Design for Finite Strains

Chapter IV: Optimal Design of 3D Printed Soft Responsive Actuators . 60

4.3 Design for Finite Strains

along direction a,

`^1/2₀ (r₀) =r^−1/6₀ (I_3x3+ (r^1/2₀ −1)a⊗a). (4.69)

`^1/2 is the deformation experienced by the material through re-orientation of the mesogens. Here, we assume the LCE to be heavily cross-linked. Thus, the re-orientation of the nematic director is described by the deformation of its orthogonal plane [16]

`^−1/2(r) =r^1/6 I_3x3+ (r^−1/2−1) F^−Ta

kF^−Tak ⊗ F^−Ta kF^−Tak

!

. (4.70)

In the transformed, isotropic state, the order parameter takes the value of unity. Thus, `<sup>−1/2</sup>(1) =I3×3, and the material experiences a spontaneous de- formation of`^−1/2₀ (r₀), which corresponds to a contraction alongabyr^−1/3 and extension along the perpendicular directions byr^1/6. We consider a structure composed of both the LCE as well as a passive, finite elastic material described by internal energy density W_p. As in (4.4), we consider a relaxed energy for displacement field u: Ω7→R³

E(φ, a, u, r) :=W(φ, a, u, S)−

Z

∂tΩ

t·u dΩ, W(φ, a, u, r) :=

Z

Ω

[(1−(f ∗φ)^p)W_p(F) + (f ∗φ)^pW_l(a, r, r₀, F)] dΩ,

(4.71)

where F(u) := ∇u+I3×3 is the deformation gradient. W_p is a poly-convex strain energy function

W_p(F) = ¯W_p(F,AdjF,DetF), (4.72) where ¯W_p is convex in each of its arguments, and satisfy the growth conditions outlined in [5].

To characterize designs, we consider deformations which minimize this energy.

Thus, we prove existence of such minimizers.

Lemma 4.3.1. Suppose Wp is polyconvex with Wp(B) ≥ C1kBk^p_F −C2, for p > 3, W_l is of the form (4.67) with W˜_l as the Mooney-Rivlin energy density from (4.68), and t∈L¹(∂_tΩ). There exists a solution u^eq∈ U such that

E(φ, a, u^eq, S) = inf

u∈UE(φ, a, u, S), U :={u∈W^1,p : u=u₀ on ∂_uΩ}.

(4.73)

Proof. It suffices to show polyconvexity of W_l(a, r, r₀, F) in F. We do so for r₀ = 1 and then extend this for arbitrary r₀ >1. Substituting M =`^−1/2F in (4.68),

W_l(a, r,1, F) = ˜W_l(`^−1/2F)

= µ 2

TrF^T`<sup>−1</sup>F−3−2 log det(`^−1/2F) +αµ

2

Tradj(`<sup>−1/2</sup>F) adj(`^−1/2F)^T−3−4 log det(`^−1/2F) + µν

1−2ν(det(`^−1/2F)−1)².

(4.74) It is a straight-forward calculations to show that

TrF^T`⁻¹F=r^1/3

kFk²_M + (r⁻¹−1)a·F⁻¹F^−Ta⁻¹

. (4.75) Recalling that adjM = (detM)M^−T when detM 6= 0,

adj(`<sup>−1/2</sup>F) = adj(`^−1/2) adj(F) = `^1/2adj(F). Further,

`<sup>1/2</sup> =r<sup>−1/6</sup>(I+ (r<sup>1/2</sup>−1)(adjF a⊗adjF a)/kadjF ak<sup>2</sup>). It follows that Tradj(`^−1/2F) adj(`^−1/2F)^T=

r^1/3





kadjFk²_M + (r−1)

(adjF)^T(adjF)a² kadjF ak²





.

(4.76)

Finally, det(`^−1/2F) = detF. Putting these together, W_l(a, r,1, F) = µr^1/3

2 W₁(F) + αµr^−1/3

2 W₂(adjF) +W₃(detF) (4.77) where

W₁(A) =kAk²_M + (r⁻¹−1)a·A⁻¹A^−Ta⁻¹−3r^−1/3, W₂(A) =kAk²_M + (r−1)

(A)^TAa²

Aa² −3r^1/3, W₃(x) =−(µ+ 2αµ) log(det(x)) + µν

1−2ν(x−1)².

(4.78)

Note thatW₁, W₂ are homogeneous of degree 2. Define m₁ = max

kAk_M=1,kBk_M=1 B · ∂²

∂A²

a·A⁻¹A^−Ta⁻¹

!

B,

m₂ = max

kAk_M=1,kBk_M=1 B ·







∂²

∂A²

A^TAa²

Aa²





B.

