Micromechanics & emergence in time

Micromechanics & emergence in time

van der Giessen, Erik

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European Journal of Mechanics A-Solids DOI:

10.1016/j.euromechsol.2019.01.003

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van der Giessen, E. (2019). Micromechanics & emergence in time. European Journal of Mechanics A-Solids, 75, 277-283. https://doi.org/10.1016/j.euromechsol.2019.01.003

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Erik Van der Giessen

PII: S0997-7538(19)30007-5

DOI: https://doi.org/10.1016/j.euromechsol.2019.01.003

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To appear in: European Journal of Mechanics / A Solids

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Please cite this article as: Van der Giessen, E., Micromechanics & emergence in time, European Journal

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Micromechanics & Emergence in Time

Erik Van der Giessen

Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, the Netherlands

Abstract

The thrust of this paper is that micromechanics goes beyond homogenization when it is regarded as a multiscale modeling approach. Using metal plasticity as an example, the paper illustrates how macroscopic irreversibility is a natural consequence of emergent behaviour in space and in time. Special attention is given to the currently weak link between discrete dislocation plasticity and continuum crystal plasticity, and how dislocation interactions give rise to power law viscoplasticity. As an outlook it is suggested that, by enlarging its mission to coarse graining in space `and time, micromechanics can play an important role in the understanding and description of new supramolecular materials.

Keywords: micromechanics; multiscale modelling; plasticity; homogenization;

coarse graining

1. Introduction

Micromechanics is traditionally viewed as a branch of solid mechanics in which the effective elastic properties of composite materials are derived from their composition and the properties of the constituents. However, a view in some textbooks carrying the term micromechanics in their title (see, e.g., Mura,

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1987; Nemat-Nasser and Hori, 1993; Dvorak, 2013; Li and Wang, 2018) reveals that the field is more embracing than composites. In the authors’ opinion, mi-cromechanics is a branch of multiscale materials modeling which focuses on the relationship between property and material substructure. In fact, I think that the adjective ‘micro’ should not be taken literally. In a micromechanical

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proach, one zooms into the material structure to the level where the pertinent physical phenomenon takes place; develops a mathematical model of the mech-anism(s) at this scale and predicts what the response is at a larger length scale. For classical elastic composites, this is a single upscaling step but when fracture is involved, all scales above the one where the actual material separation takes

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place are generally important in determining the overall toughness.

Arguably, the most important tool in micromechanics is homogenization. It is well developed for linear elastic materials leading to the classical results for the effective moduli for composites (see, e.g., Nemat-Nasser and Hori, 1993; Dvo-rak, 2013). Homogenization in the geometrically and/or physically nonlinear

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regime is significantly more difficult (e.g., Ponte Casta˜neda and Suquet, 1998),

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and this has led to several computational homogenization techniques (see Geers et al., 2010, for a review). Although the precise interpretation is (or, should be) somewhat different, the term homogenization is sometimes used interchangeably with upscaling or coarse graining.

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In this article, I will argue that coarse graining is, in fact, distinct from homogenization in the sense that the latter refers only to upscaling in space. From a multiscale materials modeling perspective, I will use the example of dislocation plasticity in metals to emphasize that upscaling in space should go hand-in-hand with upscaling in time. The deeper reason for this is Emergence.

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When zooming out from the scale of the periodic arrangement of atoms in crystals to macroscopic metal plasticity, one will come across familiar features and phenomena; yet, the connection between them as emergent phenomena brings insight that, in my experience, is not appreciated widely. A particularly insightful observation is that emergence implies coarse graining in time and

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space, which necessarily leads to irreversibility. In this respect, a few new results will be presented to illustrate how power law viscoplasticity emerges from the collective behaviour of interacting dislocations.

After a brief summary of generic methods from statistical approaches on coarse graining in time, the paper closes with some personal views on challenges

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and opportunities offered when broadening the mission of micromechanics to explicitly incorporate emergence, time coarse graining and irreversibility. These aspects are critically important in biomaterials as well as in promising new materials inspired by nature: so-called supramolecular polymers.

