Rubber-like Materials Research Articles

Torsion and Extension of Functionally Graded Mooney–Rivlin Cylinders

We analytically study finite torsional and extensional deformations of rubberlike material circular cylinders with the two material moduli in the Mooney–Rivlin relation assumed to be continuous functions of the undeformed radius. It is shown that under null resultant axial load on the end faces the cylinder length increases upon twisting. Furthermore, when the two moduli are affine functions of the radius the inhomogeneity parameters can be found to have the maximum shear stress occur at a pre-determined interior point. Whereas the radial stress is finite at the center of a cross-section of a homogeneous material cylinder, it may have large values for an inhomogeneous material cylinder. The closed-form solutions provided herein for the two moduli having affine, power-law and exponential functions of the radius should benefit numerical analysts verify their algorithms and engineers design soft material robots for improving their performance under torsional loads.

Influence of temperature and crack-tip speed on crack propagation in elastic solids.

I study the influence of temperature and the crack-tip velocity of bond breaking at the crack tip in rubber-like materials. Bond breaking is considered as a stress-aided thermally activated process and results in an effective crack propagation energy, which increases strongly with decreasing temperature or increasing crack-tip speed. This effect is particularly important for adhesive (interfacial) crack propagation but less important for cohesive (bulk) crack propagation owing to the much larger bond-breaking energies in the latter case. For adhesive cracks, the theory results are consistent with adhesion measurements for silicone rubber polydimethylsiloxane (PDMS) in contact with silica glass surfaces. For cohesive cracks, the theory agrees well with experimental results PDMS films chemically bound to silanized glass.

A numerical study of Mullins softening effects on mode I crack propagation in viscoelastic solids

This paper presents a finite element analysis of steady-state crack propagation in viscoelastic soft solids exhibiting Mullins softening. A cohesive-zone model is employed to simulate the localized processes at the tip of a Mode I crack in materials governed by viscoelastic behavior and damage-induced Mullins effects. The study numerically evaluates the intrinsic dissipation characteristics of typical rubber-like materials, focusing on the influence of key factors such as Mullins damage, relaxation modulus, and relaxation time. The impact of these factors on material toughening is examined, with particular emphasis on their role in crack propagation. The results reveal that crack propagation velocity is highly sensitive to the interplay between energy dissipation mechanisms. Specifically, Mullins damage parameters are shown to increase fracture toughness by raising the local energy release rate threshold at the crack tip. Additionally, the relaxation modulus enhances viscous dissipation, further elevating this threshold and subsequently reducing crack propagation velocity. Interestingly, an inverse relationship between relaxation time and crack propagation velocity is observed. The study provides a detailed analysis of the dissipation mechanisms at the crack tip, offering valuable insights for improving material toughness.

Local measurement on oedometric compression tests: Time and temperature ageing effects on a fluorosilicone

O-ring durability is a key issue for engineering structures that require sealing over a long service life. Rubber-like materials are used for this type of component because of the assumption of incompressibility, i.e. a high bulk modulus K. However, this assumption has been called into question in the literature, particularly for elastomers under high hydrostatic pressures. This work examines the compressibility of rubber-like materials, with a particular focus on fluorosilicone elastomers (FVMQ). A method of confined compression using a transparent crucible is presented, which allows local measurement of the displacement field. This technique provides a better understanding of the analysis of oedometric compression data and a reliable way to determine the K value. Furthermore, this approach allows following the evolution of K over time with accelerated ageing. Three ageing temperatures - 200, 220 and 250 °C - were tested up to 34, 15 and 1 weeks respectively. For the three ageing temperatures, the FVMQ results show a decrease in K values with ageing. In particular, at 250 °C, a turning point was observed after 72 h of ageing. These results highlight the influence of ageing on the compressibility of the FVMQ and the presence of two different ageing mechanisms affecting the K evolution.

