An Atomistic-based Cohesive Zone Model for Quasi-continua

Transcription

1 An Atomistic-based Cohesive Zone Model for Quasi-continua By Xiaowei Zeng and Shaofan Li Department of Civil and Environmental Engineering, University of California, Berkeley, CA94720, USA Extended Abstract 1. Introduction Since Xu and Needleman [1] first proposed the cohesive zone model for finite element in 1994, the phenomenological cohesive zone model has been extensively applied to solve engineering fracture mechanics problems in many different disciplines. However, most of these applications are macro-scale material failure problems, and when the crack size becomes very small, say at about micron or sub-micron level, the conventional cohesive zone model may reach to its limitation, because irreversible plasticity theory is highly size-dependent. In this extended abstract, we shall report our latest development on construction cohesive zone model for a class of quasi-continuum media [2], i.e. the medium whose constitutive relations are enriched by its lattice or atomistic structures or constituents. The objectives of the research is to establish a small scale coarse grained cohesive zone model, or multi-scale cohesive zone model, that has some atomistic features, and it is not only capable of capturing certain particular local non-uniform deformation such as fracture, but also retain the simplicity of the continuum modeling and computation efficiency of finite element methods. One of the motivations of this research is to seek a possible coarse grained micromechanics model for simulation of adiabatic shear band propagation. 2. Cohesive Zone Model for Quasi-continua Since 1996, a class of quasi-continuum (QC) methods has been proposed by Tadmor, Ortiz and Philip [2] as a multi-scale simulation method to bridge atomistic simulations and continuum simulations. The quasi-continuum method has two versions: a local version, which is suite to the continuum region whose deformation is relatively uniform, and a non-local version, which is designed to model the fine scale region that has atomistic resolution. A main challenge of the quasi-continuum method is how to couple the local and non-local quasi-continuum methods. The current approach is a so-called ghost force method, which is a very complex procedure, and it has many issues that remain to be resolved. In this work, we only adopt the local version of quasi-continuum method as the way to establish the constitutive relations for the macro-scale medium considered. By doing so, we

2 2 Xiaowei Zeng and Shaofan Li Figure 1. The Cauchy-Born rule implicitly assume that the Cauchy-Born rule is satisfied in each element, see Figure 1, and this will give us an atomistically enriched constitutive relation in continuum for efficient computations. However, it, on the other hand, prevents us to simulate material failures such as fracture and dislocation motions because of the assumption of uniform deformation in an element. To solve this dilemma, we believe that one has to construct coarse grain models for both the bulk medium and the material interfaces, or defects. And the local QC method only provides the coarse gaining model for bulk materials, which is why it cannot solve small scale defect evolution problems. From this perspective, we construct a multiscale cohesive zone model to embed into the quasi-continuum bulk medium, and the so-called multi-scale cohesive zone model is an atomistically enriched cohesive zone that is compatible with the bulk (local) quasi-continuum model. The construction is as follows: We first remodel the material interface, or we re-construct material interface such that it is a compliance cohesive zone and its microstructure reflects that fact it is the weakest link in an otherwise homogeneous medium. In Figure 2, it shows that the cohesive zone between two bulk media is remodeled as a different lattice strip region whose lattice constants and atomistic potential should be different from those of the bulk medium. We can then apply the local version of quasi-continuum method to obtain coarse grain models for both the bulk region as well as the cohesive zone region. That is, we can obtain the constitutive relations for both bulk region and cohesive element by employing the local version of quasi-continuum method. For instance, the second Piola-Kirchhoff stress can be written in the following form [3]: S(C) = 1 Ω a 0 n b i=1 φ (r i ) r i C (2.1) where Ω a 0 is the volume of the unit cell, φ(r i ) is the atomistic potential, r i,i=1, 2,,n b is the current bond length for the i-th bond in an unit cell, and C is right Cauchy-Lagrangian strain tensor. The key here is how to choose atomistic potential as well as lattice constant for the cohesive zone region. To do so, we first assume that the cohesive zone region has only one element in thickness, which agrees with the conventional macroscale cohesive zone model. From the finite element discretization view point, in two-dimensional case the multiscale cohesive zone model may be simplified as two triangle bulk elements sandwich one quadrilateral cohesive element (see Figure 3). In order to obtain the atomistic potential in the cohesive zone, we assume that the cohesive zone is a kind of compliance interface that is much weaker than the two adjacent bulk

