Bounds on Precipitate Hardening of Line and Surface Defects in Solids

The yield behavior of crystalline solids is determined by the motion of defects like dislocations, twin boundaries and coherent phase boundaries. These solids are hardened by introducing precipitates -- small particles of a second phase. It is generally observed that the motion of line defects like dislocations are strongly inhibited or pinned by precipitates while the motion of surface defects like twin and phase boundaries are minimally affected. In this article, we provide insight why line defects are more susceptible to the effect of precipitates than surface defects. Based on mathematical models that describe both types of movements, we show that for small concentrations of a nearly periodic arrangement of almost spherical precipitates, the critical force that is required for a surface defect to overcome a precipitate is smaller than that required for a line defect. In particular, the critical forces for both phenomena scale with the radius of precipitates to the second and first power, respectively.


Introduction
A crystalline solid can deform inelastically through dislocation glide, the motion of twin boundaries as well as the motion of coherent phase boundaries [4,16]. While the dislocation is a topological line defect, twin and phase boundaries are surfaces across which either the orientation or the structure of crystal changes discontinuously. Mechanical stress acts as a driving force on these defects, and their motion results in inelastic deformation. Materials often contain precipitates -small inclusions of a distinct material (either second phase particles of a different composition or foreign substance), and these affect the motion of both line and surface defects by creating an internal stress field in the material. Precipitates are often introduced into the material by heat-treatment to inhibit the motion of the defects and thereby increase the yield strength (stress required for inelastic deformation).
In this paper, we will show that the critical external force for a line defect like dislocation to propagate through an arrangement of spherical precipitates scales with the radius to the power one and while the critical external force for a surface defect like a twin or phase boundary scales with the radius to the power two. Hence, for small radii, or, equivalently for small concentrations of precipitates, the effect on surface defects is negligible compared to the effect on line defects.
This has some very interesting implications. In any crystal, the energetics and mobility of dislocations and twin boundaries depend on crystallography. In high symmetry crystals like copper or aluminum, symmetry dictates that the system with the lowest critical resolved stress is sufficient to accommodate all deformations. However, in low symmetry materials like magnesium and zirconium, this is not the case and therefore one sees multiple defects. Magnesium and its alloys have been the topic of much recent interest since have potentially the highest strength to weight ratio. However, they lack ductility. In magnesium, the so-called basal dislocation is an order of magnitude softer than other defects, but insufficient to accommodate arbitrary distortions. So it is common to see twins, especially in tension [12,17,4]. Further this significant anisotropy is believed to be ultimately responsible for the low ductility. The results here suggests that precipitate hardening can have a differential effect and this can be used to improve the strength and ductility of magnesium. Indeed, precipitate hardening is used extensively in magnesium alloys. It has been observed through neutron diffraction and modeling that the critical stress in the basal system increases three-fold while that for tensile twinning remains essentially unchanged during aging in Mg-Y-Nd-Zr alloys [1]. It is important to note here that observations in other related alloys do not show such a clear distinction due to the elongated shape and basal orientation of the precipitates as well as the fact that twin growth is accompanied by basal slip [18,19]. Precipitates play a similar role in low stacking fault steels like TWIP steels where they increase yield strength by inhibiting dislocation motion and leave hardening rate that is influenced by twinning unaffected [3].
The commonly used shape-memory alloy has two inelastic deformation modes, plasticity due to dislocations and superelasticity due to stress induced phase transformations. The widely used shape-memory alloy nickel-titanium undergoes plastic deformation at extremely low stress, and this hides its useful superelastic effect. Therefore, commercial alloys are precipitate hardened. They increase the plastic yield strength by inhibiting dislocation activity but leave superelasticity governed by phase and twin boundaries unaffected [15].

Model and Results
We describe both defects, which are one (dislocations) and two (twin boundaries) dimensional subsets of R 3 as graphs of suitable functions and then work with the evolution equations of these functions.
The evolution of a twin boundary Γ twin (t) := {(x, w(x, t)) | x ∈ R 2 } can be described by the non-local partial differential equation where ∂ t is the derivative with respect to time and −(−∆) 1/2 is the half Laplacian with respect to space (see [8]). In this work, we consider a line tension model for a dislocation [16]. A dislocation is a line confined to the glide plane π. Let I an isometry with I(π) = {x 2 = ̟}. Then the evolution of the dislocation Γ dis (t) := {I −1 (x, ̟, v(x, t)) | x ∈ R} is described by the following partial differential equation ϕ = ϕ • I −1 (·, ̟, ·) (see also [7] and [5]). Note that in both cases, the equations essentially depend of three terms. The first term is a penalty for the deviation of the geometry of the defect from a flat state (which in our setting also ensures that the graph-setting remains appropriate), the second term -where ϕ : R 3 → R is assumed to be bounded and uniformly Lipschitz-continuous function -describes the interaction of the graph with the precipitates, and the last term constitutes the external driving force. In the context of the twin boundary, the first term arises from elasticity [8] (also [14]). The actual interaction between a defect and a precipitate is nonlocal, hence the second term should be a non-local potential. However, following [8] we assume that this interaction can be approximated well by the local term (see also the discussion of M. Koslowski et al. [14]).
