Mooney-Rivlin Hyperelastic Model Research Articles

Extracting the stress–strain properties of non-linear viscoelastic materials using the bubble inflation test

In this study, an inverse method based on the Levenberg–Marquardt algorithm was evaluated in a numerical experiment to determine the large strain viscoelastic properties from the bubble inflation test. The properties were determined by iteratively matching the calculated bubble pressure–piston displacement data from finite element simulations to a single set of bubble pressure–piston displacement data. The strain-dependent behaviour was characterised by a two-parameter Mooney–Rivlin hyperelastic model, while the time-dependent behaviour was characterised by a three-parameter power law equation. Different initial guesses were used to evaluate the inverse method, and transformation functions were applied to constrain the intermediate guesses to be within bounds. It was found that estimates of the viscoelastic properties could be obtained reasonably using only one set of bubble pressure–piston displacement data. Estimates of the properties were likely affected by the limited time duration of the test, as the behaviour at shorter and particularly larger time scales was less accurately predicted.

Long-term changes in the material properties of brain tissue at the implant–tissue interface

Objective. Brain tissue undergoes dramatic molecular and cellular remodeling at the implant–tissue interface that evolves over a period of weeks after implantation. The biomechanical impact of such remodeling on the interface remains unknown. In this study, we aim to assess the changes in the mechanical properties of the brain–electrode interface after chronic implantation of a microelectrode. Approach. Microelectrodes were implanted in the rodent cortex at a depth of 1 mm for different durations—1 day (n = 4), 10–14 days (n = 4), 4 weeks (n = 4) and 6–8 weeks (n = 7). After the initial duration of implantation, the microelectrodes were moved an additional 1 mm downward at a constant speed of 10 µm s−1. Forces experienced by the microelectrode were measured during movement and after termination of movement. The biomechanical properties of the interfacial brain tissue were assessed from measured force–displacement curves using two separate models—a two-parameter Mooney–Rivlin hyperelastic model and a viscoelastic model with a second-order Prony series. Main results. Estimated shear moduli using a second-order viscoelastic model increased from 0.5–2.6 kPa (day 1 of implantation) to 25.7–59.3 kPa (after 4 weeks of implantation) and subsequently decreased to 0.8–7.9 kPa after 6–8 weeks of implantation in 6 of the 7 animals. The estimated elastic modulus increased from 4.1–7.8 kPa on the day of implantation to 24–44.9 kPa after 4 weeks. The elastic modulus was estimated to be 6.8–33.3 kPa in 6 of the 7 animals after 6–8 weeks of implantation. The above estimates suggest that the brain tissue surrounding the microelectrode evolves from a stiff matrix with maximal shear and elastic modulus after 4 weeks of implantation into a composite of two different layers with different mechanical properties—a stiff compact inner layer surrounded by softer brain tissue that is biomechanically similar to brain tissue—during the first week of implantation. Tissue micromotion-induced stresses on the microelectrode constituted 12–55% of the steady-state stresses on the microelectrode on the day of implantation (n = 4), 2–21% of the steady-state stresses after 4 weeks of implantation (n = 4), and 4–10% of the steady-state stresses after 6–8 weeks of implantation (n = 7). Significance. Understanding biomechanical behavior at the brain–microelectrode interface is necessary for the long-term success of implantable neuroprosthetics and microelectrode arrays. Such quantitative physical characterization of the dynamic changes in the electrode–tissue interface will (a) drive the design and development of more mechanically optimal, chronic brain implants, and (b) lead to new insights into key cellular and molecular events such as neuronal adhesion, migration and function in the immediate vicinity of the brain implant.

