## Abstract

The pinning of interfaces and free discontinuities by defects and heterogeneities plays an important role in a variety of phenomena, including grain growth, martensitic phase transitions, ferroelectricity, dislocations and fracture. We explore the role of length scale on the pinning of interfaces and show that the width of the interface relative to the length scale of the heterogeneity can have a profound effect on the pinning behaviour, and ultimately on hysteresis. When the heterogeneity is large, the pinning is strong and can lead to stick–slip behaviour as predicted by various models in the literature. However, when the heterogeneity is small, we find that the interface may not be pinned in a significant manner. This shows that a potential route to making materials with low hysteresis is to introduce heterogeneities at a length scale that is small compared with the width of the phase boundary. Finally, the intermediate setting where the length scale of the heterogeneity is comparable to that of the interface width is characterized by complex interactions, thereby giving rise to a non-monotone relationship between the relative heterogeneity size and the critical depinning stress.

## 1. Introduction

The pinning of phase boundaries and domain walls by heterogeneities like solutes and precipitates plays a critical role in determining the kinetics of grain growth, martensitic phase transitions, ferroelectricity and many related phenomena [1–4]. Pinning by heterogeneities also plays a critical role in dislocation dynamics [5,6], adhesion [7,8] and fracture mechanics [9]. Therefore, pinning of propagating boundaries and discontinuities has been the topic of much investigation (e.g. [10] and references therein). Much of this literature regards the interfaces as being sharp (curves or surfaces). However, in many situations, phase boundaries, domain walls and dislocation cores have a well-defined structure and a finite thickness, and this thickness may be comparable to or larger than that of the heterogeneity. For example, ferroelastic phase boundaries may have a width of the order of tens of nanometres, 10–100 times larger than pinning vacancies and dopants [11]. Dislocation cores can extend over many Burgers vectors making them much larger than solutes or solute clusters [5]. And magnetic domain walls may be as thick as micrometres [12]. This current work explores the role of relative length scales of the interface and heterogeneity on pinning.

We focus on phase transformations in an idealized one-dimensional setting. However, the model we use—and consequently the results we obtain—are relevant to the study of other phenomena, including ferromagnetism, ferroelectricity, dislocation dynamics and shock-wave propagation. We start from a widely studied model—variously described as the Landau–Ginzburg model [13,11], the augmented model [14] and the van der Waals model [15,16]. This is based on a free energy density which consists of two terms. The first is a strain-dependent term that has a non-convex double-well structure corresponding to two stable phases with distinct stress-free strains. The second is a term that depends quadratically on the strain gradient (variously described as the capillarity, exchange and interfacial energy). The dynamics of the system in the presence of dissipation (viscosity) and inertia describes the behaviour of phase-transforming materials.

The capillarity introduces a length scale in the problem, and this length scale may be viewed as the length scale of the phase boundary as we shall presently see. At scales that are large compared with this length scale, the macroscopic or outer equations admit solutions with moving discontinuity in strain where the strains jump from a value in one phase to that in another. These discontinuities may be regarded as phase boundaries. However, the velocities of these phase boundaries are not uniquely determined by the macroscopic equations, and one needs a closure relation. Abeyaratne & Knowles [17] showed that this closure relation may be specified as a *kinetic relation* between the velocity and a thermodynamic driving force across the phase boundary. This relationship describes the evolution of the phase boundary, and thus the hysteresis.

At the other limit of small length scales, the equations admit travelling wave solutions (also described as a soliton-like or kink solution) in infinite domains that connect the two stable phases through a finite transition zone [14–16]. Such solutions are interpreted as phase boundaries. Further, Abeyaratne & Knowles [14] showed that the velocity of the travelling wave depends on the end states in a very special combination, one that represents the macroscopic thermodynamic driving force. Furthermore, this relationship between the thermodynamic driving force and the velocity depends only on a particular ratio of capillarity and viscosity coefficients in the limit of vanishing capillarity and viscosity. Thus, this relation may be viewed as the kinetic relation required by the outer model.

