UDK 539.42:669.71:519.6 ISSN 1580-2949 Izvirni znanstveni članek MTAEC9, 39(3)83(2005) NUMERICAL DETERMINATION OF J-R CURVE USING VOID MODEL NUMERIČNA DOLOČITEV ODVISNOSTI J-R NA PODLAGI MODELA MIKROVOTLIN Vladislav Laš, Jan Ocenašek, Vlastimil Vacek Department of Mechanics, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 30614 Pilsen, Czech Republic lasŽkme.zcu.cz Prejem rokopisa – received: 2004-09-17; sprejem za objavo - accepted for publication: 2005-05-09 The paper presents the determination of the J-R curve for an aluminium alloy. J-integral value is determined by numerical simulation of the test prescribed in ASTM E 1820-01. The crack growth Aa corresponding to the load increment is determined numerically. The Complete Gurson Model (CGM) based upon void models is adopted as a constitutive relation. This model describes the nucleation, growth and the coalescence of voids and it considers the constraint effect as well. The advantage of this approach is when compared with the standard experimental determination of the J-R curve that the number of experiments is reasonably lower in this case since only the experiments for the determination of the CGM parameters are needed. These experiments require much less effort and are less time demanding. Numerical simulations for the determination of the J-R curve were performed for the CT specimen. The finite element method (FEM) code ABAQUS was used for the computation. A subroutine including the CGM was implemented into the code. Static loading and room temperature were assumed in all simulations. Key words: aluminium, J-R curve, numerical simulation, Complete Gurson Model, CT specimen Predstavljena je določitev odvisnosti J-R za aluminijevo zlitino. J-integral je bil določen z numerično simulacijo preizkusa po ASTM E 1820-01. Napredovanje razpoke Aa pri povečanju obremenitve je bilo določeno numerično. Kot konstitutivna odvisnost je uporabljen Popoln Gursonov model (CGM) na podlagi propagacije mikrovotlin. Ta model opisuje nastanek, rast in koalescenco mikrovotlin ter upošteva tudi vpliv vpetosti. Prednost tega približka v primerjavi s standardno eksperimentalno določitvijo odvisnosti J-R je pomembno manjše število preizkusov, ker so za CGM potrebni le enostavni preizkusi. Numerična simulacija je bila izvršena za CT-preizkušanec. Za izračun je uporabljena metoda končnih elementov (FEM) in program Abaqus, v katerega je vstavljena podrutina s CGM. Pri vseh preizkusih smo uporabljali statično obremenjevanje pri sobni temperaturi. Ključne besede: aluminij, krivulja J-R, numerična simulacija, popoln Gursonov model (CGM), CT-preizkušanec 1 INTRODUCTION The determination of the J-R curves based upon the ASTM E 1820-01 standard 2 is rather time consuming and requires number of experiments. A minimum of 15 specimens must be tested in order to obtain a reliable J-R curve. Effort has been made to obtain the J-R curves by a less complicated experiment combined with numerical simulations. One of these methods is the Single-sample J-integral Test. For this fracture testing the on-line or continuous crack monitoring is required. This is generally performed by the Unloading Compliance Measurement (ASTM E 813-87) or by Electric Potential Drop methods. Another possibility for the determination of the J-R curve is the Load Normalization Technique. This technique does not require the on-line monitoring and it is based on the principle of load separation. The load may be mathematically expressed as a function of the crack length and the plastic deformation. The evaluation procedure uses the load-line displacement diagrams (see e.g. 5). Another method used for the determination of the J-R curve is the Direct Method 1, which employs the load-load point displacement records of a single specimen test in the procedure. Then, the Direct Method Program, including necessary input data, calibration functions and elastic-plastic analysis, is applied. The advantages of this procedure are that it avoids the accuracy problems when measuring the crack growth, and the reduction of the testing time. Some results are presented e.g. in 1. The possibility to determine the J-R curve without testing the standard specimens is discussed in this paper. Such approach requires only simple tensile tests and proper numerical simulations. 