2. Multiscale plasticity

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Plastic deformation in metals has a very interesting history, during which it was studied at the one hand by materials scientists who aimed at unravelling the physical mechanisms and by engineers who needed mathematical models to supported design. In this respect, it is at least interesting to note that seminal works at these two extreme length scales –Von Mises’ theory of plasticity based

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on a yield function versus the profound characterization of a dislocation by Burgers– were written at roughly the same time. Since that time, and driven by requirements on product reliability, miniaturization and sustainability under increasingly harsh conditions, the two approaches developed towards each other, culminating in a comprehensive multiscale picture of metal plasticity. Figure 1

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not only illustrates the key length scales, but is also exemplary in highlighting two main views on scientific thinking about plasticity.

Starting from the macroscopic length scale of a product or device, zoom-ing into the material successively reveals that (i) most engineerzoom-ing metals are polycrystalline, where (ii) plastic deformation in all grains takes place in the

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form of crystallographic slip. Further zooming in reveals that this slip is (iii) mediated by the collective motion of dislocations, each one of which is (iv) a line defect in the atomic lattice inside the grains. This is a reductionist point of view, in which macroscopic plasticity is attributed, in the end, to the smallest entities, viz. dislocations. Satisfying as this understanding may be for a

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rial scientist or a physicist, it does not help the design of a product. For the latter, a micromechanicist would traverse this reductionist’s path in the reverse direction. In the first scale transition from the bottom-up, discrete dislocations emerge out of line defects in the atomic lattice. The dislocations primarily glide

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on specific planes, thus producing slip inside grains. Finally, distributed slip

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inside grains averages out over the polycrystalline aggregate to generate plas-tic strain at the macroscopic level. The last two scale transitions –from single crystal to polycrystal plasticity, and from polycrystal to macroscopic plasticity– essentially involve homogenization of continuum descriptions of plasticity; for this purpose, inelastic versions of classical micromechanics tools such as bounds

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and self-consistent methods have been developed (see, e.g., Hill, 1965; Hutchin-son, 1970), supplemented with computational homogenization techniques, e.g. (Van Houtte et al., 2006). The other scale transitions rely on emergence.

b Burgers Besseling emergence b m(α) s(α) reductionism = (3J2/⇥Y2) 1 ˙⇤p ij := ⇧ ⇧⇥ij ⌧(↵)_{:= m}(↵) i ijs(↵)j ˙(↵) ˙0 := ✓ ⌧ ⌧0 ◆n ˙"pij:= ˙(↵)m (↵) i s (↵) j m(I)

f(I):= b(I)i ijm(I)j

v(I)_{:= f}(I)_/B

Figure 1: Metal plasticity seen at various length scales (starting from the bottom): a single dislocation as a line defect in an atomic lattice; the dynamics of multiple interacting disloca-tions; continuum crystal plasticity; polycrystal plasticity and, finally, macroscopic plasticity. A few length scale cartoons are augmented with key constitutive equations, to emphasize the importance of emergent behaviour. Adapted from (Van der Giessen, 2012, Figure 1).

Since it is not a common notion in Solid Mechanics, emergence may seem somewhat elusive for some readers. In the overview of Fig. 1, however,

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gence can easily be recognized in the way the behaviour at the various scales is described by means of different degrees of freedom. At the smallest scale in our picture, the positions of all atoms are the degrees of freedom, whereas at the next higher level, the atoms have been averaged out to an elastic background continuum and it is the positions of dislocations, as discrete entities, that

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stitute the degrees of freedom. At the next higher length scale, of continuum crystal plasticity, the amount of slip γ on each of the slip systems determines the degrees of freedom, while finally at the macroscopic scale plasticity is de-scribed by a plastic strain tensor. The constitutive equations for the degrees of freedom –as the generalization of the equations of motion of particles in classical

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mechanics– are distinctly different at each of the levels of emergent behavior, as illustrated by the insets in Fig. 1.

