Computational Mechanics 33 (2004) 292–298 Springer-Verlag 2003 DOI 10.1007/s00466-003-0531-3
Discrete dislocation simulation of nanoindentation H. G. M. Kreuzer, R. Pippan
292 region by means of generation and annihilation of dislocations. The approach of Pippan and Riemelmoser [8–10], used for the simulation of fatigue crack growth, was the main inspiration to apply this model to the simulation of a nanoindentation process. A significant difference between their approach and the present investigation lies in the formulation of the boundary value problem BVP. Whilst for the crack problem one has to solve the BVP of a semiinfinite slit along the negative x1 -axis, here we have to deal with a BVP of an infinite half space. A similar method (discrete dislocations embedded in a FE model) was very effectively used for a description of plasticity during bending of a strip in plane strain by [11]. The aim of the paper is to present a method, which permits the description of effects which are associated with Keywords Discrete dislocation simulation, Nanoindenta- the discrete nature of plasticity and arise from the dislocation microstructure interaction, or in other words, the tion, Local plasticity effect of microstructure on nanoindentation. For simplicity, we use a 2D-description of the problem that corre1 sponds to an indentation of a wedge in a crystal with an Introduction Whenever the plastic deformation is small i.e. in the order idealized orientation. Of course, this is far from real nanoindentation, however, a certain class of effects can be of some Burgers vectors it appears to be reasonable to studied very clearly and it should help to understand describe plasticity occurring in a solid by means of mathematical dislocations. As the crystallography and the selected effects of the complex phenomena involved in nanoindentation. microstructure of a material becomes important at this length scale, a standard elasto plastic continuum 2 mechanics approach, e.g. a FE simulation, is no longer sufficient anymore. Therefore a discrete dislocation model Preliminary and assumptions which takes into account the motion of dislocations con- In our simulations the free surface of the solid is considfined to a few, favored regions, can be used to describe the ered as the x1 -axis of the lower infinite half space. The coordinate system used throughout this paper is shown in plastic behavior in this case. As there are many studies using quite similar disloca- Fig. 1. The stresses, r, and displacements, u, for the infinite halfspace system are given by the equations of tion models (see for example [1–4]) it seems to be Kolosov [5]: reasonable to mention only those papers, which had a h i decicive influence on the present investigation. A vast 0 0 r þ r ¼ 2 / ðzÞ þ / ðzÞ 11 22 amount of studies deal with the problem of semi-infinite cracks and the developing local plasticity in the near-tip Abstract A methodology to describe nanoindentation by means of discrete dislocations is presented. A collocation method is used to calculate the arising contact stresses at each indentation step, which permits to realize an arbitrary shape of the indenter. Distributed dislocation sources are allowed to emit dislocations on predefined slip planes, when the critical value of the local shear stress for the emission is reached. After each indentation step, the newly emitted dislocations are brought to their equilibrium positions under the influence of the stresses induced by the contact stresses and the dislocations. As an application of our model, the plastic behavior of two materials with different densities of dislocation sources will be studied in detail.
r22 ir12 ¼ /0 ðzÞ /0 ðzÞ þ ðz zÞ/00 ðzÞ
Received: 26 May 2003 / Accepted: 11 November 2003 Published online: 22 December 2003
H. G. M. Kreuzer (&), R. Pippan Erich Schmid Institute for Materials Science, Austrian Academy of Sciences, Jahnstrasse 12, A-8700 Leoben, Austria E-mail:
[email protected] This work was financially supported by the FWF (Fonds zur Fo¨rderung der wissenschaftlichen Forschung) Project P13908N07.
u ¼ u1 þ iu2 ¼
1 ðj/ðzÞ þ /ðzÞ ðz zÞ/0 ðzÞÞ 2l
ð1Þ
Here /ðzÞ is the Goursat complex function, which is defined to be analytically in the lower and the upper half space. z ¼ x1 þ ix2 is the complex variable, i is the complex unity. A bar denotes the complex conjugate and a prime the complex derivation with respect to z. l and j are the shear modulus and the Muskhelishvili constant, respectively. We confine our considerations to plane strain conditions, thus j ¼ 3 4m where m is the Poisson’s ratio.