Cavitation damage in rubber-like silicone adhesives

In this work, the failure of rubber-like materials by cavitation is investigated. Under local triaxial loading states, as they can occur in confined geometries of adhesive bonds, the instantaneous formation of voids can be observed. Analytical results of cavitation in a commercial silicone adhesive are provided. The results show that cavitation voids can be distinguished from manufacturing voids by the inner surface of the void. The surface of the cavitation void is rugged and shows a complex fracture pattern. Pancake experiments have been conducted for different ratios of radius and thickness. To assess the fracture process of cavitation onset one macroscopic and two asymptotic solutions of the energy release rate of cavitation voids are proposed. The solutions are compared and their limitations for the analysis of cavitation onset are discussed.

Coupled torsion and inflation of functionally graded and residually stressed Mooney–Rivlin hollow cylinders

Functionally graded structures have material properties continuously varying in one or more directions. Examples include human teeth, sea shells, bamboo stems, and mammal organs in which the varying volume fraction and orientation of fibers optimize their functionalities. Here we analytically study large deformations due to combined torsion and inflation of residually stressed and radially graded Mooney–Rivlin hollow circular cylinders to provide insights into how grading their material moduli according to a power law function of the radius can be advantageously used. We simulate residual stresses in a thin wall hollow cylinder by inverting it inside out and in a thick wall cylinder by assuming that a longitudinal wedge opening parallel to the cylinder axis is closed by deforming it axisymmetrically. It is found that positive integer values of the power law exponent are generally advantageous but its negative values can have deleterious effects on stress distributions. Furthermore, a hollow cylinder with residual stresses generated by one of the foregoing methods in the reference configuration should be modeled as orthotropic rather than transversely isotropic for subsequent deformations. The analytical solutions provided here should help numerical analysts verify their algorithms for large deformations of rubber-like materials that are modeled by the Mooney–Rivlin relation, and design engineers for exploiting the radial gradation of the two material moduli to reduce structure’s mass without sacrificing its performance.

Application of PolyJet 3D Printing in Production of Flexographic Printing Plates

The aim of this study was to investigate whether PolyJet technology, which uses rubber-like materials for printing and is known for its high resolution and performance, could be suitable for producing flexographic printing plates. In our research, we designed test plates that were printed using PolyJet technology with TangoBlackPlus FLX9870-DM resin. These 3D-printed plates were evaluated for their resistance to various flexographic inks and solvents, and their contact angles were measured. Subsequently, the prints were made on a Flexiproof device using water-based ink with both the test plates and traditional photopolymer plates across six different substrates. The print quality was assessed using densitometry and spectrophotometry. Our findings indicate that the 3D-printed plates are suitable for printing solid areas and lines with water-based inks. However, the print quality of the 3D-printed plates is slightly lower than that of the photopolymer plates, with the optical density values for the high-quality prints on coated papers being approximately 10% lower. Additionally, the plates printed with TangoBlack Plus resin appear to be suitable for UV inks due to their high resistance, but they are not resistant to the solvents used in solvent-based inks.

On the experimental identification of equilibrium relations and the separation of inelastic effects in soft biological tissues

The mechanical characterization of vascular tissues has been mainly focused on the measurement of elastic properties, while the investigation of inelastic effects has received comparatively little attention. Even the relatively simple, purely elastic description of the material behavior requires an appropriate set of experimental data that cannot be easily isolated using standard testing procedures. The presence of viscous and damage-related phenomena poses some challenges in the definition of appropriate testing protocols capable of identifying an equilibrium response, which in general does not solely represent the elastic material behavior. The primary goal of the present study is therefore to devise an experimental procedure that can distinguish and evaluate the different constitutive phenomena separately. To this end, we apply methodologies widely used in the mechanical testing of rubber-like materials and transfer them to the field of biomechanics. We performed two types of experiments in equibiaxial extension on porcine thoracic aorta: a continuous cyclic test followed by a single-step relaxation test and a cyclic multi-step relaxation test, each at varying stretch rates. We demonstrate that the approximation of quasi-stationarity through continuous testing at slow rates is inadequate for the identification of an equilibrium relation. Alternatively, a step-wise protocol allows for the separation of equilibrium and viscous effects. This motivates a thermodynamic discussion of the experimental results in terms of energy dissipation and a closer look at the interplay of inelastic phenomena.

