XXIV ICTAM, 21-26 August 2016, Montreal, Canada
MODELING OF BIAXIAL RESPONSE OF POLYVINYLEDENE FLUORIDE (PVDF) 1,2
Harish Lambadi1, Lakshmana Rao C 2 Department of Applied Mechanics, Indian Institute of Technology Madras
Summary Polyvinylidene Fluoride (PVDF) is a piezo polymer and exhibits both sensor and actuator effects for different potential applications in industries. In this study the experimental and modelling results of monotonic biaxial response of PVDF will be presented. A non contact speckle monitoring method was used to monitor point wise finite deformation in the specimen. A hyper elastic constitutive model using four invariants was used to capture the anisotropy in the mechanical response. A paper reports a good correlation between the proposed model and the observed experimental data. INTRODUCTION Polyvinylidene Fluoride has a basic building block of -CH2-CF2- monomer. The microstructure of the PVDF is semi crystalline. In the manufacturing process, the PVDF polymers have been stretched in one direction and a high electric field is applied in the thickness direction to amplify the polling and improve the piezoelectricity to the PVDF. The PVDF film sheets are transparent have silver coating on top and bottom of the film to capture and transmit the conduction of electric charges across the material. The mechanical behavior of the PVDF film sheets has been addressed in literature using conventional tensile test setup to see mechanical response. Vinogradov et al. [1] have observed the mechanical anisotropic on to the PVDF film. Due to a mechanical stretch applied at the time of manufacture time, the PVDF has been observed to have a high strength in that direction compared to orthogonal (unstretched) direction. Due to semi crystalline nature and alignment of fiber in direction of stretch, we assume PVDF as a transversely isotropic in the direction perpendicular to the poling direction. A detailed investigation of the nature of anisotropy in finite deformation and its modeling is attempted in this paper. In this study, 128µm thin metalized PVDF sheets were used to do biaxial tests. The samples were cut into cruiceform shape to do biaxial static tests and a black speckle was markers on these white films to track the displacements locally. The biaxial test setup has been used to carry out the experiment and measure the strains locally using a stingray digital camera, which is fixed in a direction orthogonally to the testing sample. The stress-strain plots which were observed from the image displacements were observed to yield better estimates of the stress-strain plots compared to the plots that are generated by using data from cross head movement of the test setup. In this study we attempt to do constitutive model of the nonlinear elastic response using a transversely isotropic finite deformation constitutive model. BIAXIAL TESTING The Bi-axial monotonic static tests were carried out using a biaxial test machine. Cruciform shaped samples are cut as per the dimensions shown in Figure1 (a). Black speckle mark points are applied on the sample to monitor the displacements using a non contact optical measurement technique as shown in the figure 1(b). A low speed camera has been used to capture the images while conducting the experiment at 15fps (frames per second). The black mark dots on the captured images are tracked using a MATLAB code that was written for this purpose. Strains and stretches are calculated by measuring the displacements of the markers. Nominal stress was calculated from the loads by using the cross sectional area of the specimens. The stress-strain plots were made for the biaxial experiment (under equi biaxial loading) at a cross-head displacement rate of 0.5mm/sec. The stressstrain curves obtained using the speckle method yield a more accurate point wise strains, in comparison to the average strain estimates that were obtained by many researchers [1] using cross-head movement data. The stress-strain plots of the material are plotted in Figures 2(a) and (b). The difference between the two graphs clearly establishes the anisotropy in the mechanical response at finite strains in this material.
a
b
Figure 1: (a) PVDF cruciform specimen and corresponding dimensions (b) One of the captured image to measure the strains using black marks. a)
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CONSTITUTIVE MODELLING Usually PVDF will undergo moderate to large deformation (finite deformation) which also vary non-linearly with stresses, as observed in other uniaxial or biaxial tests. Hence a hyper elastic model was used to capture the mechanical response under monotonic loading of the material. With the transversely isotropic assumption and incompressibility constraints, we can choose a polynomial expansion for the strain energy density function, as suggested Humphrey [2] for transversely isotropic materials undergoing finite deformations. The strain energy function is proposed by Humphrey to be a function of the strain invariants I1 and I4 and is of the form given below.
W ( I1 , I 4 ) C1
2
I 4 1 C2
I 4 1 C3 I1 3 C4 I1 3 3
I 4 1 C5 I1 3
2
Where I1 is trace of the stretch ratios in the three directions, and I4 is square of the stretch ratio in the polling of stretch direction, C1 to C5 are the parameters which can be found using optimization of the experimental data and modal stress components. From the continuum literature [4], stress tensor is a partial derivative of strain energy function with respect to right Cauchy green deformation tensor (C). After simplification and using the chain rule of the derivative the following stress strain relations were derived by Humphrey[2] for transversely isotropic materials.
𝝈 = −𝑝𝑰 + 2𝑊1 𝑩 + 2𝑊4 𝑭𝑒 ⊗ 𝑒𝑭𝑇 Where p is a Lagrangian multiplier, W1 and W4 are partial derivative of W with respect to I1 and I4 invariants, bold letters refers to second order tensor, e is a unit vector in the polling of stretched direction. RESULTS Levenberg-Marquardt algorithm [3] has been used to optimise the parameters. Differences between model and experimental stresses are found as an error and sum of the squares of the error is considered as an objective function to carry out this optimization. The optimised parameters obtained using the above algorithm, stress- stretch curves are obtained and are plotted as shown in the figure 2(a) and 2(b). The optimised parameters are given below. 𝐶1 = 1.6 × 105 , 𝐶2 = −2.1 × 105 , 𝐶3 = 684, 𝐶4 = −43808, 𝐶5 = 40590,
a
b
Figure 1: Biaxial test plot (a) in a polling of stretched direction (b) in an orthogonal direction or longitudinal direction
References [1] Vinogradov, Aleksandra, and Frank Holloway. "Electro-mechanical properties of the piezoelectric polymer PVDF." Ferroelectrics 226.1 (1999): 169-181. [2] Humphrey, J. D., R. K. Strumpf, and F. C. P. Yin. "Determination of a constitutive relation for passive myocardium: I. A new functional form."Journal of biomechanical engineering 112.3 (1990): 333-339.
[3] Moré, Jorge J. "The Levenberg-Marquardt algorithm: implementation and theory." Numerical analysis. Springer Berlin Heidelberg, 1978. 105-116. [4] Lakshmana rao C., Abhijit P Deshpande., Modeling of engineering materials. Ane Books Pvt. Ltd.
