Challenges and Modeling Approaches

a term was artificially added by introducing a bodyforce that led to an energetic pun-ishment of |ϕt−F¯tX| to include this frustration, yielding the desired formation of fine microstructure. Finally, in Section 6.3.2, it was observed that in the three-dimensional problem (and equally in two dimensions), the decreasing frustration enters the problem due to the incompatibility of ∇^Xϕ_t ∈ {A,B} (modulo rotations due to material frame invariance and material symmetry) with the boundary condition ¯F_tX on∂B that results in an energetically unfavorable boundary layer B \ Bⁿf where ˆψ(∇^Xϕ_t) >ψ(A) = ˆˆ ψ(B) that decreases in size as n → ∞. Mathematically, this “decreasing frustration” means that EB(ϕⁿ_t) asymptotically approaches infEB for n→ ∞, but never fully attains it,

nlim→∞EB(ϕⁿ_t) = infEB(ϕ_t) and EB(ϕ_t)>infEB(ϕ_t) ∀ϕ_t, (6.90) where in both considered examples, we had infEB(ϕ_t) = 0. Together with the loss of sequentially weak lower semicontinuity and the average compatibility, the notion of de-creasing frustration finally explains the origin of infinitely fine microstructure.

6.5 Challenges and Modeling Approaches 85

n^∗ n

E^B+Γ(ϕⁿ_t)

E^B(ϕⁿ_t) gΓsurf(Γ)

Figure 6.9: Adding the interfacial energy contributiongΓ·surf(Γ) to the potential energy E^B(ϕⁿ_t) yields a potential energy E^B+Γ(ϕⁿ_t) that is not minimized by an infinite number of interfaces 2n−1, but by a finite number 2n^∗−1, yielding a length scale of 1/n^∗.

Murr [126, p. 142],Porter & Easterling [139, p. 123]. However, the idealized theory presented in this chapter disregards interfacial energy and does not associate any energy with discontinuities of the deformation gradient, thus allowing for infinitely fine laminates.

In reality, the presence of interface energy leads to finite length scales, see e.g. Arlt [9].

To get a first idea of the underlying energetic effect, recall the one-dimensional example treated in Section 6.3.1.2 with the minimizing sequence ϕⁿ_t(X) defined by equation (6.38) and the resulting potential energy given as EB(ϕⁿ_t) =λ(B−A)²ζ²(1−ζ)²/(3n²) (without interface energy), see (6.39). If we add an interfacial energy contributiongΓat every point of discontinuity of ∂_Xϕⁿ_t(X), then we obtain the modified potential energy

EB+Γ(ϕⁿ_t) = λ(B−A)²ζ²(1−ζ)² 3n²

| {z }

E^B(ϕⁿ_t)

+gΓ(2n−1)

| {z }

gΓsurf(Γ)

, A < B ∈ R+, ζ ∈]0,1[, (6.91)

where we see that in contrast toEB(ϕⁿ_t), the potential energy with interfacial contributions EB+Γ(ϕⁿ_t) is not minimized by limn→∞ϕⁿ_t but byϕⁿ_t^∗wheren^∗ ∈ N+is the natural number that is closest to p³

[λ(B−A)²ζ²(1−ζ)²]/[3gΓ], see Figure 6.9. This tells us that in the one-dimensional problem, the introduction of interface energy yields a finite length scale of 1/n^∗ of the now finitely fine microstructure. Note that a generalization of the concept used in (6.91) to three dimensions is generally not possible, as a finite microstructure in three dimensions does not satisfy average compatibility since vol(B \ Bⁿf)6= 0 and hence λA 6= ζ and λB 6= 1−ζ, see Section 6.4.3.2, and since even if we can estimate the total interface surf(Γ), it will not be possible to realistically estimate the energy inB\Bfⁿ, where the construction used in (6.58) by Chipot & Kinderlehrer [30] is not a solution for all n, but a construction that is a solution only in the limit n → ∞. The prediction of length scales in three dimensional microstructure will be treated in Chapters 8 and 9.

6.5.3. Modeling Approaches for Microstructure

In the following, we briefly summarize possible approaches to the modeling of microstruc-ture. We thereby particularly highlight the capability of the approaches to deal with the challenges of dissipation and interface energy outlined in Sections 6.5.1 and 6.5.2. Both effects will be important for the explicit description of microstructure and the prediction of related length scales in the context of phase transitions and plastic laminates.