In this paper, we adapt this model to heterogeneous media by assuming that the free energy, the capillarity and viscosity coefficients, and density depend on position in a periodic manner. We seek to understand the kinetics of phase transformations at length scales large compared with both the width of the phase boundary and the length scale of the heterogeneities. We naturally have three cases depending on the ratio of the two material length scales (phase boundary width and heterogeneity).

In one extreme situation—*Case L*—where the scale of the heterogeneity is large compared with the scale of the phase boundary (equivalently, sharp interface), we find through matched asymptotic analysis, that the limiting model is indeed the widely studied case of interface pinning. Specifically, we obtain the model studied by Bhattacharya [18] (also [19]). The phase boundary sees each heterogeneity, and thus is pinned till the driving force is large enough to overcome the worst heterogeneity. This in turn results in a stick–slip behaviour with significant hysteresis.

In the other extreme situation—*Case S*—where the scale of the heterogeneity is small compared with the scale of the phase boundary (equivalently, wide interface), we find through homogenization, that the limiting model is very different. The interface averages over a number of heterogeneities and thus may not be pinned in a significant manner. Thus, the resulting macroscopic behaviour may not display any stick–slip behaviour and can have extremely small hysteresis.

In the intermediate case—*Case G*—where the two length scales are comparable, there is close interaction between the phase boundary and the heterogeneity resulting in a range of behaviours. We show through numerical simulation that the amount of hysteresis can vary widely, and relation between the relative heterogeneity size and the critical depinning stress can in fact be *non-monotone*.

An insight that comes from this analysis is that a potential route to making materials with low hysteresis is to introduce heterogeneities at a length scale that is small compared with the width of the phase boundary. This is seemingly counterintuitive because pinning and increased hysteresis is well documented in the widely studied sharp interface (large heterogeneity) case. Our result shows that this is not universal and engineering the appropriate combination of length scales provides an important opportunity.

While a direct comparison with experimental observations is beyond the scope of this idealized model, the results are consistent with various experimental observations. The role of pinning in the large heterogeneity case is well documented [10]. Recently, Lisfi *et al.* [20] observed almost negligible ferromagnetic hysteresis in an annealed Co_{65}Fe_{35} alloy with heterogeneities (precipitates) at the scale of nanometres and ferromagnetic domain walls at the scale of 100 nm. The recently discovered shape-memory alloy [21] that shows remarkably low hysteresis and functional fatigue (stress–strain curve unchanged over 10^{7} cycles) contains nano-precipitates. In dislocations, it has long been known that solution hardening is effective in body-centric cubic alloys with compact cores, but less so face-centred cubic alloys with an extended core [5]. Finally, Rauls [22] studying a model particulate composite found that inclusions much smaller than the natural shock front width have little effect on shock structure and speed while comparable or large inclusions increased the rise time, increased the internal reflection and slowed the shock.

We provide a detailed description of the model in §2a and the scalings in §2b. We recall the results for the homogeneous case in §2c. We discuss the Case L of large heterogeneity or sharp interface in §3, the Case S of small heterogeneity or wide interface in §4 and the general Case G in §5. We obtain stress–strain hysteresis curves for these cases in §6. We conclude in §7 with a brief summary and discussion.

## 2. Model

### (a) Model

Let *u*(*x*,*t*) denote the longitudinal displacement of the particle *x* at time *t* in the elastic bar with inhomogeneities. The strain *γ* and the particle velocity *v* are defined as *γ*=*u*_{x} and *v*=*u*_{t}, respectively. The balance of momentum, assuming a uniform cross section, is given by
2.1
where *ρ* (≥0) is the reference density (mass per unit reference volume) while *σ* is the stress (force per unit area). The stress is related to the kinematic quantities like strain, strain rate and strain gradient by the constitutive relation
2.2
where describes the nonlinear elasticity, *ν* (≥0) is the viscosity and *κ* (≥0) is the capillarity or strain-gradient coefficient. We can combine these equations above into one governing equation,
2.3
We note before proceeding that these equations are identical to the time-dependent Landau–Ginzburg equations with inertia if we regard that the is obtained from a Landau potential *W*: [13,11].

