The concept of Representative Volume for elastic, hardening and softening materials

Transcription

1 The concept of Representative Volume for elastic, hardening and softening materials Inna M. Gitman Harm Askes Lambertus J. Sluys Oriol Lloberas Valls Abstract The concept of the Representative Volume Element (RVE) is analysed in the present paper. For elastic materials, one can determine the size of the RVE. However, for other applications, such as the case of softening materials, the RVE size determination problem can become extremely difficult or, sometimes, even impossible to solve. In the present work the RVE has been investigated for different material behaviour, namely elastic, hardening and softening. Results were based on a statistical analysis of numerical experiments, where tests have been performed on a random heterogeneous material. Introduction Representative Volume Element (RVE) is a statistical representation of typical material properties. RVE is widely used in nowadays mechanics: Aidun et al. [], Ashihmin and Povyshev [2], Behrens et al. [3], Fraldi and Guarracino [7], van Mier [6] Stroeven et al. [4], etc. were using the concept of RVE for theoretical, numerical and experimental purposes. Several attempts have been made in literature to develop a procedure to determine the representative size. Drugan and Willis [6] proposed quantitative estimates of minimum RVE size for elastic composites. Also Borbely et al. [4], Bulsara et al. [5], Ashihmin and Povyshev [2] suggest a way to define the size of the RVE. An objective method to determine the size of the RVE was proposed in Gitman et al. [8]. An overview of this method is presented in Section 2. In section 3 questions of an RVE existence and possibility to find its size are studied on the base of statistical analysis for different material behaviour: liner elastic, hardening and softening. 2 RVE size determination In order to start developing the procedure to find a representative size, an RVE should be properly defined. First some definitions of the RVE, used by scientists for different purposes follow. 2. Definitions An RVE is the minimal material volume, which contains enough statistically mechanisms of deformation processes. The increasing of this volume should not lead to changes of evolution equations for field-values, describing these mechanisms (Trusov and Keller [5]). The RVE must be chosen sufficiently large compared to the microstructural size for the approach to be valid. The RVE is the smallest material volume element of the composite for which the usual spatially constant overall modulus macroscopic constitutive representation is a sufficiently accurate model to represent mean constitutive response (Drugan and Willis [6]). The RVE is a model of the material to be used to determine the corresponding effective properties for the homogenised macroscopic model. The RVE should be large enough to contain sufficient information about the microstructure in order to be representative, however it should be much smaller than the macroscopic body. (This is known as the Micro-Meso-Macro principle (Hashin [9])). The RVE is defined as the minimum volume of laboratory scale specimen, such that the results obtained from this specimen can still be regarded as representative for a continuum (van Mier [6]). The RVE is very clearly defined in two situations only: i) unit cell in a periodic microstructure, and ii) volume containing a very large (mathematically infinite) set of microscale elements (e.g. grains), possessing statistically homogeneous and ergodic properties (Ostoja-Starzewski []). Brining together all these definitions, one can define RVE as a representation of the material to be used to determine the corresponding effective properties for the homogenised macroscopic model with a size which is small enough compared to the macroscopic body and large enough compared to the microstructural size. An RVE should contain sufficient information about the microstructure and be a good representation of a continuum. 8 Proceedings of XXXII International Summer School Conference Advanced Problems in Mechanics

2 2.2 Procedure description. Application for linear-elastic case Several methods are available in the literature in order to determine the RVE size. Bulsara et al. [5] in their work used a simulation scheme which generated statistically similar realizations of the actual microstructure of a ceramic-matrix composite. This was done on the basis of a radial distribution function which was obtained by a stereological method and image analysis. They conducted a systematic investigation of the RVE size with respect to the transverse damage initiation for one fiber volume fraction. Ashihmin and Povyshev [2] determined the statistical properties of stress using an imitation model. The model is based on finite-element simulations. They obtained the statistical criterion for metals representative volume determination. U Figure : tension test Here we propose a method to determine the size of the RVE (Gitman et al. [8]). The idea of this method is as follows: for each value of aggregate density distribution, a series of different sample size are made, and for each sample size different aggregate locations (with the given value of aggregate density distribution) are considered. Tension tests (figure ) are performed for all samples. Then a statistical analysis, which is based on the Chi-square criterion (equation ), is used to determine the size of the RVE. n χ 2 (σ i <σ>) 2 = () <σ> i= where σ i is the normalized average value of the stress in the current unit cell; <σ>is average of σ i; n is the number of realisations for the current size..5 TENSION TEST add = 3 % CHI - SQUARE VALUE add = 45 % add = 6 % table (error 5 %) SIZE OF MESOSTRUCTURAL CELL Figure 2: a) Chi-square values b) RVE size versus aggregate densities Results of the above analysis are presented in figure (2). In figure (2-a) Chi-square values for different aggregate densities and different sizes are presented together with the table value, which was found according to the prescribed accuracy (95 %) and the number of numerical tests performed (five realisations for each aggregate density and each size). Figure (2-b), derived from figure (2-a), shows the dependence of the RVE size on the aggregate density. It should be noted, that the range of aggregates sizes was the same for the complete series. 3 Statistical analysis: linear elasticity, hardening and softening Although, results mentioned in the previous section were made for the case of a linear elastic material, the procedure is unique and can be used for any material type. APM 24. June 24 July, 24, Saint Petersburg (Repino), Russia 8