Towards a rational approach for multi-axial experimental campaigns for rubberlike material

This work takes up the developments around the logarithmic strain tensor and uses the invariants of this tensor to propose a new approach to multi-axiality of fatigue experiments for elastomers. This study leads to the introduction of a new notion, modality, which is intended as the microscopic counterpart of uni- and multi-axiality. This notion is quantified by the K3 invariant (mode of deformation) of the logarithmic strain tensor, and is used to rationalize tension–torsion experimental campaigns. It is illustrated using two examples: the perfect cylinder and the AE2 “diabolo” sample. We then propose a methodology for building a test campaign based on this new definition.

New two-parameter constitutive models for rubber-like materials: Revisiting the relationship between single chain stretch and continuum deformation

The connection between macroscopic deformation and microscopic chain stretch is a key element in constitutive models for rubber-like materials that are based on the statistical mechanics of polymer chains. A new micro-macro chain stretch relation is proposed, using the Irving–Kirkwood–Noll procedure to construct a Cauchy stress tensor from forces along polymer chains. This construction assumes that the deformed polymer network remains approximately isotropic for low to moderate macroscopic stretches, a starting point recently adopted in the literature to propose a non-affine micro-macro chain stretch relation (Amores et al., 2021). Requiring the constructed Cauchy stress to be consistent with the stress tensor derived from the strain energy density results in a new chain stretch relation involving the exponential function. A hybrid chain stretch relation combining the new chain stretch with the well-known affine relation is then proposed to account for the whole range of stretches in experimental datasets. Comparison of the model predictions to experimental data in the literature shows that the two new micro-macro chain stretch relations in this work result in two-parameter constitutive models that outperform those based on existing chain stretches with no increase in the number of fitting parameters used.

Mooney-Rivlin Parameter Determination Model as a Function of Temperature in Vulcanized Rubber Based on Molecular Dynamics Simulations.

The automotive industry is entering a digital revolution, driven by the need to develop new products in less time that are high-quality and environmentally friendly. A proper manufacturing process influences the performance of the door grommet during its lifetime. In this work, uniaxial tensile tests based on molecular dynamics simulations have been performed on an ethylene-propylene-diene monomer (EPDM) material to investigate the effect of the crosslink density and its variation with temperature. The Mooney-Rivlin (MR) model is used to fit the results of molecular dynamics (MD) simulations in this paper and an exponential-type model is proposed to calculate the parameters C1(T) and C2T. The experimental results, confirmed by hardness tests of the cured part according to ASTM 1415-88, show that the free volume fraction and the crosslink density have a significant effect on the stiffness of the EPDM material in a deformed state. The results of molecular dynamics superposition on the MR model agree reasonably well with the macroscopically observed mechanical behavior and tensile stress of the EPDM at the molecular level. This work allows the accurate characterization of the stress-strain behavior of rubber-like materials subjected to deformation and can provide valuable information for their widespread application in the injection molding industry.

Finite Elastic Deformations of Initially Flat Thin Rubber-Like Materials

This contribution aims to derive a nonlinear FE formulation for large elastic deformation of hyperelastic thin rubber-like elastomeric materials. The formulation accounts for isotropic as well as transversely isotropic rubber-like materials. Besides, with the developed formulation, deformation of incompressible as well as compressible materials can be investigated. The formulation is applicable for in-plane and out-of-plane deformation of thin membranes. Several problems are inspected including uniaxial and equibiaxial extension of rubber, axisymmetric deformation of annular sheet, stretching a perforated rubber, extension of an anisotropic rubber, and inflation of an initially-flat circular and a square membrane. The results show very good agreement compared with those reported in the literature. Moreover, this model is able to model the inflation of initially flat membranes which is not possible to be done by commercial software.