We are interested in heterogeneous media, and we assume for convenience that it is periodic with some period *h*. So *ρ*, , *ν* and *κ* are taken to be 1-periodic functions of *x*/*h*, and our governing equation becomes
2.4

We are specifically interested in phase-transforming materials. So we take (with *y* fixed) to be an increasing–decreasing–increasing function with the increasing branches describing the two phases as illustrated in figure 1. This is equivalent to assuming that the Landau potential has two wells. We use the following two examples to illustrate our results.

*Material 1*. Set
2.5
where *s* is a material modulus and *a*_{0},*a*_{1},*a*_{2} are dimensionless quantities representing the stress-free strains of the phases. If *a*_{1}=*a*_{2}=0, then
2.6
We recognize the term in parenthesis on the right as the widely used fourth-order Landau energy with two stable phases with stress-free strain equal to ±*a*_{0}. The terms with coefficients *a*_{1},*a*_{2} modify the stress-free strains depending on position *y*. At position *y*, the material has two stable phases with stress-free strains
2.7
Note that *a*_{1} modifies the stress-free strain in a symmetric manner while *a*_{2} does so in an asymmetric manner.

*Material 2*. Let *r*(*y*)=*y* mod 1 be the difference between *y* and the largest integer smaller than *y*. It is clearly periodic with unit period. Set
2.8
where the material modulus *s* sets the energy scale. At position *y*, the material has two stable phases with stress-free strains
2.9

### (b) Scaling

It is convenient to non-dimensionalize this equation as follows:
2.10
for dimensional constants λ, *c*, *s*, *r*, *n* and *k*. It is also convenient to divide by *R* and define
2.11
and introduce
2.12
We obtain
2.13
It is convenient to change the notation and write
2.14
The non-dimensional constant *ρ*_{0} determines the importance of inertia. In particular, *ρ*_{0}=0 is the overdamped or quasi-static case. *h* is the periodicity of the microstructure or size of the heterogeneity. *w* determines the width of a stationary phase boundary and the non-dimensional ratio *α* determines how the phase boundary width changes with propagation [14]. We take this last ratio to be of order unity for simplicity.

We are interested in understanding the behaviour of the body that is large compared with the heterogeneity and for the propagation of the phase boundary over distances much larger than its width. In other words, we are interested in the asymptotic limit . However, they can do so at different rates so we have the two asymptotic cases,

*Case L*. Large heterogeneity or sharp interface,*h*≫*w*;*Case S*. Small heterogeneity or wide interface,*h*≪*w*;and the general case,

*Case G*. Comparable heterogeneity and interface,*h*∼*w*.

### (c) Homogeneous material

We recall the results of the homogeneous case
2.15
following Abeyaratne & Knowles [14] (also [15,16]). We note that letting gives a problem of singular perturbation. Formally, letting yields
2.16
which has solutions *u* that are continuous in space and time, but suffer discontinuities in its derivatives *γ* and *v* along *x*=*s*(*t*). These discontinuities can be either elastic waves where end states *γ*^{±} stay along the same rising branch of the stress–strain curve, or phase boundaries where the end states *γ*^{±} are along different rising branches of the stress–strain curve (figure 1). It turns out that the jump conditions (the weak form) are insufficient to specify the propagation speed of a phase boundary and one needs additional information. It is possible to show that the rate of dissipation across the shock is given by the expression
2.17
is called the *thermodynamic driving force* and is the propagation velocity of the phase boundary. Thus, the thermodynamic driving force is the force conjugate to the phase boundary velocity. Therefore, the additional information is postulated to be prescribed in the form of a kinetic relation,
2.18
It is possible to derive the kinetic relation by looking at the inner expansion by setting *ω*/λ=1 in (2.15), i.e.
2.19
Now, given any *γ*^{±}, *V* that satisfy the jump condition, (2.19) has travelling wave solutions *u*=*u*(*x*−*V* *t*) where *γ* is monotone and as . Remarkably, the velocity *V* depends only on the ends states through the thermodynamic driving force defined by the end states *γ*^{±} [17]. Further resulting kinetic relation *V* (*f*) is monotone, passes smoothly through the origin and takes values only in the first and third quadrants (i.e. *f* and *V* share a sign).

