Nonlinear Stress-strain Response Research Articles

Fiber Crimp Confers Matrix Mechanical Nonlinearity, Regulates Endothelial Cell Mechanosensing, and Promotes Microvascular Network Formation.

Mechanical interactions between cells and their surrounding extracellular matrix (ECM) guide many fundamental cell behaviors. Native connective tissue consists of highly organized, 3D networks of ECM fibers with complex, nonlinear mechanical properties. The most abundant stromal matrix component is fibrillar type I collagen, which often possesses a wavy, crimped morphology that confers strain- and load-dependent nonlinear mechanical behavior. Here, we established a new and simple method for engineering electrospun fibrous matrices composed of dextran vinyl sulfone (DexVS) with controllable crimped structure. A hydrophilic peptide was functionalized to DexVS matrices to trigger swelling of individual hydrogel fibers, resulting in crimped microstructure due to the fixed anchorage of fibers. Mechanical characterization of these matrices under tension confirmed orthogonal control over nonlinear stress-strain responses and matrix stiffness. We next examined ECM mechanosensing of individual endothelial cells (ECs) and found that fiber crimp promoted physical matrix remodeling alongside decreases in cell spreading, focal adhesion area, and nuclear localization of Yes-associated protein (YAP). These changes corresponded to an increase in migration speed, along with evidence for long-range interactions between neighboring cells in crimped matrices. Interestingly, when ECs were seeded at high density in crimped matrices, capillary-like networks rapidly assembled and contained tube-like cellular structures wrapped around bundles of synthetic matrix fibers due to increased physical reorganization of matrix fibers. Our work provides an additional level of mechanical and architectural tunability to synthetic fibrous matrices and implicates a critical role for mechanical nonlinearity in EC mechanosensing and network formation.

Inverse material characterisation of human aortic tissue for traumatic injury in motor vehicle crashes

Traumatic aortic rupture (TAR) resulting due to motor vehicle crashes (MVC) accounts for about 10%–20% fatalities. This article presents the outcome of our findings of the non-linear stress–strain response and strain rate-dependent behaviour of the aortic tissue under crash like impacts. This study uses both finite element (FE) modelling and experimental testing to enhance the understanding of injury mechanisms associated with TAR. Accurate material properties are essential for correct FE model predictions. Therefore, the objective of the current study was to experimentally characterise and identify a suitable constitutive model and model parameters for aortic tissue that can be incorporated into FE human body models for studying aortic rupture. Inverse characterisation and genetic algorithms (GA) were used to train the FE model to simulate real-life scenarios applying loading in multiple directions. A total of 32 uniaxial tensile tests were conducted on human aortic tissues along with the longitudinal and circumferential directions by loading at different strain rates ranging up to 200/s. The engineering stress–strain relationship obtained for human aorta in the longitudinal and circumferential directions via uniaxial tensile tests demonstrated an approximately bilinear behaviour and strain rate dependency in both directions. The higher stresses and modulus in the circumferential direction as compared to longitudinal direction demonstrates the anisotropic behaviour of the tissue. A constitutive model has been developed and implemented via user subroutine (UMAT) in LS-DYNA that accounts for the strain rate-dependent effects and bilinear behaviour observed in the aortic tissue. A FE model of the experimental set-up was developed, and the parameters of the model were estimated using a GA-based inverse mapping technique. The developed model enables investigation of the mechanical response of the aortic tissue under crashes and other high rate loading conditions. The work is based on the premise that a reliable constitutive model coupled with a FE model of the aorta shall help predict TARs.

3D printed tubular lattice metamaterials with engineered mechanical performance

We numerically and experimentally report 3D tubular lattice metamaterials with sinusoidal ligaments, which enable the deformation pattern to switch from bending to stretching. A non-linear J-shape stress-strain response and auxetic behavior over a large deformation range are observed due to the intrinsic deformation mechanism. Numerical analyses further demonstrate that the proposed tubular lattice metamaterials exhibit a high degree of tunability for Poisson's ratio and mechanical responses. In particular, the design flexibility can be harnessed to create heterogeneous tubular structures that can morph into target shapes. The findings presented here provide insights to develop 3D architected metamaterials with various potential applications, such as tissue engineering, biomedical devices, and stretchable electronics.

Influence of Network Topology on the Viscoelastic Properties of Dynamically Crosslinked Hydrogels.

