Towards the Development of a Strategy to Characterize and Model the Rheological Behavior of Filled, Uncured Rubber Compounds.

In this paper, an experimental strategy is presented to characterize the rheological behavior of filled, uncured rubber compounds. Oscillatory shear experiments on a regular plate-plate rheometer are combined with a phenomenological thixotropy model to obtain model parameters that can be used to describe the steady shear behavior. We compare rate- and stress-controlled kinetic equations for a structure parameter that determines the deformation history-dependent spectrum and, thus, the dynamic thixotropic behavior of the material. We keep the models as simple as possible and the characterization straightforward to maximize applicability. The model can be implemented in a finite element framework as a tool to simulate realistic rubber processing. This will be the topic of another work, currently under preparation. In shaping processes, such as rubber- and polymer extrusion, with realistic processing conditions, the range of shear rates is far outside the range obtained during rheological characterization. Based on some motivated choices, we will present an approach to extend this range.

Optical characterization of acceleration-induced strain fields in inhomogeneous brain slices.

The aim of this study was to measure high-resolution strain fields in planar sections of brain tissue during translational acceleration to obtain validation data for numerical simulations. Slices were made from fresh, porcine brain tissue, and contained both grey and white matter as well as the complex folding structure of the cortex. The brain slices were immersed in artificial cerebrospinal fluid (aCSF) and were encapsulated in a rigid cavity representing the actual shape of the skull. The rigid cavity sustained an acceleration of about 900m/s(2) to a velocity of 4m/s followed by a deceleration of more than 2000m/s(2). During the experiment, images were taken using a high-speed video camera and Von Mises strains were calculated using a digital image correlation technique. The acceleration of the sampleholder was determined using the same digital image correlation technique. A rotational motion of the brain slice relative to the sampleholder was observed, which may have been caused by a thicker posterior part of the slice. Local variations in the displacement field were found, which were related to the sulci and the grey and white matter composition of the slice. Furthermore, higher Von Mises strains were seen in the areas around the sulci.

Characterisation of the mechanical behaviour of brain tissue in compression and shear.

No validated, generally accepted data set on the mechanical properties of brain tissue exists, not even for small strains. Most of the experimental and methodological issues have previously been addressed for linear shear loading. The objective of this work was to obtain a consistent data set for the mechanical response of brain tissue to either compression or shear. Results for these two deformation modes were obtained from the same samples to reduce the effect of inter-sample variation. Since compression tests are not very common, the influence of several experimental conditions for the compression measurements was analysed in detail. Results with and without initial contact of the sample with the loading plate were compared. The influence of a fluid layer surrounding the sample and the effect of friction were examined and were found to play an important role during compression measurements.To validate the non-linear viscoelastic constitutive model of brain tissue that was developed in Hrapko et al. (Biorheology 43 (2006), 623-636) and has shown to provide a good prediction of the shear response, the model has been implemented in the explicit Finite Element code MADYMO. The model predictions were compared to compression relaxation results up to 15% strain of porcine brain tissue samples. Model simulations with boundary conditions varying within the physical ranges of friction, initial contact and compression rate are used to interpret the compression results.

A model of human skin under large amplitude oscillatory shear.

Skin mechanics is of importance in various fields of research when accurate predictions of the mechanical response of skin is essential. This study aims to develop a new constitutive model for human skin that is capable of describing the heterogeneous, nonlinear viscoelastic mechanical response of human skin under shear deformation. This complex mechanical response was determined by performing large amplitude oscillatory shear (LAOS) experiments on ex vivo human skin samples. It was combined with digital image correlation (DIC) on the cross-sectional area to assess heterogeneity. The skin is modeled as a one-dimensional layered structure, with every sublayer behaving as a nonlinear viscoelastic material. Heterogeneity is implemented by varying the stiffness with skin depth. Using an iterative parameter estimation method all model parameters were optimized simultaneously. The model accurately captures strain stiffening, shear thinning, softening effect and nonlinear viscous dissipation, as experimentally observed in the mechanical response to LAOS. The heterogeneous properties described by the model were in good agreement with the experimental DIC results. The presented mathematical description forms the basis for a future constitutive model definition that, by implementation in a finite element method, has the capability of describing the full 3D mechanical behavior of human skin.

Towards a reliable characterisation of the mechanical behaviour of brain tissue: The effects of post-mortem time and sample preparation.