In the overdamped situation when *ρ*_{0}=0, we conclude from (2.19) that . Therefore, for our stress–strain curve, *γ*^{±} are determined by *σ* (figure 1), and
2.20
It is also true that *f* is monotone in *σ*. We call the value of *σ* for which *f*(*σ*)=0 the *Maxwell stress* *σ*_{M}. Geometrically, *f* is simply the difference between the shaded areas A and B, while *σ*=*σ*_{M} when the areas of A and B are equal. It follows then that the phase boundary velocity is determined by the stress, *V* =*V* (*σ*) with
2.21
For the examples described in §2a, the Maxwell stresses are given below.

*Material 1.* In the special case when *a*_{2}=0, is an odd function satisfying for any fixed *y*. It follows then that *σ*_{M}=0. If *a*_{2}≠0, then *σ*_{M} is non-zero and depends on *y*. Unfortunately, the explicit relation is too cumbersome to calculate explicitly, but is readily computed numerically. The results are plotted in figure 2 for *a*_{0}=0.06, *a*_{1}=0.01 and *a*_{2}=0.01; here, *σ*_{M} ranges from −0.261 to 0.187.

*Material 2*. is an odd function satisfying for any fixed *y*. So, *σ*_{M}=0.

## 3. Case L: large heterogeneity or sharp interface, *h*≫*w*

We use matched asymptotic analysis to study (2.14) in this case.

First we examine the inner-most problem λ=*w*. Since *h*≫*w*=λ,
3.1
where the macroscopic position *x*/(*h*/*w*) is a parameter denoted by ⋅. We look for travelling wave solutions analogous to (2.19) and obtain a kinetic relation
3.2
for each mesoscale position *x*.

We then examine the situation λ=*h*. Since *w*≪*h*=λ, we obtain
3.3
This equation, along with the kinetic relation (3.2), describes the behaviour of phase boundaries at this scale.

Finally, we look to the macroscopic scale λ≫*h*≫*w*. We obtain this by homogenizing equations (3.3) and (3.2). This remains open in the general case with inertia. However, the overdamped case *ρ*_{0}=0 has been studied by Bhattacharya [18]. We obtain a macroscopic behaviour which can be quite different from that of the mesoscale behaviour. Specifically, the interface can get pinned and one can obtain stick–slip behaviour. To see this, recall that for an applied stress *σ*, the propagation velocity is *V* (*σ*,*x*) and the Maxwell stress is *σ*_{M}(*x*) at the point *x*. Therefore, if or , then the overall phase boundary velocity is the harmonic mean of the point-wise velocity. If these conditions are not met, then the phase boundary becomes stuck and the overall velocity is zero. We conclude that
3.4
Thus heterogeneities that are large compared with the phase boundary can increase the amount of hysteresis.

*Material 1*. Recall Material 1 described in (2.5) and the corresponding Maxwell stress described in §2c. When *a*_{0}=0.06, *a*_{1}=0.01 and *a*_{2}=0.01, the effective velocity is zero in the range *σ*∈[−0.261,0.187].

## 4. Case S: small heterogeneity or wide interface, *h*≪*w*

In this case, it is natural to study the homogenization at the scale λ=*w*. The resulting equation is given in (4.19). Set *ε*=*h*/*w* so that (2.14) becomes
4.1
We introduce a fast variable *y*=*x*/*ε* and look for a solution *u*^{ε} of the form
4.2
where the first term *u*^{0} normally represents the homogeneous part of the solution and all further terms *u*^{i}, *i*=1,2,3,…, describe local variation on the scale of heterogeneities.