3 An Atomistic-based Cohesive Zone Model for Quasi-continua 3 Figure 2. The reconstruction of interface model: the atomistic cohesive zone model elements. The cohesive strength of the cohesive zone is soely determined by the intermolecular forces between the cohesive zone and the bulk medium, and it is a kind of the Van der Waals interaction between non-covalent bonds. When we consider the interaction between any material point inside the cohesive zone and the bulk medium, we may assume that the bulk medium is rigid with almost no deformation, so the two bulk elements adjacent to the compliant cohesive zone may be assumed as two rigid body half spaces (see Figure 3). Suppose that we know the atomistic potential inside the bulk medium, which we also know, and it can be a pair potential or potentials from embedded atom method (EAM), we can obtain the atomistic potential of the cohesive zone by integrate the bulk potential over the rigid bulk medium half space. For instance, if the bulk potential is the Lennard-Jones (LJ) potential as Eq.(2-2), a coarse graining potential can be obtained by analytical integration, e.g. [4] which has the close form expression as Eq.(2-3). (( σ ) 12 ( σ ) 6 ) φ bulk = 4ɛ r r φ cohesive = πɛ 2 ( 1 45 ( r0 r ) ( r0 ) 3 ) r (2.2) (2.3) where ɛ is the depth of the potential well, and σ is the (finite) distance at which the bulk atomistic potential is zero. The equilibrium bond distance in the bulk material is r 0 = σ2 1/6. Figure 3. Cohesive zone model (left) and cohesive zone potential integration scheme (right). Another important assumption is that the deformation inside the cohesive zone is also uniform, but it is discontinuous from the bulk deformation. One can then calculate the stress inside the cohesive element. By doing so, the cohesive zone is constitutively consistent with the bulk material. In fact, before we calculate the stress inside the cohesive zone, we have

4 4 Xiaowei Zeng and Shaofan Li to first calculate its deformation, which is completely determined by the four FEM nodal points displacement value, if we assume that inside the element the deformation gradient FC is constant. Consider the finite deformation in a plane strain case, the constant deformation gradient can explicitly determined as follows (see Figure 4): F c 11 F c 12 F c 21 F c 22 = 1 (ad cb) d 0 b 0 c 0 a 0 0 d 0 b 0 c 0 a x + l+1 x l y + l+1 y l x + l x l+1 y + l x l+1 where a = X + l+1 X l,b= Y + l+1 Y l,c= X+ l X l+1,d= Y + l Y l+1. (2.4) Figure 4. Deformation gradient in cohesive zone In Figure 5(left), we compare the force-displacement relations between the bulk medium (red lines) and that of the cohesive zone (blue lines). One can simplify the 2D bulk-cohesive zone sandwich may be viewed basically as an one-dimensional three spring model with two red springs and one blue spring in the middle in series connection. Among all three springs, the tensile force should be equal, because of the series connection. Therefore as the tensile force increases the displacements in both bulk medium and cohesive zone will increases until the force in cohesive zone reaches to its maximum. After the cohesive force in the cohesive zone reaches to maximum, it will unload as the displacement inside the cohesive zone increases, which will eventually lead to surface separation, whereas in the bulk region, the force and displacement will be restricted in the black color region, which is almost a linear elastic relation, and the magnitude of displacement field in the bulk (red spring) is limited in a very small range as shown in Figure 5(left). We also plot the normal cohesive force against normal separation distance from our simulation results, it is observed that as expected the normal traction shows a similar pattern as the interatomic force as shown in Figure Numerical Simulations We have applied such multiscale cohesive zone model to simulate crack propagation. In the following is a 2D example of our simulation. Consider a 2D plate under unilateral tension

5 An Atomistic-based Cohesive Zone Model for Quasi-continua 5 Figure 5. Comparison between bulk zone and cohesive zone force-displacement relation (left) and normal traction forve vs. normal separation distance in cohesive zone (right). Figure 6. Crack morphology for three different interface lattice orientations. (see Figure 6), and there is a pre-crack at the left side of the plate. The simulation results are shown in Figure 6(a-c) with respect to different orientations of lattice structures inside the cohesive zones. One may find that crack branching and void formation are also possible for certain material parameters and lattice orientations. References [1]. Xu, X. P. and A. Needleman (1994), Numerical Simulations of Fast Crack-Growth in Brittle Solids, Journal of the Mechanics and Physics of Solids 42(9): ; [2]. Tadmor, E.B., Ortiz, M., and Philips, R (1996), Quasicontinuum analysis of defects in solids, Philosophical Magazine A, 73, ; [3]. Martin, CR (1994), Nanomaterials-a membrane-based synthetic approach, Science, 266(5193): ; [4] Israelachvili, J. (1991), Intermolecular & Surface Forces, Second Edition, Academic Press.