Multi-scale Modeling of the Mechanical Response of Disordered Materials with Molecular Dynamics Simulations

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1 Multi-scale Modeling of the Mechanical Response of Disordered Materials with Molecular Dynamics Simulations A. TANGUY a, T. ALBARET a, C. FUSCO a, B. MANTISI b, M. TSAMADOS c a. Institut Lumière Matière (ILM), université Lyon 1, Domaine Scientifique de La Doua, Villeurbanne cedex, France b. Institut Jean Le Rond D'Alembert, 4 place Jussieu, Paris cedex 05, France c. Center for Polar Observation and Modeling, Pearson Building, University College London, Gower Street, London WC1E 6BT, UK Résumé : L'étude par simulations de Dynamique Moléculaire de la réponse mécanique de matériaux désordonnés permet d'avoir accès à la réponse quasi-statique du matériau, de l'échelle atomique jusqu'au dixième de micron. L'influence du taux de déformation peut aussi être étudié, mais soit sur de très faibles intervalles de déformation, soit pour des taux très élevés. Dans le régime quasi-statique, cette étude nous a permis d'étudier le comportement mécanique de matériaux désordonnés modèles (du verre modèle de Lennard- Jones au verre de silice et au silicium amorphe) à différentes échelles: l'analyse des modules d'élasticité locaux permet de mettre en évidence différentes tailles caractéristiques associées à différentes réponses élastiques locales, et aussi d'identifier les zones susceptibles de donner lieu à un endommagement d'origine plastique. L'analyse de la dynamique locale permet de donner une interprétation microscopique aux anomalies de modules d'élasticité mesurés à grande échelle, et aussi de comprendre l'origine des comportements rhéologiques anormaux (Non Newtoniens). Enfin, les simulations à l'échelle submicrométrique permettent de construire les courbes de charge de ces matériaux atypiques, nécessaires pour les calculs à plus grande échelle (par éléments finis). Abstract : The numerical study of the mechanical response of disordered materials with Molecular Dynamics simulations gives easy access to the quasi-static response of materials samples whose size ranges from the atomic scale to tens of microns. The role of shear rates can also be studied, but only for very small strain steps, or very high shear rates (~10 8 s -1 up to few 100% strain). We have studied the mechanical behavior of model disordered materials, at different length-scales and in the quasi-static regime. The materials studied are Model Lennard-Jones glasses, silica glasses, or amorphous silicon samples. The analysis of the local Elastic Moduli shows different characteristic lengths for different elastic behaviors. It also allows determining the locations giving rise to a plastic rearrangement. The local dynamics analysis allows understanding of the large scale anomalies of elastic moduli, and also of the anomalous (non Newtonian) rheological behavior. Finally, sub-micrometric numerical simulations allow constructing the loading curves of these very specific materials. These loading curves can then be used as input for large scales finite element calculations. Mots clefs: Elasticité, Plasticité, Rhéologie, Verres, Amorphes 1 Introduction The small scale mechanical response of disordered materials is difficult to measure experimentally, due to the lack of long range order [1]. However, numerical simulations are an alternative way to get insight into the atomistic displacements [2]. The first limitation of the classical numerical simulations is the small time scale (τ= (m.a 2 )/ε at ~10-12 s, where m is the atomic mass, a is the average interatomic distance, and ε at is the typical energy barrier). For N=10 6 1

2 particles (size L 10nm), the number of operation is about 10 8 at each time step δt~0.01τ=10 14 s. The total simulated time in a typical molecular dynamics (MD) run will thus be of the order of 10 6 x10-14 s=10 ns. It will correspond to quenching rates of the order of 1000 K/10 ns = K/s, or to shear rates of the order of 10 8 s -1. A way to circumvent this limitation is to deal with quasi-static simulations by imposing the strain step and not the time step. With a strain step δγ=10-6 corresponding to an elementary elastic deformation, 10 6 quasi-static strain steps will allow to reach 100% strain. The simulation procedure in this case, will be an energy relaxation at each strain step to reach a local mechanical equilibrium. This corresponds to an athermal quasi-static simulation. The second limitation of classical simulations [3] for a realistic description of the mechanical behavior of a specific