Fig. 1. Coordinate system of the indentation problem
As we model the nonlinear, i.e. plastic, material response in two dimensions by generation and motion of parallel edge dislocations with parallel Burgers vectors in the linear elastic, isotropic body, it is important to note, that as a consequence of this linear description of the plastified system, the linear theory of elasticity can be applied even to the plastified region below the indent.
2.1 The framework of the discrete dislocation model Due to the fact that it is impossible to describe the real material behavior during the nanoindentation process on the basis of real crystal dislocations, a reasonable compromise between computability and closeness to reality is the simulation of plasticity as motion of mathematical dislocations. These dislocations are generally restricted in their freedom of movement. In the following set, the properties of the mathematical dislocations are described and the basic assumptions for the simulation of a nanoindentation process are introduced: – Distribution of slip planes: In the present investigation one slip system, where the slip planes are distributed in the lower half space at regular intervals parallel to the free surface, is chosen. For simplicity one pointtype Frank–Read source is placed in each slip plane at a random x1 -Position within a certain range determined by the distance between the slip planes. This procedure results in a reasonable density of dislocation sources. – Dislocation nucleation is modeled by simulating the operation of point-type Frank–Read sources. When the local shear stress, rloc 12 , which is the sum of the shear stresses induced by the indenter and all the dislocations at the position of the dislocation source, exceeds a critical value for the emission, rem , the source becomes active and a dislocation dipole is generated. For a specified source strength, rem , the dipole separating distance, dem , is given via [11]:
dem ¼
E b 2 4pð1 m Þ rem
ð2Þ
– We allow only parallel edge dislocations with parallel Burgers vectors, thus, we have just to distinguish between positive and negative dislocations. Here, ‘‘positive’’ means that its Burgers vector points to the positive x1 -direction, whereas ‘‘negative’’ means that it points to the negative x1 -direction. – A dislocation is immobile, if the sum of the shear stresses induced by the indenter and all the other dislocations at the position of the dislocation is smaller than the friction stress, rf . Otherwise the dislocation tends to move in the direction, where it will contribute to a decrease of rloc 12 . If the value of the local shear stress at the new position of the dislocation is smaller than rf , the dislocation becomes immobile again. – Two dislocations with opposite Burgers vectors on the same slip plane annihilate, if their mutual distance becomes less than 50jbj. As the methodology we present here, is not limited to the case where only one type of slip system (parallel to the free surface) can be studied, we will discuss in brief, the modifications in our algorithm, to extend it to slip planes with oblique angles and to multiple slip systems. If we assume an arbitrary angle of the slip planes, one has to take into account that one type of dislocations which move to the free surface, will finally rest at the surface and produce steps there. This results in a different surface contour and will therefore give rise to contact stresses, which can easily be calculated via the method described in this paper. In the case of multiple slip systems one has to be careful during the movement of dislocations: It is important to avoid that dislocations on different slip planes can occupy exactly the same position. Otherwise one would get serious computational problems with the arising singularity of the stress field at the position of the dislocation itself.
3 Simulation of the indentation process In this section the program for the simulation of the nanoindentation process will be introduced on the basis of a flow diagram. We will discuss the loading and unloading procedure in some more detail, where our main focus will be attached to the procedure ‘‘Calculation of contact stresses from the surface contour’’ which is one of the key routines besides the ‘‘Equilibrium arrangement’’ procedure. In the latter, especially the equations that we use for the calculation of the shear stress and the displacement fields for an edge dislocation in a semi-infinite solid are presented. At the end of this section a convenient equation is given, to compute the local shear stress at any point in the material. The various computational steps of the flow diagram in Fig. 2 are discussed in the following subsections.
where E is the Young’s modulus and b the magnitude of the Burgers vector. At this distance, the shear stress of 3.1 one dislocation acting on the other is balanced by the Loading procedure slip plane shear stress. In our simulation, the indentation depth is increased in – The generated dislocations are allowed to move on the small steps, typically in the order of jbj. From the indenpredefined slip planes only and no cross slip is taken tation contour, the contact stresses and also the local shear stresses are calculated in each indentation step. Then into account.