Since the early seventies, the material properties of brain tissue have been studied using a variety of testing techniques. However, data reported in literature show large discrepancies even in the linear viscoelastic regime. In the current study, the effect of the sample preparation procedure and of post-mortem time on the mechanical response of porcine brain tissue is examined. Samples from the thalamus region were prepared with different techniques and were tested for different loading histories. Each sample was tested in oscillatory shear tests (1% strain amplitude, 1-10 Hz frequencies) followed by sequences of 5% strain loading-unloading cycles. The stress response to the loading-unloading cycles showed a clear dependency on post-mortem time, becoming more stiff with increasing time. This dependency was affected by the mechanical history induced by the preparation procedure.

Full Characterization of Multiphase, Multimorphological Kinetics in Flow-Induced Crystallization of IPP at Elevated Pressure.

Understanding the complex crystallization behavior of isotactic polypropylene (iPP) in conditions comparable to those found in polymer processing, where the polymer melt experiences a combination of high shear rates and elevated pressures, is key for modeling and therefore predicting the final structure and properties of iPP products. Coupling a unique experimental setup, capable to apply wall shear rates similar to those experienced during processing and carefully control the pressure before and after flow is imposed, with in situ X-ray scattering and diffraction techniques (SAXS and WAXD) at fast acquisition rates (up to 30 Hz), a well-defined series of short-term flow experiments are carried out using 16 different combinations of wall shear rates (ranging from 110 to 440 s-1) and pressures (100-400 bar). A complete overview on the kinetics of structure development during and after flow is presented. Information about shish formation and growth of α-phase parents lamellae from the shish backbones is extracted from SAXS; the overall apparent crystallinity evolution, amounts of different phases (α, ß, and γ), and morphologies developing in the shear layer (parent and daughter lamellae both in α and γ phase) are fully quantified from the analysis of WAXD data. Both flow rate and pressure were found to have a significant influence on the nucleation and the growth process of oriented and isotropic structures. Flow affects shish formation and the growth of α-parents; pressure acts on relaxation times, enhancing the effect of flow, and (mainly) on the growth rate of γ-phase. The remarkably high amount of γ-lamellae found in the oriented layer strongly indicates the nucleation of γ directly from the shish backbone. All the observations were conceptually in agreement with the flow-induced crystallization model framework developed in our group and represent a unique and valuable data set that will be used to further validate and implement our numerical modeling, filling the gap for quantitatively modeling crystallization during complicated processing operations like injection molding.

Film drainage and interfacial instabilities in polymeric systems with diffuse interfaces.

We report an experimental investigation on the effect of mutual diffusion in polymeric systems on film drainage between two captive drops. The main objective is to study the influence of diffuse interfaces on film drainage. This is done by using material combinations with different interfacial properties and interferometric visualization of the film between two interacting drops. For highly diffusive systems film drainage is observed to be, in contrast to immiscible systems, non-axisymmetric and unstable immediately after the film formation (at a few micrometers film thickness). Depending on whether the total thickness of the diffusion layers in the film is smaller or larger than the thickness of the film, Marangoni convection is found to enhance or delay film drainage. Enhanced film drainage is determined to be in order of 100 times faster than predicted by the current models, while delayed film drainage is observed after a drainage period where experimental and predicted results (assuming, a partially mobile interface) are in close agreement.

The mechanical behaviour of brain tissue: large strain response and constitutive modelling.

The non-linear mechanical behaviour of porcine brain tissue in large shear deformations is determined. An improved method for rotational shear experiments is used, producing an approximately homogeneous strain field and leading to an enhanced accuracy. Results from oscillatory shear experiments with a strain amplitude of 0.01 and frequencies ranging from 0.04 to 16 Hz are given. The immediate loss of structural integrity, due to large deformations, influencing the mechanical behaviour of brain tissue, at the time scale of loading, is investigated. No significant immediate mechanical damage is observed for these shear deformations up to strains of 0.45. Moreover, the material behaviour during complex loading histories (loading-unloading) is investigated. Stress relaxation experiments for strains up to 0.2 and constant strain rate experiments for shear rates from 0.01 to 1 s(-1) and strains up to 0.15 are presented. A new differential viscoelastic model is used to describe the mechanical response of brain tissue. The model is formulated in terms of a large strain viscoelastic framework and considers non-linear viscous deformations in combination with non-linear elastic behaviour. This constitutive model is readily applicable in three-dimensional head models in order to predict the mechanical response of the intra-cranial contents due to an impact.

Collagen fibers reduce stresses and stabilize motion of aortic valve leaflets during systole.

