Mechanical analysis of PDMS films based on different hyperelastic numerical constitutive models

PDMS(polydimethylsiloxane) is widely employed as a substrate material in flexible electronic devices, necessitating its ability to undergo camber deformation while maintaining excellent ductility and flexibility. Understanding the mechanical behavior of PDMS is imperative for its practical applications. Consequently, to numerically investigate the tensile behavior of PDMS, two commonly used hyperelastic constitutive models, the Mooney-Rivlin model and the 3-term Ogden model, have been employed to describe its mechanical characteristics. The material coefficients were determined by fitting the uniaxial tensile experimental data. Subsequently, separate finite element models were developed for the PDMS membrane and the Cu-PDMS composite layer structure. The numerical findings demonstrate that both the Mooney-Rivlin model and the 3-term Ogden model adequately fit the experimental data. Nevertheless, in comparison to the 3-term Ogden model, the PDMS stress distribution exhibits higher values at the same tensile rate, consequently resulting in the fracture of the Cu layer first.


Introduction
Due to material and design constraints, traditional electronic devices face challenges in meeting the demands of emerging fields, including biomedical applications and wearable electronics.Flexible electronics exhibit distinct characteristics compared to their rigid counterparts, as they possess both excellent flexibility and ductility [1].Despite being utilized in diverse domains, flexible electronics maintain a fundamental structural similarity.However, the key differentiating factor lies in the flexible substrate, which plays a crucial role in distinguishing flexible electronics technology from conventional counterparts [2].PDMS (polydimethylsiloxane) films have gained significant attention as a preferred choice for substrate materials [3].
In order to meet the demands of flexible electronics, the PDMS film is required to undergo inclination deformation.The relationship between stress and strain in PDMS is not simply linear, and the mechanical response behavior of PDMS films is commonly described using various hyperelastic material isomorphic models, such as the Mooney-Rivlin model and the 3-term Ogden model [4,5] .The choice of constitutive model has a significant impact on the mechanical simulation and analysis of PDMS.
In this work, uniaxial stretching experiments are conducted on PDMS films to gather stress-strain data.These experimental data are then fitted to the 3-term Ogden model and the Mooney-Rivlin model, both of which are suitable for simulating the mechanical behavior of PDMS during stretching.Finite element simulations are subsequently carried out to compare the effects of different constitutive models on the distribution of stresses and strains during the PDMS stretching process.

Hyperelastic Models
The intrinsic model of hyperelastic materials expresses the stress-strain relationship through the strain energy density function, which can be divided into deviatoric strain and volumetric strain.
is the total strain energy function and   denotes the strain energy density of the bias strain.  denotes the strain energy density of the volume strain.
denotes the strain energy density for volumetric strain,   represents the bulk modulus parameter of the i-th superellipsoid function, and   is the volumetric resistance criterion (Jacobian), N is the strain energy potential order.Most of the hyperelastic materials are considered incompressible, so the vast majority of the different constitutive models have the same expression for the bulk strain and different mathematical expressions for the bias strain.proposed by Mooney and Rivlin et al. in the 1950s, based on the invariance-based strain-energy model [6], the basic form of the Mooney-Rivlin model is as follows: where C 10 and C 01 are the material parameters of the model, and I ̅ 1 and I ̅ 2 are the invariants of the strain tensor.I ̅ 1 denotes the trace of the principal strains, and I ̅ 2 denotes the sum of the squares of the principal strains minus the squares of the traces.
The 3-term Ogden model is based on the strain energy density function and introduces multiple high-order strain invariants [7].The expression for the model's bias strain energy density function can be represented as follows: Where:  ̅  represents the principal stretch ratio.  and   denote material constants that describe the shear behavior.
For the purpose of investigating the impact of different hyperelasticity principal models on the mechanical simulation of PDMS, this study conducted experiments using dumbbell-shaped PDMS samples.Uniaxial stretching was performed utilizing a mechanical testing machine, Subsequently, the stress and strain were calculated.
It was observed that the experimental data exhibited a better fit with the 3-term Ogden model and the Mooney-Rivlin model.The material coefficients and stability tests resulting from the fitting process are presented in Table 1

Simulation
Finite element modeling of PDMS was performed using the 3-term Ogden model and the Mooney-Rivlin model.Displacement constraints were applied to both sides of the model [8].Subsequently, stress and strain distribution cloud diagrams were generated as output, as shown in Figure 1 and Figure 2.  The stress distribution graphs demonstrate that, under the same tensile rate, both intrinsic models exhibit higher stress concentrations at the sides and in the middle of the PDMS film.However, the stress values predicted by the 3-term Ogden model are smaller compared to those obtained from the Mooney-Rivlin model simulation.To gain a more detailed understanding of the stress and strain distribution in PDMS films, a further investigation was carried out at a 120% stretch rate.Specifically, the most representative films were selected from the lateral and centerline units, and their stress distribution is presented in Figure 3.Both models show a similar trend in terms of stress variation in PDMS.However, the Mooney-Rivlin model simulation yields higher stress values compared to the predictions obtained from Ogden's third-order model.The simulation results reveal that stretching the PDMS based on the Mooney-Rivlin model leads to higher stress compared to the same stretching distance.This work further investigates the impact of different constitutive models on PDMS, which is commonly used as a flexible substrate in flexible electronic devices.In such devices, an ultra-thin copper film is often utilized as a conductor for connecting different structures.

Figure1.
Figure1.The stress distribution of the PDMS film at a 40% stretch rate.(a) 3-term Ogden model (b) Mooney-Rivlin model.

Figure2.
Figure2.The stress distribution of the PDMS film at a 120% stretch rate.(a) Mooney-Rivlin model (b) the 3-term Ogden model.The stress distribution graphs demonstrate that, under the same tensile rate, both intrinsic models exhibit higher stress concentrations at the sides and in the middle of the PDMS film.However, the stress values predicted by the 3-term Ogden model are smaller compared to those obtained from the Mooney-Rivlin model simulation.To gain a more detailed understanding of the stress and strain distribution in PDMS films, a further investigation was carried out at a 120% stretch rate.Specifically, the most representative films were selected from the lateral and centerline units, and their stress distribution is presented in Figure3.Both models show a similar trend in terms of stress variation in PDMS.However, the Mooney-Rivlin model simulation yields higher stress values compared to the predictions obtained from Ogden's third-order model.

Figure 3 .
Figure 3. Stress in different parts.(a) center (b) sideThe simulation results reveal that stretching the PDMS based on the Mooney-Rivlin model leads to higher stress compared to the same stretching distance.This work further investigates the impact of different constitutive models on PDMS, which is commonly used as a flexible substrate in flexible electronic devices.In such devices, an ultra-thin copper film is often utilized as a conductor for connecting different structures.

Figure4.
Figure4.Stress distribution.(a) Mooney-Rivlin model (b) 3-term Ogden model.To construct the finite element simulation, the PDMS was modeled using both the Mooney-Rivlin model and Ogden's third-order model.The copper film layer was constrained to the PDMS layer, and displacement loads were applied to the common nodes on both sides[9].The stress diagram is presented in Figure4.Previous experiments had demonstrated that the yield stress of the 250nm-thick copper film was approximately 340 MPa[10].According to the simulation results obtained using the Mooney-Rivlin model, the copper layer experiences higher stress and is more susceptible to fracture. .