Note, that ϕ : R 3 → R is assumed to be bounded and uniformly Lipschitzcontinuous which allows to use exponential time scaling, i.e. replace a subsolution v and a supersolution v of (2) by V := e −λt v and V := e −λt v respectively, to derive a comparison principle. These new functions are sub-and supersolution to an equation ∂ t W = H(x, t, W, ∇W, D 2 W ) with an appropriate choice of H. By choosing λ wisely, the righthandside will be -in the nomenclature of [6] -proper and hence we have a comparison principle for this equation. This does imply a comparison principle for (2) and we can conclude that there exists an unique viscosity solution provided that the initial datum is smooth enough, see [6, Theorem 8.2, Theorem 4.1]. For equation (1) a similar argument can be found in [13,Theorem 2] and for further details we refer to [11]. We conclude, that both equations (1), (2) satisfy a comparison principle and hence unique viscosity solutions exist.
Further note, that in the governing equations (1) and (2), constants such as elastic parameters, line tension, etc. have been suppressed. As we are merely interested in a scaling result, this suppression does not hinder us to compare both critical forces. For the two equations, we will assume, respectively, w = 0 and v = 0 as initial conditions. For both models, we furthermore assume F ≥ 0 and ϕ ≥ 0, which in our model implies that the precipitates always impede the motion in the (positive) x 3 -direction that the external driving force is favoring.
We are interested in the pinning of defects by precipitates, i.e, the question whether 1. there exists stationary supersolutions w : 2. or whether w or v are unbounded as t → ∞ due to the existence of propagating subsolutions (e.g., While the question whether the two points above form a dichotomy is open in the general setting [9,2], the following simple statement follows immediately from the comparison principle using the assumptions on ϕ made above. Proposition 2.1. There are critical forces F twin ≥ 0, F twin ≥ 0 such that for all F < F twin the interface Γ twin (·) gets pinned, i.e. for all F < F twin , there is a stationary supersolution. Moreover, for all F > F twin the interface does not get pinned, i.e. there is a propagating subsolution. The same result holds with a critical forces F dis , F dis for dislocations.
Our main strategy to prove the result that the pinning threshold for twin boundaries is lower than the pinning threshold for dislocations involves obtaining a lower bound for F dis and an upper bound for F twin and then comparing these bounds to conclude. To retrieve the bounds, we are going to construct worst case scenarios for pinning (in the case of dislocations) and depinning (in the case of twin boundaries).
So far, we did not make any assumptions on the distribution of the precipitates or on their shape. In real crystals, we see a lot of different arrangements and we notice that the shapes can range from rod-like objects to spheres. We thus need to fix some general assumptions on the distribution (periodic orthogonal to the propagation direction and well spaced in the propagation direction) and on the shape (bounded by a sphere from the inside and the outside) of the precipitates.
We first assume that the domain is an infinite strip Ω := T 2 × R, i.e., we assume periodicity orthogonal to the propagation direction. Moreover, let β > 0, we consider a distribution of precipitates in the strip.
be a family of random variables, that represent the centers of the precipitates. We will assume that a) (X i 1 ) i∈N , (X i 2 ) i∈N are identically and independently distributed with Furthermore, we will also make the following assumption on the shape of the precipitates.
i.e. the precipitates contain a ball of radius λR =: r, are bounded by a rectangle of sidelength R and have a pinning strength which is bounded by ϕ * and ϕ * .
We assume that the resistance provided by the precipitates is given by Remark 2.2. The assumptions are chosen in such a way, that the x 1 and x 2 components of the centers of the contained balls are iid. and that the distance between two precipitates cannot approach zero to fast. Our proofs will work for any configuration of precipitates that satisfy these conditions. Even though not included in our assumptions, one could image toroidal precipitates for which our results will also hold. For clarity, we chose to formulate our assumptions in the way above and they do include most of the physical cases as spherical, elliptical and rod-shaped precipitates.
