Density-based crystal plasticity : from the discrete to the continuum
Because of the enormous range of time and space scales involved in dislocation dynamics, plastic modeling at macroscale requires a continuous formulation. In this paper, we present a rigorous formulation of the transition between the discrete, where plastic flow is resolved at the scale of individual dislocations, and the continuum, where dislocations are represented by densities. First, we focus on the underlying coarse-graining procedure and show that the emerging correlation-induced stresses are scale-dependent. Each of these stresses can be expanded into the sum of two components. The first one depends on the local values of the dislocation densities and always opposes the sum of the applied stress and long-range mean field stress generated by the geometrically necessary dislocation (GND) density; this stress acts as a friction stress. The second component depends on the local gradients of the dislocation densities and is inherently associated to a translation of the elastic domain; therefore, it acts as a back-stress. We also show that these friction and back- stresses contain symmetry-breaking components that make the local stress experienced by dislocations to depend on the sign of their Burgers vector.
pacs:61.72.Bd, 61.72.Lk, 62.20.fq, 05.20.Dd
Plasticity of crystalline solids involves the notion of dislocations. However, even today, conventional plasticity theories use mesoscopic variables and evolution equations that do not involve dislocations. This paradoxical situation is due to the enormous length and time scales that separate the description of plasticity at the level of individual dislocations and the macroscopic scale of engineering materials. This huge space and time separation renders the hope to use a discrete dislocation based approach out of reach for treating engineering problems. It could be argued that conventional or phenomenological plasticity theories are justified because, at the macroscopic scale, engineering materials always display some sort of disorder that gives to any macroscopic property or measure an inevitable averaging character. Hence, at macroscale, plastic strain may be seen as resulting from a space and time average over a huge number of individual dislocation glide events.
Nevertheless, conventional plasticity theories rely on strong approximations and on phenomenological laws that must be calibrated for each material or for each specific applications. Therefore, it is desirable to make a link between the micro and macro scales and to develop a mesoscopic plasticity theory that relies on a sound physical basis, i.e. that at least incorporates dislocation glide. The development of such a mesoscale theory is also crucial to better understand and simulate the materials behavior at length scales where the elastic interaction between dislocations becomes of the order of the interaction between dislocations and obstacles, such as precipitates in a matrix, small grains in a polycrystal or interfaces in nano-materials. At these scales, dislocations display collective phenomena that result in patterning and complex dynamic regimes. In these situations, plasticity cannot be described by a simple averaged plastic strain that obeys local time-dependent equations. Size-dependent effects and, most importantly, transport become fondamental. Conventional theories of plasticity are no longer valid and are unable to account for the complexity of the plastic activity because they lack the relevant internal length scale and do not incorporate transport.
These considerations motivate the development of continuum models in which dislocations are represented by continuous densities and in which the dynamics has conserved the transport character of the underlying dislocation glide.
Continuum dislocation representations often start from the Nye NY1953 and Kröner KR1959 representation of dislocations. This is the case of the Field Dislocation Model (FDM) proposed by Acharya AC2001 ; AC2003 and developed subsequently by various authors RO2005 ; FR2011 ; TAU2013 ; FR2014 . The basic equations have been in fact known as early as the ’s KO1962 ; MU1963 (see also Ref. LA1970, ; NA1979, ). The basic ingredient of the FDM is the dislocation density tensor , where is the plastic distorsion tensor. When envisaged at the smallest scale, the tensor represents all the dislocations and there is no need to introduce the concept of ”geometrically necessary” or ”statistically stored” dislocations (GND and SSD, respectively). The model is then exact, regardless of the atomic nature of the dislocations and provided that we accept that the dislocation velocity is simply proportional to the local resolved shear stress. However, being continuous by nature, the implementation of the model requires the use of a computational grid with a grid step significantly smaller than the Burgers vector length. This drastically limits the spatial length scale that can be investigated. Therefore, in order to reach a convenient macro scale, a change of scale must be performed to bridge the gap between the singular density tensor introduced above and a continuous one defined at an intermediate scale. There is of course no unique way to select this so-called ”mesoscale”. Obviously, the mesoscale must be larger than the average distance between dislocations and smaller than the characteristic length scale we want to investigate (average grain size in polycrystals, average distance between interfaces in multiphase alloys, etc.). The underlying averaging or ”coarse-graining” procedure has of course been already mentioned in the context of the FDM AC2006 ; FR2010 .