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2.1. From discrete dislocation plasticity to continuum slip

While other scale transitions have been studied for a long time and are largely resolved, the transition from the collective behaviour of many dislocations to continuum slip still holds many challenges because it involves true and poorly understood emergence. One of the deep reasons for this, is that this scale transition marks the point where plastic deformation at larger length scales is irreversible and history dependent, whereas dislocations are entities in an elastic background which give rise to emergent inelastic deformations because of the associated displacement discontinuity. A volume average of the total area

Aslip on planes with unit normal m containing a displacement discontinuity

characterized by Burgers vector b gives rise to a plastic strain tensor εpthrough

(Rice, 1970) εp= 1 V Z Aslip 1 2(m⊗ b + b ⊗ m)dA . (1)

The history dependence of plasticity emerges from the fact that the slipped area

Aslip evolves as the dislocations move. The motion of dislocations is, in general,

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a tremendously complex process because of the variety of physical mechanisms. Even the simplest mechanism of dislocation glide leads to a highly non-trivial evolution because of the long-range interactions between dislocations. Impor-tant progress has been made, however, for systems of straight edge dislocations. In two dimensions, edge dislocation lines appear as point objects, typically

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denoted with the symbol > for a negative dislocation and ⊥ for a positive

dislocation. Their ‘charge’ reflects the direction of the actual dislocation lines –into or out of the plane of consideration, respectively– that bound the edges of the slipped area. Thus, a dislocation loop with Burgers vector b bounded

by two infinitely long straight edges is represented in 2D by a pair (−b, +b).

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The analogy with charges extends to a similar 1/r-decay of the dislocation-dislocation interaction forces with distance r as in electrostatics, albeit that the interaction between dislocations is not spherically symmetric but depends on orientation. This orientation dependence makes it possible that screening of dislocations can lead to the formation of patterns such as dislocation walls. That

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is, if dislocations would only be governed by energy minimization; yet, there is kinetics as well which leads to correlations in the dynamics of dislocations.

The most basic source for kinetics in dislocation mechanics is that moving dislocations are subject to a drag force, caused by phonons in the crystal lattice, see e.g. (Hirth and Lothe, 1982). In a simple linear description, a dislocation

gliding with velocity v is subject to a drag force−Bv, determined by the drag

coefficient B. When inertia effects are ignored, the equation of motion for this dislocation then reduces to

v= B−1f, (2)

where f is the so-called Peach-Koehler force acting on the dislocation. The latter is a configurational force that includes the interaction forces with any other dislocation in the system as well as any externally applied force. When

the dislocation under consideration is labelled I _{∈ [1, N], the component of the}

Peach-Koehler force on I along its glide plane m(I) _{is given by}

f(I) = b(I)_i  σext ij + N X J6=I σ(J)_ij   m(I) j , (3)

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where σ_ij(J) is the (long-range) stress field associated with dislocation J. Since

the Burgers vector of edge dislocations is parallel to their slip direction, the

Burgers vector of I can be expressed as b(I) _{= bs}(I)_{, where b is the magnitude}

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of the Burgers vector.

In discrete dislocation plasticity, the dynamics of every dislocation, as gov-erned by Eq. (2), is computed incrementally using (3), with continuous update of the dislocation position. When desired, the instantaneous plastic strain can be computed at any moment from (1). So far, in this discussion, the number

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of dislocation pairs (or dipoles) is constant. In reality, however, dislocations of opposite sign can annihilate when they come within a distance of a few b, and dislocations get generated by the Frank-Read mechanism. Van der Giessen and Needleman (1995) proposed a 2D version of a Frank-Read source as a point from which a dislocation dipole is created when the resolved shear stress at the

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location of this source is equal or larger than its strength τnuc for a sufficiently

long time. The source strength is generally taken to be randomly selected from a normal or a log-normal distribution (Shishvan and Van der Giessen, 2010),

the nucleation time tnuc is often assumed to be a constant on the order of

nanoseconds. The second timescale in discrete dislocation dynamics, is the

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laxation time associated with viscous drag, tdrag = B/τnuc (using the mean

source strength τnuc as a reference stress).