To compare twin boundaries and dislocations, that evolve in the same direction, we will moreover only consider dislocations that propagate in the same direction as the twin boundaries, i.e. they glide along planes π := span{v, e y } with v ⊥ e y . In this case, we can find a lower bound for the critical force for dislocations. where h(s) := 1−2s −β 3+Cϕ * −2s −β and C > 0 is a geometric constant. The proofs of the two statements above are presented in the next section. Using these theorems, one immediately obtains the following result. This means that if the concentration of the precipitates is small enough, there are external forces F , such that dislocation boundaries get blocked, but twin boundaries can move freely throughout the crystal.
Proof. If R is small enough, we can assume that h(R) > 1 and ηr ≤ 1 We note, that the first term is a constant, the second term converges to 1 and as r = λR, the last term goes to zero as R → 0. Hence, as R converges to zero, the quotient goes to zero.

Proofs
Lower bound for the pinning threshold of dislocations We want to construct a stationary supersolution for equation (2). Therefore, we have to know if and how a random slip plane intersects the precipitates.
Lemma 3.1. Let η ∈ (0, 1), i ∈ N and ̟ ∈ [−1, 1] be fixed. Denote by A i the event the plane {x 2 = ̟} intersects B r (X i 1 , X i 2 , Y i ) and that the intersection contains a circle of radius ηr. Then we have The event A i occurs if and only if the plane {x 2 = ̟} intersects B r √ 1−η 2 (p i ). We can calculate this probability, knowing the distribution of X i 1 and X i 2 , Lemma 3.2. Let η ∈ (0, 1) be fixed and π := span{v, e y } with v ⊥ e y be any plane. Then almost surely the plane π intersects at least a precipitate with an intersection containing a circle of radius ηr.
Proof. If we apply a rotation, we can assume that π = {x 2 = ̟} and that the X i 1 , X i 2 are still independent and identically distributed according to the uniform distribution. The probability that the plane π intersects at least a precipitate with an intersection containing a circle of radius ηr, is bigger then the probability that the plane π intersects infinitely many precipitates with an intersection containing a circle of radius ηr. Let us denote this event by A. As the X i 2 are independent so are the events A i . Moreover, we have Hence the Borel-Cantelli Lemma applies and it follows that P (A) = 1.
Knowing that we almost surely intersect a precipitate, we can start with the proof of theorem 2.3. The proof is based on the ideas in [7].
Proof of theorem 2.3. Using lemma 3.2, we know that we almost surely find a precipitate in the slip plane, that contains a sphere of radius ηr and center (x 1 , x 2 , y), such that ηr ≤ 1 ϕ * . If we only focus on this one precipitate, we can treat it as if it was a periodic arrangement of precipitates with radius ηr and a fixed spacing s ∈ [1, √ 2] depending on the rotation of the plane. We are now going to construct a periodic, stationary supersolution v : [−s, s] → R with periodic boundary conditions for all F ≤ F 0 . In return, this gives us a lower bound to the critical pinning threshold F dis ≥ F 0 . We know that if v satisfies the inequality if the function v is inside (−ηr, ηr) bounded by ηr (this is guaranteed by the fact, that ηr ≤ 1 ϕ * ), then after applying a rotation R : R 2 → π, we have This construction works if ϕ * − F ≤ 1 ηr , F ≤ 1 s−ηr and if the mean curvature at x = ±ηr is negative. Therefore we get a third condition, namely By choosing F , such that max{ϕ * − 1 ηr , 0} ≤ F ≤ min{ϕ * ηr s , 1 1−ηr }, we have a well defined v, which solves equation (3) in the viscosity sense. As this construction works for any η ∈ (0, 1), we conclude that the critical pinning force satisfies F dis ≥ min{ √ 2 −1 ϕ * r, 1 1−r } by using the worst estimate on the spacing.

Upper bound for the pinning threshold of twin boundaries
We want to establish an upper bound for the pinning threshold of twin boundaries. To achieve this goal, we are going to find a lower bound for the forces F , such that the interface does not get pinned. In contrast to equation (1), we are looking at the following more general partial differential equation This nonlocal partial differential equation obeys a comparison principle [11]. Based on ideas established in [10], we are going to construct a solution to the problem 0 ≤ −(−∆) α w 0 (x) + min y ϕ(x, y) + F − τ, for some τ > 0. Then the function w = τ t + w 0 is a non stationary subsolution, which we are going to use to establish lower bounds for the critical pinning force. We start by constructing an auxiliary function, which we will use to define w 0 .  Proof. For every n, m ∈ N denote by s n,m := sin(πnx + πmy) and c n,m := cos(πnx + πmy) the eigenfunctions of the negative Laplacian in [−1, 1] 2 with periodic boundary conditions. Moreover we define the eigenvalue to s n,m and c n,m as λ n,m := π 2 (n 2 + m 2 ). We are now going to compute the Fourier series of g. As g is a radially symmetric function, we have g, s n,m L 2 = 0. Moreover, note that a −a a −a c n,m (x, y) dx dy = 2 π 2 nm (cos(πan − πam) − cos(πan + πam)) = 4 π 2 nm sin(πan) sin(πam), which allows us to compute Hence the Fourier series of g is given by Assume that u ∈ L 2 ((−1, 1) 2 ) with periodic boundary data. Hence, we can represent u by its Fourier series u = n,m∈N u c n,m c n,m + u s n,m s n,m . This leads to (−∆) α u(x) = n,m∈N λ α n,m u c n,m c n,m + λ α n,m u s n,m s n,m .
