Nonlinear Stress-strain Curves Research Articles

Monotonic and fatigue behaviour of chopped-strand-mat/polyester composites with rigid and flexibilised matrix

Abstract Monotonic and tension–tension fatigue tests were carried out on E-glass chopped-strand-mat/polyester composites, varying the flexibiliser content by weight in the matrix in the range 0–30%. The flexibilising action was due to the adipic acid monomers present in the flexibiliser. In monotonic tests, the most marked effect of resin flexibility was in the transverse cracks formed during loading, whose critical density (i.e., the density at failure) was very high for the rigid matrix, resulting in a highly non-linear stress–strain curve, and in the largest apparent strain to failure. With suitably increasing the flexibiliser content, the transverse crack formation was nearly suppressed, and the overall stress–strain curve approached linearity. In fatigue, the critical crack density decreased with increasing fatigue life in the case of the rigid matrix. For the flexibilised resins, the crack density at failure was independent of the maximum applied stress, larger than observed in monotonic tests, and higher the higher was the flexibiliser content, up to about 80% of the tensile strength. Beyond this limit, it converged through the material monotonic behaviour. The evolution of the residual elastic modulus with elapsing fatigue cycles was qualitatively consistent with the number of transverse cracks observed. The more flexible the matrix, the lower was the fractional modulus loss in fatigue. However, the highest elastic modulus along all the fatigue life pertained to the composite with rigid matrix, due to the flexibiliser adversely affecting the initial rigidity. Despite the differences in monotonic response and crack formation features, all the materials tested exhibited very similar S – N curves at moderately high fatigue lives. Nevertheless, appropriately treating the experimental results in terms of fatigue sensitivity, it was found that this parameter tends to increase with increasing matrix flexibility.

A soft computing based approach for the prediction of ultimate strength of metal plates in compression

This paper presents two plate strength formulations applicable to metals with nonlinear stress–strain curves, such as aluminum and stainless steel alloys, obtained by soft computing techniques, namely Neural Networks (NN) and Genetic Programming (GP). The proposed soft computing formulations are based on well-defined FE results available in the literature. The proposed formulations enable determination of the buckling strength of rectangular plates in terms of Ramberg–Osgood parameters. The strength curves obtained by the proposed soft computing formulations show perfect agreement with FE results. The formulations are later compared with related codes and results are found to be quite satisfactory.

Elasticity of the Human False Vocal Fold

Very little is known about the elasticity of the human ventricular fold (false vocal fold). To better understand the potential role of the false fold in the fluid dynamics and aeroacoustics of phonation, we made some measurements on the elastic properties of human ventricular fold tissues in vitro. Uniaxial tensile stress-strain characteristics of 6 male and 6 female false fold specimens were quantified with sinusoidal stretch-release deformation. Midcoronal sections of 3 specimens were examined to quantify the relative densities of collagen, elastin, seromucous glandular tissue, and adipose tissue by digital image analysis. Nonlinear stress-strain curves with hysteresis (viscous energy loss) were observed, with large interindividual differences. A hybrid linear-exponential model was used to determine the elastic modulus (tangent Young's modulus) of the false fold. On average, the male false fold was twice as stiff as the female at a tensile strain of 20% to 30%. This preliminary gender-related difference in elasticity could be attributed to a higher proportion of glandular tissue in the female false fold, due to the lower elastic modulus of glands. The present data allow one to develop a more comprehensive biomechanical model of phonation, for optimizing postoperative voice production following laryngeal reconstruction procedures.

Overall elastoplastic property for micropolar composites with randomly oriented ellipsoidal inclusions

Overall linear and non-linear properties for micropolar composites containing 3D and in-plane randomly oriented inclusions are examined with an analytical micromechanical method. This method is based on Eshelby solution for a general ellipsoidal inclusion in a micropolar media and secant moduli method. The influence of inclusion’s shape, size and orientation on the classical effective moduli, yielding surface and non-linear stress and strain relation are examined. The results show that the effective moduli and non-linear stress–strain curves are always higher for micropolar composites than the corresponding classical composites. When the inclusion’s size is sufficiently large, the classical results can be recovered.