The first step in making a formal scale transition was proposed by Groma (1997) for single slip on parallel slip planes driven by an external shear stress τ . The procedure is rooted in the statistical mechanics of particles and involved (i) the homogenization of the the discrete dislocation distribution into density

fields of dislocations with positive or negative signs –ρ+ and ρ−, respectively;

(ii) the introduction of the total dislocation density, ρ = ρ++ ρ−, and the

density of geometrically necessary dislocations (GND) κ = ρ+− ρ−; and, most

importantly, (iii) incorporation of numerical results about dislocation shielding to simplify the dislocation-dislocation correlations. The result is a set of coupled evolution equations for ρ and κ. This theory was subsequently transformed into a crystal plasticity theory by Yefimov et al. (2004) by adopting Orowan’s law to determine the (continuum) slip rate ˙γ from the mean dislocation velocity, yielding

˙γ = B−1b2ρ(τ_{− τ}s) , (4)

where the back stress τs follows from Groma’s procedure as the gradient of the

density of geometrically necessary dislocations along the slip direction, τs=

µb

2π(1_{− ν)ρ}

∂κ

∂s. (5)

Here, µ is the shear modulus and ν is Poisson’s ratio of the isotropic elastic background medium. To incorporate the nucleation of dislocations, Yefimov et al. (2004) added a production term f (ρ, κ, ...) to the evolution equations of total and GND dislocation densities, which eventually took the form

˙ρ + B−1b ∂

∂s[κ(τ− τs)] = f (ρ, κ, ...) , ˙κ + B−1b ∂

∂s[ρ(τ− τs)] = 0 .

(6)

As a consequence of the way it is constructed, this crystal plasticity theory is a nonlocal theory with additional boundary conditions associated with the

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tion equations (6). This is one of the notable distinctions from phenomenological

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strain gradient crystal plasticity theories like that of Gurtin (2002) which also have additional boundary conditions but require higher-order stresses. Another one is that the associated material length scale is not a user-chosen fixed value but is a variable determined through the dislocation density 1/√ρ.

The predictions of the Yefimov et al. (2004) crystal plasticity theory have

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been compared directly with the results from discrete dislocation simulations for a number of boundary-value problems. Here, we only briefly review a com-parison of the behaviour of a model composite material with micrometer-size reinforcing particles, suggested by Cleveringa et al. (1997), as illustrated in Fig. 2(a). Under shear parallel to the slip systems in the matrix, discrete

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location plasticity computations reveal localized slip above and below the re-inforcing particles, which is blocked by the impenetrable particles. Figure 2(c) shows that this characteristic flow pattern shown in (b) is reproduced rather well by the crystal plasticity theory summarized above. By contrast, the theory

2w=2 3 h 2h 2hf 2w_f U·=hΓ· U·=hΓ· x₁ x₂ 0.040 0.036 0.031 0.027 0.022 0.018 0.014 0.009 0.005 γ 2.4E-02 2.1E-02 1.7E-02 1.4E-02 1.0E-02 6.9E-03 3.4E-03 0.0E+00 γ(1) (a) (b) (c) (d)

Figure 2: Simple shear of a model composite material with impenetrable elastic reinforcing particles in a hexagonal stacking. The matrix allows for single slip on planes parallel to the

shearing direction x1. (a) The 2w × 2h unit cell is periodic in x1 and contains two particles

of size 2wf × 2hf with hf = 2wf = 0.588h. Plastic slip distribution inside the unit cell

with h = 1 µm at an overall shear strain of Γ = 0.6% according to: (b) discrete dislocation plasticity (visualized through distorted mesh with displacements magnified by a factor of 20)

using a density ρnuc = 55.4 µm−2; (c) statistical-mechanics based theory of Yefimov et al.