The effect of collagen fibers on the mechanics and hemodynamics of a trileaflet aortic valve contained in a rigid aortic root is investigated in a numerical analysis of the systolic phase. Collagen fibers are known to reduce stresses in the leaflets during diastole, but their role during systole has not been investigated in detail yet. It is demonstrated that also during systole these fibers substantially reduce stresses in the leaflets and provide smoother opening and closing. Compared to isotropic leaflets, collagen reinforcement reduces the fluttering motion of the leaflets. Due to the exponential stress-strain behavior of collagen, the fibers have little influence on the initial phase of the valve opening, which occurs at low strains, and therefore have little impact on the transvalvular pressure drop.

Design and numerical implementation of a 3-D non-linear viscoelastic constitutive model for brain tissue during impact.

Finite Element (FE) head models are often used to understand mechanical response of the head and its contents during impact loading in the head. Current FE models do not account for non-linear viscoelastic material behavior of brain tissue. We developed a new non-linear viscoelastic material model for brain tissue and implemented it in an explicit FE code. To obtain sufficient numerical accuracy for modeling the nearly incompressible brain tissue, deviatoric and volumetric stress contributions are separated. Deviatoric stress is modeled in a non-linear viscoelastic differential form. Volumetric behavior is assumed linearly elastic. Linear viscoelastic material parameters were derived from published data on oscillatory experiments, and from ultrasonic experiments. Additionally, non-linear parameters were derived from stress relaxation (SR) experiments at shear strains up to 20%. The model was tested by simulating the transient phase in the SR experiments not used in parameter determination (strains up to 20%, strain rates up to 8s(-1)). Both time- and strain-dependent behavior were predicted accurately (R2>0.96) for strain and strain rates applied. However, the stress was overestimated systematically by approximately 31% independent of strain(rate) applied. This is probably caused by limitations of the experimental data at hand.

A computational fluid-structure interaction analysis of a fiber-reinforced stentless aortic valve.

The importance of the aortic root compliance in the aortic valve performance has most frequently been ignored in computational valve modeling, although it has a significant contribution to the functionality of the valve. Aortic root aneurysm or (calcific) stiffening severely affects the aortic valve behavior and, consequently, the cardiovascular regulation. The compromised mechanical and hemodynamical performance of the valve are difficult to study both 'in vivo' and 'in vitro'. Computational analysis of the valve enables a study on system responses that are difficult to obtain otherwise. In this paper a numerical model of a fiber-reinforced stentless aortic valve is presented. In the computational evaluation of its clinical functioning the interaction of the valve with the blood is essential. Hence, the blood-tissue interaction is incorporated in the model using a combined fictitious domain/arbitrary Lagrange-Euler formulation, which is integrated within the Galerkin finite element method. The model can serve as a diagnostic tool for clinical purposes and as a design tool for improving existing valve prostheses or developing new concepts. Structural mechanical and fluid dynamical aspects are analyzed during the systolic course of the cardiac cycle. Results show that aortic root compliance largely influences the valve opening and closing configurations. Stresses in the delicate parts of the leaflets are substantially reduced if fiber-reinforcement is applied and the aortic root is able to expand.

A three-dimensional computational analysis of fluid-structure interaction in the aortic valve.

Numerical analysis of the aortic valve has mainly been focused on the closing behaviour during the diastolic phase rather than the kinematic opening and closing behaviour during the systolic phase of the cardiac cycle. Moreover, the fluid-structure interaction in the aortic valve system is most frequently ignored in numerical modelling. The effect of this interaction on the valve's behaviour during systolic functioning is investigated. The large differences in material properties of fluid and structure and the finite motion of the leaflets complicate blood-valve interaction modelling. This has impeded numerical analyses of valves operating under physiological conditions. A numerical method, known as the Lagrange multiplier based fictitious domain method, is used to describe the large leaflet motion within the computational fluid domain. This method is applied to a three-dimensional finite element model of a stented aortic valve. The model provides both the mechanical behaviour of the valve and the blood flow through it. Results show that during systole the leaflets of the stented valve appear to be moving with the fluid in an essentially kinematical process governed by the fluid motion.

Remodelling of continuously distributed collagen fibres in soft connective tissues.

Extracellular matrix remodelling plays an essential role in tissue engineering of load-bearing structures. The goal of this study is to model changes in collagen fibre content and orientation in soft connective tissues due to mechanical stimuli. A theory is presented describing the mechanical condition within the tissue and accounting for the effects of collagen fibre alignment and changes in fibre content. A fibre orientation tensor is defined to represent the continuous distribution of collagen fibre directions. A constitutive model is introduced to relate the fibre configuration to the macroscopic stress within the material. The constitutive model is extended with a structural parameter, the fibre volume fraction, to account for the amount of fibres present within the material. It is hypothesised that collagen fibre reorientation is induced by macroscopic deformations and the amount of collagen fibres is assumed to increase with the mean fibre stretch. The capabilities of the model are demonstrated by considering remodelling within a biaxially stretched cube. The model is then applied to analyse remodelling within a closed stented aortic heart valve. The computed preferred fibre orientation runs from commissure to commissure and resembles the fibre directions in the native aortic valve.