2 EXPERIMENTAL DETERMINATION OF J-R CURVE The J-R curve can be determined experimentally according to the ASTM standard 2. This procedure requires large number of experiments fulfilling the assumptions for the specimens used and for the measurement procedures. Three types of specimens are used: single edge bend specimen, compact specimen and disc-shaped compact specimen with fatigue pre-crack. The J-R curve determination procedure consists of loading the specimen to a given level and then determining the J-integral value and the crack length MATERIALI IN TEHNOLOGIJE 39 (2005) 3 83 V. LAŠ ET AL.: NUMERICAL DETERMINATION OF J-R CURVE USING VOID MODEL increment Aa. Thus, one point of the J-R curve is obtained. The minimum number of points to be determined is 15, whereas each point must be obtained from measurement on a new specimen subjected to higher load level, thus exhibiting longer crack length. In order to measure the crack length, each specimen must be broken using brittle fracture after cooling. Since the numerical determination of the J-R curve in this work is performed by the simulation of experiment, the basic relations for the J-integral determination and the procedure of the relevant crack length increment Aax determination necessary for the experiment evaluation are presented below. CT specimen is used. For the J-integral determination so-called load-line displacement curve (P versus v) is used. A typical curve is shown in Figure 1a, where Apl represents the area corresponding to plastic work. The J-integral is then determined as: 0.01B J K2(1-v2) E + J (1) where the first part of equation (1) represents the elastic value of J-integral, K is the stress intensity factor, v is Poisson’s ratio, E is Young’s modulus, and Jpl represents the J-integral plastic component. The stress intensity factor is determined for the relevant force Pi as: Pi (ai\ (BBNW) 1/2 fŠW) Ki = (2) where f(ai/W) is the compliance function, ai is the crack length, W is the specimen width, B is the specimen thickness and BN is the specimen thickness measured at side grooves. The J-integral plastic component is determined from the loading-force plastic work as: b ( i -1)J BN where *7(i -1) = 2.0 +0522b (i-1)/W y(i -1) = 1.0 +0.76b ( i -1 )/W 1 - y(i) (3) (4) The crack length increment is measured optically. After loading to the given level the specimen is unloaded and the free crack surfaces are heat tinted. Then the Č vpl vpii) vii) Figure 1: a) Load – line displacement curve (P versus v), b) Plastic work ?Apl(i) increment Slika 1: a) Odvisnost obremenitev-premik obremenitvene črte (P v odvisnosti od v); b) Poveček plastičnega dela ?Apl(i) A t A V _i_ B \0.01B Figure 2: Fracture surfaces (A - brittle breaking surface, B - crack increment, C - pre-cracked surface, D - notch surface) and 9 points for the crack length increment measurement Slika 2: Površine preloma (A - površina krhkega preloma, B -poveček propagacije razpoke, C - začetna površina razpoke, D -površina zareze) in 9 točk meritev povečka dolžine razpoke specimen is cooled and brittle broken. Figure 2 shows the surfaces of the CT specimen after brittle fracture and 9 locations for the crack increment measurement. The crack length increment value is determined as: Aa 1f Aa +Aa + ČAa i=2 (5) where Aai = ai - a0. The corresponding values of J and Aa are recorded into the diagram together with the construction lines, offset line and two exclusion lines, which define points from which the J-R curve will be determined (see Figure 3). The way of drawing the particular lines is given by the standard 2. The J-R curve is then the power law regression line: lnJ = lnC +C ln k ) (6) determined from points between two exclusion lines, where k = 1 mm (or k = 0.0394 in). Constants C1 and C2 are determined with the regression analysis. Value JQ is given by the J-R curve and the offset line crossing point, ?amin and ?alimit are given by the J-R curve and the exclusion lines crossing points. The values ?amin and ?alimit and the exclusion lines then determine