The crucial point is that the application of the coarse-graining procedure to the FDM equations leads to transport equations for the averaged one-body GND density in which the plastic strain rate inevitably depends on the correlations between the lower scale GND and velocity fields. This closure problem is often resolved by using a phenomenological velocity law borrowed from macroscopic plasticity models leading to the so-called Phenomenological Mesoscopic Field Dislocation Model (PMFDM) AC2006 ; RO2006 . The actual implementation of the mesoscale FDM thus suffers from the lack of a mathematically justified mesoscale plastic strain rate.
A more recent formulation of a Continuum Dislocation Dynamics (CDD) has been proposed by Hochrainer and its collaborators HO2006 ; HO2007 . It is based on a modified definition of the dislocation density tensor, in order to keep at mesoscale information concerning the geometry of the dislocations (in particular, line directions and curvatures). The necessity of using an averaging procedure to obtain a meaningful continuum model has also be pointed out in the context of the CDD formulation SA2011 (see also AZ2000 ; AZ2006 ), but a rigorous mathematical formulation of this coarse-graining procedure has not yet been proposed.
The first attempt to better treat the closure problem has been proposed by Groma and its collaborators 1GR1997 ; ZA2001 ; 2GR2002 . This is the route that we follow below. A particular attention will be paid to the nature of the coarse-graining procedure and its consequences on the local stress fields that emerge from the averaging process. We show that the emerging local friction and back-stresses, which are reminiscent of the dislocation-dislocation correlations, depend on the length scale associated to the averaging process required by the coarse-graining procedure. We also show that these correlation-induced stresses contain symmetry-breaking components that make the local stress experienced by dislocations to depend on the sign of their Burgers vector. Finally, we find that the emerging back-stress depends on the gradients of both the geometrically necessary and total dislocation densities. A brief version of these results has been presented in Ref. FI2014, .
Ii Mesoscale density-based theory
We first clarify the mathematics and physical aspects of the coarse-graining procedure that must be used to coarse-grain the dislocation dynamics from the discrete to the continuum. We consider the simplest situation, namely a 2D dislocation system with edge dislocation lines parallel to the -axis restricted to glide along the -axis. The Burgers vector of dislocation , to , is noted , where is the sign of the dislocation and . We assume an overdamped motion: the glide velocity of the dislocation along the -axis is simply proportional to the resolved Peach-Koehler force acting on the dislocation ,
where is the mobility coefficient equal to the inverse of the dislocation drag coefficient, the external stress resolved in the slip system and the resolved shear stress at position generated by a positive dislocation located at the origin:
where is the shear modulus and the Poisson ratio.
The first step is to define discrete dislocation densities:
where refers to the initial positions of the dislocations, is the Kronecker symbol and the 2D Dirac function. The notation means that the trajectory of dislocation depends on the initial dislocation positions .