Effects of Cyclic Motion on Coronary Blood Flow

The goal of this study was to establish a computational fluid dynamics model to investigate the effect of cyclic motion (i.e., bending and stretching) on coronary blood flow. The three-dimensional (3D) geometry of a 50-mm section of the left anterior descending artery (normal or with a 60% stenosis) was constructed based on anatomical studies. To describe the bending motion of the blood vessel wall, arbitrary Lagrangian-Eularian methods were used. To simulate artery bending and blood pressure change induced stretching, the arterial wall was modeled as an anisotropic nonlinear elastic solid using the five-parameter Mooney-Rivlin hyperelastic model. Employing a laminar model, the flow field was solved using the continuity equations and Navier-Stokes equations. Blood was modeled as an incompressible Newtonian fluid. A fluid-structure interaction approach was used to couple the fluid domain and the solid domain iteratively, allowing force and total mesh displacement to be transferred between the two domains. The results demonstrated that even though the bending motion of the coronary artery could significantly affect blood cell trajectory, it had little effect on flow parameters, i.e., blood flow velocity, blood shear stress, and wall shear stress. The shape of the stenosis (asymmetric or symmetric) hardly affected flow parameters either. However, wall normal stresses (axial, circumferential, and radial stress) can be greatly affected by the blood vessel wall motion. The axial wall stress was significantly higher than the circumferential and radial stresses, as well as wall shear stress. Therefore, investigation on effects of wall stress on blood vessel wall cellular functions may help us better understand the mechanism of mechanical stress induced cardiovascular disease.

Rheological model optimization using advanced evolutionary computation for the analysis of the influence of recycled rubber on rubber blend dynamical behaviour

Rheological models of tested rubber blend samples, with different ratios of smut/recycled material, were developed, analyzed and discussed according to experimental data. The experimental measurements show the dynamical behavior of these samples after a short-term axial load. The rheological model of samples was assumed as a viscoelastic Voigt-Maxwell and hyperelastic Mooney-Rivlin model. Real coded genetic algorithms were implemented for optimal model parameters identification in order to minimize the difference between the real sample behavior and dynamical model output. Numerical results obtained for analyzed mixture were compared with the original sample to verify the presented method. In purpose of predicting the dynamical characteristics for desirable rubber blends, an interrelation was formed based on the concentration of recycled material in sample blends and dynamical model parameters dependence.

Numerical and experimental analysis of wall thickness variation of a hemispherical PMMA sheet in thermoforming process

In thermoforming, a heated plastic sheet is stretched into a mold cavity by applying pressure, eventually assisted by direct mechanical loading. Since upon its contact with the cold surface of the mold the sheet is prevented from undertaking any further deformation, the forming sequence induces a thickness variation in the final part. This fundamental inherent defect of thermoforming technology highly affects the optical characteristics of optical products. Therefore, the more uniform the wall thickness, the less chance optical defects will occur. In this research, the production process of a hemispherical transparent PMMA sheet as an optical product was numerically simulated. The simulated process is a two-step process comprising a combination of free forming and plug-assisted forming. In the simulation, the acrylic sheet is assumed to undergo a nonlinear and large elastic deformation which merits application of hyperelastic models. Mooney–Rivlin hyperelastic model is used as the constitutive equation. The obtained numerical results are validated with those achieved from the experiments. Different combinations of free forming and plug-assisted forming methods are studied based on what percentage of total height of the final part is produced by each method. Finally, an optimum combination of the two-step forming process is proposed. With this optimum combination, satisfactorily uniform wall thickness and minimal mold marks on the product surface will be achieved.

A Comparison of Back Grinding Processes for Bare Silicon and Through-Silicon Via Wafers Using Numerical Simulations

The optimization of grinding parameters for silicon wafers is necessary in order to reduce the likelihood of residual stresses and crack nucleation in the machined surface and improve the reliability of electronic packages. This paper describes numerical simulations performed to characterize the back grinding process for bare silicon wafers and through-silicon via (TSV) wafers using the finite element code ABAQUS. The grinding of the two wafer types was performed by simulating the motion of a diamond particle cutting through successive silicon and TSV layers. The silicon material was modeled using orthotropic elasticity and isotropic plasticity, while the copper vias were modeled using isotropic elasticity and Johnson-Cook plasticity. The polyethylene terephthalate (PET) material used as a backing tape for the silicon wafer was modeled using the Mooney-Rivlin hyperelastic model. The computed residual stresses and the plastic deformation in the superficial layer of the ground wafer were compared with experimental values and good correlation was observed. The simulation results can be used to better understand the local stresses and strain fields in both bare silicon and TSV wafers, and to quantify the effect of the copper vias on the wafer properties during and after grinding.