This proves the lemma.
Lemma 3.4. The function u from lemma 3.3 has the following L ∞ bounds depending on α, where C(α) is a constant depending badly on α. This means in our case, that C(α) → ∞ as α → 0 and α → 1.
Lemma 3.5. Let F > 4R 2 ϕ * and τ := F − 4R 2 ϕ * > 0, then there exists a stationary solution w 0 : which is maximal among all solutions with w 0 ≤ 0. Moreover, we get Proof. Let g(x) := −ϕ * χ [−R,R] 2 (x)+ F − τ , then g has zero average over [−1, 1] 2 and we can apply Lemma 3.3 with Therefore there exists a solution u 0 that is bounded and has vanishing average. Now, we set w 0 := u 0 − sup{u 0 }, which leads to desired solution. The estimate on the minimum of w 0 is a consequence of lemma 3.4.
Then we have In particular, we have w(t, ·) ≥ 2R, i.e. the solution exited the precipitate, if and at this timepoint t 0 , we get an estimate on the distortion of the solution caused by the precipitate Proof. Note that the function w(x, t) := w 0 (x) + tτ satisfies w(x, 0) ≤ 0 and is therefore a subsolution by construction. Hence, by the comparison principle we get To get an upper bound, we note that the function w(x, t) := F t satisfies w(·, 0) = 0 and Again, by the comparison principle, we obtain w ≤ w and therefore we get Hence, the claim follows.
As we know how much a precipitate distorts the solution, we can find a necessary condition on F such that the solution does not enter the next precipitate before completely leaving the previous one.
Proof. Let us start by showing ∂ t w ≥ 0. Note that at t = 0 we have a flat interface, that does not intersect any precipitates for a short amount of time. Therefore we know that, ∂ t w = F in [0, ǫ]. Denote by t 0 and x 0 the first time and point, such that ∂ t w(x 0 , t 0 ) = 0. Then we know that ∂ t w(t 0 , ·) has a minimum at x 0 . On the other hand the second derivative at this point satisfies by the maximum principle for the fractional Laplacian. We are now going to argue, why a solution exists. Let t 1 be the first time w intersects a precipitate, by lemma 3.6 the applied force is big enough that we can move trough this precipitate. Let t 2 be the first time, when w does not intersect the precipitate anymore, i.e. w(·, t 2 ) ≥ 2R+w(·, t 1 ). By corollary 3.7, we know that our solution w is confined between the planes {y = 2R + w(·, t 1 )} and {y = 2R + w(·, t 1 ) + 2R 1−β − 2R}. According to assumption 2.1, we know that almost surely the next precipitate lies behind {y = 2R+w(·, t 1 )+2R 1−β −2R}. Now we define flat function w f (·, t 1 ) := 2R + w(·, t 1 ) and note that w f (·, t 1 ) ≤ w(·, t 1 ). We let w f evolve according to our partial differential equation and can conclude, using the comparison principle, that w f ≤ w. Applying an isometry to the whole torus, we can use the above argument to conclude that w f and therefore w will pass trough the next precipitate. Iteratively this argument leads to a non stationary subsolution.
Theorem 2.4 is now a simple consequence of the previous theorem.
Proof of theorem 2.4. In the driving equation for twin boundaries (1), the fractional Laplacian is the half Laplacian. This means that α = 1/2 and our results above are valid. Hence, we know, that the interface does not get pinned if F twin ≥ 4ϕ * max{h(R), 1}R 2 .

Conclusion and Future Work
We showed that dislocations are more likely to get pinned than twin boundaries in crystals with a well spaced, quasi periodic arrangement of spherical precipitates. Future work would entail looking at more general arrangements. It is a hard problem to create non stationary subsolutions to a random arrangement of precipitates (for instance generated by a Poisson process), as we cannot ignore precipitates that lie outside of a given grid, like N. Dirr, P. Dondl and M. Scheutzow did in [10]. One possible solution to this would be to show that for a given concentration there is an arrangement of precipitates that maximizes the required critical force F twin . We can then only retrieve an estimate for the upper bound of F twin for this arrangement.