By multiplying Eq. (1) by the Dirac function and taking its derivative with respect to , we get the following transport equation for the discrete densities:
where, to simplify the notation, we write for . Obviously, these transport equations link the time-dependence of the one-body densities to the products of two one-body densities, which is a direct consequence of the pairwise dislocation interactions. At this stage, the dislocation densities are highly singular. The next step is to introduce a coarse-graining procedure.
ii.1 Coarse-graining procedure
We introduce now a coarse-graining procedure commonly used in statistical physics (see, for exemple, Ref. MA2003, ). We first define a space and time convolution process that we use to coarse-grain microscopic fields to mesoscopic ones:
where the weighting function is normalized. For simplicity, and without loss of generality, we choose to be separable:
where the functions and are separately normalized:
The spatial linear dimension of should be of the order of the spatial resolution of the continuous model we seek and, obviously, significantly larger than the average distance between dislocations. The temporal width of the time window should, in all generality, depend on . In fact, the appropriate choice of is linked to the kinetic behaviour of the degrees of freedom that, inevitably, we will have to average out in order to close the theory: should be defined in such a way that the correlations we want to average out have the time to reach a stationary state at scale . We comment on that point in section II.4. Here, we just mention that, for convenience, we choose to be non-zero only for :
Mesoscopic density fields may be defined through Eq. (5), but this is not enough to get a consistent continuous transport theory. First, we expect that the time evolution of the mesoscopic dislocation densities will be given by first-order transport (i.e. hyperbolic) equations. These equations must be supplemented by initial conditions at which, of course, must be defined at mesoscale. In other words, the coarse-graining procedure should be such that, when applied to Eq. (4) and its initial condition given by the dislocation positions at , we end up with a set of mesoscopic transport equations supplemented by continuous initial conditions that do not depend on any specific initial set . Therefore, if , , are given initial continuous densities, we must introduce a -body probability density distribution on the (discrete) initial positions which is linked to the initial mesoscopic densities in a way that we discuss below. The distribution , where refers to the predefined (and fixed) signs of the dislocations, introduces a statistical ensemble on the initial discrete dislocation positions: is the probability to have an initial dislocation configuration with dislocation , whose sign is , in a small volume around position , dislocation , whose sign is , in a small volume around position , etc.
Now, the overall coarse-graining procedure is defined as the conjugate action of the space-time convolution window and the ensemble average defined by the probability density . The mesoscopic field associated with the discrete field is therefore defined by:
We refer to this coarse-graining procedure by the following short-hand notation:
where the double brakets refer to the space and time convolution and the lower index to the ensemble average. The mesoscopic one-body and two-body densities are therefore defined by:
We mention that the two-body densities defined in Eq. (12) are continuous function of and . This would not be the case if the coarse-graining procedure was limited to a space and time convolution. This is the second reason why we need to consider an average over a statistical ensemble.
We can now precise the link, mentioned above, between the probability density , that defines the statistical ensemble, and the continuous dislocation densities that will be used as initial conditions for the mesoscopic kinetic equations. We consider that any discrete initial condition on the dislocation positions is extended to :
This equation constitutes a constraint that must fulfill for a given set of initial mesoscopic densities . However, this is not enough to completely define the probability density . In order to proceed, supplemental properties must be assigned to . As in Ref. ZA2015, , we argue that, in order to use no more information than the one actually embedded into the mesoscopic initial densities, which in principle are meant to reflect a realistic experimental situation, the supplemental rule needed to completely define should simply invoke the maximum entropy principle. This is equivalent to impose that no other information, besides that given by the constraint of Eq. (15), should be used to define the statistical ensemble associated to . This implies that the stochastic variables , to , must be considered as statistically independent. Therefore, they must follow one-body distribution functions , that depend only on their sign , over which the density is factorized:
Of course, the distribution functions , , are separately normalized:
which may be written as
where is the number of dislocations of sign . Up to the coefficient , the initial condition is simply equal to the convolution of , the distribution of initial positions of the discrete dislocations of sign , with the convolution window . For given set of initial conditions , , and a given convolution window , Eq. (19) defines a unique set of functions , and, therefore, a unique probability density . Thus, for prescribed initial mesoscopic dislocation densities and a given spatial convolution window , the coarse-graining procedure introduced in Eq. (9) is completely and uniquely defined.
ii.2 Coarse-grained kinetic equations
where the mesoscopic one-body and two-body densities and have been defined in Eq. (11) and (12).
