Transactions on Modelling and Simulation vol 10, 1995 WIT Press, ISSN X

Transcription

1 Parameters controlling the numerical simulation validity of damageable composite toughness testing S. Yotte, C. Currit, E. Lacoste, J.M. Quenisset Laboratoire de Genie Meanique - IUT 'A\ Domaine Universitaire, Talence cedex, France Abstract Assuming that CMC's toughness can be assessed through crack growth resistance curves, a procedure of testing has been proposed. Despite various approximations which are pointed out, the use of the finite element method is shown to be helpful to define the specimen shape and loading conditions. More particularly, an initial procedure based on the CT test is modified to prevent various perturbations related to brinelling effects or compressive damage and to generate steady state crack propagations. 1. introduction Ceramic matrix composites are now well known for their quasi damageable elastic behaviour. This fact renders difficult their toughness characterisation through an intrinsic characteristic. As a matter of fact a damage zone whose size and shape depend on the specimen type and loading conditions develops in front of the macrocrack during toughness tests. As a consequence the stress intensity factor K commonly used for characterising brittle or quasi brittle materials cannot be considered anymore as a representative fracture parameter. An energetical approach seems to be more suitable so that the energy release rate G can be chosen for assessing the ability of the material to dissipate its energy during the propagation of a process zone consisting in mic roc racking and in fibre/matrix sliding and friction. Due to the difficulty of determining intrinsic toughness characteristics, CMC's toughness requires the standardisation of a procedure usable with every type of materials. One objective of the procedure is the development of a state of propagation giving rise to a plateau on the R curve. Providing we propose specimen shapes and conditions of loading which prevent any interaction between the propagating damage zone and the specimen edges, the G values related to R curve plateau could be considered as an acceptable representation of CMC's toughness. The choice of a method among various testing procedures can be guided either by performing experimental tests which is time and material consuming or by simulating the toughness tests with the help of

2 738 Computational Methods and Experimental Measurements numerical approaches. Although the exactitude of such computations of process zone growth is questionable, the approximate results are able to drive a choice of procedure. The aim of the present contribution is to point out the various sources of uncertainties related to the numerical aspects of a procedure determination. First the conditions of obtaining CMC's crack growth resistance curves are investigated prior to the presentation of the parameters controlling the validity of numerical tests. Then, an analysis of the loading conditions in relation with specimen shapes allows the procedure initially based on a CT test (figure 1), to be proposed. Figure 1 : CT test 2. Determination of CMC crack growth resistance curves R curve in term of crack growth release rate G can be obtained in the case of quasi linear elastic CMC's with the following equation [1] : G = (p2/2b)dc/da (1) where P is the applied load, B the specimen width, C the specimen compliance which is the displacement obtained for a unit load, and a the crack length which defines the fracture area in the case of brittle materials. Relation (1) shows that the main difficulty in the calculation of G versus a can be related to the determination of a which is not precisely defined. Crack length definition In the case of CMC materials, crack growth occurs by propagating a damage zone made of numerous microcracks and fibre/matrix debonding rather than an undeniable crack. Various studies have attempted to measure a macrocrack length from microscopic observation but the uncertainties assigned to the crack length values are very significant while the 3C/9a values and as a consequence G are very sensitive to a. Thus a determination of a through a crack growth calibration curve has been preferred, and requires focusing on the methods of calibration. Compliance calibration The calibration of compliances versus crack length can be performed either by loading specimens with various notch lengths in the elastic domain or by computing with a finite element programme specimen compliances, with material characteristics derived from tensile and shear tests. In the latter case, the numerical approach of compliance can be performed by varying either a notch length or an ideal crack length from the initial notch as illustrated in figure 2. On the one hand, the first case which corresponds to the conditions of experimental calibrations, allows a validation of the numerical approach, while on the other hand, the second case satisfies the need of a calibration curve corresponding to an ideal crack. The results reported in figure 2 point out significant deviations between the two calibration curves as already noted elsewhere [2]. Thus after validating the computation with various notch lengths, the use of calibration curves computed with crack lengths has allowed equivalent crack length a to be

