On the numerical implementation of these theories The Koiter theoryThe Koiter theory

MECHANICAL RESPONSE AND INSTABILITIES OF THIN SHEETS

3.9 On the numerical implementation of these theories The Koiter theoryThe Koiter theory

(3.213) hold. Combining all this, we conclude that (∇y_k^h)^Te₁ = (∇y˜ k|bk)^Te₁ = λ¯Me₁ onΩ_h, (∇y_k^h)^Tγ⁰_k = (∇y˜ k|bk)^Tγ⁰_k +O(x₃)(∇˜bk|0)^Tγ⁰_k

=

λ¯^−1/2_M +O(k x₃)

e₂ onΩ_h,

(∇y_k^h)^TN_yk = (∇y˜ k|bk)^TN_yk +O(k x₃)(e₃⊗bk)N_yk +O(k²x²₃)(e₂⊗bk)N_yk

=

λ¯^−1/2_M +O(k x₃)

e₃+O(k²x²₃)e₂ onΩh,

(3.256) where the first result uses thate₁is orthogonal tobk and ˜∇bk, the second result uses that γ⁰_k is orthogonal to bk and the last result uses that N_yk is orthogonal the ˜∇bk. Since ¯λM > λ¯^−1/2_M , the first term dominates the other two terms for k hsufficiently small, and we readily conclude that

λM(∇y_k^h) = λM(∇y˜ k|bk) = λ¯M > r^1/3 (3.257) as desired.

Now to show theλM(cof(∇y˜ k|bk))= λ¯^−1/2_M , we note thatλM(cofF) = λM(F)λ₂(F) for any F ∈ R^3×3 whereλ₂denotes the middle singular value. Thus, the result fol- lows from the fact that onω, (∇y˜ k|bk)^Tγ⁰_k = λ¯^−1/2_M e₂, (∇y˜ k|bk)^TN_yk = λ¯^−1/2_M e₃and (∇y˜ k|bk)^Te₁ = λ¯Me₁. This proves the last indentity in (3.214).

Now given (3.257) we know that ∇y_k^h cannot lie in L on the energy landscape for W_3D^qc. So to complete the proof, we simply have to conclude that∇y_k^his not in Mon this landscape. This is assured if

λ¯M >r^1/2λ₂(∇y_k^h), (3.258) again given (3.257) and the definition of the set M in (2.26). It is easy to see that from (3.256) that

λ₂(∇y_k^h)= λ¯^−1/2_M +O(k x₃). (3.259) In addition, since ¯λM >r^1/3, we know that ¯λM > r^1/2λ¯^−1/2_M . Thus, (3.258) follows from (3.259) for k h sufficiently small, and therefore ∇y_k^h lies in S on the energy

landscape. This completes the proof.

3.9 On the numerical implementation of these theories

a function of the 2×2 surface deformation gradient ˜Fs ∈R^2×2as well as the consistent tangent modulus derived from this Cauchy stress (see, for instance, the Abaqus User Subroutines Reference Manual, the section on UMAT, and the subsection on large volume changes with geometric nonlinearity [90]). For this formulation, we take the energy as in (3.184) restricted to 2×2 matrices ˜F ≡ F˜s where ˜F31 = F˜32 = 0.

This corresponds to fixing a frame. Also, δ(F)˜ = det ˜Fs, λM(F)˜ = λM(F˜s) and following the formalism established at the end of Section 3.6, the Cauchy stress is given by

σ_r^K(F˜s):= (W_ps),F˜_s F˜_s^T, (3.260) where the subscript r is used to emphasize the dependence of this stress on the anisotropy parameter.

Now, the membrane stress σ_r^K in (3.260) inherits soft elasticity, a feature distinct to nematic elastomers as compared to purely elastic materials. This poses some numerical challenges. At small strains, much of the deformation of these elas- tomers can be accommodated by either soft or very lightly stressed microstructure (regions L_m and M on the energy landscape in Figure 3.5). For these regions, the tangent modulus derived from σ_r^K in (3.260) is positive semi-definite and not strictly positive definite: a nematic elastomer has zero stiffness in region L_m and it has no stiffness against shear in region M. These features lead to difficulties in numerical convergence at small strains. To combat this, we introduce a small energy regularizer whose Cauchy stress has the form

σ^{r eg} (F˜s) := µ

F˜sF˜_s^T −I_2×2

(3.261) for 0 ≤ 1. The consistent tangent modulus derived from σ^{r eg} is strictly positive definite for all surface deformation gradients ˜Fs ∈ R^2×2 with det ˜Fs > 0.