material is the choice of the empirical potential [4]. For simplicity this potential contains a small number of parameters, thus fitted on a small number of physical properties [5]. It can be inappropriate for the combined description of the elastic, of the plastic, and of the rheological response of a given system. Usually, the elastic description is well described, but the plastic response is more approximate since it involves small scale dissipative processes of various origins [6]. The exact description of the vibration modes at intermediate frequencies is also a difficult task. It is thus important to check the mechanical response also for large deformations. Finally, the numerical simulations will allow a generic description of the mechanical response at the submicrometer scale, with a local measurement of atomic displacements and stresses [3]. From the displacements, the local strains can be computed with different methods [7], for example with the help of a coarse-grained derivable expression of the displacements [8]. This will allow to compute the local Elastic Moduli [8, 9] and to compare the mechanical responses measured at different length-scales. In this paper, we will give some examples of numerical measurements obtained in three different model materials: a pure silica glass, a sample of amorphous silicon with different angular interactions between atoms, and a model two-dimensional Lennard-Jones glass. 2 Local Elasticity From the joined measurement of atomic displacements and forces, we can measure the local strains and stresses centered on each atom, or on grid points [8]. Two interesting results emerge for disordered materials: 2.1 Heterogeneous strain (non-affine displacement field). 3 4 FIG. 1 Total displacement (left) and non-affine displacement (right), in a 2D Lennard-Jones glass (N= particles, L=10 nm) submitted to a quasi-static uniaxial deformation with δγ=<δε xx >=10-4. During an elementary strain step δγ, three different kinds of displacement fields can appear: either a heterogeneous displacement field with vortices distributed in the system and superimposed to the usual continuous response, either a localized Shear Transformation Zone when a dissipative plastic rearrangement of low intensity occurs, either an elementary shear band for large dissipative events [10]. In case of a reversible and elastic response, only the first kind of displacements occur. The disorder then contributes by v adding a heterogeneous but not localized displacement u rn. a.( r ) to a homogeneous deformation imposed for example through the deformation of the simulation box. 2

3 v u rn a..( r) is called the non-affine displacement field by opposition to the affine field ε. r obtained from the continuous homogeneous strain ε, where. is a spatial average over the positions r. The non-affine r v r r r displacement can then be defined as u ( r) = u ( ).. The relative intensity r r u n / u n. a. tot ε. a. tot of this heterogeneous displacement depends on the details of the interatomic interactions [6]. The heterogeneous strain contributes to decrease the apparent elastic moduli measured at the scale of the whole system, in agreement with the well known contribution of heterogeneous strain to the Reuss bound of Elastic Moduli [11]. 4.1 Elasticity Map. Different definitions can be used for the stress and the strain components at the atomic level [12]. We used the coarse-grained definition proposed by I. Goldhirsch et al. [8], that preserves the mass conservation equation, and the Newton equation for atomistic motion. This definition allows a coherent measurement of stresses and strains at different length scales. From the measurement of local stresses and strains (3 components in 2D and 6 components in 3D), for respectively 3 different imposed deformations in 2D and 4 deformations in 3D, we can get the elastic moduli (6 in 2D and 21 in 3D) by solving a system of linear equations. The result is shown in Fig. 2, for 2D Lennard-Jones glasses, at a length-scale w=2 Ả. It shows that the system is heterogeneous and anisotropic at small scale. Interestingly, we have shown that the spatial distribution of the elastic moduli in disordered materials evolves during the deformation of the system. Especially before a plastic instability, the lowest modulus decreases strongly at the place where the plastic deformation will take place, allowing for the prediction of plastic damage. Moreover, before the occurrence of a plastic shear band, an alignment of moduli is shown, corresponding to a percolation of soft zones inside the system [13]. FIG. 2 Elastic Moduli C 1 (left), C 2 (middle) and C 3 (right) obtained as the eigenvalues of the 3x3 matrix of the 6 elastic moduli, in a 2D Lennard-Jones glass (N= particles, L=48,3 nm). C 1 =2µ, C 2 =2µ, C 3 =2(λ a +µ) in isotropic materials, where µ is the shear modulus, and λ a is the Lamé coefficient. Here the material is heterogeneous and anisotropic at small scales [13] 5 Plastic and Micro-plastic mechanical behavior The measurement of the quasi-static mechanical response allows also getting access to a detailed description of the plastic rearrangements, i.e. to identify the irreversible motion of the particles. A detailed description of the dissipative processes would need of course a more accurate description of the atomic bonding. However, the classical simulations allow already the identification of the residual plastic strain through the dissipation of the total energy. We show in this part two examples of results obtained in this context: the case of quasistatic compression of a silica glass, and the measurement of the yield stress for amorphous silicon submitted to simple shear. 3

4 5.1 Microplastic behavior of a silica glass FIG. 3 Left : Pressure as a function of the relative volume variation V/V in a model silica glass. Right: Shear Transformation Zone appearing during a cycle aperture (micro-plasticity). In this example, a sample of pure silica glass described by BKS interactions [5] is obtained after numerical melting and quenching of a crystobalite sample. The sample is then relaxed to zero pressure, and submitted to a volume controlled compression (red curve in Fig.3). At different pressures Pmax, the volume is increased again (the pressure decreases) up to a state with zero pressure. The residual plastic strain can then be measured from the difference between the initial volume at zero pressure, and the volume reached after the go and reverse deformation. The value of Pmax giving rise to a significant residual strain can be compared to experimental measurements [14]. But the simulations give also additional informations: it is possible for example to have access to the small pressure variations occurring during the deformation at imposed volume, and to the corresponding atomistic motion. It is shown that these pressure drops are related to irreversible atomistic motion preserving the total volume of the system (Fig. 3-b), but giving rise to energy dissipation. These dissipative rearrangements are responsible not only for cycle s aperture, but also for the apparent anomalous pressure dependence of the tangent moduli [15]. This effect was explained long time ago by a double well potential model [16]. 5.2 Size of Platic Rearrangements FIG. 4 Left: Shear Transformation Zone in a-si (N= atoms). Middle: Amplitude of the corresponding plastic energy. Right: Table of the maximum width W of the plastic rearrangement, the yield stress σ Y, and the parameter b appearing in the Peierls stress, for different prefactors λ of the 3-body interactions. The atomistic motion during an irreversible rearrangement can be analyzed in more details. This was done for example in an amorphous silicon-like sample submitted to simple shear. In this case, two kinds of dissipative rearrangements occur [6]: either local shear transformation zones involving a tens of atoms (Fig.4), either elementary shear bands consisting in an alignment of shear zones [17]. The width of these transformations can be measured by looking at the spatial variation of the amplitude of the non-affine field, or (as shown in Fig. 4b), by looking at the spatial variation of the dissipated energy. The later shows an exponential decay with a characteristic length W, identified as the width of the plastic rearrangement. Interestingly, the relation between the width W and the tensile stress σ Y (preceding the formation of an elementary shear band) follows a Peierls formula (1) suggesting an analogy between the formation of a shear band, and the depinning of a dislocation: 4

5 σ Y 2C 44 2πW / b e (1) (1 ν ) 6 Rheological behavior The preceding results were obtained in quasi-static simulations. It is also possible to study the response to a mechanical deformation applied at finite rate. In this case, numerical limitations impose to start at a rate of 10 8 s -1. This rate is not very far from the athermal quasi-static behavior at small temperature. Indeed, the athermal regime may be defined as the regime where the typical relative displacement between particles due to the external strain aγ t (where a is the interatomic distance) is larger than the typical vibration of an atom due to its local thermal activation, (k B T/K) with K = mω 2 D and t = 2π/ω D. This results in a condition on T: 2 2 4π ma 2 T < & γ that is T< 10-6 K for γ& =10 8 s -1, showing that the temperature has less effect on the k B dynamics for high shear rates, but can be determinant for low shear rates. In this part, we will comment two results obtained in this context by Molecular Dynamics simulations. The first result is a measurement of the yield stress for different pressures imposed, and its atomistic counterpart. The second is a measurement of the shear rate dependence of the flow stress, with comment on its possible atomistic origin. 6.1 Elastic limit. Loading curve: the example of a silica glass. The measurement of the stress/strain relationship for shear deformation at constant pressure allows the construction the loading curve of the system (Fig. 6). The shear stress separating the apparent elastic regime to the apparent plastic regime is obtained by looking for example to the departure from the linear response in the stress/strain curve, when a residual strain becomes measurable at the scale of the total sample. For a pure silica glass, it is shown that the yield stress tends to decrease linearly with pressure, in opposition to the behavior of granular materials. This behavior can be related to the collective motion of particles inside the sample [18]. FIG. 6 Loading curve for a model silica glass (N= atoms), obtained by applying a simple shear at constant pressure and zero temperature (black). The colored plots correspond to samples densified previously at different pressures Pmax as indicated in the legend. 6.2 Flow curve and relaxation times. The stress/strain relationship of disordered materials submitted to a simple shear is strongly dependent on the shear rate (Fig.3). In the plastic flow regime, the dependence of the flow stress with the shear rate is nonlinear, following a Herschel-Buckley type of law σ σ & β F = 0 + A. γ [19]. The exponent β of this law can be compared to the exponent α relating the relaxation time τ α of particle motion to the shear rate [20]. This exponent can also be compared to that obtained by the number of simultaneous plastic rearrangements, number that increases as a function of the shear rate. 5

6 FIG. 3 (Left) Stress/strain behavior of amorphous Si samples, with Stillinger-Weber empirical interaction, and submitted to simple shear with different shear ratesγ&. (Right) Average flow stress as a function of the shear rate. 7 Conclusion As illustrated in this short article, numerical simulations can provide useful data (atomistic displacements, local elastic moduli, dissipative energy and their spatial distribution) at length-scales that are actually unreachable in the experiments. However the parameters of the simulations (as those of the empirical interatomic potentials used) must be validated by a solid comparison with accessible experimental results. The simulations then provide complementary tools to analyze the physical origins of the mechanical response. References [1] C.A. Schuh, TC. Hufnagel and U. Ramamurty, Acta Materialia 55, 4067 (2007) [2] D. Rodney, A. Tanguy and D. Vandembroucq, Model. Simul. Mater. Sci. Eng., 19, (2011) [3] D. Frenkel and B. Smit, Understanding Molecular Simulation, Academic Press (2002) [4] J. Godet, L. Pizzagalli, S. Brochard and P. Beauchamp J. Phys.: Condens. Mater., 15, 6943 (2003) [5] A. Carré, J. Horbach, S. Ispas and W. Kob Eur. Phys. Lett (2008) [6] C. Fusco, T. Albaret and A. Tanguy, Phys. Rev. E, 82, (2010) [7] B. Dollet, M. Aubouy and F. Graner, Phys. Rev. Lett. 95, (2005) [8] I. Goldhirsch and C. Goldenberg, Eur. Phys. J. E 9, (2002) [9] H. Mizuno, S. Mossa and J.-L. Barrat, Condmat arxiv: (2013) [10] A. Tanguy, F. Leonforte and J.-L. Barrat Eur. Phys. J. E (2006) [11] S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer- Verlag (2001) [12] S. Alexander, Physics Report, 296, (1998) [13] M. Tsamados, A. Tanguy, C. Goldenberg and J.-L. Barrat, Phys. Rev. E, 80, (2009) [14] D. Vandembroucq, T. Deschamps, C. Coussa, A. Perriot, E. Barthel, B. Champagnon, C. Martinet, J. Phys. : Condens. Matter 20, (2008). [15] B. Mantisi, A. Tanguy, G. Kermouche and E. Barthel, Eur. Phys. J. B, 85, 204 (2012) [16] D. Tielbürger, R. Merz, R. Ehrenfels and S. Hunklinger, Phys. Rev. B, 45, (1992) [17]T. Albaret and A. Tanguy, (2013) to be submitted. [18] B. Mantisi, G. Kermouche and E. Barthel and A. Tanguy, (2013) to be submitted. [19] M. Tsamados, Eur. Phys. J. E, 32, (2010) [20]L. Berthier and J.-L. Barrat, Journal of Chemical Physics, 116, (2002) 6