3 Computational Methods and Experimental Measurements 739 determined and 3C/3a to be derived providing we are able to measure precisely the load displacement. notch length variation^ crack length Figure 2 : Compliance curves in the case of notch length increase and crack length increase Load displacement measurement In addition to the need of measuring load displacements for equivalent crack length determination, 8 is an important parameter of the energy balance resulting from the use of the equation (1). In order to avoid taking into account in the energy balance, the dissipation induced by brinelling effects occurring in the vicinity of the load application point, load displacements have been measured at a point located slightly under the pin holes both for the experimental and the numerical tests. All the precautions previously noticed can be considered as essential requirements whatever the type of specimen and loading conditions expected to be used to ensure a good interlaboratory reproductibility. 3. Numerical simulation of a process zone development The aim of numerical simulated tests consists in the assessment and representation of the damage zone development while saving experiments and

4 740 Computational Methods and Experimental Measurements more particularly expensive composite consumption. They enable various problems to be pointed out in relation with the specimen shape and the loading conditions. Damage model The simulation is based on a damageable elastic model of the composites. The effect of damage is depicted by a decrease of the elastic constants [3]. Although such an approximate approach of constitutive laws already used by many authors, is rather simple and could be more realistic with more sophisticated models, it has been assumed to be sufficient for the purpose of the study all the more so since the related CMC's exhibit some variability in properties. The programme schematically illustrated in figure 3 is made with the f.e. code MODULEF. [ Mesh X K Materials^ properties j jf (Displacemtmt applied to thesp>ecimen ]^_ C Increment oh ^ J^ "1 displacement! ^ V (Stress in every elements ^^ (Determination of ^ J ^ the new properties J A <Stress deviationv^.or. > First iteration/^ /"Damage in all the"\ 1 elements J yes _^ Figure 3: Flowchart of the test numerical simulation of the D D easy convergence <r difficulty of convergence <7 Figure 4 : Influence of the material behaviour on the numerical convergence

5 Computational Methods and Experimental Measurements 741 The material characteristics of each element vary with the local state of stress, which is calculated through an iterative procedure, allowing the material damage to be increased or decreased during each iteration. Some difficulties of convergence arise when the D-a curvature is small and negative as illustrated in figure 4. For instance a material such as C/SiC composite, whose damage behaviour can be represented by a damage curve exhibiting this feature, induces a longer calculation time. Although the unicity of the resulting process zone development has not been established the stability of the solution for different mesh and loading increments has been considered as a first argument of validation. The deviations of the elastic properties versus stress level were derived from tensile tests performed on unnotched specimens. It is noteworthy that these properties concern loading conditions for which the principal axes correspond to the orthotropic axes of the materials while the finite element programme treats each element in the material orthotropic axes rather than in each element principal axes of loading. However, the analysis of the first results shows a test for which the damage occurs mainly in the vicinity of the ligament that is the specimen symmetry axis where the loading principal axes are the orthotropic axes of the material. Also this result shows that the only case of interference between damage zone and specimen edges are related to a correspondence between the axes. Thus, the deviations between loading and material axes are expected to induce only small errors. Rupture criterion A major difficulty related to numerical simulation is the stress state that causes the occurrence of local fractures. As an accurate fracture criterion is not available, the fracture is assumed to happen when the maximum stress value is reached in one direction of stress. This assumption is not a source of error since fractures develop in the direction perpendicular to the ligament, for which the stresses are the most significant compared to other directions. The difficulties could arise in the case of macrocrack bifurcation for which the validity of the result is questionable and has to be checked. First results In spite of these approximations in the numerical simulation, the correlation between the plateau of the R curves and the stabilisation of the damage zone has been established. It is worthy of note that the damage does not extend anymore toward the two symmetric specimen edges after the macrocrack initiation, but it develops along the ligament toward the specimen backedge (figure 5). i microcrack tmm damage zone Figure 5 : Damage zone extension during a CT test