At this stage, no approximation has been introduced. Eq. (20) is exact and contains the same information and complexity as Eq. (4) and, therefore, as Eq. (1). However, the time evolution of one-body densities is linked to the two-body dislocation densities . It is straightforward to realize that the time evolution of these two-body densities are themselves linked to the three-body densities, and so forth. Obviously, we are faced by the classical problem of closure that we meet in statistical physics when we try to replace a set of discrete degrees of freedom by a set of continuous densities.
The next step is to solve the closure problem. This of course requires the introduction of some approximations. One way to do that is to analyse and possibly approximate the two-body correlations, defined by:
where the local stresses and are defined by:
ii.3 Mean field stress
The local stress exerted on dislocations of sign does not depend on and is simply the sum of the external stress and the stress generated by all the one-body densities and defined in Eq. (23):
where we have introduced the polar or GND (Geometrically Necessary Dislocation) density:
As does not incorporate any correlation effects, it may be called a mean field stress or, as it closes the theory, a self-consistent stress 1GR1997 .
ii.4 Correlation-induced local stresses
We want now to go beyond the mean field approximation and incorporate the correlations. In other words, the correlation stress defined in Eq. (24) is now taken into account. These correlations should be approximated in order to close the theory.
For that purpose, we need to discuss the time and spatial variations of the correlation functions . It has already been observed ZA2001 ; 2GR2002 that the correlation length of is finite and of the order of a few average dislocation spacings. Consequently, if the width of the convolution window is sufficiently larger than the mean dislocation spacing, the correlations , for a fixed point and as a function of , decrease to zero before the one-body densities vary significantly. Therefore, within the domain around point where they are non-zero, may be considered as a function of and of the local one-body densities :
where the notation refers to . Now, we comment on the time dependence of the correlations. We recall that the coarse-graining procedure introduced above (see Eqs. (6) and (9)) involves a time convolution. A width for the time window must be selected.
Our present purpose is to close the theory at the order of the two-body correlations. In other words, we want to incorporate two-body correlations in such a way that their time dependence is formally linked to the time dependence of the one-body densities, which themselves are defined at scale . Therefore, the time convolution should be such that the averaging process incorporates all the time scales associated to the kinetics up to spatial scale . This is essential for capturing and embedding properly the lower scale kinematics and configurational dislocation properties into a physically sensitive theory where the correlations are expressed as local functionals of one-body dislocation densities defined at scale . In physical terms, this requires to select a time window such that the coarse-grained correlations reach a steady state at scale .
This point should be analysed in light of the very complex spatio-temporal behaviour that dislocations often display. Their dynamics is in particular characterised by the existence of a yielding transition when they are subject to an increasing stress. Both below the yielding point and in the subsequent flowing regime, the collective dislocation motion exhibits strongly intermittent avalanche-like dynamics characterised by a slow relaxation process. It has been in particular observed IS2011 ; IS2014 that, close to the yielding point but also far below, the dynamics is characterised by power laws and, therefore, is essentially scale-free up to a cut-off time that depends essentially on the system size . This size-dependent relaxation time marks a cross-over from a regime where the strain rate follows a power law, , to a regime where the strain rate decays exponentially to zero or reaches a steady value, depending on whether the stress is below or above the yielding point. Therefore, a convenient choice for the time convolution window is to select a width of the order of the relaxation time . Under this condition, the overall coarse-graining procedure will generate correlations which are dependent on the local one-body densities only: the explicit time dependence in disappears and shows up only implicitly through the time dependence of the one-body densities . In short, Eq. (28) becomes:
Now, using again the short-range nature of the correlations discussed above, we note that in Eq. (24) may be expanded to -order around . The local stress defined in Eq. (24) is then split into two terms:
At this stage, the coarse-grained kinetic equation given in Eq. (22) reads:
ii.5 Physical meaning of the correlation-induced stresses and
The physical meaning and properties of these local stresses will of course be inherited from the symmetry properties of the correlations. It should also be clear that these correlations depend on the stress experienced by the dislocations. Within the spirit of the present coarse-graining procedure, which inevitably leads to a hierarchy of independent and successive many-body densities, we consider that the stress dependence of the -body densities is due to the stress generated by the correlations up to order . Therefore, the stress dependence of the correlations is due to the sum of the external stress and the mean-field stress . We note this low-order stress: .
