter-Point Elements as Elastic and Perfectly-Plastic Crack Tip Elements", Int. J. for Num. Meth. Engng., 11, 1977, pp. 85-98. [45] Atluri, S.N., "Higher Order, Special, and Singular Elements, Chapter 4," State of the Art Surveys in the Finite Element Method, Eds., A.K. Noor and W. Pilkey, ASME special pub., to appear in 1981. [46] Ying, L.A., "A Note on the Singularity and the Strain Energy of Singular Elements", Int. J. for Num. Meth. Engng., :981, (in press). [47] Tracey, D.M. and Cook, T.S., "Analysis of Power Type Singularities Using Finite Elements", Int. J. for Num. Meth. Engng., 11, 1977, pp. 1225-1233. [48] Stern, M., "Families of Consistent Conforming Elements with Singular Derivative Fields", Int. J. for Num. Meth. Engng., 14, 1979, pp. 409-421. [49] Srawley, J.F., "Wide Range Stress Intensity Factor Expressions for ASTM E399 Standard Fracture Toughness Specimens", Intl. J. Fracture, 12, 1976, pp. 475-476. [50] Rice, J.R., "A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks", J. Appl. Mech., 1968, pp. 379-386. [51] Koterazawa, R. and Iwata, Y., "Fracture Mechanics and Fractography of Creep and Fatigue Crack. Propagation at Elevated Temperature", J. of Engng. and Technology, 98, 1976, pp. 296-304. [52] Tata, H., Paris, P.C. and Irwin, G.R., The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, Pennsylvania, 1973. [53] Nishioka, T. and Atluri, S.N., "Numerical Modelling of Dynamic Crack Propagation in Finite Bodies, by Moving Singular Elements, Part I -Theory", J. Appl. Mech., ASME, Vol. 47, No. 3, Sept. 1980, pp. 570-576. [54] Nishioka, T. and Atluri, S.N., "Numerical Modeling of Dynamic Crack Propagation in Finite Bodies, by Moving Singular Elements, Part II -Results", J. Appl. Mech., ASME, Vol. 47, No 3, Sept. 1980, pp. 577-583. March 1981 GEORGIA INSTITUTE OF TECHNOLOGY Atlanta, Georgia 30332 BRIEF BIOGRAPHICAL SKETCH ATLURI, SATYA N. - Regents' Professor of Mechanics Center for the Advancement of Computational Mechanics School of Civil Engineering Formal Education Sc. D. Aeronautics and Astronautics, M.I.T., Cambridge, MA 1969 Academic Experience. Teaching and Professional Experience Georgia Institute of Technology a) Regents' Professor of Mechanics b) Professor of Engineering 1979 - present Science & Mechanics c) Associate Professor of Engineering 1977 - 1979 Science & Mechanics 1974 - 1977 University of Washington Assistant Professor of Aeronautics & Astronautics 1971 - 1974 Boston University Invited guest lecturer, Aerospace Engineering 1970 Massachusetts Institute of Technology 1966 - 1971 Additional Academic Experience. Jan. 1979 - Aug. 1971 --- Lecturer (continuum vibrations) Boston University June 1971 - Aug. 1971 --- Visiting Scientist, M.I.T. June 1971 - Sept. 1971 -- Faculty participant, Cardiovascular Research Programme, Center for Bioengineering, College of Medicine, University of Washington ATLURI, SATYA N. Other Professional Activities Reviewer for various academic journals (such as the AIAA Journal; ASME Journal of Applied Mechanics; ASTMSpecial Technical Publications; Int. Jnl. of Solids & Structures; Int. Jnl. Num.-Meth. in Engg; Journal of Sound & Vibration; Jnl. of Computers & Structures; Int. Jnl. of Fracture and others); and reviewer for research proposals to agencies such as NSF, ARO-NRC, etc. Scientific and Professional Societies American Academy of Mechanics (Nominated to be a Fellow, 1981) American Society of Mechanical Engineers American Institute of Aeronautics and Astronautics American Association for the Advancement of Science Sigma Xi Society for Natural Philosophy Chairman Elect, ASME Committee on Cpmputation in Applied Mechanics Member, Panel of Independent Experts on Structures, Federal Aviation Admin. Honors and Awards Gold Medal for Excellence in Engineering College 1964 Andhra University Roll of Honors of the Indian Institute of Science 1966 Bangalore, India Mentioned in several Who's Who Publications (A) Books (as editor/contributor) (1) N. Perrone and