(2004) and (d) strain-gradient theory of Gurtin (2002) with a value of the material length fit to the overall stress-strain response obtained from discrete dislocation plasticity (Bittencourt et al., 2003). The dashed rectangles in (c) and (d) trace the boundaries of the elastic particles. Reprinted with permission from (Yefimov et al., 2004; Bittencourt et al., 2003).

by Gurtin (2002) predicts wider slip regions plus narrow zones of slip next to the

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rection of slip yet elongated perpendicular to the slip planes, they resemble kink bands rather than slip bands. These zones are also found with classical crystal plasticity with no effect of GNDs (see, e.g., Bassani et al., 2001), but are not seen in Figs. 2(b)-(c). In the author’s opinion, these kink-like bands are

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facts born out of the the inherent assumption in phenomenological continuum theories that dislocations are available whenever and wherever they are needed to enable plastic flow; in reality, however, dislocations need to be generated. 2.2. Rate dependence

The observation that the flow pattern induced by discrete dislocations is retained in the homogenization approach proposed by Groma (1997) makes it a promising candidate for a proper scale transition from discrete dislocations to crystal plasticity. That is, in space. The approach leaves the kinetics unchanged: the evolution of dislocation densities, see Eq. (6), is governed by the same drag coefficient B as that of individual dislocations. As a consequence, the slip rate is linear in stress, cf. Eq. (4). This is in sharp contrast to the usual framework of

continuum crystal plasticity where the slip rate ˙γ(α)_{is often assumed to depend}

on the resolved shear stress τ(α) _{in a power law of the form}

˙γ(α) ˙γ0 = _τ(α) τ0 n (7)

where ˙γ0 and τ0 are reference strain rate and stress, respectively, and with an

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exponent n that is usually much larger than unity. In fact, when τ0is identified

with the flow shear strength, the exponent n is the reciprocal of the so-called rate-sensitivity parameter m which for typical FCC metals is in the range of 0.03 to 0.3, corresponding to values of n between 3 and 30 (see, e.g., Klopp et al., 1985). In order to approximate time-independent plasticity, researchers

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use values of n > 50 or so (see, e.g., Asaro and Needleman, 1985).

The gap between the linear time response of individual dislocations and the nonlinear response at the crystal plasticity level has only begun to be closed. The first approach by El-Azab (2006) draws on the analogy between systems of interacting dislocations and classical kinetic theory of interacting particles.

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Spatial averaging of edge dislocations in his approach is similar to that proposed by Groma (1997), and El-Azab (2006) proposes to introduce a phase density of

dislocations averaged over a time interval tCG. This coarse-graining time scale is

larger than the “reaction” timescale of the direct interaction between individual dislocations, determined in Eq. (4)–(6) by the drag coefficient B, but smaller

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than the macroscopic timescale. The coarse-graining time scale plays a similar role in time as correlation length does in space. Deng and El-Azab (2010) analyzed the correlation time in results of 3D dislocation dynamics simulations with pre-existing dislocation loops and arrived at a value for the coarse-graining

time tCG ≈ 10 to 30 µs. This is several orders of magnitude longer than the

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relaxation timescale of individual dislocations, tdrag= B/τY= 5.5fs when using

their assumed value B = 5.5× 10−5_{Pa s and using the initial yield strength}

τY = 10 MPa obtained from their simulations. These and the more recent

findings (Xia et al., 2016) cannot be connected directly to the simpler 2D model discussed above, because they are essentially 3D and highlight cross-slip of screw

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dislocation which is not possible in 2D, yet they do demonstrate that the purpose of time coarse graining is to suppress short-time fluctuations.

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Kooiman et al. (2016) did consider a 2D model comprising straight edge dislocations, yet they adopted a different coarse-graining technique. In their approach, which is based on the generic framework (more details in Sec. 3),

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dislocation velocities and their associated (Peach-Koehler) driving forces are sys-tematically coarse grained in space and in time simultaneously. The resulting macroscopic dislocation flux, or slip rate, is of Green–Kubo type and strongly related to the fluctuation theorem of non-equilibrium statistical mechanics. The corresponding macroscopic driving force is the gradient of a free energy

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tive that is stress dependent in a non-linear manner. In fact, Kooiman et al. (2016) were the first to show that the emerging slip rate in single slip at rea-sonable stress levels can actually be fit to a power law of the form (7) with an exponent larger than unity: n = 3.7. Even though they did not pursue this, they

could have extracted a value of ˙γ0 (after selection of an appropriate reference

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stress τ0).