the feasible domain of the J values. Figure 3: Determination of the J-R curve according to ASTM standard Slika 3: Določitev krivulje J-R po ASTM standardu b 84 MATERIALI IN TEHNOLOGIJE 39 (2005) 3 V. LAŠ ET AL.: NUMERICAL DETERMINATION OF J-R CURVE USING VOID MODEL 3 COMPUTATIONAL MODEL Since the experimental determination of the J-R curve is not simple, other possibilities for the determination of this curve are sought. One of these possibilities is a numerical simulation of the test prescribed in ASTM standard 2. A proper model describing the real behaviour of the material during damage (the crack propagation) is necessary in order to obtain a reliable J-R curve. Since the stress-strain field near the crack tip cannot be described by common constitutive relations the material model used must include e.g. the effect of triaxiality, the nucleation, growth and coalescence of voids existing in the zone with high stress concentration. The above requirements are fulfilled by certain so-called void models, where the dilatation plasticity is used. This means that the plasticity condition is also function of the mean normal stress and that the volume change takes place, which is not considered in classical plasticity models. One of the real plasticity models is the Complete Gurson Model (CGM), which eliminates the disadvantages of the Gurson Model (GM) and also of the modified Gurson Model (GTN) implemented by Tvergaard and Needleman 13. The CGM yield function has the same form as that of GTN model: Č(Če,Čk,ČM,f*) - = Č- + 2q1f*cosh 0\ q2akk\ 2o 1-(q1f*) = 0 (7) M ) where ae is the macroscopic von Misses equivalent stress, aM is the actual yield stress of the matrix material, okk is the trace of the macroscopic Cauchy stress tensor,f* is the modified void volume fraction, q1, q2 are constants introduced by Tvergaard. This model involves phases of nucleation, growth and coalescence of voids 13. Contrary to the GTN model, for which 6 parameters must be determined experimentally, only 4 parameters (f0 initial void volume fraction, ŁN mean nucleation equivalent plastic strain, s standard deviation andfN volume fraction of void nucleating particles) are to be determined. It is particularly important that it is not necessary to determine experimentally the critical void volume fracture f C, which was considered to be a material constant in GTN and, as a matter of fact, it depends also on the stress state. Zhang eliminated this disadvantage 15 by the introduction of the Thomason plastic limit load model into GTN and by this way the CGM was created. Parameters f0, eN, s and fN are determined by numerical fitting to the experimental data, where for the notched bar subjected to tension the relation load P versus notch diameter reduction Ad is recorded. Other CGM parameters are assumed to be q1 = 1.5,q2= 1. CGM enables the numerical simulation of the crack growth and the determination of the crack length Figure 4: CT specimen dimensions Slika 4: Dimenzije CT-preizkušanca increment Aa as well as the simulation of the load-line displacement curve P versus v (Figure 1), from which the relevant value of the J-integral can be determined using relations in equation (1) to equation (4). 4 J-R CURVE NUMERICAL SIMULATION The problem was solved using the finite element method programme ABAQUS, into which the CGM was implemented as a subroutine 15. Static loading and room temperature were assumed in the analysis. The tested material was aluminium alloy with material constants: Young’s modulus E = 6.9122·104 MPa, Poisson’s ratio v = 0.315, yield stress ay = 280 MPa. The true stress-strain diagram was available, too. For the CGM the following parameter values were used: q1 = 1.5, q2 = 1,f 0 = 0.02, 6N = 0.1, s = 0.1,fn = 0.01. As stated above, these parameters were obtained by the numerical simulation of a tensile test of a notched bar of circular cross-section. Numerical fitting to the experimental data of the loading force P versus diameter reduction Ad was carried out. Figure 5: Computational model mesh Slika 5: Model izračuna MATERIALI IN TEHNOLOGIJE 39 (2005) 3 85 V. LAŠ ET AL.: NUMERICAL DETERMINATION OF J-R CURVE USING VOID MODEL 0.2