S.N. Atluri, Editors, Nonlinear and Dynamic Frac-ture Mechanics AMD Vol. 35, ASME, N.Y., 1979, 215 p. (Also contrib-utor of article no. 3, pp 37-67). (2) S.N. Atluri, "Higher-order, Special, and Singular Finite Elements" Chapter 4 in Finite Element Methods (Ed. by A.K. Noor & W. Pilkey) Published by the American Society of Mechanical Engineers, 1980, 84 p. (3) S.N. Atluri, "On Rate Principles for Finite Strain Analysis of Elastic and Inelastic Nonlinear Solids", in Recent Researches on Mechanical Behavior of Solids, (Edited by the Committee on Recent Researches) Univ. of Tokyo Press, 1979, pp 79-109. (4) S.N. Atluri, (Editor and Contributor) Part II, Finite Element Methods in Handbook of Finite Elements,McGraw-Hill, N.Y., (In Preparation). (5) S.N. Atluri, O.C. Zienkiewicz, and R.H. Gallagher, Hybrid and Mixed Finite Element Methods, John Wiley & Sons, London, (In Preparation). (B) Papers Published in Journals and Other Archival Literature (1) Atluri, S.N., "A New Assumed Stress Hybrid Finite Element Model for Solid Continuua", AIAA Journal, Vol. 9, No. 8, pp 1647-1649, 1971. (2) Atluri, S.N., "A. Perturbation Analysis of Nonlinear Free Flexural Vibrations_of a Circular Cylindrical Shell", Int. Jnl. of Solids & Structures, Vol. 8, pp 549-571, 1972. (3) Atluri, S.N., and Pian, T.H.H., "Theoretical Formulations of Finite Element Methods in Linear-Elastic Analysis of General Shells", Journal of Structural Mechanics, Vol. 1, No. 1, pp 1-43, 1972. (4) Atluri, S.N., "Further Studies on an Assumed Stress Hybrid Finite Element Model for Solid Continuua", Zentralblatt Fur Mathematik, Vol. 269, p. 521, 1972. (5) Atluri, S.N., "On the Inverse Problem of Natural Vibration of Elastic Shells", Shock and Vibration Digest, Vol. 6, pp 380-383, 1972. (6) Atluri, S.N., and Pian, T.H.H., "Finite Element Analysis of Shells of Revolution by Two Doubly-Curved Quadrilateral Elements", Journal of Structural Mechanics, Vol. 1, No. 3, pp 393-416, 1973. (7) Atluri, S.N., Kobayashi, A.S., and Cheng, J.S., "Mechanics of Brain- Tissue Fragility", 1973 Biomechanics Symposium, AND Vol. 2, ASME, N.Y., pp 103-105, 1973. (8) Atluri, S.N., "On the Hybrid Stress Finite Element Model in the Incremental Analysis of Large Deflection Probelms", Intl. Jnl. of Solids & Structures, Vol. 9, pp 1177•1191, 1973. (9) Atluri, S.N., "Non-Linear Vibrations of a Hinged Beam Including Nonlinear Inertia Effects", Journal of Applied Mechanics, ASME, Vol. 40, No. 1, pp 120-127, 1973. (10) Atluri, S.N., and Gordon, J.T., "Influence of Large Amplitudes and Boundary Conditions on the Supersonic. Flutter of Cylindrical Shells", Proceedings of the 4th Canadian Congress of Applied Mechanics, May 1973, Montreal, Canada, pp 555-559. (11) Atluri, S.N., "Advances in Applied Mechanics", Journal of the Franklin Institute, (C.S. Yih, Editor), Academic Press, May 1973, (a review). (12) Atluri, S.N., "On. the Hybrid Stress Model in Linear Elastic Dynamic Analysis", AIAA Journal, Vol. 11, No. 7, pp 1028-1032, 1972. (13) Atluri, S.N., "Author's Closure: Non-linear Vibrations of a Hinged Beam Including Non-linear Inertia Effects", Journal of Applied Mechanics, 40(3), Series E, pp 221-223, 1973. (14) Atluri, S.N., and Gordon, J.T., "Non-linear Flutter of a Circular Cylindrical Shell", Developments in Theoretical and Applied Mech-anics, Vol. 7, Catholic University of American Press, (Chi and Heller, Editors) pp 285-304, 1973. (15) Atluri, S.N., Kobayashi, A.S., and Nakagaki, M., "Application of an Assumed Displacement Hybrid Finite-Element Procedure to Two-Dimensional Probelms in Fracture Mechanics, AIAA Paper No. 74-390, Proceedings of AIAA/ASME/SAE 15th SDM Conference, Las Vegas, Nevada, pp 220-231, 1974. (16) Deak, A.L., and Atluri, S.N., "The Non-linear Stress Analysis of Laminated Shells by the Hybrid Stress Finite Element Methods", Computational Methods in Non-linear Mechanics, University of Texas at Austin Press, pp 79-89, 1974. (17) Atluri, S.N., "Finite-Element-Perturbation Analysis of Non-linear