Biological materials combine stress relaxation and self-healing with non-linear stress-strain responses. These characteristic features are a direct result of hierarchical self-assembly, which often results in fiber-like architectures. Even though structural knowledge is rapidly increasing, it has remained a challenge to establish relationships between microscopic and macroscopic structure and function. Here, we focus on understanding how network topology determines the viscoelastic properties, i.e., stress relaxation, of biomimetic hydrogels. We have dynamically crosslinked two different synthetic polymers with one and the same crosslink. The first polymer, a polyisocyanopeptide (PIC), self-assembles into semi-flexible, fiber-like bundles, and thus displays stress-stiffening, similar to many biopolymer networks. The second polymer, 4-arm poly(ethylene glycol) (starPEG), serves as a reference network with well-characterized structural and viscoelastic properties. Using one and the same coiled coil crosslink allows us to decouple the effects of crosslink kinetics and network topology on the stress relaxation behavior of the resulting hydrogel networks. We show that the fiber-containing PIC network displays a relaxation time approximately two orders of magnitude slower than the starPEG network. This reveals that crosslink kinetics is not the only determinant for stress relaxation. Instead, we propose that the different network topologies determine the ability of elastically active network chains to relax stress. In the starPEG network, each elastically active chain contains exactly one crosslink. In the absence of entanglements, crosslink dissociation thus relaxes the entire chain. In contrast, each polymer is crosslinked to the fiber bundle in multiple positions in the PIC hydrogel. The dissociation of a single crosslink is thus not sufficient for chain relaxation. This suggests that tuning the number of crosslinks per elastically active chain in combination with crosslink kinetics is a powerful design principle for tuning stress relaxation in polymeric materials. The presence of a higher number of crosslinks per elastically active chain thus yields materials with a slow macroscopic relaxation time but fast dynamics at the microscopic level. Using this principle for the design of synthetic cell culture matrices will yield materials with excellent long-term stability combined with the ability to locally reorganize, thus facilitating cell motility, spreading, and growth.

Hyperelastic constitutive modeling of hydrogels based on primary deformation modes and validation under 3D stress states

In isotropic linear elasticity, a constitutive model calibrated using a single deformation mode (e.g., compression or tension or shear) is sufficient to describe a complex three-dimensional (3D) stress state. Such an approach, however, is likely inadequate for modeling hydrogels, which exhibit a nonlinear stress-strain response that varies significantly between deformation modes and is also sensitive to microstructure via gel concentration. In this study, a combined experimental and constitutive modeling framework is proposed for the development and validation of concentration-dependent 3D hyperelastic models for hydrogels. Agarose hydrogel in a concentration range of 0.4–4% w/v is chosen as the model material. Uniaxial compression, uniaxial tension, and simple shear (three primary deformation modes) experiments are conducted. The small strain elastic modulus-gel concentration relationships obtained from experiments are compared with those predicted by the molecular theory of rigid polymer networks (Jones-Marques theory) to identify the concentration range in which entropic elastic (hyperelastic) response dominates. In this range (1.5–4% w/v), four hyperelastic constitutive models are fit to the combined compression-tension-shear stress-strain data: Mooney-Rivlin, three-parameter generalized Rivlin, Gent, and Gent-Gent models. It is demonstrated that the generalized Rivlin model offers the best overall accuracy, and the variation of its model parameters with gel concentration is consistent with the Jones-Marques theory. The resulting concentration-dependent Extended Generalized Rivlin model is employed in finite element simulations of the non-homogeneous 3D stress state of wedge indentation. Simulated load versus depth and strain field predictions show very good agreement with experimental wedge indentation results. Finally, it is shown that a hyperelastic model calibrated using only a single deformation mode yields poor results for other primary and 3D deformations, and thus multiple primary deformation modes (preferably all three) should be considered.