Film drainage between two captive drops: PEO-water in silicon oil.

An experimental study of the deformation and drainage of a Newtonian liquid film trapped between two drops is performed for the cases of a constant and slightly rising interaction force. Series of polyethylene oxide (PEO) water solutions are used for the dispersed and polydimethylsiloxane (PDMS) for the continuous phase. The film evolution is observed by an interferometric technique. Experimental data for the film thinning rate and for the film profile allow quantitative comparison with the available drainage models.

A novel method for visualising and quantifying through-plane skin layer deformations.

Skin is a multilayer composite and exhibits highly non-linear, viscoelastic, anisotropic material properties. In many consumer product and medical applications (e.g. during shaving, needle insertion, patient re-positioning), large tissue displacements and deformations are involved; consequently large local strains in the skin tissue can occur. Here, we present a novel imaging-based method to study skin deformations and the mechanics of interacting skin layers of full-thickness skin. Shear experiments and real-time video recording were combined with digital image correlation and strain field analysis to visualise and quantify skin layer deformations during dynamic mechanical testing. A global shear strain of 10% was applied to airbrush-patterned porcine skin (thickness: 1.2-1.6mm) using a rotational rheometer. The recordings were analysed with ARAMIS image correlation software, and local skin displacement, strain and stiffness profiles through the skin layers determined. The results of this pilot study revealed inhomogeneous skin deformation, characterised by a gradual transition from a low (2.0-5.0%; epidermis) to high (10-22%; dermis) shear strain regime. Shear moduli ranged from 20 to 130kPa. The herein presented method will be used for more extended studies on viable human skin, and is considered a valuable foundation for further development of constitutive models which can be used in advanced finite element analyses of skin.

Mechanical properties of brain tissue by indentation: interregional variation.

Although many studies on the mechanical properties of brain tissue exist, some controversy concerning the possible differences in mechanical properties of white and gray matter tissues remains. Indentation experiments are conducted on white and gray matter tissues of various regions of the cerebrum and on tissue from the thalamus and the midbrain to study interregional differences. An advantage of indentation, when compared to standard rheological tests as often used for the characterization of brain tissue, is that it is a local test, requiring only a small volume of tissue to be homogeneous. Indentation tests are performed at different speeds and the force relaxation after a step indent is measured as well. White matter tissue is found to be stiffer than gray matter and to show more variation in response between different samples which is consistent with structural differences between white matter and gray matter. In addition to differences between white matter and gray matter, also different regions of brain tissue are compared.

Confined flow of polymer blends.

The influence of confinement on the steady-state morphology of two different emulsions is investigated. The blends, made from polybutene (PB) in polydimethylsiloxane (PDMS) and polybutadiene (PBD) in PDMS, are sheared between two parallel plates, mostly with a standard gap spacing of 40 microm, in the range of shear rates at which the transition from "bulk" behavior toward "confined" behavior is observed. For both cases, the influence of the concentration was systematically investigated, as well as the shear rate effects on the final steady-state morphology. By decreasing the shear rate, for each blend, the increasing droplets, i.e., increasing confinement for a fixed gap spacing, arrange themselves first into two layers, and when the degree of confinement reaches an even higher value, a single layer of droplets is formed. The ratio between the drop diameters and the gap spacing at which this transition occurs is always lower than 0.5. While decreasing the shear rate, the degree of confinement increases due to drop coalescence. Droplets arrange themselves in superstructures like ordered pearl necklaces and, at the lower shear rates, strings. The aspect ratio and the width of the droplet obtained from optical micrographs are compared to predictions of the single droplet Maffettone-Minale model (MM model(1)). It is found that the theory, meant for unconfined shear flow, is not able to predict the drop deformation when the degree of confinement is above a critical value that depends on the blends considered and the shear rate applied. A recently developed extension of the MM model is reported by Minale (M model(2)) where the effect of the confinement is included by using the Shapira-Haber correction.3 Further extending this M model, by incorporating an effective viscosity as originally proposed by Choi and Showalter,4 we arrive at the mM model that accurately describes the experiments of blends in confined flow.

The influence of test conditions on characterization of the mechanical properties of brain tissue.