6.5.3.1. Relaxation. The first approach we want to briefly discuss is directly based upon the ideas presented in Sections 6.3 and 6.4 and allows the modeling of materials with non-quasiconvex free energies ˆψ that form microstructure without constructing new minimizing sequences for every different problem. The basic assumption is thereby that at each point of amacroscopic problem, the materialmicroscopically is subject to a bound-ary value problem of the type considered in Section 6.3.2, which requires the assumption of scale separation. The idea is then to solve such a microscopic problem with bound-ary deformation FtX for every Ft and to replace the previously non-quasiconvex free energy ˆψ(Ft) by the average energy of the solution infEB/vol(B). This is equivalent to using (6.74) and (6.75) to construct aquasiconvexified free energy ψˆ_QC(F_t) from the non-quasiconvex free energy ˆψ(Ft) by computing (for every Ft) the deformation mapϕ_t(X) in a domainBP with ϕ_t=F_tX on∂BP that minimizes the total energy,

ψˆQC(Ft) = 1

vol(BP) inf

ϕ_t∈W^ϕ

Z

B^P

ψ(ˆ ∇Xϕ_t)dV , Wϕ = ϕ_t

ϕ_t=FtX on∂BP }, (6.92) which will satisfy (6.74) and thereby be quasiconvex, see e.g. Dacorogna [38, p. 201].

Insertion back into the variational problem given in Section 6.2 will lead to a sequentially weak lower semicontinuous functional EB, for which the existence of solutions is guaran-teed, see e.g. Ball [14] or Marsden & Hughes [106, p. 377 ff.]. Obviously, it will be very hard, if not impossible to solve (6.92) for every Ft, such that very often, an alter-native approach is used. We have seen in Section 6.3.2 that the assumption of a two-well free energy function with rank-one connected minima A−B = ˆa⊗Mˆ led to a lami-nate microstructure with deformation amplitude ˆa (see (6.54)), normal ˆM and volume fractions ζ and 1−ζ for a homogeneous deformation of ∂BP by ζA+ (1−ζ)B. Fur-thermore, we know from Section 6.4.2.2 that rank-one convexity is a notion very close to quasiconvexity. This both motivates the construction of afirst order rank-one-convexified free energy ψˆ1RC(Ft) from a non-rank-one-convex free energy ˆψ(Ft) by computing (again for every F_t) the laminate parameters a, M and ξ that minimize

ψˆ1RC(Ft) = inf

a,M,ξ

ξψˆ Ft(1 + (1−ξ)a⊗M)

+ (1−ξ) ˆψ Ft(1−ξa⊗M) , (6.93) see e.g. Kohn & Strang [89, 90, 91], where also higher order (sequential) rank-one convexification is discussed. Note that insertion back into a variational problem as given in Section 6.2 will not necessarily lead to a sequentially weak lower semicontinuous functional EB, but will be a suitable approximation in numerical applications. Approaches that follow (6.92) or (6.93) are referred to as relaxation approaches, see e.g. Luskin [105], Bhattacharya & Dolzmann [23] and Aubry et al. [13]. However, as mentioned, they are not suited for the modeling of neither dissipative effects such as hysteresis nor of effects related to interfacial energy such as length scale effects.

6.5.3.2. Partial Relaxation. An alternative approach that conceptually consists in a modification of (6.93) to allow for the modeling of dissipation is the approach of partial relaxationof non-rank-one-convex free energy functions ˆψ(Ft). The idea is to parametrize the microstructure and to separate the parameters into elastic parameters q_el and dissi-pative parameters q_diss based on whether a change of these parameters is connected to an interface motion and hence connected with dissipation. The partial relaxation then consists in a relaxation with respect toq_eland a dissipative evolution of theq_diss. In case

6.5 Challenges and Modeling Approaches 87

of a first order laminate parametrized by a,M and ξ (see (6.93)), one hasq_el ={a}and q_diss={M, ξ}and can then use the partially first order rank-one convexified free energy

ψˆP1RC(Ft,M, ξ) = inf

a

ξψˆ Ft(1+(1−ξ)a⊗M)

+(1−ξ) ˆψ Ft(1−ξa⊗M) , (6.94) where M and ξ take the role of internal variables, whose evolution can be given by a suitable dissipation potential φ, such that the problem is fully prescribed by

ψˆP1RC(Ft,M, ξ) and φ( ˙M,ξ)˙ , (6.95) see Section 3.2.7. For applications of approaches in the spirit of (6.94) and (6.95) to laminate formation in crystal plasticity (via relaxation of the condensed energy), see e.g. Ortiz & Repetto [131], Miehe et al. [115, 116, 114], Kochmann [85] and Kochmann & Hackl [86], for partial relaxation in the context of martensitic phase transitions see e.g. Bartel & Hackl [19, 20] and Bartel [18]. Note that partial relaxation accounts for dissipation, but does not allow for an incorporation of interface energy. Furthermore, note that partial relaxation cannot be easily combined with other length scale effects such as e.g. GNDs in case of plastic laminates.