(4.79)

It is easy to see that m₁, m₂ exist and are finite. Therefore, there existsr^∗ >1 such that the second derivatives of W₁, W₂ are positive definite and therefore W₁, W₂ are convex for any r ∈ [1, r^∗]. Finally, W₃ is convex, and therefore W_l(a, r,1, F) is polyconvex in F for any given a, r ∈ [1, r^∗]. Therefore, exis- tence follows in light of the growth conditions [5, 13].

Turning now to the general case r₀ >1, observe that

Wl(a, r, r₀, F) = Wl(a, r,1, F `^1/2₀ ). (4.80) The polyconvexity and the growth conditions are preserved, and therefore existence follows.

4.3.2 Optimization

To characterize the design, we consider the stimulus dependent compliance.

In the small strain setting this is^R_∂

tΩt·u dΩ, whereuis the energy minimizer.

However, because the possible multiplicity of the solutions, we instead consider C(φ, a, r) := inf

u∈U_min(φ,a,r)

Z

∂tΩ

t·u dΩ, (4.81)

where U_min(φ, a, r) is the set of energy minimizing configurations. This is the technique used in [24]. While we might be concerned with the worst- case compliance, that is the maximum over equilibrium solutions, this lacks sequential weak lower semicontinuity. Thus, we will consider the minimum compliance over equilibrium solutions, as we will see it allows us to prove existence to the optimal design problem.

We must now choose a suitable objective to optimize. In [2], it is shown that minimizing the ratio of the stimulated to un-stimulated compliance in the small-strain setting is equivalent maximizing the blocking load. However, optimizing for blocking load does not exploit the large deformations and shape- change the structure may undergo through actuation. Thus, we choose to consider the compliance ratio,

φ∈Dinf_φ, a∈N

C(φ, a,1)

C(φ, a, r), (4.82)

where N is the space of unit vector fields on Ω,

N :={a:a(x)∈ Sⁿ⁻¹ a.e. on Ω}. (4.83)

In the small strain setting, we proved that additional regularization was not necessary. However, in the current setting there may be fine structure forma- tion ina; the nonlinearity in the kinematics and material model now allow fine mixtures of actuation strains to be favorable. To address this, we consider a modified problem where we regularize a through penalizing the gradients of the director field,

φ∈D_φinf, a∈Da

O(φ, a) := C(φ, a,1)

C(φ, a, r)+P(a), (4.84) where

P(a) :=

Z

Ω

r_g

4k∇ak⁴

dΩ, (4.85)

Da:={a∈W^1,4(Ω), a(x)∈ Sⁿ⁻¹ a.e. on Ω}, (4.86) and we reuse the definition of D_φ from (4.64). We may prove existence to the optimization problem under some suitable assumptions. Following a sim- ilar study which analyzed existence for finite deformation structures [24], we require two lemmas to support this. The first being the sequential lower- semicontinuity of the energy function, and the second being its Γ−convergence.

This enables us to establish lower-semincontinuity of the compliancs, which we then use to prove existence.

Lemma 4.3.2. SupposeW_p is isotropic and polyconvex withW_p(B)≥C₁kBk^p_F− C₂, for p > 3, W_l is of the form (4.67) with W˜_l as the Mooney-Rivlin en- ergy density from (4.68), and t ∈ L¹(∂_tΩ). Then W(φ, a, u, S) is sequen- tially lower semi-continuous along sequences {φk, ak, uk} ⊂ Dφ × Dn × U with φk * φ¯in L²(Ω), ak * a¯ in W^1,s(Ω), s > 3, uk * u¯ in W^1,p(Ω) with(F(u_k),Adj F(u_k),DetF(u_k))*(F(¯u),Adj F(¯u), Det F(¯u)) in L^p(Ω)× L^q(Ω)×L^r(Ω) for q, r > 1.

Proof. Notice that since ˜W_l is isotropic,

W_l(a, r, r₀, F) =W_l(e, r, r₀, F R(a)) (4.87) whereR(a)∈SO(3) is the minimal rotation that maps a constant unit vector e ∈ R³ to a. As {a_k} is a bounded sequence in W^1,s(Ω), the compact em- bedding of W^1,s(Ω) in L^∞(Ω) gives a_k → ¯a strongly in L^∞(Ω). Additionally,

R(a_k)→R(¯a) strongly in L^∞(Ω). Then,

(F(u_k)R(a_k), Adj(F(u_k)R(a_k)), Det(F(u_k)R(a_k))) * (F(¯u)R(¯a)), Adj(F(¯u)R(¯a)), Det(F(¯u)R(¯a)))

in L^p(Ω)×L^q(Ω)×L^r(Ω).