Tricuspid Chordae Tendineae Mechanics: Insertion Site, Leaflet, and Size-Specific Analysis and Constitutive Modelling

Background: Tricuspid valve chordae tendineae play a vital role in our cardiovascular system. They function as “parachute cords” to the tricuspid leaflets to prevent prolapse during systole. However, in contrast to the tricuspid annulus and leaflets, the tricuspid chordae tendineae have received little attention. Few previous studies have described their mechanics and their structure-function relationship. Objective: In this study, we aimed to quantify the mechanics of tricuspid chordae tendineae based on their leaflet of origin, insertion site, and size. Methods: Specifically, we uniaxially stretched 53 tricuspid chordae tendineae from sheep and recorded their stress-strain behavior. We also analyzed the microstructure of the tricuspid chordae tendineae based on two-photon microscopy and histology. Finally, we compared eight different hyperelastic constitutive models and their ability to fit our data. Results: We found that tricuspid chordae tendineae are highly organized collageneous tissues, which are populated with cells throughout their thickness. In uniaxial stretching, this microstructure causes the classic J-shaped nonlinear stress-strain response known from other collageneous tissues. We found differences in stiffness between tricuspid chordae tendineae from the anterior, posterior, or septal leaflets only at small strains. Similarly, we found significant differences based on their insertion site or size also only at small strains. Of the models we fit to our data, we recommend the Ogden two-parameter model. This model fit the data excellently and required a minimal number of parameters. For future use, we identified and reported the Ogden material parameters for an average data set. Conclusion: The data presented in this study help to explain the mechanics and structure-function relationship of tricuspid chordae tendineae and provide a model recommendation (with parameters) for use in computational simulations of the tricuspid valve.

Micromechanical model of time-dependent damage and deformation behavior for an orthogonal 3D-woven SiC/SiC composite at elevated temperature in vacuum

This paper presents a micromechanical model to predict the time-dependent damage and deformation behavior of an orthogonal 3-D woven SiC fiber/BN interface/SiC matrix composite under constant tensile loading at elevated temperature in vacuum. In-situ observation under monotonic tensile loading at room temperature, load–unload tensile testing at 1200 °C in argon, and constant load tensile testing at 1200 °C in vacuum were conducted to investigate the effects of microscopic damage on deformation behavior. The experimentally obtained results led to production of a time-dependent nonlinear stress–strain response model for the orthogonal 3-D woven SiC/SiC. It was established using the linear viscoelastic model, micro-damage propagation model, and a shear-lag model. The predicted creep deformation was found to agree well with the experimentally obtained results.

Model for description of nonlinear unloading-reloading stress-strain response with special reference to plastic-strain dependent chord modulus

To describe the nonlinear stress-strain response in the unloading-reloading process, a model was proposed for the variation of the elastic modulus during the process. This model is directly associated with the widely used equation for the plastic-strain dependent chord modulus, and no additional material parameters are needed for modeling. In this model, nonlinearly changing elastic modulus is explicitly determined from the current stress state, instead of using an accumulated-strain measure. Therefore, it is robust against the numerical noise (e.g., stress oscillation) in computation. Based on the same elastic modulus model, two types of calculation models were constructed, namely, a micro-plasticity model and nonlinear-elasticity model. From the numerical simulations of the non-proportional unloading-reloading processes, it was found that the stress-strain responses calculated by the two models were very similar to each other. The model was validated by comparing the calculated stress-strain results with the corresponding experimental observations on a DP980 advanced high-strength steel sheet (AHSS). The influence of the nonlinear unloading behavior on the springback prediction was discussed by performing two cases of springback simulations (a uniform bending-unbending and a hat-shaped draw-bending) on two types of AHSSs (dual-phase DP980 and precipitation strengthen 780 R), by using several types of elastic modulus models. The best choice for springback simulation is to use the nonlinear elastic model to describe the realistic stress-strain responses in the yield surface, along with an appropriate kinematic hardening model of plasticity to express smooth elastic-plastic transition behavior. In comparison with the springback calculation by the nonlinear model, a certain amount of errors were found when using the plastic-strain dependent linear models, however, they were not very large. Thus the second best option in springback simulation would be to use the chord modulus model because of its simple modeling and high computation efficiency.

A Pilot Study on Biaxial Mechanical, Collagen Microstructural, and Morphological Characterizations of a Resected Human Intracranial Aneurysm Tissue

Intracranial aneurysms (ICAs) are focal dilatations that imply a weakening of the brain artery. Incidental rupture of an ICA is increasingly responsible for significant mortality and morbidity in the American's aging population. Previous studies have quantified the pressure-volume characteristics, uniaxial mechanical properties, and morphological features of the human aneurysms. In this pilot study, for the first time, we comprehensively quantified the mechanical, collagen fiber microstructural, and morphological properties of a resected human posterior inferior cerebellar artery aneurysm. The tissue from the dome of a right posterior inferior cerebral aneurysm was first mechanically characterized using biaxial tension and stress relaxation tests. Then, the load-dependent collagen fiber architecture of the aneurysm tissue was quantified using an in-house polarized spatial frequency domain imaging system. Finally, optical coherence tomography and histological procedures were used to quantify the tissue's microstructural morphology. Mechanically, the tissue was shown to exhibit hysteresis, a nonlinear stress-strain response, and material anisotropy. Moreover, the unloaded collagen fiber architecture of the tissue was predominantly aligned with the testing Y-direction and rotated towards the X-direction under increasing equibiaxial loading. Furthermore, our histological analysis showed a considerable damage to the morphological integrity of the tissue, including the lack of elastin, intimal delamination, and calcium deposition. This pilot study has provided new insights into the multiscale properties of aneurysm tissues. The method presented herein, together with the novel results, can be used to develop accurate constitutive models to use in computational models of the aneurysm hemodynamics, growth, and remodeling.

Effect of porosity on the nonlinear and time-dependent behavior of Ceramic Matrix Composites

The effect of porosity on the non-linear stress-strain response and time-dependent strain response under constant stress has been investigated. The material used in the study is a woven composite of CG-Nicalon™ fiber and SiNC matrix. The composite has been modeled using a model for spheroidal inclusions in a medium in combination with a micromechanics-based progressive damage and constituent inelasticity model for CMCs. Numerical estimates of deformation and failure are shown to compare favorably with experimental measurements. It has also been shown that porosity affects through-thickness properties more as compared to the in-plane properties and that creep rupture strength is significantly degraded by increased porosity in the material.

Descriptions of stress-strain responses of non-linear unloading and closure of stress-strain hysteresis loop based on the Yoshida-Uemori model

This paper proposes models to describe a non-linear unloading-reloading stress-strain response and closure of a cyclic stress-strain hysteresis loop based on the Yoshida-Uemori (Y-U) model. The average stress-strain slope given by the non-linear model is consistent with the Y-U model of Young’s modulus degradation. The closure of a cyclic stress-strain curve is described by automatically changing one of the Y-U material parameters. An advantage of the above approaches is that no additional parameters are needed for modeling.

A molecular perspective to analytical modeling that reveals new instabilities in dielectric elastomer transducers

We use a four-parameter material model for polymers - the Edward–Vilgis (EV) model, which models polymeric crosslinks, sliplinks, slippage and inextensibility to analyze dielectric elastomers (DEs). The EV model allows us to investigate the DE response from a molecular perspective, revealing new instabilities in DEs, primarily due to sliplinks. We compare the EV model with the Gent model – a crosslink-only model commonly used in the field of DE, on its ability to fit the non-linear stress–strain response of an acrylic elastomer (VHB 4905), which contains a significant amount of entanglements. We found that the EV model provided an excellent fit to the strain softening phenomenon at small to modest strains, which is missed by the Gent model. Slippage of sliplinks at these regions of strain produces slip resistance that contributed to enhance the resistance to applied forces. Subsequent slip relaxation due to the uncoiling of entanglements will lead to significant strain softening, giving the DE a higher susceptibility to mechanically and electro-mechanically-induced instabilities. From our analyses, we produced mechanical stability phase plots and electromechanical stability design plots, for elastomers of different sliplink-to-crosslink compositions. Such plots may serve as guides to develop dielectric elastomers with desired material properties and performances, and tailor to specific applications.

Investigation of the stress-strain constitutive behavior of ±55° filament wound GFRP pipes in compression and tension

The behavior of hollow ±55° filament wound glass fiber-reinforced polymer (GFRP) pipes under longitudinal compressive and tensile loading is studied experimentally using 31 pipe specimens. The main parameters were the nominal pipe pressure rating, namely 350 kPa, 700 kPa and 1050 kPa, and the inner diameter, namely 76 mm and 203 mm. For the compression tests, each specimen was cut to a length of twice the outer diameter and the ends were protected from premature failure using a basalt fiber-reinforced polymer (FRP) wrap. A spherical platen was used to ensure concentric loading. For the tensile tests, a novel test set-up with a specialized gripping mechanism was used to test the full pipe in lieu of the standard coupons extracted from the pipe which suffer from fiber discontinuity, thereby underestimating the strength. The pipes exhibited a nonlinear stress-strain response under both tension and compression loads and exhibited a higher strength in compression than in tension. Additionally, the pipes showed a significant post-peak behavior when loaded in tension. An analytical model capable of representing the behavior of FRPs under axial tension and compression using simplified constitutive relations was adopted and applied. The proposed model captures the nonlinear stress-strain behavior of the hollow pipes in tension and compression as well as the post-peak behavior of the pipes in tension. The effects of fiber angle error are also explored. Understanding the material behavior is important when analyzing these pipes under flexural or axial loading or when analyzing structures using these pipes, such as concrete-filled FRP tubes (CFFTs).

A method analyzing deformation of anchor foundations in soft clay under static and cyclic loads

The paper presents a constitutive model to describe undrained cyclic stress-strain responses of soft clays based on the equivalent visco-elastic and creep theories. The hysteretic and nonlinear stress-strain responses of soft clays are described using the equivalent visco-elastic relationship and variations of the cyclic modulus and the damping ratio with the octahedral shear strain, respectively in the model. The cyclic accumulative strain is described using the Mises creeping potential function and the associated flow rule. The method determining the model parameters is given by static and cyclic triaxial tests. The finite element method to analyze deformation of anchor foundation in soft clay under static and cyclic loads is developed based on the model. For the method, a cyclic loading time history is divided into a series of incremental loading sub-processes which include one load cycle at least. The cyclic stress-strain responses of soil elements at any time are not tracked in detail and determined by the equivalent visco-elastic calculations for every loading sub-process. The accumulative deformation of anchor foundations is calculated using the initial strain algorithm. The method has been implemented in ABAQUS Software by developing interface programs. Model tests of the suction anchors are conducted and predicted using the method. Comparisons of predicted and model test results show that the method can be used to evaluate cyclic stability and reveal the failure process and mechanism of anchor foundations by analyzing deformation time-histories.

Agonistic and antagonistic roles of fibroblasts and cardiomyocytes on viscoelastic stiffening of engineered human myocardium

Cardiomyocyte and stroma cell cross-talk is essential for the formation of collagen-based engineered heart muscle, including engineered human myocardium (EHM). Fibroblasts are a main component of the myocardial stroma. We hypothesize that fibroblasts, by compacting the surrounding collagen network, support the self-organization of cardiomyocytes into a functional syncytium. With a focus on early self-organization processes in EHM, we studied the molecular and biophysical adaptations mediated by defined populations of fibroblasts and embryonic stem cell-derived cardiomyocytes in a collagen type I hydrogel. After a short phase of cell-independent collagen gelation (30 min), tissue compaction was progressively mediated by fibroblasts. Fibroblast-mediated tissue stiffening was attenuated in the presence of cardiomyocytes allowing for the assembly of stably contracting, force-generating EHM within 4 weeks. Comparative RNA-sequencing data corroborated that fibroblasts are particularly sensitive to the tissue compaction process, resulting in the fast activation of transcription profiles, supporting heart muscle development and extracellular matrix synthesis. Large amplitude oscillatory shear (LAOS) measurements revealed nonlinear strain stiffening at physiological strain amplitudes (>2%), which was reduced in the presence of cells. The nonlinear stress-strain response could be characterized by a mathematical model. Collectively, our study defines the interplay between fibroblasts and cardiomyocytes during human heart muscle self-organization in vitro and underscores the relevance of fibroblasts in the biological engineering of a cardiomyogenesis-supporting viscoelastic stroma. We anticipate that the established mathematical model will facilitate future attempts to optimize EHM for in vitro (disease modelling) and in vivo applications (heart repair).

An efficient homogenization technique for fiber tows in textile composites with emphasis on directionally dependent nonlinear stress–strain behavior

Now it is common to utilize a FE-based representative volume element (RVE) model to study the mechanical performances of textile composites. Complex architecture of fiber tows can be explicitly reflected in the RVE model, but a homogenization process for the tows, bundle of micrometer-scale fibers bound by a matrix material, is still required for the meso-scale model. We present a systematic yet simple approach to homogenize the fiber tow including directionally dependent nonlinear stress–strain responses. Nonlinear behavior of the matrix material is considered in the present homogenization method through the well defined Hill’s anisotropic plasticity model. Parameters required for the anisotropic plasticity model are obtained from virtual test results at the fiber/matrix level. Tension and shear tests on plain weave textile composites are performed to validate the present numerical approach. It is also demonstrated that the homogenization technique can also be utilized to isolate the matrix stress–strain curve from composite shear stress-shear strain data.