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Fig. 2. Flow diagram of the procedure for simulating a nanoindentation process by means of mathematical dislocations
immediately the procedure to determine the equilibrium position of the dislocations is carried out. This means, that the local shear stress on the position of each dislocation is computed. As already explained above, those dislocations which experience a shear stress larger than the friction stress are moved stepwise, until they are in their equilibrium position. The equilibrium procedure is repeated until 98% of the dislocations are in rest. Then it is controlled for which dislocation source the local shear stress reaches a maximum. If the emission criterion for this dislocation source is fulfilled, a dislocation dipole is emitted and the equilibrium procedure is carried out again. This part of the program is repeated as long as none of the Frank–Read sources becomes active anymore. Then the next loading step is performed and the entire procedure is repeated until the maximum indentation depth is reached.
dislocation arrangement. The entire unloading procedure is repeated until the indenter is not in contact with the surface anymore.
3.3 Evaluation of the contact stresses from the surface contour To calculate the arising contact stresses when an indenter with a given geometry is pressed into a specimen, we use the collocation method, which is described for the crack face contact problem during fatigue crack growth simulation in [8]. The basic idea is to divide the overlap interval I1 caused by the indenter and the surface of the specimen, which is usually pre-deformed by the displacement field of the emitted dislocations (except in the purely elastic calculation where no dislocations are involved), into small elements. Now a contact stress distribution is sought, so that the indentation contour can be reproduced. Since we do not know, a priori, the surface contour after applying the contact 3.2 stresses and, as a consequence, we do not know the Unloading procedure boundaries of the physical contact interval, In (which is In the unloading sequence the indentation depth is decreased stepwise. At each unloading step the contact equal to the interval of stress transfer), we have to estimate In stresses and the local shear stresses are calculated, in the and the corresponding contact stresses, PðxÞ, iteratively1 : same way, as already described in the loading part of the In the first step, we assume that the interval of stress program. Then the equilibrium procedure for the transfer is equal to the initial overlap interval, I1 . The unloading sequence is performed, which differs from its contact stress distribution, PðxÞ, is obtained by solving the loading counterpart only in the point, that a moving following integral equation: dislocation comes to rest when the local shear stress at the position of the dislocation becomes larger than the 1 That is why we denoted the physical contact interval ‘‘In ’’. In is negative friction stress. If the annihilation condition for a the integration interval of the n-th itertion. I1 (initial overlap dislocation pair is satisfied, the dipole is removed from the interval) is the integration interval of the first iteration.
Z
Pðx0 Þgðx0 ; xÞdx0 ¼ u2 ðxÞ x; x0 I1
ð3Þ
I1
u2 ðxÞ is the prescribed displacement in the interval I1 . The kernel, gðx; x0 Þ, gives the displacement of the surface contour at position x due to a unit load at position x0 . A convenient way to get an approximate solution of Eq. (3), is to divide the interval I1 into a number of finite subdomains (elements). Next we have to assume a ‘‘shapefunction’’, i.e. we have to prescribe a polynomial, which describes the variation of the stresses within an element. Doubtless, the mechanical stress problem is the better solved the higher the degree of the polynomial is. But one should keep in mind that the aim of the procedure is, above all, the computation of dislocation motion caused by the contact. To get mathematically tractable equations for the dislocation motion, we have choosen contact elements with a constant stress within the element (= uniform load element). In the next step several points are choosen within the overlap interval, where the approximate solution u^2 ðxÞ has to fulfill the exact solution u2 ðxÞ. For convenience, in our procedure, these points coincide with the midpoints of the elements. With these assumptions we get a system of linear algebraic equations (4), which can be solved by an ordinary equation solver:
P^j g^ij ¼ u^i
i; j ¼ 1 . . . n
ð4Þ
where the summation convention applies to repeated indices. P^j is the stress within the j-th uniform load element. The function g^ij is the displacement u2 at position xi caused by a uniform load element with the midpoint xj . For a uniform load element the complex function /ðzÞ is known as [5]:
/ðzÞ ¼
1 ½ða1 zÞ logðz a1 Þ ða2 zÞ logðz a2 Þ 2pi ð5Þ
a1 and a2 are the coordinates of the left and right boundary of a contact element, respectively. The displacement u2 and therefore the function g^ij can now be easily calculated via Eq. (1). In problems like the one under consideration, the overlap interval I1 is always larger than the ‘‘true’’ physical contact interval. Therefore Eq. (3) provides not only contact stresses (which are compressive stresses) but also tension stresses. As in our indentation problem only compressive stresses may develop, the solution of the first iteration is an unreasonable one. A mechanically more suitable solution of the given contact problem is obtained by canceling the rows and columns in the system of algebraic equations (4) which correspond to elements with tension stresses2 . The reduced algebraic equation system is solved once more, and the tension elements are again canceled. This procedure is repeated until all elements are compression elements. In this case an approximate solution has been obtained for both, the size of physical contact interval and the corresponding contact stresses. 2
Consider, that through this reduction procedure also the interval boundaries shrink.
3.4 Stress and displacement fields of an edge dislocation in a semi-infinite solid To obtain the stress and displacement fields of an edge dislocation near a free surface, we choose the convenient approach to solve the boundary value problem by the method of complex functions. Therefore, one has to take the complex potentials for an edge dislocation in an infinite solid and apply to them the boundary conditions. The complex potentials, /00 and x00 , for an edge dislocation with the Burgers vector b ¼ b1 þ ib2 at the position z0 in an infinite homogeneous medium, can be found in literature[6]: 2A 2A 2Aðz0 z0 Þ /00 ¼ ; x00 ¼ ð6Þ z z0 z z0 ðz z0 Þ2 where the constant A ¼ lb=ð2piðj þ 1ÞÞ characterizes the material and the strength of the dislocation. For an edge dislocation in the lower half plane, the complex potential /0 ðzÞ can be evaluated by the method of Muskhelishvili [5], which is based on an analytic continuation of the stress fields into the upper half plane to satisfy the boundary condition of a traction free surface. This yields:
/0 ðzÞ ¼
0 z0 Þ 2A 2A 2Aðz z z0 z z0 ðz z0 Þ2
ð7Þ
From this result the shear stress and the displacement field induced by an edge dislocation in a semi-infinite medium can easily be obtained with the help of Eq. (1)3 . It is important to mention, that the above solution for the u2 displacement of an edge dislocation in the lower half space, is also valid at the free surface. Therefore we can use this result to compute the deformation of the surface contour induced by the dislocations during the indentation process after each loading step.
3.5 Calculation of the local shear stress To decide, whether a dislocation source gets activated or a dislocation becomes mobile/immobile, we have to calculate the local shear stress at the position of the Frank-Read source or at the position of the dislocation, respectively. Thus the local shear stress is computed via: X X rloc rdisl relem ð8Þ 12 ¼ 12 þ 12 dislocations
elements
As a dislocation does not contribute to the resolved shear stress at its position, the first sum in Eq. (8) has to be taken over all the other dislocations, except the one of interest4 . The second sum is simply the contribution of the indentation contour (interval of contact), where each single uniform load element contributes to the local shear stress with its particular shear stress field, evaluated with Eq. (1) and Eq. (5). 3 It can be shown, that this solution is equivalent to the result obtained by Head, [7] who solved the standard boundary value problem of potential theory for the three types of boundaries, ‘‘Free boundary’’, ‘‘Welded boundary’’, ‘‘Slipping boundary’’ by the method of images. 4 The dislocation cannot move itself.