To understand brain injuries better, the mechanical properties of brain tissue have been studied for 50 years; however, no universally accepted data set exists. The variation in material properties reported may be caused by differences in testing methods and protocols used. An overview of studies on the mechanical properties of brain tissue is given, focusing on testing methods. Moreover, the influence of important test conditions, such as temperature, anisotropy, and precompression was experimentally determined for shear deformation. The results measured at room temperature show a stiffer response than those measured at body temperature. By applying the time-temperature superposition, a horizontal shift factor a(T)=8.5-11 was found, which is in agreement with the values found in literature. Anisotropy of samples from the corona radiata was investigated by measuring the shear resistance for different directions in the sagittal, the coronal, and the transverse plane. The results measured in the coronal and the transverse plane were 1.3 and 1.25 times stiffer than the results obtained from the sagittal plane. The variation caused by anisotropy within the same plane of individual samples was found to range from 25% to 54%. The effect of precompression on shear results was investigated and was found to stiffen the sample response. Combinations of these and other factors (postmortem time, donor age, donor type, etc.) lead to large differences among different studies, depending on the different test conditions.

Transient interfacial tension and morphology evolution in partially miscible polymer blends.

The influence of molecular weight asymmetry across an interface on the transient behavior of the interfacial tension is investigated for two different polymer combinations, polybutadiene (PBD)/polydimethylsiloxane (PDMS) and polybutene (PB)/PDMS. This choice ensures a minor diffuse interface using the first combination and a very diffuse interface in the latter case. Measurements of the interfacial tension as a function of time are carried out using a pendent/sessile drop apparatus at different temperatures ranging from 0 degrees C to 80 degrees C. Variations in the transient interfacial tension are attributed to diffusion of the lower molecular weight components from one phase into the other and the most pronounced changes are measured for the most diffusive systems (low molecular weight and high polydispersity) when diffusion goes from the drop into the matrix. By reversing the phases, only minor changes in the transient interfacial tension are measured. This is due to a fast saturation of the drop-phase since the drop volume is much smaller than that of the continuous phase. In all cases investigated, after a sufficient time a steady value of the interfacial tension is reached. In order to estimate the characteristic diffusion times of the migrating species, a discrete solution of the diffusion equation and a kinetic model from literature are applied. Results obtained are in line with the experimental observations. The importance of a changing interfacial tension on morphology development is studied on dilute (1%) blends, using two in situ techniques: small angle light scattering (SALS) and optical microscopy (OM). The SALS patterns yield the time evolution of the drop size, which is subsequently compared with the morphology following from OM. Depending on the diffusivity of the system, the morphology development is dominated by either diffusion or coalescence. Existing sharp-interface drainage models indeed do not apply for the diffuse blends and an improved quantitative estimation of the value of the critical film thickness is needed.

Transient interfacial tension of partially miscible polymers.

The interfacial tension of three different binary polymer blends has been measured as function of time by means of a pendent drop apparatus, at temperatures ranging from 24 degrees C to 80 degrees C. Three grades of polybutene (PB), differing in average molecular weight and polydispersity, are used as dispersed phase, the continuous phase is kept polydimethylsiloxane (PDMS), ensuring different asymmetry in molecular weight across the interface. The interfacial tension changes with time and, therefore, this polymer blends can not be considered fully immiscible. Changes in interfacial tension are attributed to the migration of low-molecular weight components from the source phase into the interphase and, from there, into the receiving phase. In the early stages of the experiments, just after the contact between the two phases has been established, the formation of an interphase occurs and the interfacial tension decreases with time. As time proceeds, the migration process slows down given the decrease in driving force which is the concentration gradient and, at the same time, molecules accumulated in the interphase start to migrate into the "infinite" matrix phase. A quasi-stationary state is found before depletion of the low-molecular weight fraction in the drop occurs and causes the interfacial tension sigma(t) to increase. The time required to reach the final stationary value, sigma(stat), increases with molecular weight and is a function of temperature. Higher polydispersity leads to lower sigma(stat) and a weaker dependence of sigma(stat) on temperature is found. A model coupling the diffusion equation in the different regimes is applied in order to interpret the experimental results. Numerical solutions of the diffusion equation are proposed in the cases of a constant and a changing interphase thickness. In the latter case, the interphase is defined by tracking with time a fixed limiting concentration in the transient concentration profiles and the variations found in sigma(t) are attributed to the changes in the interphase thickness. A discrete version of this continuous model is proposed and scaling arguments are reported in order to compare the results obtained with the predictions of the continuous model. The kinetic model as proposed by Shi et al. [T. Shi, V.E. Ziegler, I.C. Welge, L. An, B.A. Wolf, Macromolecules 37 (2007) 1591-1599] appears as a special case of the discrete model, when depletion is not taken into account. Using the models, time scales for the diffusion process can be derived, which fit the experimental results quite well.