Dynamic Response of Elastic Continuua", Proceedings of the 1974 International Conference on Finite Element Methods in Engineering, University of New South Wales, Sydney, Australia, pp 581-597, 1974. (18) Atluri, S.N., Cheng, J.S., and Kobayashi, A.S., "Brain Tissue Fragility", A Finite Strain Analysis by a Hybrid Finite Element Model", Journal of Applied Mechanics, Trans. ASME, Vol. 42, Series E, No. 3, June 1975, pp 263-269. (19) Atluri, S.N., Kobayashi, A.S., and Nakagaki, M., "Stress Intensity Factors of Cracked Orthotoropic Plates", in Proceedings of the Conference on the Fundamental Aspects of the Deformation and Fracture of Composite Materials, Battle Seattle Research Center, Seattle, February 1975. (20) Atluri, S.N., "Rotationally Symmetric Bending of Orthotropic Conical Shells: Transverse Shear and Couple-Stress Stress-Couple Effects", Developments in Mechanics, Vol. 8, University of Oklahoma Press, pp 53-71, 1975. (21) Atluri, S.N., Kobayashi, A.S., and Nakagaki, M., "An Assumed Dis-placement Hybrid Finite Element Model for Linear Fracture Mechanics", International Journal of Fracture, Vol. 11, No. 2, pp 257-271, 1975. (22) Atluri, S.N., and Deak, A.L., "Nonlinear Stress Analysis of Pneumatic Structures", Proceedings of the 5th Canadian Conference on Applied Mechanics, University of New Brunswick, Canada, pp 645-647, 1975. (23) Atluri, S.N., and Deak, A.L., "Integral Equation Solution of Three-Dimensional Nonhomogeneous Elasticity Problems", Proceedings of 5th Canadian Congress of Applied Mechanics, University of New Brunswick, Canada, pp 41-43, 1975. (24) Atluri, S.N., "On Hybrid Finite Element Models in Solid Mechanics", in Advances in Computer Methods for Partial Differential Equations, (R. Vishnevetsky, Editor), AICA, Rutgers University, pp 346-356, 1975 (invited article).. (25) Atluri, S.N., Kathiresan, K., and Kobayshi, A.S., Three-Dimen-sional Linear Fracture Mechanics Analysis by a Displacement Hybrid Finite Element Model", Paper No. L-7/3, Proceedings of 3rd International Conference on Structural Reactor Mechanics in Technology, University of London, Vol. 1, 13, pp 1975. (26) Atluri, S.N., Kobayashi, A.S., and Nakgaki, M., "A Finite Program for Fracture Mechanics Analysis of Composite Material", Fracture Mechanics of Composites, ASTM STP 593, American Society for Testing and Materials, pp 86-98, 1975. (27) Atluri, S.N., Kobayashi, A.S., and Nakagaki, M., "Fracture Mechanics Application of a Hybrid Displacement Finite Element Model", AIAA Journal, Vol. 13, No. 6, pp 734-739, 1975. (28) Crawford, J.G., and Atluri, S.N., "Nonlinear-Oscillations of an Initially Stressed Flat Plate", Journal of Sound and Vibration, Vol. 42, No. 4, pp 22-44, 1975'. (29) Atluri, S.N., and Nakagaki, N., "Analysis of Plane Fracture Problems Involving Large-Scale Yielding: A Displacement Hybrid Method", Proceedings of 12th Annual Meeting of the Society of Engineering Science, University of Texas at Austin, pp 66-67, 1975. (30) Atluri, S.N., and Kathiresan, K., "An Assumed Displacement Hybrid Finite Element Model for Three-Dimensional Linear Fracture Mech-anics Analysis", Proceedings of 12th Annual Meeting of the Society of Engineering Science, University of Texas, Austin, pp 77-87, 1975. (31) Atluri, S.N., and Nakgaki, M., "Post-Yield Analysis of a 3-Point Bend Fracture Test Specimen: An Embedded Singularity Finite Ele-ment Incremental Method", Developments-in Theoretical and Applied Mechanics, Vol. 8, Proceedings of the 8th SECTAM, Virginia Polytech-nic and State University, Blacksburg, Va., pp 206-224, 1976. (32) Atluri, S.N., and Kathiresan, K., "J-Integral Estimation Proceedures for Strain Hardening Materials in Ductile Fracture Problems", Proceedings of the 17th AIAA/ASME/SAE Structures, Structural Dynamics and Materials Conference, Valley Forge, Pa., pp 140-152, 1976. (33) Atluri, S.N., and Kathiresan, K., "On a 3-D 'Singularity-Element' for Computation of Combined Mode Stress Intensities, Advances in Engineering