Biomechanical wall properties of human intracranial aneurysms resected following surgical clipping (IRRAs Project)

Background and purpose Individual rupture risk assessment of intracranial aneurysms is a major issue in the clinical management of asymptomatic aneurysms. Aneurysm rupture occurs when wall tension exceeds the strength limit of the wall tissue. At present, aneurysmal wall mechanics are poorly understood and thus, risk assessment involving mechanical properties is inexistent. Aneurysm computational hemodynamics studies make the assumption of rigid walls, an arguable simplification. We therefore aim to assess mechanical properties of ruptured and unruptured intracranial aneurysms in order to provide the foundation for future patient-specific aneurysmal risk assessment. This work also challenges some of the currently held hypotheses in computational flow hemodynamics research. Methods A specific conservation protocol was applied to aneurysmal tissues following clipping and resection in order to preserve their mechanical properties . Sixteen intracranial aneurysms (11 female, 5 male) underwent mechanical uniaxial stress tests under physiological conditions, temperature, and saline isotonic solution. These represented 11 unruptured and 5 ruptured aneurysms. Stress/strain curves were then obtained for each sample, and a fitting algorithm was applied following a 3-parameter ( C 10, C 01, C 11) Mooney–Rivlin hyperelastic model. Each aneurysm was classified according to its biomechanical properties and (un)rupture status. Results Tissue testing demonstrated three main tissue classes: Soft, Rigid, and Intermediate. All unruptured aneurysms presented a more Rigid tissue than ruptured or pre-ruptured aneurysms within each gender subgroup. Wall thickness was not correlated to aneurysmal status (ruptured/unruptured). An Intermediate subgroup of unruptured aneurysms with softer tissue characteristic was identified and correlated with multiple documented risk factors of rupture. Conclusion There is a significant modification in biomechanical properties between ruptured aneurysm, presenting a soft tissue and unruptured aneurysms, presenting a rigid material. This finding strongly supports the idea that a biomechanical risk factor based assessment should be utilized in the to improve the therapeutic decision making.

Photopatterning the mechanical properties of polydimethylsiloxane films

Silicone rubber films with graded and localized mechanical properties are prepared using two-part polydimethylsiloxane (PDMS) elastomer, photoinhibitor compounds and conventional photolithography. First the un-cross-linked PDMS is mixed with benzophenone. The resulting positive photosensitive material is then exposed through a mask to UV light from a conventional mask aligner. Cross-linking of the UV exposed elastomer is inhibited, leading to softer regions than the surrounding unexposed matrix. By empirically fitting the nonlinear, hyperelastic Mooney–Rivlin model to experimentally measured stress–strain curves we determine the equivalent tensile modulus (E) of the rubber film. We show the PDMS tensile modulus can then be adjusted in the 0.65–2.9 MPa range by decreasing the UV exposure dose (from 24 000 to 0 mJ cm−2). Further, using a patterned UV mask, we can locally define differential regions of tensile modulus within a single PDMS rubber film. We demonstrate that “hard islands” (E ≈ 2.9 MPa) of 100 μm minimum diameter can be patterned within a 100-μm-thick, single “soft” PDMS rubber membrane (E ≈ 0.65 MPa) cured at 150 °C for 24 h. Thin gold film conductors patterned directly onto the photopatterned PDMS are stretchable and withstand uniaxial cycling to tens of percent strain. The mechanically “pixellated” PDMS rubber film provides an improved substrate with built-in strain relief for stretchable electronics.

Influence of arterial wall-stenosis compliance on the coronary diagnostic parameters