Discussion on “A Study on the Beginning of Secondary Compression of Soils” by R. G. Robinson

R. G. Robinson has suggested an approach for studying the beginning of secondary consolidation. Secondary compression is defined as any extra compression other than primary compression [1]. The primary consolidation occurs due to the dissipation of excess pore water pressure. Terzaghi's one-dimensional consolidation theory [2] has been used widely to analyze the time-compression data but is valid only in the primary consolidation phase. Non-linear stress-strain curves, time-dependent loading, impeded drainage, stratified soils multi-dimensional flow, large strains, effective stress-dependent (void ratio dependent) hydraulic conductivities, anisotropy, and other such effects are included and can lead to a response that deviates widely from that predicted using Terzaghi's theory. Remaining effects may loosely be termed initial and secondary. Secondary effects are then defined as all effects except these classified as initial and primary [1].

Selected macroscopic properties of liquid crystalline elastomers.

In this short review we give an overview of selected macroscopic properties of sidechain liquid crystalline elastomers (LCEs) focusing on three closely related topics (a) the influence of relative rotations between the director and the strain field on various reorientation instabilities, (b) the nonlinear stress-strain curves for the polydomain-monodomain transition and for the reorientation transition in LCE monodomains and (c) the shear mechanical response of LCEs in the linear regime. We consider only already existing real materials and do not discuss hypothetical "ideal" systems. We conclude that all observations reported to date can be accounted for without invoking the concept of soft elasticity, but instead relying on macroscopic dynamics in the linear and the nonlinear domain.

Dynamic Viscoelastic Behavior of Dental Composites Measured by Split Hopkinson Pressure Bar

The present study evaluated the dynamic viscoelastic behavior of commercially available dental composites by a Split Hopkinson pressure bar (SHPB) test machine. Five commercially available composite resins--namely, two conventional hybrid composites (Filtek Z100, Z100; Filtek Z250, Z250), a packable composite (Filtek P60, P60), a flowable composite (Filtek Flow, FL), and a nanofill composite (Filtek Supreme, SU)--were evaluated. By means of SHPB technique, the dynamic stress-strain curve, storage modulus, and loss tangent of the five dental composites were calculated. All specimens exhibited a nonlinear stress-strain curve in the loading process, which resulted not only from the viscoelasticity--but also from the plasticity--of matrix. In terms of storage modulus, no significant differences were exhibited among the five dental composites (p > 0.05). In terms of loss tangent, Z100 showed a significantly higher value than P60, FL, and SU (p < 0.05). Within the limitations of this investigation using SHPB, it was indicated that the loss tangent increased with increasing filler content.

Study of the micro-mechanical behaviour of the Opalinus Clay: an example of co-operation across the ground engineering disciplines

The high degree of scientific cross-fertilisation possible between the three geo-engineering disciplines soil mechanics, rock mechanics and engineering geology, is demonstrated by means of a micro-mechanical model of the Opalinus Clay. After a brief review of Terzaghi’s effective stress principle and the importance of micro-mechanical models in general, a conceptual study of a micro-mechanical model of a claystone is presented in some detail. The model is based on the Particle Flow Code (PFC) developed by Itasca Corp. It introduces into the model the pertinent composition and structure of the Opalinus Claystone established in the local engineering geology of Switzerland and SW Germany. This includes elongated clay platelets, various layers of densified water around the platelets, free water in the pores and a specific texture of the platelets after consolidation. The model is numerically subjected to a series of loading stages. It is shown that the micro-mechanical model reproduces a number of features which have been known for a long time in soil and rock mechanics but which are often intractable in conventional generic models. The features include non-linear stress–strain curves with pre-failure damage and post-failure strain softening, a non-linear increase of the particle contacts with loading, distinct clustering of deformations, clustering of micro cracks leading to the development of shear bands and hysteresis in cyclic loading. It is concluded that micro-mechanical models are promising tools for further development of our understanding of the mechanical behaviour of geological materials. They offer an excellent opportunity for scientific co-operation between engineering geologists and soil and rock mechanics engineers.

