Shape Memory Polymeric Metasurfaces
3.2. Shape memory polymer synthesis
3.2.1. Introduction
3.2.2.3. Characterizations
Fourier transform infrared spectroscopy (FTIR) was employed to characterize chemical composition of the SMPs. The spectral data of the samples were acquired using an attenuated total Reflectance-Fourier transform infrared (ATR-FTIR) mode. All data were obtained at resolution of 4 cm-1 in the mid infrared region from 4000 to 650 cm-1 with 64 scans using a Varian 660 FTIR spectrometer. The data were normalized for comparison
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between the SMP samples. The Varian Resolutions Pro software was employed to analyze the data.
X-ray photoelectron spectroscopy (XPS) analysis was conducted by using an AXIS- His electron spectrophotometer (Kratos Ltd., United Kingdom) with a monochromatic Mg Ka X-ray source (hυ = 1253.6 eV) of 450 W power. The SMP samples were prepared by coating a film on the Al substrate placed on the XPS sample holder. The samples were scanned under a high vacuum pressure of 5 x 10-10 torr at room temperature. A survey spectrum was recorded over a binding energy range from 0 to 1500 eV by using a pass energy of 300 eV. Detailed analysis of the C1s region for the samples was carried out over a binding energy range from 280 to 300 eV by using a pass energy of 150 eV. The corresponding data was analyzed by using the CasaXPS program for deconvolution of the C1s spectra, background subtraction, fitting, peak integration, and quantification of chemical elements. Since C is a dominant element in the spectra, the charging effect induced by the C 1s peak of hydrocarbons was considered for the results. Details on this methodology is explained in the reference [199]. The curve fitting of spectrum peaks was carried out by combining the Gaussian with the Lorentzian peak shapes.
Thermomechanical properties of the SMP samples were characterized by running a dynamic mechanical thermal analysis machine (DMTA machine, TA Q800 DMA, United States). The samples were cut into a dimension of 1 x 5.3 x 40 mm3 and held in the clamp.
Storage and loss moduli of the samples were measured in the single frequency tensile loading mode at 1 Hz. The conditions of 0.2% strain, 5℃/min thermal heating rate, 125 % force track, and 0.001 N preloading were applied while heating of the samples.
Transmittance of the samples was measured by using a UV-vis spectrophotometer (Cary 100, Agilent Technologies, United States). The used optical configuration and monochromator were double beam and the Czerny-Turner type, respectively. The wavelength range of light was from 380 nm to 900 nm.
Transient shape recovery ratio ( ) was evaluated by measuring the change of angle during the shape recovery [93,200]. The tests were conducted at 80℃, 40℃, and 0℃. In the shape programming step, the SMP samples were heated up to 20℃ above , bended, and cooled down to 20℃ below . The initial angle ( ) and the recovered angle ( )
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were examined at 80℃ , 40℃, and 0℃ . The shape recovery ratio was calculated by following definition.
= − ( )
− x 100 (3.2.1) where , ( ) , and are the initial angle, the deformed angle at time t, and the recovered angle.
Cyclic stress-strain-temperature (SST) curves of S-MMA/BMA and S-BA/BMA were obtained by using the TA Q800 DMA with a 3-point bending mode clamp. The samples were bended by 2 mm deflection ( =3%) at 40℃ for S-MMA/BMA and 0℃ for S- BA/BMA, respectively. The cooling step was conducted at 0℃ for S-MMA/BMA and - 40℃ for S-BA/BMA, while maintaining the applied deformation strain. After the unloading step, the heating step was applied at 40℃ for S-MMA/BMA and 0℃ for S- BA/BMA, for shape-recovery of the original permanent shape.
Figure 3.2.1. Fabrication scheme of the SMP samples.
148 3.2.3. Numerical analysis
Numerical analysis for modeling shape memory behavior of the samples was studied by using ABAQUS/CAE, which is based on a finite element method (FEM) code, with user subroutines UMAT and SDVINI. Neo-Hookean model, the hyperelastic model, was used in the simulation by assuming the existence of strain energy as a form of Helmholtz potential for deformation of the material. Also, an isothermal, compressible, and hyperelastic material was assumed. In the simulation, the material properties were modeled as continuously changed dependent on the applied temperature condition. In the hyperelastic model, non-linear material properties are defined and modeled by using stored strain energy function ( ). The stored strain energy function consists of a glassy phase part ( ) and a rubbery phase part (1 − ). The constitutive equations are expressed as
= (1 − ) + (3.2.2)
= − 3 + 1
− 1 (3.2.3)
= ̅
( )− 3 + 1
( )− 1 (3.2.4) , where is the total stored strain energy function, is the stored strain energy function for the rubbery region, is the stored strain energy function for the glassy region, ̅ and ̅ ( ) are the first invariant of the isochoric part of the right Cauchy- Green deformation tensors, and ( )are the Jacobian matrices for each region at time
. The subscripts “r” and “g” refer to the rubbery and glassy phases, respectively.
(= ⁄ ) and 2 = ⁄2 are the material input parameters which are related to shear modulus ( ) . Similarly, (= 2⁄ ) and = 2⁄ are the material input parameters related to bulk modulus ( ). These parameters are continuously updated while running the simulation according to the changes of phase content and deformation gradients at the current state.