An exponent 3.7 in (7) is a definite improvement over the linear relation (4), but is on the very low end of what seems to be realistic and is lower than what is typically used in crystal plasticity computations. Kooiman et al. (2016) expected that the incorporation of additional mechanisms, such as annihilation

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and generation of dislocations, would increase the value of n. I will now briefly present the results of a preliminary study which seem to corroborate this.

The model, illustrated in Fig. 3a, is similar to that used in Fig. 2a–b, but without reinforcing particles yet with a distribution of point obstacles, which mimic very small precipitates or forest dislocations. Just like the strength of the

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dislocation sources, the obstacle strengths are taken from a normal distribution,

yet with mean values τnuc= 75 MPa and τobs = 100 MPa, respectively. An

as-sumed obstacle density that is twice the source density together with τobs > τnuc

ensures that dislocations can spread out over the entire computational cell, thus preventing localized slip on the plane with the weakest source. Moreover,

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stead of prescribing the overall shear at a certain rate, we impose a certain overall resolved shear stress τ and compute the shear strain rate from the rate form of Eq. (1) and average over a certain time period. This coarse-graining timescale is determined such that the average strain rate has converged. Fig-ure 3b shows the computed average strain rates for eight different realizations

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(that is, different positions of sources and obstacles as well different strengths, though from the same distributions). Focusing attention to the behaviour under

applied shear stresses in the range of τnuc plus/minus the standard deviation

of the distribution (taken to be 0.1τnuc), the stress-dependence of the average

strain rate is found to be rather strongly non-linear, see Fig. 3b. The best fit

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in this stress range to a power law relationship of the type (7) with τ0 ≡ τnuc

yields an exponent of roughly n = 12, see Fig. 3c. This approach allows for the identification of the reference strain rate associated with this reference stress to

be ˙γ0= 3× 104s−1. The reciprocal value of around 3× 10−5s is large compared

to the characteristic time scale tdrag= B/τnuc= 10−4/(75×10−6)≈ 1fs for

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location drag as well as the nucleation time tnuc= 10 ns used in the dislocation

dynamics. On the other hand, it is significantly smaller than the correlation timescale that was obtained in (Deng and El-Azab, 2010). It is not clear at this point whether this distinction in “coarse-grained” timescales is primarily due to differences in the physical mechanisms studied (nucleation and glide of 2D

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edge dislocations versus glide and cross-slip in 3D), a fundamental conceptual difference or both.

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Resolved shear stress (MPa)

Strain rate (1/s) τnuc (b) 60 62 64 66 68 70 72 74 76 78 101 102 103 104

Resolved shear stress (MPa) (log scale)

Strain rate (1/s) (log scale)

˙p ˙0 = ✓ _⌧ ⌧nuc ◆n n⇡ 12 ˙0⇡ 3 ⇥ 104s 1 (c)

Figure 3: Determination of mean plastic strain rate from dislocation dynamics simulations. (a) periodic computational cell containing 50 parallel slip planes, with a distribution of dislocation

sources (blue ellipses) of strength τnucand obstacles (short black lines perpendicular to the slip

planes) with strength τobs(as shown in more detail in the inset). (b) Computed plastic strain

rate as function of the applied (resolved) shear τ : blue curves are the results for each of the eight random realizations of sources and obstacles and the black curve represents the average

including standard deviation. The red curve is a power-law fit over a stress range of ±0.1τnuc

around the mean nucleation strength τnuc= 75 MPa. Graph (c) shows the same results on

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3. Time coarse graining and irreversibility

Multiscale plasticity, as discussed in the foregoing, is an appealing example of emergent behaviour in the context of the mechanics of solids. It tells us that

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upscaling is more than homogenization (in the sense of classical micromechan-ics). The key is that degrees of freedom are “condensed out” into a smaller

number of (coarse-grained) degrees of freedom `and that the information

con-tained in the small-scale fluctuations is not lost but transferred to the larger scale in the form of entropy. This implies that coarse graining induces

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versibility, whereas homogenization does not. Basic examples of the same basic notion include diffusion and viscosity, which were shown in 1905 by Einstein to originate from microscopic, random walks and collisions of particles.