6 742 Computational Methods and Experimental Measurements The macrocrack initiation can be distinguished from R curves as schematically illustrated in figure 6 : the plateau of R curves corresponds to the macrocrack growth that is the fibre fracture. The transitory part of the R curves before the plateau, corresponds to the development of the damage zone by matrix microcracking. G A fdamage zone development macrocrack appearance crack length Figure 6 : Schematic of an R curve illustrating a damage zone development and a steady state macrocrack growth The previously described numerical simulation of toughness tests is able to generate informations similar to observations derived from experimental tests while saving CMC materials. However experimentation cannot be avoided since the constitutive laws representative of CMC's behaviour are only approximate and do not depict material damage in the vicinity of macrocrack tips. In contrast complementary indications can be expected from elastic analyses of toughness tests. 4 Analysis of the loading mode During the numerical simulation of tests, the complexity of phenomena involved in the development of damage zones renders difficult the analysis of the loading mode in the vicinity of the singularity. With this goal, it is easier and sufficient to visualise the distribution of stress or specimen deformations with the help of a mere elastic approach. The isostatic lines show what can be the influence of the loading conditions on the development of the damage zone independently of the damage material behaviour. isostatic lines related _ damage zone isostatic lines related damage zone Figure 7 : Isostatic lines with the damage zone related to the same test before macrocracking initiation.

7 Computational Methods and Experimental Measurements 743 Comparisons between the indications derived from both the elastic analyses and numerical test simulations show that the damage zone begins to develop in the direction of the isostatic lines. This correlation allows an interaction of damage zones with specimen edges to be prevented thanks to an adjustment of the loading conditions. As an example it has been possible to shift from the configuration illustrated in figure 7a toward that of figure 7b. Pointing out the deformation of the specimen helps to understand this distribution of the isostatic lines and to modify the specimen geometry for a better stress distribution. From the isostatic lines the contribution of bending and tension to the loading can be assessed depending on for instance the notch length as shown in figure 8. Figure 8 : Tensile bending repartition for a test with grips ; this curve is obtained by comparing the load distribution on the nodes along the ligament: Since the tensile/bending contribution tends to prevent damage zone/specimen edge interaction, as schematically illustrated in figure 9, such elastic analyses constitute guides for the definition of a test procedure prior to numerical simulations and experimental investigations. - a- -b- Figure 9 : Influence of the loading conditions on the damage zone shape : a : in the case of a tensile dominated test; b : in the case of a bending dominated test. The contribution of (1) the objectives of a toughness test concerning the steady state growth of a damage zone in thin or thick plates without any specimen edge interaction, and (2) the indications derived from the previous numerical approaches, gave rise to evolutions of the test procedure initially based on a CT configuration. 5. Evolution of the test The first chosen specimen was identical to that defined by the ASTM 399 standard, (specimen 1, figure 10) but the small material thickness caused damages around the pin holes. As a consequence steel grips were added to

8 744 Computational Methods and Experimental Measurements distribute applied loads (specimen 2, figure 10), but lateral buckling and compression appeared at the backedge of specimens, mainly due to the grips stiffness. Preventing these phenomena leads to extend the glued grip all around the unnotched edges (specimen 3, figure 10). Then a study of the influence of the distance between the centre of the pin hole and the position of the notch tip shows the influence of the loading position and allowed the bending contribution to be adjusted in the conditions of loading. It shows that shifting the pin holes ahead of specimens tends to restrict the risks of damage zone/specimen interaction (specimen 4, figure 10). 1C} v/v \ \ N f S S S S N ) \ ' N X \ f*s S S S S 4f S S SV c) specimen 1 specimen 2 specimen 3 specimen 4 Conclusion steel grips k\v CMC material to be tested Figure 10 : Evolution of test conditions The need for comparing CMC's toughness and the difficulty of obtaining an intrinsic characteristic of toughness for damageable materials leads to search for a testing procedure allowing a steady state propagation of macrocracks and related damage zone. Such an approach of toughness would not be necessary if representative constitutive laws of the concerned materials could be identified, which requires a large amount of testing. However the availability of some CMC's approximate constitutive laws have allowed numerical simulation of toughness tests to be performed. Despite many approximations and a lack of precision, which were investigated, such approaches completed by elastic analyses are able to guide the design of a testing procedure. As an example the evolution of a CT test has been described and has pointed out the main criteria to be satisfied for generating steady state crack propagations that is plateaux on the related R curves whose corresponding G values could be used as comparative data. References 1. Bouquet, M., Birbis J. M. & Quenisset J. M. Toughness assessment of ceramic matrix composites, Composites Science and Technology, 1990, 37, Kachanov Y. N., Time of the rupture process under creep conditions, Izv. Akad. Nauk S.S.R. Otd. Tekh Nauk, 1958, 8, Rouillon, M.H. Resistance a la propagation de fissure de materiaux composites ceramiques SiC/C/SiC 2D, PhD thesis, Caen University, France 1993.