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4 Simulation and results In the following section, we study the influence of the density of dislocation sources in a model material. Therefore the indentation procedure we previously described, is performed by means of two concrete examples for different densities of sources. As a consequence of the method of our contact stress evaluation, the shape of the indenter is in principle arbitrary. Therefore we choose a V-shaped indenter with an angle of 140, which seems to be a reasonable choice, regarding to an indentation test with a wedge-shaped indenter. The parameters we use in our simulations are summarized in Table. 1. In the present study, a simple case is examined where only one type of slip planes can generate dislocations.The technique itself is not limited to this case and so the effect of the type of slip planes and also the influence of microstructural obstacles will be presented in a forthcoming paper. In the following, the development of the local plasticity during the indentation process is studied in detail for two different densities of sources. For both simulations, (a) and (b), 1500 dislocation sources were randomly distributed within a range of (a) ð4000 7500Þjbj and (b) ð5000 7500Þjbj (the distance between the glide planes is 5jbj in both cases). Figure 3 shows the equilibrium dislocation arrangement for the two densities of dislocation sources below the indent after the chosen maximum indentation depth of 300jbj is reached. Due to the shape of the shear stress field induced by the indenter and the dislocations, it can be observed that the negative dislocations prefer to pile up in a center region below the indenter tip, whilst the positive dislocations tend to move further away from the tip region. Analyzing the positions of the positive dislocations, forming the outer range of the plastic zone, one can see, that they arrange in approximately vertical lines. This phenomenon occurs, because the compressive stress field of one dislocation overlaps with the tensile stress field of its above neighbor and therefore they attract each other, respectively. In Fig. 4 the surface deformation resulting from those dislocations and the indentation contour are plotted for simulation (a) exemplarily. Here one can see, that the negative dislocations lower the surface contour, whereas the positive ones contribute to a pile up at both sides of the indent, due to their displacement fields. Each point in the load–displacement curves, as they are shown for simulation (a) and (b) in Fig. 5, is represented by the sum of the contact stresses, rcontact , of the compression contact elements (total load) and the x2 -displacement of the indenter tip ( = predetermined indentation depth, plus the contribution of the ‘‘deformed’’ surface contour of the dislocations) for each single loading/unloading step. In order to get an impression, at which point in our simulation we reach the continuum mechanics material behavior, the mean stress ( = sum of the contact stresses of Table 1. Parameters of the simulation l
m
rem
rf
b
80 GPa
0.3
500 MPa
100 MPa
2.5 · 10)10 m
Fig. 3. Positions of the dislocations at the maximum indentation depth for a the higher and b the lower density of dislocation sources (see text for details)
Fig. 4. Displacement of the surface, induced by the indenter and by the dislocations presented in Fig. 7 for simulation a
the compression contact elements, divided by the number of compression contact elements, Ncontact ) is plotted versus the x2 -displacement of the indenter tip in Fig. 6. The so defined mean stress, takes on the meaning of a nominal
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Fig. 5. Load–displacement curves of the simulations a and b
Fig. 7. Positions of the dislocations after complete unloading for a the higher and b the lower density of dislocation sources Fig. 6. ‘‘Hardness’’–displacement curves of the simulations a and material, forming the plastic zone and the remaining b
indentation contour after complete unloading. hardness. It can be observed in Fig. 6, that at a very small indentation depth, the mean stress is equal to the elastic solution, then it decreases and approaches, at larger indentations, a constant value. This can be interpreted as the transition from the discrete material behavior to the plastic continuum one. For the two different densities of dislocation sources, one can see that the continuum value is reached at a smaller indentation, depth for the higher density of dislocation sources. It is a remarkable result, that the indentation size effect (decreasing hardness with increasing indentation depth), as it has been observed in nanoindentantion experiments many times (see for example [12]), also occurs in the early stage of the simulated indentation process, where the discrete nature of plasticity is predominant. The positions of the dislocations after complete unloading is illustrated in Fig. 7. Compared to the arrangement at the maximum indentation depth (Fig. 3) one can see, that the outermost dislocations nearly stay at their positions, whilst the innermost come close enough so that they can annihilate. It is observed, that about a half of the originally emitted dislocation dipoles remain in the
5 General remarks on the different types of modeling of nanoindentation It is a well known fact, that length scales play an important role if one wants to study the mechanical behavior of a material. Therefore the selection of a suitable model strongly depends on the information we want to get about the particularly material behavior. If the interest is focused on probing incipient plasticity at the early stage of nanoindentation, one needs to understand such points as the mechanics governing defect nucleation and the evolution of the defects. At this length scale, an atomistic study of the locally concentrated stress levels in a crystal can bring light in this case [13, 14]. As in this work, we are mainly interested in the evolution of plasticity in a material and the lessons we can learn from modeling nanoindentation with a discrete dislocation model (microstructural influence factors, like: the density of dislocation sources, strength of sources, friction stress, dislocation obstacles, grain and phase boundaries etc.), we are somewhere in between an atomistic and a continuum mechanics description, such as different types of FE
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studies (e.g. [15]). These are often used to act as a bridge from the microscopic to the macroscopic length scale and are a powerful tool to study contact problems, at least in purely elastic cases. Up to now, looking at all this approaches, still a compromise between limited computational power and a model, which can describe real material behavior at the best possible rate, has to be made. Therefore a hybrid model, where a clever combination, depending on the understanding of each of the above models itself and how they can be coupled together, can be applied to make predictions about how a particular material behaves in an actual experiment.