Science, Vol. I, NASA CP-2001, pp 267-276, Nov. 1976. (34) Atluri, S.N., Kathiresan, K. , Nakagaki, M., and Kobayshi, A.S., "Inner Surface Cracks in an Internally Pressurized Cylinder Analyzed by a Three-Dimensional Displacement Hybrid Finite Element Method", in Pressure Vessel Technology, (Part II, Materials and Fabrication), ASME, New York, N.Y., pp 527-535, 1977. (35) Atluri, S.N., Nakagaki, M., and Chen, W.H., "Fracture-Initiation in Plane Ductile Fracture Problems: An Elastic-Plastic Finite Element Incremental Analysis", Pressure Vessel Technology, (Part II, Materials and Fabrication), ASME, pp 579-593, 1977. (36) Tong, P., and Atluri, S.N., "On Hybrid Finite Element Techniques in Crack Analysis",'PrOteedingS of International Conference on Fracture Mechanics and Technology, Hong Kong, pp 56-81, 1977, (invited paper). (37) Atluri, S.N., "On the Formulation and Application of Refined Num-erical Methods for Problems with Non-Removable Singularities", Proceedings of International Symposium on Innovative Numerical Analysis in Applied Science, Versailles, France, pp 19-28, 1977. (38) Quinlan, P.M., Fitzgerald, J.E., and Atluri, S.N., "The Edge Function Method", Ibid., loc. cit., pp 1-15, 1977 (invited feature article). (39) Atluri, S.N., and Nakagaki, M., "Stress Analysis of Cracks in Elastoplastic Range", Fracture 1977, Vol. 3, (D.M.R. Talpin, Editor), University of Waterloo Press, pp 45-463, 1977. (40) Atluri, S.N., "Hybrid Crack Elements", Proceedings of Joint Army, Navy and Air Force Conference on Propellants, Edwards Air Force Base, California, March 1977, (extended abstract only), (invited paper). (41) Cheng, J.S., Atluri, S.N., and Kobayshi, A.S., "Assumed Displace-ment Finite Element Model for Nonlinear-Elastic and Elastoplastic Analyses", Proceedings AIAA 18th Structures, Structural Dynamics and Materials Conference, San Diego, California, pp 240-252, 1977. (42) Atluri, S.N., andNakagaki, M., "J-Integral Estimates for Strain HardeningMaterialsm, AIAA Journal, Vol. 15, No. 7, pp 842-851, 1977. (43) Atluri, S.N., Nakagaki, M., Kathiresan, K., and Rhee, H.C., Fracture Analysis of Structures Under Combined Mode Loadings", in Advances in Civil Engineering Through Engineering Mechanics, Proceedings of 2nd ASCE, New York, N.Y., pp 89-93, 1977, (invited paper). (44) Atluri, S.N.,Nakagaki, M., and Cheng, W.H., "Fracture Analysis of Structures Under Large Scale Yielding: A Finite Deformation, Embedded Singularity Elasto-plastic Incremental Finite-Element Solution", in Flaw Growth and Fracture, ASRM STP 631, American Society for Testing and Materials, pp 43-61, 1977. (45) Atluri, S.N., and Kathiresan, K., "Outer and Inner Surface Flaws in Thick-Walled Pressure Vessels", Paper G 5/4, Transaction of 4th International Conference on Structural Mechanics in Reactor Technology, 13 pages, 1977. (46) Rhee, H.C., Atluri, S.N., Moriya, K., and Pian, T.H.H., "Hybrid Finite Elements Procedures for Analyzing Through Flaws in Plates in Bending", Paper M 2/4,. Transactions of 4th 'International Conference on Structural Mechanics in'Reactor. Technology, 14 pages, 1977. (47) Atluri, S.N., and Murakawa, H., "Hybrid Finite Element Models in Nonlinear Solid Mechanics", in Finite Elements in Nonlinear Mechanics,_ Vol. 1, (Ed. by P.G. Bergan, et. al.), Tapir Press, Norway, pp 3-41, 1977, (invited feature article). (48) Atluri, S.N., and Nakagaki, "An Analysis of Stable Crack Growth in Ductile Materials", in Materials and Processes - In Service Performance, National SAMPE Technical Conference Series, Vol. 9, pp 48-57, 1977, (invited paper). (49) Atluri, S.N., and Kathiresan, K., "Surface Flaws in Thick Plates", in Proceedings of 14th Annual Society of Engineering Science Meeting, Lehigh University, Bethlehem, Pa., pp 761-771, 1977. (50) Atluri, S.N., Nakagaki, M., 1

Crack growth in creep brittle materials

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Crack growth in creep brittle materials

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