Furthermore, we have
4.3
We substitute (4.2) in (4.1) and collect terms in powers of *ε*. Some of terms are easy: up to ,
4.4
The evaluation of requires some care. To evaluate this, we use the Taylor expansion of in the variable *γ*, which leads to
4.5
where
4.6
and prime denotes differentiation with respect to *γ*. It follows:
4.7
where
4.8
Putting all these together, up to ,
4.9
We substitute (4.4), (4.9) in (4.1). Gathering terms with similar powers of *ε*, we obtain the following:
4.10
4.11
4.12
From (4.10) to (4.12), we conclude
4.13
Thus (4.9) becomes
4.14
So continuing to collect terms in (4.1), we have
4.15
and
4.16
Since is periodic in *y*, averaging (4.15) over *Y* results in
4.17
From (4.16), we observe that and are periodic functions of *y*, so the averages of and over *Y* are equal to zero. Moreover, , and are functions of *x* and *t* only, i.e. independent of *y*. Averaging of the last two terms in (4.16) over *Y* gives
4.18
where we use (4.17) for the final identity.

Therefore, averaging (4.16) over *Y* , we obtain our homogenized equation for *u*^{0}
4.19
where the effective constitutive quantities are
4.20

Notice that the homogenized equation (4.19) is exactly like the equation for a homogeneous material (2.15) but with an effective stress–strain relation which is the spatial average of the stress–strain relations and with new effective viscosity and capillarity coefficients. In particular, since the effective stress–strain relation is the average of many stress–strain relations, it can have a range of behaviors including an up–down–up like the point-wise stress–strain relation or even a monotone stress–strain relation. This has many important consequences at the macroscopic scale.

If the effective stress–strain relation remains up–down–up, then one has phase boundaries at the macroscopic scale (λ≫*h*) and travelling wave solutions to (4.19) determine the kinetic relation. However, the nature of this kinetic relation can be quite different from the point-wise ones, resulting in either more or less mobile phase boundaries and either decreased or increased hysteresis, respectively. If the effective stress–strain relation is monotone, then the homogenized equations have no phase boundaries and the material becomes *supercritical* and can go smoothly from one phase to another at the macroscopic scale. A wide range of other behaviours are also possible. In short, heterogeneities on a small scale provide a means for tuning the hysteresis of the material.

We demonstrate this with the following examples.

*Material 1*. Recall Material 1 described in (2.5). For this material,
4.21
This remains up–down–up and one has macroscopic phase boundaries.

*Material 2*. Recall Material 2 described in (2.8). For this material,
4.22
Note that
4.23
and thus, the stress–strain relation is monotone. Thus, one does not expect phase boundaries: instead one transforms smoothly from one phase to another.

## 5. Case G: comparable heterogeneity and interface, *h*∼*w*

We examine this case through numerical simulation without inertia, *ρ*_{0}=0. We choose λ to be some macroscopic scale and set *h*↦*h*/λ, *w*↦*w*/λ. We also take *κ*=*ν*=1 and integrate over *x* so that the governing equation is
5.1

### (a) Phase boundary evolution

We consider a domain (0, 1) and apply a prescribed stress *σ* at the boundary along with the natural boundary conditions *γ*_{x}=0. We start with an initial condition with a phase boundary at the centre of the domain, i.e. *γ*(*x*)=sgn(*x*−0.5)*a*_{0}. We use an implicit finite difference method to solve (5.1).

*Material 1*. Recall Material 1 from (2.5). We take *a*_{0}=0.06, *a*_{1}=0.03, *a*_{2}=0. The applied stress is set to be *σ*=0.2315. *α* and *w* are taken to be 0.215 and 4.303, respectively. Figure 3 shows snapshots in time for strain versus position for various sizes of heterogeneity *h*. For the larger heterogeneity, the phase boundary remains stuck close to the original position, but develops oscillations in space corresponding to those in stress-free strains of the underlying material. Thus, even though a stress is applied that favours one phase to another, the heterogeneities pin the phase boundary. For smaller heterogeneities, the phase boundary propagates in a pulsatile manner (*γ*(*x*−*h*,*t*)=*γ*(*x*,*t*+*T*) for some fixed *T*). The oscillations decrease with decreasing heterogeneity size and the speed increases but ultimately saturates. Figure 4 shows the corresponding snap-shots for the homogenized material (4.21) derived in §4. Here one has steady propagation with a smooth profile. Importantly, we see that results for the small heterogeneities converge to that of the homogenized model.