6.5.3.3. Strain Gradient Approaches. A modification of (6.16) in a different direction is achieved by using strain gradient approaches. These approaches replace discontinuities of the deformation gradient Ft =∇Xϕ_t by smooth transitions with a large change of the deformation gradientF_t in a small region and allocate interfacial energy to these regions by combining the non-quasiconvexity of ˆψ(Ft) with an additional term proportional to the squared norm of the gradient of the deformation gradient ∇XFt=∇²Xϕ_t,

ψˆε(Ft,∇^XF_t) = ˆψ(Ft) +ε²k∇^XF_tk². (6.96) The resulting smooth, relatively localized transition is caused by the energetic compe-tition of the non-convex energy that wants to force a value of Ft corresponding to the energy wells, and hence tries to minimize the transition region between the associated states on the one hand and the gradient term that tries to minimize the gradient in the transition region, thus spreading it out to a width proportional to ε. It can be shown that for the case of a two-well free energy with rank-one connected minima (as in Section 6.3.2), the variational problems resulting from (6.96) Γ-converge to sharp interface prob-lems similar to those discussed in Section 6.1 but with interface energy, see Conti et al. [35]. Generally, Γ-convergence means that a family of functionals Fε converges to a limit functional F forε→0, seeDal Maso[40] orBraides [26] for a general treatment.

For the application of the strain gradient approach (6.96) to phase transitions, see e.g.

Barsch & Krumhansl [17] or Finel et al. [52]. The general shortcoming of strain gradient approaches in the modeling of microstructure is that the energetics of interfaces cannot be separated from the energetics of deformation, such that the strain gradient term punishes any inhomogeneous deformation states (also away from the interface re-gions) and such that the dissipation related to interface motion can only be included if a punishment of any strain rate is accepted (thus leading to the prescription of effectifely viscoelastic behavior). Another drawback of these models specifically with respect to a numerical implementation is the necessity to discretize second gradients.

6.5.3.4. Phase Field Approaches. After our very brief (and by no means compre-hensive) survey of modeling approaches to microstructure, we are still lacking a method

that allows the incorporation of both interfacial energy and dissipation. This is achieved by using a so-called Ginzburg-Landau or Allen-Cahn type approach that goes back to Ginzburg & Landau [61] and Allen & Cahn [8]. The basic idea is to introduce an internal variable (or several internal variables) that takes the role of an order parameter orphase field and can thus be used to describe microstructure. Assuming the order pa-rameter is denoted by Iand the two constituents of the microstructure are associated to I=I₁ and I =I₂, a purely energetic problem in the spirit of (6.96) can be constructed by introducing a free energy of the form

ψˆε(Ft,I,∇^XI) = ¯ψ(Ft,I) + 1

εg(I) +εk∇^XIk², (6.97) where ¯ψ(F_t,I₁) = ˆψ₁(F_t) and ¯ψ(F_t,I₂) = ˆψ₂(F_t) recovers the single-well energies asso-ciated with the minima of ˆψ(Ft) (AandB),g(I) is a non-convex two-well function with minima atI₁ and I₂ > I₁ that tries to minimize the transition region where I ∈]I₁,I₂[ and where the gradient term tries to minimize the gradient in the transition region, thus spreading it out to a width proportional to ε. It can be shown (at least for the small strain case) that the variational problem of the minimization of a potential energy given by (6.97) with respect to ϕ_t and I Γ-converges to a sharp interface problem similar to that discussed in Section 6.1 but with interface energy, see e.g. Garcke[60]. Note that interestingly, (6.97)replaces the non-quasiconvexity ofψ(Fˆ t)with a non-convexity ofg(I).

So far, we have made plausible that the phase field approach (6.97) is similar to the strain gradient approach (6.96) in the sense that it allows a smooth approximation of the sharp interfaces in a localized region proportional to some parameter ε and as its variational representation Γ-converges to a sharp interface problem that contains interface energy.

The main advantage of (6.97) compared to (6.96) is now that by introduction of the order parameterIas an additional field, the description of the interfaces is decoupled from the description of deformation. On the one hand, this means that (6.97) does not punish strain gradients within one constituent of the microstructure as was the case for (6.96).

On the other hand, it means that dissipation can be included without inducing a viscous deformation behavior. To do so, the dissipative evolution of I is specified by use of a dissipation potential φ( ˙I) such that the problem is completely specified by

ψ˜ε(Ft,I,∇XI) and φ( ˙I). (6.98) By comparison with Chapter 3 we see that the phase field approach as specified by (6.97) and (6.98) can be suitably modeled within the framework of gradient-extended standard-dissipative solids. Note that for a quadratic dissipation potential of the formφ( ˙I) = ¹₂ηI˙², the evolution equation of the order parameter Ifollows as

−δIψ˜ε ∈∂I˙φ ⇒ I˙ =−1

ηδIψ˜ε =−1

η ∂Iψ˜ε−Div(∂_∇XIψ˜ε)

, (6.99)

which is an evolution equation of generalized Ginzburg-Landau-type, seeGurtin[65]. It can be shown by means of asymptotic analysis, that (6.99) approaches a sharp interface problem with interface energy for ε → 0, see e.g. Fried & Gurtin [59], Fried &

Grach [56] orAlber & Zhu [5]. For applications of the phase field approach to phase transitions, see Chapter 8. If the evolution equation (6.99) is altered, the conceptual idea of a combination of an energy that is non-convex in terms of one or more internal variables with gradient terms in those variables and a dissipative evolution of those variables can also be applied to simulate microstructure in crystal plasticity, see Chapter 9.

89