Using the discrete kinetic equation (1) and its symmetry properties, it is easy to show that the correlations display the following property:
where the dependence of the correlations on the low-order stress has been explicitly pointed out. Also, according to their very definition (Eq. 12), we obviously have:
For later use, we also note that, if the local GND density is equal to zero, correlations and display the following symmetry :
where the one-body dislocation densities and local stress dependencies have been explicitly added and the explicit time dependence suppressed, because and inherit this time dependence precisely through and . These properties clarify the physical meaning of the local stresses and . The stresses change their signs with the sign of the local low-order stress and, as shown below in section IV, they are positive when is positive. In contrast, the stresses are invariant upon a change of sign of . As a consequence, the stresses , which always oppose the low-order stress (see Eq. (33)), play the role of friction stresses whereas the stresses , which are invariant upon a reversal of the local stress , may generate a Bauschinger effect and a translation of the elastic domain. Therefore, the stresses play the role of back-stresses.
Iii Broken symmetry in the kinetics of the coarse-grained signed dislocation densities
It is important to note that, according to Eq. (33), the local stress fields experienced respectively by the positive and negative dislocation densities are different: the correlation-induced stress components and depend on the sign . In other words, the symmetry that exists at the discrete scale (positive and negative discrete dislocations at the same point have opposite velocities) is broken at mesoscale: the velocities of positive and negative dislocation densities are not simply of opposite sign. This broken symmetry is the direct consequence of a mesoscale description and its associated coarse-graining procedure: the averaging process required to build a continuous description generates kinetic equations for one-body densities that inevitably incorporate two-body correlations which, in all generality, break the lower-scale symmetry.
In order to be more specific, we analyse explicitly the friction stresses and experienced by the positive and negative dislocation densities, respectively. According to Eqs. (31), we have:
where, because they are not needed for the present argument, the low-order stress and dislocation density dependencies of the correlations and friction stresses have been omitted, as well as the time dependencies. Using the symmetry property given in Eq. (35), it is easy to show that the terms that depend on and are equal to zero. Therefore, the previous equations reduce to:
When the signed densities and are different, which is the generic situation, the friction stresses and are different, which is sufficient to break the symmetry between the velocities of the positive and negative dislocation densities. To better understand this broken symmetry in physical terms, we note that may be interpreted as the excess (with respect to the uncorrelated state) of negative dislocations in the surrounding of a positive dislocation that sits at point . Equation (41) tells us that this excess of negative dislocations at is at the origin of the friction stress experienced by a positive dislocation. There is of course no reason for this excess of negative dislocations around a positive dislocation to be exactly the opposite of the excess of positive dislocations around a negative one. Therefore, the friction stresses and ought to be different.111In fact, this broken symmetry could already have been pointed out earlier when we wrote the coarse-grained kinetic equations in the form of equation (22), where the dislocation-induced stress was split into the mean field stress and the correlation-induced stress defined in Eq. (24). Using the fact that the stress function is odd with respect to the coordinate (see Eq. (2)), it is easy to realize that would be independent of if and only if the correlations are such that the product is equal to the opposite of , where and are symmetric points with respect to . There is of course absolutely no reason for this to be fulfilled, even if, due to the short range nature of the correlations, point and may be restricted to be very close to each other.
Now, to better visualize this broken symmetry in the signed kinetic equations, we introduce the half sums and half differences of the friction and back-stresses: