Heterogeneous Strain Fields Research Articles

EDITORIAL

EDITORIAL

Neuronal Response to High Rate Shear Deformation Depends on Heterogeneity of the Local Strain Field

Many cellular models of traumatic brain injury (TBI) deform cells in a planar (2-D) configuration, a contrast from the three-dimensional (3-D) architecture of the brain, resulting in strain fields that may fail to represent the complex deformation patterns seen in vivo. Cells cultured in 3-D may more accurately represent in vivo cellular behavior than planar models due to differences in cytostructure, cell-cell/cell-matrix interactions and access to trophic factors; however, the effects of culture configuration on the response to high rate deformation have not been evaluated. We examined cell viability following a defined mechanical insult to primary cortical neurons distributed throughout a bioactive matrix (3-D) or in a monolayer sandwiched between layers of a bioactive matrix (2-D). After high rate loading (20 or 30 sec(1); 0.50 strain), there was a significant decrease in neuron viability for both configurations versus unloaded control cultures; however, neurons in 3-D presented greater cell death based on matched bulk loading parameters. Computer simulations of bulk loading predicted local cellular strains, revealing that neurons in 3-D were subjected to a heterogeneous strain field simultaneously consisting of tensile, compressive and shear strains; conversely, neurons in 2-D experienced a less complex deformation regime varying mainly based on shear strains. These results show differential susceptibility to mechanical loading between neurons cultured in 2-D and 3-D that may be due to differences in cellular strain manifestation. Models of TBI that accurately represent the related cellular biomechanics and pathophysiology are important for the elucidation of cellular tolerances and the development of mechanistically driven intervention strategies.

Stress Analysis and Identification with Full-Field Measurements

The wealth of information provided by full-field measurement techniques is very useful in experimental mechanics. Among different possible applications, full-field measurements can be used to identify parameters governing constitutive equations from heterogeneous strain fields. This keynote lecture first describes the different possible uses of such measurements. It then focuses on the virtual fields method which has been proposed to extract constitutive parameters from full-field measurements. Finally, the method is compared with the finite element model updating technique which is usually used for solving such a problem.

Chondrocyte Deformation Induces Mitochondrial Distortion and Heterogeneous Intracellular Strain Fields

Chondrocyte mechanotransduction is poorly understood but may involve cell deformation and associated distortion of intracellular structures and organelles. This study quantifies the intracellular displacement and strain fields associated with chondrocyte deformation and in particular the distortion of the mitochondria network, which may have a role in mechanotransduction. Isolated articular chondrocytes were compressed in agarose constructs and simultaneously visualised using confocal microscopy. An optimised digital image correlation technique was developed to calculate the local intracellular displacement and strain fields using confocal images of fluorescently labelled mitochondria. The mitochondria formed a dynamic fibrous network or reticulum, which co-localised with microtubules and vimentin intermediate filaments. Cell deformation induced distortion of the mitochondria, which collapsed in the axis of compression with a resulting loss of volume. Compression generated heterogeneous intracellular strain fields indicating mechanical heterogeneity within the cytoplasm. The study provides evidence supporting the potential involvement of mitochondrial deformation in chondrocyte mechanotransduction, possibly involving strain-mediated release of reactive oxygen species. Furthermore the heterogeneous strain fields, which appear to be influenced by intracellular structure and organisation, may generate significant heterogeneity in mechanotransduction behaviour for cells subjected to identical levels of deformation.

Mechanical Strains Induced in Osteoblasts by Use of Point Femtosecond Laser Targeting

A study demonstrating how ultrafast laser radiation stimulates osteoblasts is presented. The study employed a custom made optical system that allowed for simultaneous confocal cell imaging and targeted femtosecond pulse laser irradiation. When femtosecond laser light was focused onto a single cell, a rise in intracellular Ca2+ levels was observed followed by contraction of the targeted cell. This contraction caused deformation of neighbouring cells leading to a heterogeneous strain field throughout the monolayer. Quantification of the strain fields in the monolayer using digital image correlation revealed local strains much higher than threshold values typically reported to stimulate extracellular bone matrix production in vitro. This use of point targeting with femtosecond pulse lasers could provide a new method for stimulating cell activity in orthopaedic tissue engineering.

Heterogeneous three-dimensional strain fields during unconfined cyclic compression in bovine articular cartilage explants

Articular cartilage provides critical load-bearing and tribological properties to the normal function of diarthrodial joints. The unique properties of cartilage, as well as heterogeneous deformations during mechanical compression, are due to the nonuniform microstructural organization of tissue components such as collagens and proteoglycans. A new cartilage deformation by tag registration (CDTR) technique has been developed by the authors to determine heterogeneous deformations in articular cartilage explants. The technique uses a combination of specialized MRI methods, a custom cyclic loading apparatus, and image processing software. The objective of this study was to use the CDTR technique to document strain patterns throughout the volume of normal bovine articular cartilage explants during cyclic unconfined compression at two physiologically-relevant applied normal stress levels (1.29 and 2.57 MPa). Despite simple uniaxial cyclic compressive loading with a flat, nonporous indenter, strain patterns were heterogeneous. Strains in the thickness direction (E(yy)) were compressive, varied nonlinearly with depth from the articular surface from a maximum magnitude of 11% at the articular surface, and were comparable despite a 2-fold increase in applied normal stress. Strains perpendicular to the thickness direction (E(xx) and E(zz)) were tensile, decreased linearly with depth from the articular surface from a maximum of 7%, and increased in magnitude 2.5-fold with a 2-fold increase in applied normal stress. Shear strains in the transverse plane (E(xz)) were approximately zero while shear strains in the other two planes were much larger and increased in magnitude with depth from the articular surface, reaching maximum magnitudes of 2% at the articular cartilage-subchondral bone interface. In general, strain patterns indicated that cartilage osteochondral explants exhibited depth-dependent nonisotropic behavior during uniaxial cyclic loading. These results are useful in verifying constitutive formulations of articular cartilage during cyclic unconfined compression and in characterizing the micromechanical environment likely experienced by individual chondrocytes throughout the tissue volume.

Phenomenological analysis of heterogeneous strain fields in epitaxial thin films using x-ray scattering

A model that allows the quantitative analysis of heterogeneous strain fields in epitaxial thin films using x-ray diffraction (XRD) is presented. Particular emphasis is laid on the modelling of the two-component XRD profiles (i.e. profiles made of the superposition of a narrow coherent Bragg peak and a broad diffuse scattering profile) encountered in the XRD investigation of epitaxial thin films containing localized strain fields. The spatial properties of the strain field are included in a correlation function based on phenomenological parameters such as the defect correlation length ξ and the level of disorder σ∞. No assumption regarding the nature of the defect is hence required. The statistical properties of the strain field are described by means of Lévy-stable distributions which allow us to account for profile shapes ranging between the Gaussian and profiles exhibiting pronounced power law-type tails, as well as for asymmetrical profiles. The effects of finite size of the domains (crystallites) over which diffraction is coherent are rigorously taken into account by calculating the auto-correlation function of the crystallite shape including the size distribution effects. The effects of each parameter are presented and discussed in detail and the applicability of the model is illustrated with two examples.

Heterogeneous three-dimensional strain fields during unconfined cyclic compression in bovine articular cartilage explants

Articular cartilage provides critical load-bearing and tribological properties to the normal function of diarthrodial joints. The unique properties of cartilage, as well as heterogeneous deformations during mechanical compression, are due to the nonuniform microstructural organization of tissue components such as collagens and proteoglycans. A new cartilage deformation by tag registration (CDTR) technique has been developed by the authors to determine heterogeneous deformations in articular cartilage explants. The technique uses a combination of specialized MRI methods, a custom cyclic loading apparatus, and image processing software. The objective of this study was to use the CDTR technique to document strain patterns throughout the volume of normal bovine articular cartilage explants during cyclic unconfined compression at two physiologically-relevant applied normal stress levels (1.29 and 2.57 MPa). Despite simple uniaxial cyclic compressive loading with a flat, nonporous indenter, strain patterns were heterogeneous. Strains in the thickness direction ( E yy ) were compressive, varied nonlinearly with depth from the articular surface from a maximum magnitude of 11% at the articular surface, and were comparable despite a 2-fold increase in applied normal stress. Strains perpendicular to the thickness direction ( E xx and E zz ) were tensile, decreased linearly with depth from the articular surface from a maximum of 7%, and increased in magnitude 2.5-fold with a 2-fold increase in applied normal stress. Shear strains in the transverse plane ( E xz ) were approximately zero while shear strains in the other two planes were much larger and increased in magnitude with depth from the articular surface, reaching maximum magnitudes of 2% at the articular cartilage–subchondral bone interface. In general, strain patterns indicated that cartilage osteochondral explants exhibited depth-dependent nonisotropic behavior during uniaxial cyclic loading. These results are useful in verifying constitutive formulations of articular cartilage during cyclic unconfined compression and in characterizing the micromechanical environment likely experienced by individual chondrocytes throughout the tissue volume.

Highly localized strain fields due to planar defects in epitaxial SrBi2Nb2O9 thin films

Thin films of (00l) oriented SrBi2Nb2O9 epitaxially grown on SrTiO3 by sol-gel spin coating have been studied by means of high-resolution x-ray diffraction reciprocal space mapping. It is shown that these materials contain highly localized heterogeneous strain fields due to imperfect stacking faults (i.e., faults that do not propagate throughout the crystallites building up the film). In the film plane, the strain fields are confined to 11 nm wide regions and characterized by a vertical displacement of 0.18c (where c is the cell parameter) showing that the stacking faults are mainly composed of one additional (or missing) perovskite layer. Prolonged thermal annealing at 700 °C strongly reduces the density of stacking faults and yields a more uniform strain distribution within the film volume without inducing significant grain growth.

Using self-assembling monolayers to study crack initiation in epoxy/silicon joints

The effect of the density and in-plane distribution of interfacial interactions on crack initiation in an epoxy-silicon joint was studied in nominally pure shear loading. Well-defined combinations of strong (specific) and weak (nonspecific) interactions were created using self-assembling monolayers. The in-plane distribution of strong and weak interactions was varied by employing two deposition methods: depositing mixtures of molecules with different terminal groups resulting in a nominally random distribution, and depositing methyl-terminated molecules in domains defined lithographically with the remaining area interacting through strong acid-base interactions. The two distributions lead to very different fracture behavior. For the case of the methyl-terminated domains (50 μm on a side) fabricated lithographically, the joint shear strength varies almost linearly with the area fraction of strongly interacting sites. From this we infer that cracks nucleate on or near the interface over nearly the entire range of bonded area fraction and do so at nearly the same value of local stress (load/bonded area). We postulate that the imposed heterogeneity in interfacial interactions results in heterogeneous stress and strain fields within the epoxy in close proximity to the interface. Simply, the bonded areas carry load while the methyl terminated domains carry negligible load. Stress is amplified adjacent to the well-bonded regions (and reduced adjacent to the poorly bonded regions), and this leads to crack initiation by plastic deformation and chain scission within the epoxy near the interface. For the case of mixed monolayers, the dependence is entirely different. At low areal density of strongly interacting sites, the joint shear strength is below the detection limit of our transducer for a significant range of mixed monolayer composition. With increasing density of strongly interacting sites, a sharp increase in joint shear strength occurs at a methyl terminated area fraction of roughly 0.90. We postulate that this coincides with the onset of yielding in the epoxy. For methyl-terminated area fractions less than 0.85, the joint shear strength becomes independent of the interfacial interactions. This indicates that fracture no longer initiates on the interface but away from the interface by a competing mechanism, likely plastic deformation and chain scission within the bulk epoxy. The data demonstrate that the in-plane distribution of interaction sites alone can affect the location of crack nucleation and the far-field stress required.

Numerical issues in the virtual fields method

AbstractThis paper deals with the direct identification of parameters governing anisotropic elastic constitutive equations. These parameters are identified from heterogeneous strain fields with the virtual fields method. This method is based on a relevant use of the principle of virtual work. Different numerical aspects of the implementation of the method are discussed in the paper, mainly in terms of stability of the identified parameters when noisy data are processed. It is shown that the sensitivity of the method to noisy data is compatible with a practical use during experiments. Copyright © 2004 John Wiley & Sons, Ltd.

Special virtual fields for the direct determination of material parameters with the virtual fields method. 3. Application to the bending rigidities of anisotropic plates

This paper deals with the direct identification of bending rigidities of thin anisotropic plates. These parameters are extracted from an heterogeneous strain field which takes place onto the top surface of a bent plate. The loading conditions are such that no closed-form solution is available for the deflection/slope/curvature fields. The procedure presently used is the virtual fields method with “special” virtual fields. It is shown that the unknown parameters are directly extracted with this method since no iterative calculations are required. The parameters are in fact directly equal to the virtual work of the applied loading with the special virtual displacement fields. The headlines of the method are recalled in the first part of the paper. They are then applied in the case of anisotropic bent plates. The accuracy and the stability of the procedure are finally discussed through some relevant examples.

Simulation of inclusion trail patterns within rotating synkinematic porphyroblasts

The paper presents two-dimensional numerical simulations of inclusion trail patterns of spherical and non-spherical synkinematic porphyroblasts rotating and growing over a pre-existing, passive foliation during a single progressive deformation taking into account the effects of heterogeneous strain field around the rigid porphyroblast. The analysis is based on the velocity functions around a rigid object hosted in a Newtonian viscous medium. Using Visual Basic software, the trail patterns have been simulated in a PC assuming that the porphyroblast grows by increments of constant radius, over a pre-existing foliation, which simultaneously gets distorted due to heterogeneous strain field around the porphyroblast and included within the porphyroblast. The numerical simulations show that the trail patterns in spherical porphyroblasts are controlled by two factors: (1) the initial orientation of the foliation marker relative to the shear direction ( θ) and (2) the ratio of pure shear and simple shear rates in the bulk deformation ( S r ). Complex trail structures form when θ<0° and S r ⩽0.75. The ratio of growth rates along axial directions of non-spherical porphyroblasts is an additional factor that controls the trail patterns.

Special virtual fields for the direct determination of material parameters with the virtual fields method. 2––Application to in-plane properties

This paper deals with the direct identification of mechanical parameters that govern the in-plane constitutive law of orthotropic materials. Those parameters are extracted from heterogeneous strain fields that occur in a short beam specimen tested in a Iosipescu fixture. The procedure used is the virtual fields method with special virtual fields. The case of linear elasticity is first addressed. It is shown that the parameters are directly extracted with this method: no iterative calculations are required. The stability is also discussed in different cases. A non-linear shear response is then considered. The parameter that governs this non-linearity is also directly identified with the special virtual fields.

Tools for multiaxial validation of behavior laws chosen for modeling hyper‐elasticity of rubber‐like materials

AbstractWe present an experimental approach to discriminate hyper‐elastic models describing the mechanical behavior of rubber‐like materials. An evaluation of the displacement field obtained by digital image correlation allows us to evaluate the heterogeneous strain field observed during these tests. We focus on the particular case of hyper‐elastic models to simulate the behavior of some rubber‐like materials. Assuming incompressibility of the material, the hyper‐elastic potential is determined from tension and compression tests. A biaxial loading condition is obtained in a multiaxial testing machine and model predictions are compared with experimental results.

L'identification des propriétés mécaniques de matériaux avec la méthode des champs virtuels, une alternative au recalage par éléments finis

A method allowing the identification of parameters that govern constitutive equations of materials is proposed in this Note. These parameters are identified from heterogeneous strain fields. The method is direct so that iterative calculations based on finite element model updating are avoided. To cite this article: M. Grédiac et al., C. R. Mecanique 330 (2002) 107–112.

Numerical simulation of the observed strain field in the central-eastern Mediterranean region

The highly heterogeneous strain field indicated by neotectonic and seismological data in the central-eastern Mediterranean region has been reproduced, at a first approximation, by finite element modelling, of a 2D elastic thin plate. The zone considered is modelled as a mosaic of poorly deformable zones decoupled by highly deformable belts, simulating the major tectonic structures indicated by geological and geophysical evidence. The deformation of the model is obtained by imposing kinematic boundary conditions, representative of the motion of Africa and eastern Anatolia relative to Eurasia. Experiments carried out with different boundary conditions and model parameterisations have provided information on the sensitivity of the model and some insights into the geodynamic behavior of the study area. The deformation pattern of the central Mediterranean area is strongly conditioned by the mechanical properties assumed in the border zones between the Aegean and Adriatic systems. The match of the complex strain pattern observed in the western Anatolian–Aegean–Balkan zones is significantly favoured if high rigidity is assigned to the inner part of this structural system. A motion of Africa with respect to Eurasia compatible with an Eulerian pole located offshore Portugal best accounts for the observed strains in the central Mediterranean region. The match of the strongly heterogeneous strain field observed in the study area can hardly be achieved by simplified models not including major tectonic features and lateral heterogeneity of mechanical properties. The kinematic field resulting from the model configuration which best simulates the observed strain field presents some differences with respect to geodetic measurements in the Aegean–Western Anatolian area, where the computed velocities are systematically lower than the geodetic ones. It is suggested that the most plausible explanation of such differences is related to the fact that the present deformation pattern, inferred from geodetic data, may be different from the middle–long term one, inferred from seismological and geological data.

Digital image correlation used to analyze the multiaxial behavior of rubber-like materials

We present an experimental approach to discriminate models describing the mechanical behavior of polymeric materials. A biaxial loading condition is obtained in a multiaxial testing machine. An evaluation of the displacement field obtained by digital image correlation allows us to evaluate the heterogeneous strain field observed during these tests. We focus on the particular case of hyper-elastic models to simulate the behavior of a rubber-like material. Different expressions of hyper-elastic potential are used to model experiments under uniaxial and biaxial loading conditions.

Couplings between plasticity and martensitic phase transformation: overall behavior of polycrystalline TRIP steels

Abstract Several couplings between plasticity and martensitic phase transformation are at the origin of remarkable properties of ductility and toughness in the case of TRIP steels. A micromechanical model is developed to predict the conditions of nucleation and growth of a martensitic microdomain inside an inhomogeneous plastic strain field. More explicit relations are developed in the case of a simple shear test where a heterogeneous plastic strain field leads to a significant decrease of the critical stress for martensitic transformation. The obtained results are combined with a kinetics and kinematics studies to derive the constitutive equation of an austenitic single crystal from which the overall behavior of a polycrystalline steel is deduced using the self-consistent scale transition method. Comparison with experimental data shows a good agreement.

Elastic-plastic indentation creep of glassy poly(methyl methacrylate) and polystyrene: characterization using uniaxial compression and indentation tests

This paper discusses the time-dependent mechanical response of poly(methyl methacrylate) (PMMA) and polystyrene (PS) subjected to a shallow indentation made by a 750 µm radius steel ball at 25 °C. The penetration depth was measured as a time-response to a prescribed loading history of the indenter. The tests were performed in the intermediate elastic-plastic regime, which is of paramount importance when dealing with the mechanics of a multicontact interface between rough bodies. The creep characteristics under constant load were analysed with an Eyring law relating the average stress and the strain rate in the deformed volume. The results were compared to data obtained by a standard uniaxial compression test on the bulk material. We have deduced a simple empirical rule which accounts for the heterogeneous stress and strain fields under the indenter and reduces the intricate problem of elastic-plastic indentation creep to an effective scalar rule.