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The temperature-dependent glass content function ( ) was defined as a linearized temperature variation like
= −
− (3.2.5) , where is the current temperature, is the temperature at the fully glassy state and is the temperature at the fully rubbery state. The four simulation steps of loading, cooling, unloading, and heating were modeled and coded in ABAQUS/CAE program, in the same way as the experimental shape memory test using the DMTA machine.
The defined non-linear material behaviors were programmed in the user subroutine code UMAT. A stiffness tensor matrix (ℂ) should be mathematically derived to be built and used in a finite element method (FEM) program. The process to derive the tangent stiffness matrix of glass shape memory polymers from the stored energy function ( ) is briefly suggested below based on the Barot’s dissertation [197]. A more detailed description of the term definition and modeling is provided in this reference, beyond the scope of this dissertation. The ABAQUS program requires the stiffness matrix defined as the following relationship.
ℂ =1
(3.2.6)
, where is the Jawman rate of Kirchoff stress defined as
= ̇ − −
= ̇ + ̇ − −
= ̇ + (3.2.7) And is the symmetric part of the spatial gradient of velocity. Therefore, the stiffness matrix can be transformed by putting Eqn. 3.2.7 into Eqn. 3.2.6 as
ℂ = ⊗ + (3.2.8)
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To calculate the stiffness matrix, the Cauchy stress tensor ( ) and the Jawman rate of Cauchy stress ( ) should be obtained first. The Cauchy stress tensor ( ) for hyperelastic materials is defined as
=2
(3.2.9)
, where is the deformation gradient with respect to reference configuration. The stored energy function ( ) for the glassy phase at each time increment (used in Eqn. 3.2.9) can be written as
= (1 − ) − 3 + 1
− 1
+ − 3 + 1
( )− 1 (3.2.10) The derivative of stored energy function with (the right Cauchy stretch tensor) on the right-hand-side of Eqn. 3.2.9 is derived as
= (1 − ) det −1
3 + 1
− 1
+( ) det ( ) +1
3 ( ) ( ) + 1
( ) ( )− 1 ( ) (3.2.11) , where is the identity tensor. From Eqns. 3.2.9 and 3.2.11, the Cauchy stress tensor ( ) can be calculated by the following relationship, where is the isochoric part of the deformation component for each phase.
= (1 − ) 2
−1
3 + 1
− 1 +( ) 2
( )−1
3 ( ) + 2
( )− 1 (3.2.12) The material derivative of the Cauchy stress tensor ( ̇) is defined as
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̇ = (1 − ) 2
+ −2
3 −2
3 ∙ +2
9
− 1
−1
3 + 2
+ ( ) 2
( )
( )+ ( ) −2
3 ( )−2
3 ( )∙ +2
9 ( ) − 1
( )
( )−1
3 ( )
+ 2
( ) (3.2.13) , where is the spatial gradient of velocity ( ⁄ ). Therefore, the Jawman rate of the Cauchy stress tensor ( ) is obtained by the following relationship.
= (1 − ) 2
+ −2
3 −2
3 ∙ +2
9
+ 2
2 − 1
+ ( ) 2
( )
( )+ ( ) −2
3 ( )−2
3 ( )∙ +2
9 ( ) + 2
2 ( )− 1 − (3.2.14)
By using Eqns. 3.2.12 and 3.2.14, the stiffness matrix (ℂ) is derived by the Kirchhoff stress- Newman strain rate relationship as an essential input in the ABAQUS/CAE program.
ℂ =1
2 ℂ + ℂ (3.2.15)
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ℂ = (1 − ) 2
+ + −4
3 +
+ 4
9 ( ) + 2
2 − 1
+ ( ) 2
( )
+ + −4
3 ( ) +
+ 4
9 ( ) + 2
2 ( )− 1 (3.2.16)
, where is Kronecker delta. The stiffness matrix was used in the ABAQUS/CAE program as a linearized weak formulation of principle of virtual work (P.V.W.).
∆ = 2 (ℂ Δ ∙ ) + ( Δ ∙ ) (3.2.17)
At the end of every time increment, the UMAT code is invoked for each nodal point of meshes, and updates stresses and solution-dependent state variables with the SDIVINI subroutine.
The input material parameters for the simulation were obtained by performing isothermal tensile tests at some temperature points [49]. The obtained material parameters are listed in Table 3.2.1.
A geometry for the simulation was constructed based on the real measurement specimen, 1 x 5.3 x 40 mm3 (Fig. 3.2.2(a)). The C3D8IH elements with 8-node linear brick, hybrid, linear pressure, incompatible modes were used when generating meshes. A loading boundary condition was applied at the specimen center by using an analytical rigid plate with a reference point. A constrained boundary condition was set at the two side pins to prevent detaching after the unloading step (Fig. 3.2.2(b)). As a simulation result, the von Mises stress contour at the end of each step was calculated and visualized.
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Table 3.2.1. Material parameters of the SMCPAs employed in the simulation.
S-MMA/BMA S-BA/BMA
(℃) 60 10
(℃) -20 -60
(MPa) 6 6
(MPa) 370 398
(1/MPa) 0.06 0.05
(1/MPa) 0.001 0.0008
ν 0.35 0.35
* : Temperature on fully rubbery phase.
** : Temperature on fully glassy phase.
***ν: Poisson ratio.
Figure 3.2.2. Constructed geometry for the shape memory simulation. (a) The modeled specimen. (b) The applied boundary conditions.
154 3.2.4. Results and discussion