A hyperelastic strain energy function for isotropic rubberlike materials

A three parameter novel hyperelastic strain energy function is introduced in this paper for soft and rubber-like materials. The function integrates a non-separable exponential component with a single term Ogden-type polynomial-like function, resulting in an exponential-polynomial based strain energy function. This helps in capturing both small and large deformation (stretch) behaviours of hyperelastic materials. The structure of the model is simple and validated against several experimental datasets including rubbers, hydrogel, and soft tissues. The model is reported to capture key material behaviors, including strain stiffening and various deformation paths. Through comparative studies with well-known models like the Ogden (six parameters) and Yeoh (three parameters), the model’s effectiveness is established. Furthermore, the model successfully addresses pressure-inflation instability in thin spherical balloons. It’s applicability extends to biological materials, as evidenced by its effectiveness in characterizing porcine brain tissue and a monkey’s bladder.

Rediscovering the Mullins effect with deep symbolic regression

The Mullins effect represents a softening phenomenon observed in rubber-like materials and soft biological tissues. It is usually accompanied by many other inelastic effects like for example residual strain and induced anisotropy. In spite of the long term research and many material models proposed in literature, accurate modeling and prediction of this complex phenomenon still remain a challenging task.In this work, we present a novel approach using deep symbolic regression (DSR) to generate material models describing the Mullins effect in the context of nearly incompressible hyperelastic materials. The two step framework first identifies a strain energy function describing the primary loading. Subsequently, a damage function characterizing the softening behavior under cyclic loading is identified. The efficiency of the proposed approach is demonstrated through benchmark tests using the generalized the Mooney–Rivlin and the Ogden–Roxburgh model. The generalizability and robustness of the presented framework are thoroughly studied. In addition, the proposed methodology is extensively validated on a temperature-dependent data set, which demonstrates its versatile and reliable performance.

A second gradient theory of thermoviscoelasticity

This paper is concerned with a rate-type theory of thermoviscoelasticity in which the second gradient of the displacement and the second temperature gradient are added to the classical set of independent constitutive variables. Viscoelasticity and related phenomena are of great importance in the study of biological materials. An adequate modeling of rubber-like materials and of biological soft tissues requires the use of the theory of viscoelasticity. Introduction of the concept of thermal displacement and the theory of multipolar continua allows us to show that Green-Naghdi thermomechanics can be used to derive a second gradient theory. The basic equations of the theory are established and the boundary conditions associated to nonsimple materials are investigated. The stress tensor and hyperstress tensor are shown to depend on the first and second temperature gradients. For rigid heat conductors we find that the temperature satisfies a fourth order equation. The boundary-initial-value problems are formulated. A uniqueness result in the dynamic theory of thermoviscoelastic materials is presented. We establish an existence result and prove the analyticity of the solutions. As a consequence, the exponential decay of the solutions and their impossibility of localization are obtained.

Solving non-linearities of hyperelastic rubbery behaviour by the asymptotic numerical method: Highlighting with the case of a Hybrid Integral Approach (HIA) model

This paper presents mathematical modelling and simulation founded on an Asymptotic Numerical Method (ANM) for the rubber-like material hyperelastic behaviour. The basic models such as Mooney-Rivlin, and the Gent partial models and the Hybrid Integral Approach (HIA) model are considered. The HIA is a strong nonlinear model developed from the inverse Langevin function approximation is considered here for its efficiency. Explicit formulations of each model are presented as well as the associated asymptotic developments in the frame of the ANM. Various types of boundary conditions effects are analysed for each considered hyperelastic model. The effectiveness of the ANM compared to the Newton-Raphson method is demonstrated and particularly for the HIA model. We present as results stress-strain curves for the implementation of these different models where we specify the influence of the Imposed Displacement for tensile and compressive stresses when the structure is clamped and simply supported. In term of this study, we observe that the complete HIA model present the curves with the difference hardly perceptible when we vary results in term of stress-strain curves with different imposed Displacement. But when strain is high than 5%, the Mooney-Rivlin and the Gent partial models, presents the high disparities with different imposed Displacement. This observation justifying the limitations of partial models in term of characterisation of the behaviour law of hyperplastic materials.

