Formulation of the constitutive relation - A GENERAL CONSTITUTIVE MODEL FOR A NON-IDEAL ISOTROP

A GENERAL CONSTITUTIVE MODEL FOR A NON-IDEAL ISOTROPIC-GENESIS POLYDOMAIN NEMATIC ELASTOMER

4.2 Formulation of the constitutive relation

Revisiting the relaxed energy of an ideal nematic elastomer

We follow the work of DeSimone and Dolzmann [23] using the same three regions of interest, a liquid-like region 𝐿, a solid-like region 𝑆, and a region 𝑀 in which laminated microstructure occurs. Recall the generalized energy density of a nematic elastomer

𝑊_NE(F,n) = 𝑓(F^𝑇`⁻¹_n F), (4.1) where`_n =𝑟⁻¹^/³( (𝑟 −1)n⊗n+I) is the “step-length tensor" that describes the metric of the nematic elastomer in the “stress-free state". The corresponding elastic energy is

𝑊(F) =min

𝑊_{𝑁 𝐸}(F,n) = min

Q∈𝑆𝑂(3)

𝑓(F^>`⁻¹_Qn

0F) = min

Q∈𝑆𝑂(3)

𝑓(F^>Q`⁻¹_n₀Q^>F). (4.2) Now, consider a situation where the nematic elastomer has formed a fine-scale domain pattern so that the resulting “stress-free" state is described by a metric G.

We know from previous chapters that

G=QG₀(𝜆, 𝛿)Q^>, (4.3)

where

Q∈𝑆𝑂(3), (4.4)

G₀(𝜆, 𝛿) =©

𝜆² 0 0 0 ^𝛿²

𝜆² 0

0 0 ¹

𝛿²

, and (4.5)

𝜆, 𝛿 ∈ T :={(𝑠, 𝑡) :𝑡 ≤ 𝑟^1/6, 𝑡 ≤ 𝑠², 𝑡 ≥ √

𝑠}. (4.6)

The triangular regionT is depicted in Figure 4.1.

In analogy to (4.2), we may write the energy of a nematic elastomer with fine-scale domain pattern characterized by𝜆, 𝛿to be

𝑊_PNE(F, 𝜆, 𝛿) = min

Q∈𝑆𝑂(3)

𝑓(F^>QG⁻₀¹Q^>F), (4.7)

Figure 4.1: Triangular region T in (𝜆, 𝛿) space enclosed by the three constraints:

𝛿 ≤𝑟^1/6,𝛿 ≤ 𝜆², and𝛿 ≥ √ 𝜆.

so that the effective energy over arbitrary domain patterns is given by 𝑊_eff(F)= min

𝜆,𝛿∈T min

Q∈𝑆𝑂(3)

𝑓(F^>QG⁻¹₀ Q^>F). (4.8) Theorem 1. With the definitions above,

𝑊_eff(F) =𝑊^qc(F). (4.9)

Proof. Let 𝑠 be the largest singular value of F and 𝑡 the product of the largest singular values ofF. Then,

𝑊_eff(F) = min

𝜆,𝛿∈T

𝑓

𝑠 𝜆

2 𝑡 𝜆 𝛿 𝑠

2 𝛿 𝑡

. (4.10)

Now, recall that

𝑓(A) =Õ

𝑖

𝑐_𝑖(trA−3)^𝑝^𝑖 +Õ

𝑗

𝑑_𝑗(tr(cofA) −3)^𝑞^𝑗 (4.11) where 𝑝_𝑖, 𝑞_𝑗 ≥ 1 so that

𝑊_PNE(F, 𝜆, 𝛿) =Õ

𝑖

𝑐_𝑖 𝑠²

𝜆²

+ 𝑡²𝜆² 𝛿²𝑠²

+ 𝛿² 𝑡²

−3 ^𝑝^𝑖

+Õ

𝑗

𝑑_𝑗 𝜆²

𝑠²

+ 𝛿²𝑠² 𝑡²𝜆²

+ 𝑡² 𝛿²

−3 ^𝑞^𝑗

(4.12) and

𝑊_eff(F) = min

𝜆,𝛿∈T

𝑖

𝑐_𝑖 𝑠²

𝜆²

+ 𝑡²𝜆² 𝛿²𝑠²

+ 𝛿² 𝑡²

−3 ^𝑝^𝑖

+Õ

𝑗

𝑑_𝑗 𝜆²

𝑠²

+ 𝛿²𝑠² 𝑡²𝜆²

+ 𝑡² 𝛿²

−3 ^𝑞^𝑗!