6 Conclusions A method to investigate a nanoindentation process and the developing local plasticity was presented in detail. In our model, discrete dislocations are allowed to move on predefined slip planes under the influence of their shear stress field and the shear stress field induced by the indenter, and form the local plasticity in the material. Two simulations for different densities of dislocation sources were performed, to study the transition from the discrete to the continuum material behavior. We could show clearly, that this change occurs somewhat earlier for the material with the higher density of dislocation sources, compared to that with the lower one. Also the indentation size effect at small indentation depths has been verified as an intrinsic effect of the discrete nature of plasticity in our simulations. References 1. Weertman J (1996) Dislocation based fracture mechanics. World Scientific, Singapore 2. Raabe D (1998) Computational materials science: the simulation of materials microstructures and properties. WileyVCH, Weinheim
3. Kubin LP (1993) Dislocation patterning during multiple slip of F.C.C crystals. Phys Stat Sol 135: 433–443 4. Devincre B, Kubin LP (1994) Simulations of forest interactions and strain hardening in FCC crystals. Model Simul Mater Sci Eng 2: 559–570 5. Muskhelishvili NI (1963) Some basic problems of the theory of elasticity. Noordhof Groningen, the Netherlands 6. Lin LH, Thomson R (1986) Cleavage, dislocation emission, and shielding for cracks under general loading. Acta Metall 34: 187–206 7. Head AK (1953) Edge dislocations in inhomogeneous media. Proc Phys Soc 66: 793–801 8. Riemelmoser FO, Pippan R, Kolednik O (1997) Cyclic crack growth in elastic plastic solids: a description in terms of dislocation theory. Comput. Mech. 20: 139–144 9. Riemelmoser FO, Pippan R, Stu¨we HP (1997) A comparison of a discrete and a continuous model for describing cyclic crack tip plasticity. Int J Fract 85: 157–168 10. Pippan R, Riemelmoser FO (1998) Visualization of the plasticity-induced crack closure under plane strain conditions, Eng Fract Mech 60: 315–322 11. Cleveringa HHM, Van der Giessen E, Needleman A (1999) A discrete dislocation analysis of bending. Int J Plast 15: 837– 868 12. Nix WD, Gao H (1998) Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 46: 411–425 13. Lilleodden ET, Zimmerman JA, Foiles SM, Nix WD (2003) Atomistic simulations of elastic deformation and dislocation nucleation during nanoindentation. J Mech Phys Solids 51: 901–920 14. Li J, Van Vliet KJ, Zhu T, Yip S, Suresh S (2002) Atomistic mechanisms governing elastic limit and incipient plasticity in crystals. Letters Nature 418: 307–310 15. Fivel MC, Robertson CF, Canova GR, Boulanger L (1998) Three-dimensional modeling of indent-induced plastic zone at a mesoscale. Acta Mat 46: 6183–6194