(4.88)

Following the arguments of [24] and using the continuity of W with f∗φ and the strong convergence off∗φ_k →f∗φ¯gives the required result. Additionally, the linearity ofC and the convergence of{u_k}give sequential weak lower semi- continuity of the energy function E.

Lemma 4.3.3. SupposeW_p is isotropic and polyconvex withW_p(B)≥C₁kBk^p_F− C₂, for p > 3, W_l is of the form (4.67) with W˜_l as the Mooney-Rivlin energy density from (4.68), andt∈L¹(∂_tΩ). For a sequence{φ_k, a_k} ⊂L²(Ω)×W^1,s, s >3, where φk *φ¯in L²(Ω), ak *¯a in W^1,s, then

Γ− lim

k→∞E(φ_k, a_k,·, S) =E( ¯φ,a,¯ ·, S). (4.89) Proof. As in [24], we only need to show Γ-convergence ofW. To establish this, we first show the lim-inf followed by the lim-sup condition [14].

Let u_k * u¯in W^1,p(Ω) with lim sup_k→∞W(φ_k, a_k, u_k, S) < ∞. From the growth conditions of W we obtain boundedness of (Adj F(u_k),Det F(u_k)) in L^p/2(Ω)×L^p/3(Ω) and thus the weak convergence of a subsequence. Then, from [5], we find that (F(u_k),AdjF(u_k),Det F(u_k))*

(F(¯u),AdjF(¯u),Det F(¯u)) in L^p(Ω)×L^p/2(Ω)×L^p/3(Ω). Lemma 4.3.2 yields the lim inf conditionW( ¯φ,¯a,u, S)¯ ≤lim inf_k→∞W(φk, ak, uk, S).

For the lim sup condition we consider the convergence ofW(φ_k, a_k,u, S). From¯ the weak convergence of φ_k, the properties of the filter, and the strong con- vergence of a_k in L^∞, the integrand of W(φ_k, a_k,u, S) converges pointwise to¯ the integrand of W( ¯φ,¯a,u, S). Additionally,¯

(1−(f∗φ)^p)Wp(F(¯u)) + (f∗φ)^pWl(ak, r, r₀, F(¯u))

≤W_p(F(¯u)) +W_l(a_k, r, r₀, F(¯u)).

(4.90) Asa_k is uniformly bounded inL^∞(Ω), we may assume that W_l(a_k, r, r₀, F(¯u)) is uniformly bounded. Thus

(1−(f ∗φ)^p)W_p(F(¯u)) + (f ∗φ)^pW_l(a_k, r, r₀, F(¯u))≤G(¯u), (4.91)

where G is some measurable function independent of k. Then, by Lebesque dominated convergence, lim_k→∞W(φ_k, a_k,u, S) =¯ W( ¯φ,¯a,u, S). For a recov-¯ ery sequence u_k = ¯u for all k, lim sup_k→∞W(φ_k, a_k, u_k, S) = W( ¯φ,¯a,u, S).¯ This proves the lim sup condition.

Before we prove existence to the optimization problem 4.84, we require weak lower-semicontinuity of C(φ, a, r) along sequences of designs.

Theorem 4.3.4. SupposeW_p is isotropic and polyconvex withW_p(B)≥C₁kBk^p_F− C₂, for p > 3, W_l is of the form (4.67) with W˜_l as the Mooney-Rivlin energy density from (4.68), and t ∈ L¹(∂_tΩ). Then C(φ, a, r) is sequentially lower semi-continuous along sequences {φ_k, a_k} ⊂ D_φ× D_n with φ_k * φ¯ in L²(Ω), a_k *a¯ in W^1,s(Ω), s >3.

Proof. We may consider an energy minimizing solution u_k ∈ U_min(φ_k, a_k,1) associated with the minimum actuated compliance such that ^R_∂tΩt·u_k dΩ = C(φ_k, a_k, r). Then, using thatu_k is an energy minimizing solution, the growth conditions on W_p andW_l gives a uniform bound on the W^1,4(Ω) norm. Thus, there exists a ¯u∈W^1,4(Ω) such thatuk *u¯ in W^1,4(Ω) up to a subsequence.

Then, the Γ-convergence of Lemma 4.3.3 ensures that ¯u ∈ U_min( ¯φ,¯a,1). The compact embedding of W^1,4(Ω) in L^∞(Ω) gives u_k → u¯ strongly in L^∞(Ω) and L^∞(∂_tΩ). Then, C(φ_k, a_k,1) converges to ^R_∂

tΩt·u dΩ. As¯ C( ¯φ,¯a, r) ≤

R

∂tΩt·u dΩ, we recover sequential lower semi-continuity of¯ C(φ_k, a_k, r).

With the lower semi-continuity of the compliance established, we are now ready to prove existence of optimal designs to (4.84) under some suitable assumption.