. (4.13)

In light of the constraint𝜆, 𝛿 ∈ T, we have multiple cases.

Case 1: Attained minimum. We solve

𝜕𝑊_eff

𝜕𝜆

= 𝜕𝑊_eff

𝜕 𝛿

=0. (4.14)

Since𝑐_𝑖, 𝑑_𝑗 > 0, 𝑝_𝑖, 𝑞_𝑗 ≥1, it is sufficient (and necessary in a generic sense) that

𝜕

𝜕𝜆 𝑠²

𝜆²

+ 𝑡²𝜆² 𝛿²𝑠²

+ 𝛿² 𝑡²

=0, (4.15)

𝜕

𝜕 𝛿 𝑠²

𝜆²

+ 𝑡²𝜆² 𝛿²𝑠²

+ 𝛿² 𝑡²

=0, (4.16)

𝜕

𝜕𝜆 𝜆²

𝑠²

+ 𝛿²𝑠² 𝑡²𝜆²

+ 𝑡² 𝛿²

=0, (4.17)

𝜕

𝜕 𝛿 𝜆²

𝑠²

+ 𝛿²𝑠² 𝑡²𝜆²

+ 𝑡² 𝛿²

=0, (4.18)

𝜕𝑊

𝜕𝜆

=0→ 𝜆⁴ 𝑠⁴

= 𝛿² 𝑡² ,

𝜕𝑊

𝜕 𝛿

=0→ 𝜆² 𝑠²

= 𝛿⁴ 𝑡⁴

⇐⇒ 𝜆=𝑠, 𝛿=𝑡 =⇒ 𝑊^eff(𝐹) =0. (4.19) This is possible if and only if𝑠, 𝑡 ∈ T. Recalling that𝑊^qc = 0 when𝑠, 𝑡 ∈ T, we conclude

𝑊_eff(𝐹) =𝑊^qc(𝐹) in𝐿 . (4.20) Case 2: 𝑡 > 𝑟¹^/⁶. We set𝛿 =𝑟¹^/⁶and solve

𝜕𝑊_eff

𝜕𝜆

=0. (4.21)

Arguing as before, we conclude 𝜆 𝑠

= 𝑟¹^/¹² 𝑡¹^/²

(4.22) which implies

𝑊_eff(𝐹) =Õ

𝑖

𝑐_𝑖

2 𝑡 𝑟^1/6

+𝑟^1/3 𝑡²

−3 ^𝑝^𝑖

+Õ

𝑗

𝑑_𝑗

2𝑟^1/6 𝑡

+ 𝑡² 𝑟^1/3

−3 ^𝑞^𝑗

. (4.23) Note that this coincides with the expression for𝑊^qcin𝑀. However, for𝛿 =𝑟^1/6and 𝜆according to (4.22),𝜆, 𝛿 ∈ T if and only if 1 ≤ 𝑠/𝑡^1/2 ≤ 𝑟^1/4or𝑡 ≤ 𝑠² ≤ 𝑟^1/2𝑡. By assumption,𝑡 > 𝑟^1/6. So this is the region 𝑀. We conclude,

𝑊_eff(𝐹) =𝑊^qc(𝐹) in𝑀 . (4.24)

Case 3: 𝑠² > 𝑟^1/2𝑡 , 𝑡 > 𝑟^1/6. Note that this is the region 𝑆. We set𝜆 =𝑟^1/3, 𝛿 = 𝑟^1/6, and it is easy to verify that

𝑊_eff(𝐹) =𝑊^qc(𝐹) in𝑆 . (4.25)

Constitutive relation

The effective or relaxed energy (4.8) is obtained by assuming that the microstructure evolves instantaneously to minimize the energy. However, microstructure evolves according to a kinetic process which is dissipative. Further, some domains may be locally pinned, and this introduces a hardening energy. Finally, there is viscosity associated with the polymer network. All of these considerations motivate the following constitutive relation. We describe this for the isothermal situation where the temperature is fixed. However, this is easily generalized to a general temperature- dependent situation.