It is beyond the scope of this contribution to go into the technical details, but it does seem appropriate to note that the irreversibility consequence

coarse-graining is expressed in a compelling and elegant way in the generic1

frame-work of non-equilibrium thermodynamics. It was developed originally for

com-plex (polymer) liquids by Grmela and ¨Ottinger (1997) and H¨utter and Tervoort

(2008) have made the first steps towards applying the framework to solids. In essence, generic recognizes that, except at the most microscopic level of description, the time evolution of all degrees of freedom has a reversible contri-bution and an irreversible contricontri-bution to the driving force. These contricontri-butions are governed by the energy E and the entropy S, respectively, and the evolution equations for the degrees of freedom x take the generic form

∂x ∂t = L· δE δx + M· δS δx. (8)

Here, L is a Poisson operator (analogous to the Poisson brackets in classical mechanics) and M is the so-called friction operator. The degrees of freedom x

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are different at different scales, and so are the four building blocks (E, S, L, M) in Eq. (8). Although, in principle, the coarse-graining procedure of (8) is known, application of generic is far from being trivial. Part of the challenge is the selection of the proper coarse-grained variables such that timescale separation is guaranteed and derivation of the coarse-grained versions of the four

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ing blocks. H¨utter and Tervoort (2008) have demonstrated the power of the

generic approach by deriving the tensorial form of Newtonian viscous flow and of Fourier’s law of heat conduction. Subsequently they describe ideas for the derivation of viscoplasticity, but a more or less complete development for crystal plasticity starting from dislocation dynamics was achieved only in the

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work by Kooiman et al. (2016).

Emerging irreversibility plays an extremely important role in materials that, in terms of their mechanical behaviour, are in between fluids and solids, namely polymers and other forms of soft matter. i will give two examples as warming-up to the concluding section of this article.

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The first example is bio-polymer networks, such as those forming the inter-nal skeleton of cells (the so-called cytoskeleton) or the matrix in which cells are embedded in many tissues (the so-called extracellular matrix). These networks

1_{Acronym for “General Equation for the Non-Equilibrium Reversible-Irreversible}

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consist of long semi-flexible polymers (i.e., chains of relatively bulky monomers, which endows them with an axial stiffness as well as with a bending and

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sional stiffness) that are connected by crosslinking molecules. These individual ingredients can be thought of as elastic filaments and elastic connectors, but when immersed in water at a sufficiently high concentration to organize into a network, it yields a gel with viscoelastic properties. There has been much work on modeling the elastic properties of bio-polymer networks such as actin along

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these lines, (see, e.g., Huisman et al., 2007; Picu, 2011; Zagar et al., 2015). This has provided much insight in the role of architecture in the remarkable nonlinear elasticity, but the viscous properties are not well understood. A potential cause for the viscous behaviour of biophysical networks is the viscosity of the water, but this cannot explain the complex frequency dependence of the loss modulus.

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Lieleg et al. (2009) have proposed that the viscoelastic response of reconsti-tuted, or in-vitro, actin networks is largely due to the microscopic interactions between filaments and crosslinkers. We have very recently been able to confirm this by means of a minimalistic model comprising two (actin) filaments coupled by crosslinkers whose dynamics is described by a grand canonical Monte Carlo

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scheme (Boerma et al., 2018).

In Nature, biopolymer networks exhibit much richer behaviour than vis-coelasticity or even viscoplasticity due to crosslink failure (Abhilash et al., 2012) because of the biochemical adaption of the composition to the conditions and because of the presence of molecular motors. Macromolecular chemists are

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rently taking inspiration from Nature to develop supramolecular materials with remarkable mechanical properties. For example, Zheng et al. (2016) have

fabri-cated a supramolecular hydrogel by functionalizing SiO2nanoparticles such that

they co-polymerize with acrylic acid, Fig. 4a. Similar to many hydrogels, this material exhibits enormous deformability (up to a stretch ratio of 15), but it also