On modelling the constitutive and damage behaviour of highly non-linear bio-composites – Mesh sensitivity of the viscoplastic-damage law computations

The large strain fracture of non-linear complex solids concerns a wide range of applications, such as material forming, food oral processing, surgical instrumental penetration as well as more recently, the design of biodegradable composites for packaging and bio-medical use. Although simulations are a powerful tool towards understanding and designing such processes, modelling ductile fracture in materials such as soft natural composites imposes a new challenge, particularly when the fracture patterns cannot be pre-defined. Here we bring to light new information on these aspects of benefit to the multidisciplinary community, by characterising and modelling the deformation and fracture of short cellulose fibre starch extruded composites. Hyperviscoelastic-Mullins damage laws show merits in modelling such complex systems. Yet they are inferior to a viscoplastic-damage law able to capture exactly their highly non-linear, rate dependent and pressure dependent pseudo-plastic stress-strain response. The viscoplastic-damage law also predicts fracture based on experimental toughness values without pre-specifying the crack path in a Finite Element (FE) model, displaying superiority over the conventional cohesive zone approach. Yet, despite using a toughness parameter to drive crack propagation, spurious mesh dependency is still observed while other previously unreported sources of error imposed by the finite element aspect ratio are also highlighted. The latter is rectified by developing a novel numerical strategy for calculating the characteristic element length used in the damage computations. Inherent mesh dependency suggests that non-local damage models may be essential to model this newly investigated class of natural composites.

An elasto-viscoplastic model to describe the ratcheting behavior of articular cartilage.

In the present work, a constitutive model for articular cartilage is proposed in finite elasto-viscoplasticity. For simplification, articular cartilage is supposed to be a typical composite composed of a soft basis and a fiber assembly. The stress tensor and free energy function are hence accordingly divided into two components. The high nonlinear stress-strain response is assumed to be mainly related to the fiber assembly and described by an exponential-type hypoelastic relation. Ratcheting is considered according to the viscoplasticity, the evolution rule of which is deduced from the dissipative inequality by the co-directionality hypotheses. Then, the capability of the proposed model is validated by comparing its predictions with related experimental observations. Results show that the ratcheting behavior and stress-strain hysteresis loops are reasonably captured by the proposed model.

Mechanical Behavior of Silicon Carbide Under Static and Dynamic Compression

This paper compared the mechanical behavior of 6H SiC under quasi-static and dynamic compression. Rectangle specimens with a dimension of 3 × 3 × 6 mm3 were used for quasi-static compression tests under three different loading rates (i.e., 10−5/s, 10−4/s, and 10−3/s). Stress–strain response showed purely brittle behavior of the material which was further confirmed by scanning electron microscopy (SEM)/transmission electron microscopy (TEM) examinations of fractured fragments. For dynamic compression, split Hopkinson pressure bar (SHPB) tests were carried out for cubic specimens with a dimension of 6 × 6 × 4 mm3. Stress–strain curves confirmed the occurrence of plastic deformation under dynamic compression, and dislocations were identified from TEM studies of fractured pieces. Furthermore, JH2 model was used to simulate SHPB tests, with parameters calibrated against the experimental results. The model was subsequently used to predict strength and plasticity-related damage under various dynamic loading conditions. This study concluded that, under high loading rate, silicon carbide (SiC) can deform plastically as evidenced by the development of nonlinear stress–strain response and also the evolution of dislocations. These findings can be explored to control the brittle behavior of SiC and benefit end users in relevant industries.

Description of non-linear unloading curve and closure of cyclic stress-strain loop based on Y-U model

This paper proposes a cyclic elasto-plasticity model to describe non-linear unloading-reloading curves and closure of a cyclic stress-strain hysteresis loop based on the Yoshida-Uemori (Y-U) model. The nonlinear unloading stress-strain response is expressed by an equation for the stress-dependent Young’s modulus in the yield surface. The closure of a cyclic stress-strain curve is described by automatically changing Y-U material parameters. The great advantage of the above approaches is that no additional parameters are needed.