Mechanical Characterization of Vial Strain During Freezing and Thawing Operations Using Amorphous Excipients

The purpose of this study was to investigate the mechanical stresses and strains acting on pharmaceutical glass tubing vials during freezing and thawing of model pharmaceutical formulations. Strain measurements were conducted inside of a laboratory-scale freeze-dryer using a custom wireless sensor. In both sucrose and trehalose formulations at concentrations between 5 % and 20 % w/v, the strain measurements initially increased before peaking in magnitude at temperatures close to the respective glass transition temperatures of the maximally freeze concentrated solutes, Tg’. We attribute this behavior to a shift in the mechanical properties of the frozen system from a purely elastic glass below Tg’ to a viscoelastic rubber-like material above Tg’. That is, when the interstitial region becomes mechanically compliant at temperature above Tg’. The outputs were less predictable below 5 % w/v and tended to exhibit two separate peaks in strain output, one near the equilibrium melting temperature of pure ice and the other near Tg’. The peaks merged at concentrations between 4 and 5 % w/v where the largest strain magnitude was observed. The strain on primary packaging has traditionally been applied to evaluate the risk of damage or breakage due to, for example, crystallization of excipients. However, data collected during this study suggest there may be utility in formulation design or as a process analytical technology to minimize potentially destabilizing stresses and strains in the frozen formulation.

A general hyperelastic model for rubber-like materials incorporating strain-rate and temperature

A comprehensive hyperelastic model that precisely forecasts the mechanical characteristics of materials with rubber-like qualities is presented. This model relies on the well-established five-parameter Mooney-Rivlin model, which is then extended to incorporate strain rate dependent and temperature dependent term. To validate its accuracy, experimental data from ethylene propylene diene monomer (EPDM) rubber materials was utilized to compare with the model prediction. Hyperelastic stress-strain curves were collected from a variety of materials that were subjected to varying temperatures and strain rates to improve the model’s applicability. Its prediction results are compared against the collected experimental data, resulting in consistent and reliable outcomes. The simplified form of the model not only establishes an effective framework for characterizing and predicting the mechanical response of materials that resemble rubber under different working conditions but makes the coding and implementation of finite element analysis easier.

A finite strain theory for incompressible rubber-like circular arches with an application

A theory for incompressible rubber-like circular arches, undergoing finite strains and rotations, with an application to the stability of Mooney-Rivlin, doubly fixed, thin beams having initially small circular imperfections under uniformly distributed radial loads, is presented. The essence of considering transverse normal strain is emphasized for an incompressible rubber-like material. The effects of the thickness, initial curvature, and perturbation terms of the pseudo strain components on the buckling loads and central deflections are investigated. The thresholds of buckling are observed to depend on the values of the curvature of the initially imperfect beam for the chosen nonlinear material.

Experiments and modeling of the coupled viscoelasticity and Mullins effect in filled rubber materials

Filled rubber-like materials are widely used in engineering applications, and are known to exhibit a rate-dependent non-linear inelastic behavior, and stress-softening, also known as the Mullins effect is frequently encountered. In this work, we characterized and modeled the constitutive response of a handful of commercially available filled rubber-like materials. We first perform a set of large-deformation uniaxial experiments at room temperature and at multiple rates. Those experimental findings are used to develop and calibrate a thermodynamically consistent constitutive model, which is then numerically implemented in a finite element package by writing a user material subroutine. The constitutive model is validated by comparing the results of an inhomogeneous experiment and simulation. A key finding of this work is that the mechanisms that cause the Mullins effect appear to be the main drivers of viscoelasticity in the materials used here.