Thus, introducing the regularizer 1 stabilizes the numerical calculation at small strains. For our calculations we use =0.05. We have studied the role of on our simulations and only present conclusions that are independent of this regularization.

The formulas introduced in Section 3.7 for the Koiter theory take the isotropic state to be the reference. However, notice that identity is deep in the interior of the soft region and σ_r^K = 0 in a neighborhood of identity forr > 1 (identity corresponds to the intersection of the red and dark blue curves on the energy landscape forWps

in Figure 3.5(a)). This is a trivial soft regime, which is not of interest here.4 We

4We refer to Conti et al. [27, 28] for extensive investigations of soft elasticity in a similar framework.

are interested in the interplay of microstructure and tension wrinkling, an interplay which is broached in this setting when a genuine tensile response to stretch is induced in the membrane. Therefore, we take the end of this trivial soft state, one withλM = δ =r^1/6, as the reference state by setting

σ_r,^sim(F˜s) :=σ_r^K(F˜sU˜_s^r)+σ^{r eg} (F˜s), U˜_s^r :=* ,

r^1/6 0

0 1

+ -

. (3.262)

Finally, the Abaqus S4 shell element slightly modifies the bending term from that derived in our Koiter theory. Specifically, the shell element is based on a kinematic ansatz of the shell deformation through the thickness (see the Abaqus Theory Man- ual, 3.6.5 Finite-strain shell element formulation [91]). This formulation, in effect, yields a bending energy which depends on a linearization of the second fundamental form.

To study wrinkling, we use the two part procedure detailed by Wong and Pellegrino [111] and Ling Zheng [113] to capture the wrinkling geometry in Abaqus. This involves first a pre-buckling eigenvalue analysis, and then a post buckling analysis.

For the pre-buckling eigenvalue analysis, we introduce a small initial prestress in the form of an edge displacement to the membrane. Then we perform an eigen- value buckling analysis on this lightly stressed membrane to determine the likely buckling modes. After computing the buckling mode shapes, we introduce a linear combination of one or more selected eigenmodes as a geometric imperfection in the membrane at the start of the post buckling analysis. We then apply an initial pre- stress to the membrane (as in the pre-buckling analysis) in order to provide an initial out-of-plane stiffness to the membrane. Finally, we use the static stabilization proce- dure in Abaqus to compute the wrinkled shape of the membrane under the clamped stretching deformation. Our detailed implementation of this procedure follows the input file example (Ling Zheng [113] in Appendix B) modified appropriately to incorporate the user material model (3.262) for nematic elastomer sheets.

The effective membrane theory

We compare the results of simulations under the implementation of the Koiter theory described above with analogous simulations of the membrane theory for nematic elastomers (described in Section 3.2). We implement this once again into ABAQUS via UMAT for plane stress CPS4 element, where we take the Cauchy stress in the implementation to be akin to (3.262), with

σ_r^sim(F˜s) :=σ_r^mem(F˜sU˜_s^r), σ_r^mem(F˜s) := (W_2D^qc),F˜_s F˜_s^T. (3.263)

Notice here that we do not employ the use of a regularizer. For simulating the effec- tive membrane theory, there is no wrinkling bifurcation to compute, as wrinkling is treated through effective deformation. Consequently, we do not need to explore the stretch monotonically from small initial stretch. Instead, we explore a range of stretch which excludes small stretch associated to soft elasticity. This can be done without regularization (i.e., by first deforming the specimen homogeneously from the unstressed reference state to a state of stretch far away from soft elastic behavior with the top and bottom of the sheet fixed in thee₂direction, and then relaxing the boundary conditions on top and bottom to a sheet which is traction free away from the clamped ends).