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features near-perfect self-healing. The latter is a consequence of the non-covalent nature of the crosslinks in the network, where electrostatic interactions between the oppositely charged particles and the polymer network are able to restore the architecture. Restoration of these bonds is near instantaneous, but requires re-organization of the molecular architecture. This is the reason why the

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ing process illustrated in Fig. 4b took several hours, demonstrating once again a giant timescale gap between macroscopic behaviour and microscopic processes. Currently, the development of these materials is largely based on trial-and-error and time-dependent properties are measured experimentally. Material design for targeted performance requires a thorough quantitative understanding of the

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emergent behaviour across multiple length and time scales. 4. Concluding Remarks

This paper has attempted to create awareness of the role of the triptych

Emergence—Time Coarse graining—Irreversibility in the mechanics of

materi-als. The reader is undoubtedly familiar with classical examples of irreversible

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processes originating from microscopic interactions between particles, such as diffusion and heat conduction, but in solid mechanics, emergence is more hid-den. Unveiling it requires a multiscale view in space `and time, as shown by the example of dislocation-mediated plasticity.

Connecting scales in spatial dimensions has always been the mission of

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www.MaterialsViews.com © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 267

can be protonated in an acidic environment, [ 37 ] _therefore

the PDMAEMA brushes grafted onto SiO 2 nanoparticles

will be protonated during the hydrogel preparation pro-cess, leading to electrostatic interactions between SiO 2 @

PDMAEMA and PAA polymer chains. The morphology of the fabricated hydrogel was investigated by SEM and TEM, and the supramolecular hydrogel shows typical porous structure (Figure S4, Supporting Information), and the silica nanoparticles are well dispersed in the hydrogel (Figure S5, Supporting Information).

The as-prepared supramolecular hydrogel displays a high performance in tensile strength as expected. As shown in Figure 1 , the hydrogel can be stretched to more

than 15 times and recover to its initial state (Figure 1 A,B, Movie S1, Supporting Information), and is tough enough to withstand high deformation in compression without obvious damage (Figure 1 C,D, Movie S2, Supporting Information).Tensile tests were performed to further investigate the mechanical properties of the supramolec-ular hydrogel (Figure 1 E). As the concentration of SiO 2 @

PDMAEMA increases from 0.125 to 0.5 mg mL −1 _{, the high}

elongation at break decreases from 2000% to 1250%. It is worth mentioning that an unusual yielding was observed during the tension test, this may be attributed to the breakage of the electrostatic interactions between SiO 2 @

PDMAEMA and polymer chains during the stretching Macromol. Rapid Commun. 2016, 37, 265−270

Scheme 1. A) The simple process of fabrication of SiO 2 @PDMAEMA/PAA supramolecular hydrogel. B) The as-prepared hydrogels can be

stretched to 15 times as the original length. After cutting to two parts, the hydrogels can self-heal without any external stimuli. C) The robust mechanical properties and self-healing abilities of the hydrogels are probably attributed to the supramolecular network formed by electrostatic interactions between the anionic polymer matrix and the cationic polymers grafted on the surfaces of nanoparticles.

Figure 4: Example of the remarkable mechanical properties that can be achieved by

supramolecular chemistry. (A) Schematic of the fabrication of a hydrogel with

PDMAEMA-functionalized SiO2 nanoparticles that co-polymerize with acrylic acid (AAc) in solution,

resulting in a material with (B) extreme deformability and self-healing properties. When the sample was cut, and the two pieces were brought together for half a day, the stretchability was identical to the as-prepared material. Reprinted with permission from (Zheng et al., 2016).

gence, Time Coarse graining and Irreversibility become an integral part of

micromechanics. This will require extra effort through the expansion of the micromechanics toolkit with statistical mechanics, and further development of time-coarse graining techniques. In the authors’ opinion, this is an exciting

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challenge that will keep micromechanics very much alive in the coming decades. It promises new opportunities for interdisciplinary research to help understand living materials and to aid the design of new supramolecular materials with unprecedented engineering performance.

Acknowledgement

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The results presented in Fig. 3 have been produced by Guus Winter in the context of his M.Sc. research project.

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