Functional diagnostic parameters such as Fractional Flow Reserve (FFR), which is calculated from pressure measurements across stenosed arteries, are often used to determine the functional severity of coronary artery stenosis. This study evaluated the effect of arterial wall-stenosis compliance, with limiting scenarios of stenosis severity, on the diagnostic parameters. The diagnostic parameters considered in this study include an established index, FFR and two recently developed parameters: Pressure Drop Coefficient (CDP) and Lesion Flow Coefficient (LFC).The parameters were assessed for rigid artery (RR; signifying high plaque elasticity), compliant artery with calcified plaque (CC; intermediate plaque elasticity) and compliant artery with smooth muscle cell proliferation (CS; low plaque elasticity), with varying degrees of epicardial stenosis. A hyperelastic Mooney–Rivlin model was used to model the arterial wall and plaque materials. Blood was modeled as a shear thinning, non-Newtonian fluid using the Carreau model. The arterial wall compliance was evaluated using the finite element method.The present study found that, with an increase in stenosis severity, FFR decreased whereas CDP and LFC increased. The cutoff value of 0.75 for FFR was observed at 78.7% area stenosis for RR, whereas for CC and CS the cutoff values were obtained at higher stenosis severities of 81.3% and 82.7%, respectively. For a fixed stenosis, CDP value decreased and LFC value increased with a decrease in plaque elasticity (RR to CS). We conclude that the differences in diagnostic parameters with compliance at intermediate stenosis (78.7–82.7% area blockage) could lead to misinterpretation of the stenosis severity.

Topological sensitivity analysis for a two-parameter Mooney-Rivlin hyperelastic constitutive model

The Topological Sensitivity Analysis (TSA) is represented by a scalar function, called Topological Derivative (TD), that gives for each point of the domain the sensitivity of a given cost function when an infinitesimal hole is created. Applications to the Laplace, Poisson, Helmoltz, Navier, Stokes and Navier-Stokes equations can be found in the literature. In the present work, an approximated TD expression applied to nonlinear hyperelasticity using the two parameter Mooney-Rivlin constitutive model is obtained by a numerical asymptotic analysis. The cost function is the total potential energy functional. The weak form of state equation is the constraint and the total Lagrangian formulation used. Numerical results of the presented approach are considered for hyperelastic plane problems.

Unsteady flow and mass transfer in models of stenotic arteries considering fluid-structure interaction

In this work the unsteady non-Newtonian blood flow and mass transfer in symmetric and non-symmetric stenotic arteries are numerically simulated considering the fluid-structure interaction (FSI) using the code ADINA. Blood with hyperviscosity syndrome is considered and hyperelastic Mooney–Rivlin model is used for the compliant arterial wall. The inlet boundary condition of imposed velocity or pressure is critical to obtain realistic hemodynamic results in stenotic arteries. The FSI affects significantly the hemodynamics on the stenotic arteries models, the arteries are considerably dilated and compressed due the stenosis. The stenosis severity and geometry have important influence on recirculation length, and distribution of concentration of macromolecules, such as low density lipoproteins (LDL).

Development of compliant coating system for transfer molding of sensitive silicon dice

Exposed-die type packaging requires molding process capability to provide mold flash free situation. Presently, the trend is in adopting transfer molding to encapsulate the silicon device with epoxy molding compound, which involves high transfer pressure. Hence, to avoid any epoxy bleed on the exposed die region a novel method of applying a soft coating on the mold surface is adopted. The coating is designed to act as a leakproof sealant to resist the high packing pressure during molding and can accommodate die height variations and minimizes the clamping stress passed to the silicon die, thus avoiding any chances of die cracking. This paper describes the use of Mooney-Rivlin hyperelastic model based finite element method (FEM) in the design of soft coating so as to minimize die stress under clamping and molding.

Finite-amplitude inhomogeneous waves in Mooney–Rivlin viscoelastic solids

New exact solutions are exhibited within the framework of finite viscoelasticity. More precisely, the solutions correspond to finite-amplitude, transverse, linearly polarized, inhomogeneous motions superposed upon a finite homogeneous static deformation. The viscoelastic body is composed of a Mooney–Rivlin viscoelastic solid, whose constitutive equation consists in the sum of an elastic part (Mooney–Rivlin hyperelastic model) and a viscous part (Newtonian viscous fluid model). The analysis shows that the results are similar to those obtained for the purely elastic case; inter alia, the normals to the planes of constant phase and to the planes of constant amplitude must be orthogonal and conjugate with respect to the B -ellipsoid, where B is the left Cauchy–Green strain tensor associated with the initial large static deformation. However, when the constitutive equation is specialized either to the case of a neo-Hookean viscoelastic solid or to the case of a Newtonian viscous fluid, a greater variety of solutions arises, with no counterpart in the purely elastic case. These solutions include travelling inhomogeneous finite-amplitude damped waves and standing damped waves.