Theorem 4.3.5. SupposeW_p is isotropic and polyconvex withW_p(B)≥C₁kBk^p_F− C₂, for p > 3, W_l is of the form (4.67) with W˜_l as the Mooney-Rivlin energy density from (4.68), and t∈L¹(∂_tΩ). Consider

O(φ, a) := C(φ, a,1)

C(φ, a, r)+P(a). (4.92) If the prescribed boundary displacementsu0 = 0 and there exists some {φ,˜ ˜a} ∈ D_φ× D_a such that O( ˜φ,˜a) ≤ 0, then there exists an optimal design {φ,¯ ¯a} ∈

D_φ× D_a such that

O( ¯φ,¯a) = inf

φ∈D_φ, a∈Da

O(φ, a). (4.93) Proof. We may assume that for t 6= 0 that C(φ, a, r) > 0 for all {φ, a} ∈ D_φ× D_a. AsC(φ, a,1) is bounded from below, O(a, φ) is bounded from below.

Consider a minimizing sequence {φk, ak} ⊂ Dφ× Da, and extract a subse- quence such that a_k is uniformly bounded in W^1,4(Ω). Thus there exists a

¯

a ∈ W^1,4(Ω) such that a_k * ¯a in W^1,4(Ω) up to a subsequence. From the compact embedding of L^∞(Ω) in W^1,4(Ω), a_k → ¯a strongly in L^∞(Ω). As magnitude is preserved under strong convergence, ¯a ∈ D_a. Additionally, from the definition of D_φ, there exists a ¯φ ∈ D_φ such that φ_k * φ¯ in L²(Ω) up to a subsequence. Theorem 4.3.4 gives the the sequential weak lower semiconti- nuity of the compliances C(φ_k, a_k, r) and C(φ_k, a_k,1) as φ_k * φ¯and a_k * ¯a.

From the condition that there exists of designs with non-positive objectives, we assume C(φk, ak,1) ≤ 0 for k > K. Then, since C(φk, ak,1) ≤ 0 and C(φ_k, ak, r)>0, the ratioC(φ_k, ak,1)/C(φ_k, ak,0) remains sequentially weakly lower semicontinuous. The sequential weak lower semicontinuity of P(a) im- plies that of O(φ_k, a_k), which completes the proof.

4.3.3 With Voids

We may consider the introduction of voids in the finite elastic setting in a similar manner to Section 4.2.4. Then, Theorem 4.3.5 and the associated lemmas can straightforwardly be extended to the case for voids. We introduce the additional scalar field ρ : Ω 7→[ρ_min,1] which indicates solid or void, and consider the energy

E(φ, ρ, a, u, r) := W(φ, ρ, a, u, r)−

Z

∂tΩ

t·u dΩ, W(φ, ρ, a, u, r) :=

Z

Ω

(f∗ρ)^p[(1−(f∗φ)^p)W_p(F)

+ (f ∗φ)^pW_l(a, r, r₀, F)] dΩ.

(4.94)

Again, we consider the the minimum compliance, C(φ, ρ, a, r) := inf

u∈Umin(φ,ρ,a,S)

Z

∂tΩ

t·u dΩ, (4.95) whereU_min(φ, ρ, a, r) is the set of energy minimizing displacement fields. Then, we consider minimizing the compliance ratio with the added penalty terms on

a,

φ∈D_φ, ρ∈Dinf_ρ, a∈D_a O(φ, ρ, a) := C(φ, ρ, a,1)

C(φ, ρ, a, r) +P(a), (4.96) where we re-use the definitions of the design spaces from (4.64) and (4.86).

The existence proof for two-material designs in Theorem 4.3.5 may be simply extended for the inclusion of voids.

Theorem 4.3.6. SupposeWp is isotropic and polyconvex withWp(B)≥C₁kBk^p_F− C2, for p > 3, Wl is of the form (4.67) with W˜l as the Mooney-Rivlin energy density from (4.68), and t∈L¹(∂_tΩ). Consider

O(φ, ρ, a) := C(φ, ρ, a,1)

C(φ, ρ, a, r)+P(a). (4.97) If the prescribed boundary displacementsu₀ = 0and there exists some{φ,˜ ρ,˜ ˜a} ∈ D_φ× D_ρ× D_a such that O( ˜φ,ρ,˜ ˜a) ≤ 0, then there exists an optimal design {φ,¯ ρ,¯ ¯a} ∈ D_φ× D_ρ× D_a such that

O( ¯φ,¯a) = inf

φ∈D_φ, a∈Da

O(φ, a). (4.98) Proof. The proof follows almost exactly as Theorem 4.3.5, where the associated lemmas and theorem may be simply extended to the case of the additional design variable.