3 Constitutive law analysed. In this section an elasticity based gradient damage model (Lemaitre [], Peerlings [2], Simone [3]) is σ =( ω)dε (2) where σ and ε are stresses and strains, respectively, D is the matrix of elastic stiffness and ω is historically dependent strain based softening damage evolution law. Numerical tests Tension tests have been performed for the same series of samples as in the previous section. Below (figure 3) results (stress strain dependence) are presented, corresponding to the different sizes of samples with different aggregate densities. AGGREGATE DENSITY 3 % AGGREGATE DENSITY 45 % AGGREGATE DENSITY 6 %.2 size mm size 5 mm size 2 mm.2 size mm size 5 mm size 2 mm size mm size 5 mm size 2 mm Figure 3: Sets of sample sizes for aggregate densities 3%, 45% and 6% In figure (4) aggregate density 3% is further analysed. Four pictures, corresponding to four different sizes are presented, each of them showing five different realisations. It should be mentioned, that the same analysis has been performed for aggregate densities 45% and 6%, although they are not shown here. SIZE MM AGGREGATE DENSITY 3 % SIZE 5 MM AGGREGATE DENSITY 3 % realisation realisation SIZE 2 MM AGGREGATE DENSITY 3 % SIZE 25 MM AGGREGATE DENSITY 3 % realisation realisation Figure 4: Different sizes:, 5, 2 and 25 mm Statistical analysis A statistical analysis, based on the mathematical expectation and standard deviation values has been performed on each set of results. All curves were analysed in several points, corresponding to elastic, hardening and softening regions with stiffness (slope) being a parameter of interest. Although the conclusion could be made from figure (4), that with increasing the size the difference in the slope values of different realisations is decreasing, figure (5) offers a better understanding of the situation. The three regimes presented in figure (5) 82 Proceedings of XXXII International Summer School Conference Advanced Problems in Mechanics

4 EXPECTATION AND STD. DEVIATION EXPECTATION AND STD. DEVIATION EXPECTATION AND STD. DEVIATION Figure 5: Expectation and standard deviation values for stiffness a) linear elasticity b) hardening and c) softening are linear-elastic (figure 5-a), hardening (figure 5-b) and softening (figure 5-c). All curves (figures 3, 4) are analysed by means of the mathematical expectation and standard deviation of the stiffnesses (value of slopes) in given points (corresponding to different regimes) with respect to size. In the linear elastic case (figure 5-a), the value of mathematical expectation (i.e. average slope ) is practically constant with increasing the size, the standard deviation (i.e. shifting of the slope from its average) approaches to zero with respect to size. Material in hardening (figure 5-b) shows the same trend: relatively constant mathematical expectation and approaching to zero of the standard deviation as size is increased. On the contrary, when in the softening regime (figure 5-c), the standard deviation behaves qualitatively similar to linear-elasticity and hardening (convergence to zero with respect to size), but the mathematical expectation steadily increases (it should be noted, that here all values are considered as absolute). In other words, with increasing the size the material behaves differently (here, more brittle). This statistical analysis allows to make a conclusion about an RVE existence. In linear elastic and hardening regimes, when mathematical expectations shows stable constant behaviour with respect to size and standard deviations converge while size is increased. Therefore representative volumes can be found. However in softening, when the response of the material qualitatively changes with increasing size (which is shown with the help of mathematical expectation) there is no any representative size, i.e. RVE in softening cannot be found. Figure 6: RVE size for stress and stiffness Also, a comparison has been made for RVE size dependence on the aggregate density in the elastic case for different parameters of interests, namely stress and stiffness. This comparison is presented in figure (6). In case of linear elasticity the RVE sizes for different parameters are practically the same. 4 Conclusions The issue of the representative volume is analysed for different heterogeneous materials behaviour, namely linear elasticity, hardening and softening. Following the procedure, based on the statistical analysis of numerical experiments, it has been concluded that the representative volume can be found with relatively high accuracy in cases of linear elasticity and hardening. However, in case of softening a representative volume can not be found. It has also been shown, that in case of linear elasticity the difference in RVE sizes is relatively small with respect to the parameter of interest. Acknowledgements This research is financially supported by Delft Cluster. APM 24. June 24 July, 24, Saint Petersburg (Repino), Russia 83

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