We assume that the state of a non-ideal isotropic-genesis polydomain nematic elastomer is described by the deformation gradient F, internal variables 𝜆, 𝛿, and temperature𝑇. We postulate that the stored energy density of a non-ideal isotropic- genesis polydomain nematic elastomer is given by

𝑊(F, 𝜆, 𝛿, 𝑇) =𝑊_PNE(F, 𝜆, 𝛿, 𝑇) +𝑊_h(𝜆, 𝛿, 𝑇), (4.26) where

𝑊_PNE(F, 𝜆, 𝛿, 𝑇) =Õ

𝑖

𝑐_𝑖 𝑠²

𝜆²

+ 𝑡²𝜆² 𝛿²𝑠²

+ 𝛿² 𝑡²

−3 ^𝑝^𝑖

+Õ

𝑗

𝑑_𝑗 𝜆²

𝑠²

+ 𝛿²𝑠² 𝑡²𝜆²

+ 𝑡² 𝛿²

−3 ^𝑞^𝑗

(4.27) as before, and

𝑊_h(𝜆, 𝛿, 𝑇) =𝐶

𝛿−1

(𝑟^1/6−𝛿)^𝑘 (4.28)

is the hardening energy. The form of the hardening is chosen to penalize𝛿 →𝑟^1/6, i.e. the completion of the polydomain-to-monodomain transition. The moduli𝑐_𝑖, 𝑑_𝑖 as well as the parameter𝑟 may depend on temperature.

The Cauchy stress is given by

σ(F, 𝜆, 𝛿) =−𝑝I+ 𝜕𝑊_PNE

𝜕F (F, 𝜆, 𝛿)F^>+𝛽d, (4.29)

where𝑝is an unknown pressure,d= ¹₂( ¤F F⁻¹+F^−>F¤^>)is the rate-of-deformation tensor, and 𝛽is the viscosity. The microstructure parameters𝜆, 𝛿 evolve according to the equations













𝜆¤ =−𝛼_𝜆

𝜕

𝜕𝜆

(𝑊_PNE+𝑊_h)

𝛿¤=−𝛼_𝛿

𝜕

𝜕 𝛿

(𝑊_PNE+𝑊_h)

subject to𝜆, 𝛿 ∈ T. (4.30)

Note that𝑟, and henceT, depends on temperature. It is convenient to introduce a rate-of-dissipation potential for the microstructure evolution

𝐷( ¤𝜆,𝛿, 𝑑¤ ) = 1 2

𝛼_𝜆| ¤𝜆|²+𝛼_𝛿| ¤𝛿|²

. (4.31)

If we discretize the evolution equation by a backward Euler (implicit) time dis- cretization, we can update the variables as

𝜆^𝑛⁺¹, 𝛿^𝑛⁺¹= argmin

𝜆^𝑛+1,𝛿^𝑛+1∈T

𝑊_PNE(F, 𝜆^𝑛⁺¹, 𝛿^𝑛⁺¹, 𝑇) +𝑊_h(𝜆^𝑛⁺¹, 𝛿^𝑛⁺¹, 𝑇)

+Δ𝑡 𝐷

𝜆^𝑛+1−𝜆^𝑛 Δ𝑡

𝛿^𝑛+1−𝛿^𝑛 Δ𝑡

(4.32)

Useful calculation It is useful to compute the rotation associated with the mini- mization in (4.8). Let𝜆_𝑖be the principal values ofF with𝜆₁ ≥𝜆₂ ≥ 𝜆₃. Let

C =F^>F =

𝑖=1

𝜆²

𝑖N𝑖 ⊗N𝑖, b=F F^> =

𝑖=1

𝜆²

𝑖n𝑖 ⊗n𝑖. (4.33) It follows that

𝐹 = Õ3

𝑖=1

𝜆_𝑖n𝑖 ⊗N𝑖. (4.34)