*Material 2*. Recall Material 2 from (2.8). We take *a*_{0}, *α* and *w* to be 0.06, 0.215 and 4.303, respectively as in Material 1. The applied stress is assumed linearly increasing with time, i.e. *σ*=0.023*t*. Figure 5*a*–*c* shows snapshots of the phase boundary evolution for different heterogeneities while figure 5*d* shows that for the homogenized material. Since the effective stress–strain relation in this case is monotone (cf. (4.22)), this material does not support phase boundaries. Therefore, we see that the initial discontinuity gradually decreases. Further, the strain is oscillatory for large *h* and gradually converges to the homogenized solution with decreasing *h*.

Figure 5*e*–*h* shows the evolution if we start from a different initial condition *γ*=0.

### (b) Quasi-steady propagation and effective kinetic relations

The results above show that phase boundaries potentially propagate in a heterogeneous media in the quasi-steady pulsating form satisfying
5.2
where *T* is the time period and *h*/*T* is defined as the effective interface velocity *V* . Therefore, in this section, we study solutions to (5.1) of the form (5.2).

Given a period *T*, we seek the solution *γ*, *σ* satisfying (5.2) and (5.1) as follows. We discretize the domain (0, 1) using finite differences with spacing such that each period *h* has *I* grid points and the total number of grid points is *M* over the domain. We divide our period *T* into *N* intervals. We denote to be the strain at *x*_{i} and time step *j*.

— Pick an initial guess for

*γ*(*x*_{i},0) and*σ*.— Obtain the corresponding solution for

*γ*(*x*_{i},*T*) by an implicit method: given for some*j*, we obtain by solving (5.1) or 5.3— Iterate over and

*σ*to minimize the objective function 5.4— Verify that the optimal to ensure a pulsating wave.

We repeat this procedure for various *T* to obtain a relation between the macroscopic applied stress *σ* and the average phase boundary velocity *h*/*T*. We would like to convert this relation to a kinetic relation between the average phase boundary velocity and the thermodynamic driving force. We define the latter by recalling its role in the dissipation. In (5.1), we recognize the dissipation to be due to viscosity and define the instantaneous rate of dissipation to be
5.5
Multiplying (5.1) by *γ*_{t} and integrating over the domain, we can show that
5.6
Since this varies with time, we define the time-averaged rate of dissipation to be . We obtain
5.7
We expect the second term to be small for the pulsating wave solutions, and therefore drop it. Finally, this time-averaged dissipation is set equal to the product of the effective driving force and the average velocity. Therefore, we obtain the effective driving force to be
5.8
We use this to obtain the effective kinetic relation for heterogeneous media.

We now go through a series of numerical examples, all with Material 1 defined in (2.5).

#### (i) Example of stick–slip

Set *a*_{0}=0.06, *a*_{1}=0.01 and *a*_{2}=0.01 in all the tests. Set *α*=0.215, *w*=4.303 in Case S and Case G. Set *α*=0.304, *w*=0.152 in Case L. Recall that the Maxwell stress is non-zero and depends on position (figure 2). Therefore, we need a critical force in Case L. Figure 6 shows the effective kinetic relation as well as the effective phase boundary velocity versus applied stress for the material. Note that the stick–slip behaviour persists even in Case L when *h*=1 (and the ratio of heterogeneity size and phase boundary width being 80). The critical stress equals the maximum Maxwell stress which is around 0.26 and occurs at *x*/*h*=0. Also note that this finding is consistent with the prediction in figure 2. The numerical simulation of Case S shows no stick–slip behaviour in agreement with the expectation of the homogenized model.

#### (ii) Example of anomalous stick–slip

Set *a*_{0}=0.06, *a*_{1}=0.01 and *a*_{2}=0 in all the tests. Set *α*=0.215, *w*=4.303 in Case S and Case G. Set *α*=0.304, *w*=0.152 in Case L. Note that the Maxwell stress is equal to zero at each point and also for the averaged material. Therefore, we do not expect any stick–slip behaviour in either Case L (§3) or in Case S (§4). However, figure 7 shows that when *h*=0.05, the kinetic relation does not pass through the origin and thus displays a stick–slip behaviour.