Set

G₀=

𝑖=1

𝜉_𝑖e𝑖 ⊗e𝑖, (4.35)

where𝜉₁ ≥ 𝜉₂ ≥ 𝜉₃. Therefore, F^>QG⁻¹₀ Q^>F =Õ

𝑖, 𝑗 , 𝑘

𝜆_𝑖𝜆_𝑘𝜉⁻¹

𝑗 (n𝑖·Qe𝑗) (n𝑘 ·Qe𝑗)N𝑖 ⊗N𝑘, (4.36) (F^>QG⁻¹₀ Q^>F)^−> =Õ

𝑖, 𝑗 , 𝑘

𝜆⁻¹

𝑖 𝜆⁻¹

𝑘 𝜉_𝑗(n𝑖·Qe𝑗) (n𝑘 ·Qe𝑗)N𝑖⊗N𝑘. (4.37)

Now, in light of (4.11), maximizing 𝑓(A) overAis equivalent to maximizing the trace ofAand cof(A). Examining the above, we see that we maximize the trace of F^>QG⁻¹₀ F, cof(F^>QG⁻¹₀ F) =(F^>QG⁻¹₀ F)^−>exactly when

n𝑖 =Qe𝑖, 𝑖 =1,2,3. (4.38) Thus, the maximizingQis

Q= Õ3

𝑖=1

n𝑖⊗e𝑖 (4.39)

so that for the maximizingQ, G=

𝑖=1

𝜉_𝑖n𝑖⊗n𝑖, (4.40)

i.e. Gshares an eigenbasis withband F^>QG⁻₀¹Q^>F =

𝑖=1

𝜆²

𝑖𝜉⁻¹

𝑖 N𝑖 ⊗N𝑖, (4.41)

(F^>QG⁻¹₀ Q^>F)^−>= Õ3

𝑖=1

𝜆⁻²

𝑖 𝜉_𝑖N𝑖⊗N𝑖. (4.42) 4.3 Validation of the model

We now validate the model against the experiments of Tokumoto, Takabe, and Urayama, as reported in Tokumotoet al. [71]. They subjected sheets of isotropic- genesis polydomain nematic elastomers to biaxial extension while leaving the faces of the sheet traction-free, i.e. deformations of the form

F =©

𝜆_𝑥 0 0

0 𝜆_𝑦 0

0 0 (𝜆_𝑥𝜆_𝑦)⁻¹ ª

(4.43)

where 𝜆_𝑥, 𝜆_𝑦 are imposed stretches. In all their experiments 𝜆_𝑥𝜆_𝑦 > 1, and we assume that𝜆_𝑥 > 𝜆_𝑦. Therefore,

𝑠=𝜆_𝑥, 𝑡=𝜆_𝑥𝜆_𝑦. (4.44)

We neglect viscosity (𝛽=0), and so the Cauchy stressσ is

−𝑝+𝜇₁^𝑠

𝜆² +𝜇₂

𝑠²𝛿² 𝑡²𝜆² + ^𝑡²

𝛿²

0 0

0 −𝑝+𝜇₁^𝑡

2𝜆² 𝑠²𝛿² +𝜇₂

𝜆² 𝑠² + ^𝑡²

𝛿²

0 0 −𝑝+𝜇₁^𝛿

𝑡² +𝜇₂

𝜆² 𝑠² + ^𝑠²^𝛿²

𝑡²𝜆²

¬ .

Since the faces of the sheet are traction-free,𝜎₃₃ =0. It follows, 𝑝= 𝜇₁

𝛿² 𝑡²

+𝜇₂ 𝜆²

𝑠²

+ 𝑠²𝛿² 𝑡²𝜆²

, (4.45)

and the two non-zero components of stress are 𝜎₁₁ =𝜇₁

𝑠² 𝜆²

− 𝛿² 𝑡²

+𝜇₂ 𝑡²

𝛿²

− 𝜆² 𝑠²

(4.46) 𝜎₂₂ =𝜇₁

𝑡²𝜆² 𝑠²𝛿²

− 𝛿² 𝑡²

+𝜇₂ 𝑡²

𝛿²

− 𝑠²𝛿² 𝑡²𝜆²

. (4.47)

It remains to solve for the evolution of the internal variables 𝜆, 𝛿. We do so by implementing (4.32) usingMATLAB. We use the Bladon-Warner-Terentjev form (generalization of the neo-Hookean), where 𝑐₁ = 𝜇₁/2, 𝑝₁ = 1 and 𝑐_𝑖, 𝑑_𝑗 = 0 for 𝑖 > 1, 𝑗 ≥ 1. Table 4.1 summarizes the material parameters used. In MATLAB, a gradient descent code was implemented to fit the various parameters in the theoretical model to the experimental data.