#### (iii) Parameter study

Figure 8*a*,*b* shows the dependence of the kinetic relation on the parameters *a*_{1} and *w*, respectively (*a*_{2} is zero in these cases) for Case G. An increase in the parameter *a*_{1} increases the critical driving force but apparently not the eventual slope of the kinetic relation. An increase in the parameter *w* decreases the critical driving force, but increases the limiting slope due to the increasing role of the viscous term.

Figure 9 summarizes the relation between *w* and the critical stress *σ*^{⋆} in the case of Maxwell stress being identically zero. The critical stress is zero in the two limiting cases of () and (), but non-zero for intermediate values.

## 6. Hysteresis

Hysteresis occurs as a consequence of the ‘stick–slip’ behaviour. In order to study the relation between the hysteretic behaviour in phase transitions and material heterogeneities, we minimize the inertia effect by applying a slow cyclic loading and set the viscous term to be zero. The results are plotted in figure 10, characterized by three cases in terms of the relative size of the material heterogeneity and width of phase boundary. Note that the parameters *a*_{0}, *a*_{1}, *a*_{2} and *w* in figure 10*a*,*b* are chosen to be the same as those used in figures 6 and 7, respectively. In the sharp interface region (Case L), the hysteretic behaviour is caused by the varying Maxwell stresses in the heterogeneous material, with the critical depinning stresses given by the maximum and minimum Maxwell stresses. In particular, a constant Maxwell stress will result in a hysteresis-free behaviour, as shown in figure 10*b*. On the other hand, no rate-independent hysteresis is found in the homogenized region (Case S) as the microstructure of the material becomes finer and finer. According to the homogenized result, the effective constitutive relation is simply the average of the stress function over the periodic structure. In the intermediate region (Case G), a less substantial hysteresis is observed regardless of the landscape of the Maxwell stresses in the local coordinate.

## 7. Discussion

In this work, we have used a one-dimensional model of phase transformations to study the role of length scales on the pinning of interfaces. We have shown that when the heterogeneity is large, the pinning is strong and can lead to stick–slip behaviour as predicted by various models in the literature. However, when the heterogeneity is small, we find that the interface may not be pinned in a significant manner. Finally, we have shown that the intermediate setting where the length scale of the heterogeneity is comparable to that of the interface width is characterized by complex interactions, thereby giving rise to a non-monotone relation between the relative heterogeneity size and the critical depinning stress. An insight that comes from this analysis is that a potential route to making materials with low hysteresis is to introduce heterogeneities at a length scale that is small compared with the width of the phase boundary. This is consistent with recent experimental observations.

We conclude with a comment on the relation between dynamic and quasi-static phase boundary formulations. The kinetic relations of Case S (i.e. the homogenized case) are shown in figure 11*a*. In the regime of relatively small *σ*, the kinetic relations are not distinct. In the quasi-static setting, the applied stress increases with the interface velocity and gradually converges to a critical value. While in the dynamical case, the stress can go far beyond this critical value. This finding is consistent with [17], which claims that the admissible region of the kinetic relation in the quasi-static case is a subset of the dynamically admissible region for trilinear materials based on elasticity theory. Similar results are observed in Case L where the effective kinetic relations are plotted in figure 11*b*.

## Competing interests

We declare we have no competing interests.

## Funding

This work is financially supported by the US Army Research Office through the MURI grant W911NF-07-1-0410 and the US Air Force Office of Scientific Research through the Center of Excellence in High Rate Deformation Physics of Heterogeneous Materials (grant no.: FA 9550-12-1-0091)

## Acknowledgements

We gratefully acknowledge interesting discussions with Manfred Wuttig that motivated this work.

## Footnotes

One contribution of 11 to a theme issue ‘Trends and challenges in the mechanics of complex materials’.

- Accepted November 13, 2015.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.