Table 4.1: Material properties used inMATLABsimulations

Shear modulus 𝜇₁= 4.93752·10⁴Pa

LCE anisotropy parameter 𝑟= 9.1393

Hardening coefficient 𝐶= 298 Pa

Hardening exponent 𝑘 = 2

Dissipation property 𝛼_𝛿 = 2.1838·10⁷Pa Dissipation property 𝛼_𝜆 = ₁₀₀^𝛼^𝛿 = 2.1838·10⁵Pa Exponent in dissipation potential𝐷 𝑝= 2

We explore four specific deformations in the following sections: uniaxial (U) extension, planar extension (PE), equibiaxial (EB) extension, and unequal-biaxial (UB) extension. The general deformation gradient, following from Equation 4.43 and 4.44, isF =diag(𝑠, 𝑡/𝑠,1/𝑡). Figs. 4.2 – 4.7 are plots for each individual deformation with more detail. In each figure, the top left subplot depicts the stress-stretch curve, the top right plot shows the (red) path that the internal variables take through the (black) triangular regionT, the bottom left plot shows the internal variables as a function of stretch, and the bottom right plot shows the energy density as a function of stretch.

Uniaxial extension (U) In the uniaxial case, the body is subjected to uniaxial stress in the𝑥-direction and traction-free in the 𝑦- and 𝑧-directions. Thus, 𝜎₂₂ =𝜎₃₃ =0.

The relationship between the stretches is𝑡 =√

𝑠. Thus, the deformation gradient is F =diag(𝑠,1/√

𝑠,1/√ 𝑠).

Figure 4.2: Uniaxial extension.

Planar extension (PE) In the planar extension deformation, the stretch in the 𝑦-direction is fixed at a ratio of 1, yielding 𝜆_𝑦 = 1, and the body is traction- free in the 𝑧-direction. Thus, we have 𝑡 = 𝑠, and the deformation gradient is F =diag(𝑠,1,1/𝑠).

Figure 4.3: Planar extension.

Equibiaxial extension (EB) In the equibiaxial deformation, the stretch in the𝑥- direction and 𝑦-direction are equal, and the body is traction-free in the𝑧-direction.

Thus,𝑠 = _𝑠^𝑡 or𝑡 =𝑠², and the deformation gradient isF =diag(𝑠, 𝑠,1/𝑠²).

Figure 4.4: Equibiaxial extension.

Unequal biaxial extension (UB) In the case of the unequal-biaxial extension, the body is traction-free in the 𝑧-direction. The experiments are performed by fixing the ratio of ^𝜀_𝜀^𝑥

𝑦

= ^𝜆_𝜆^𝑥⁻¹

𝑦−1 to be a constant𝛽, where𝛽has values equal to 5/3,5/2,5/1.

(Note that𝛽 =1 recovers the equibiaxial case.) This means that we have_𝑡_/^𝑠−_𝑠₋₁¹ = 𝛽, so𝑡 = _𝛽^𝑠(𝑠−1+𝛽). Thus, the deformation gradient isF =diag

𝑠,

𝑠−1+𝛽 𝛽 ,

𝛽 𝑠(𝑠−1+𝛽)

. The figures can be seen in Figures 4.5–4.7.

Figure 4.5: Unequal biaxial extension, 𝛽=5/3.

Figure 4.6: Unequal biaxial extension, 𝛽=5/2.

Figure 4.7: Unequal biaxial extension, 𝛽=5/1.

Summary: liquid-like behavior Figures 4.8 and 4.9 show a comparison of the stress-stretch data, plotted for the experimental data from Urayama’s research group [71] and the theoretical model for various deformations.

Figure 4.8: Experimental stress, plotted as a function of𝜆_𝑧.

Figure 4.9: Theoretical stress, plotted as a function of𝜆_𝑧.

Hysteresis Finally, Figures 4.10a and 4.10b show the load/unload curve for the U and PE deformations. The model is able to capture energy dissipation between the load and unload curves, depicted by the hysteresis between the dashed and solid curves.

(a) (b)

Figure 4.10: Load/unload curves for the (a) PE deformation and (b) U deformation.

Dalam dokumen Theoretical, computational, and experimental characterization of nematic elastomers (Halaman 85-96)