A nonlinear biphasic viscohyperelastic model for articular cartilage

Experiments on articular cartilage have shown nonlinear stress-strain curves under finite deformations as well as intrinsic viscous effects of the solid phase. The aim of this study was to propose a nonlinear biphasic viscohyperelastic model that combines the intrinsic viscous effects of the proteoglycan matrix with a nonlinear hyperelastic constitutive equation. The proposed equation satisfies objectivity and reduces for uniaxial loading to a solid type viscous model in which the actions of the springs are represented by the hyperelastic function proposed by Holmes and Mow [1990. J. Biomechanics 23, 1145–1156.]. Results of the model, that were efficiently implemented in an updated Lagrangian algorithm, were compared with experimental infinitesimal data reported by DiSilverstro and Suh [2001. J. Biomechanics 34, 519–525.] and showed acceptable fitting for the axial force ( R 2 = 0.991 ) and lateral displacement ( R 2 = 0.914 ) curves in unconfined compression as well as a good fitting of the axial indentation force curve ( R 2 = 0.982 ) . In addition, the model showed an excellent fitting of finite-deformation confined compression stress relaxation data reported by Ateshian et al. [1997. J. Biomechanics 30, 1157–1164.] and Huang et al. [2005. J. Biomechanics 38, 799–809.] ( R 2 = 0.993 and R 2 = 0.995 , respectively). The constitutive equation may be used to represent the mechanical behavior of the proteoglycan matrix in a fiber reinforced model of articular cartilage.

Analysis of articular cartilage as a composite using nonlinear membrane elements for collagen fibrils

To develop a composite fibre-reinforced model of the cartilage, membrane shell elements were introduced to represent collagen fibrils reinforcing the isotropic porous solid matrix filled with fluid. Nonlinear stress–strain curve of pure collagen fibres and collagen volume fraction were explicitly presented in the formulation of these membrane elements. In this composite model, in accordance with tissue structure, the matrix and fibril membrane network experienced dissimilar stresses despite identical strains in the fibre directions. Different unconfined compression and indentation case studies were performed to determine the distinct role of membrane collagen fibrils in nonlinear poroelastic mechanics of articular cartilage. The importance of nonlinear fibril membrane elements in the tissue relaxation response as well as in temporal and spatial variations of pore pressure and solid matrix stresses was demonstrated. By individual adjustments of the collagen volume fraction and collagen mechanical properties, the model allows for the simulation of alterations in the fibril network structure of the tissue towards modelling damage processes or repair attempts. The current model, which is based on a physiological description of the tissue structure, is promising in improvement of our understanding of the cartilage pathomechanics.

Flexural Strength of Reinforced and Prestressed Concrete T-Beams

This article examines the behavior of T-beams at nominal flexure strength. The authors begin by explaining the fundamental theory, then equations are derived for the various calculation methods used in the study. Explanations are provided for the differences between the various methods, with special emphasis on the difference between the LRFD (the AASHTO LRFD Bridge Design Specifications) method and the methods of other codes and references. For non-prestressed T-beams, the LRFD and STD (AASHTO Standard Specifications for Highway Bridges) methods are compared with the results of a strain compatibility analysis using nonlinear concrete compressive stress-strain curves. Prestressed beams are also evaluated. The authors conclude that the equations for calculating the flexural strength of T-beams in the current AASHTO LRFD Specifications are not consistent with the original derivation of the equivalent rectangular concrete compressive stress distribution for flanged sections. For non-prestressed T-beams of uniform strength, the equations given in the AASHTO Standard Specifications provide reasonable estimates of the flexural strength of flanged sections. For prestressed T-beams of uniform strength, a combination of the current AASHTO LRFD and Standard Specifications provides a reasonable approximation of flexural strength. In this case, the steel stress at nominal flexural strength is determined by the methods of LRFD, while the equivalent rectangular concrete compressive stress distribution of STD is used to calculate the depth to the neutral axis and flexural strength.

引張変形下のアモルファス金属における局所格子不安定性：分子動力学による検討

In order to clarify the deformation behavior of bulk metallic glasses from atomistic viewpoint, an amorphous Ni-Al binary alloy is made and subjected to tension by means of molecular dynamics simulation. The initial equilibrium is made by a melt-quench simulation of a single crystal in which Ni or Al atoms are randomly arranged on the fcc lattice point. Both the radial distribution and Voronoi analysis attests that the initial equilibrium is in amorphous state. The amorphous metal is then subjected to tension with the strain rate of 1.0×109/s under periodic boundary condition, showing nonlinear stress-strain curve from the initial stage of loading. The curve shows the peak stress of about 3.0 GPa at the strain of 0.07. After the peak the stress slightly decreases and the metal deforms with constant stress of about 2.4 GPa. There is no remarkable change in the radial distribution function and Voronoi polyhedron during the deformation so that the metal maintains the short-range order of amorphous structure. Then we have calculated the elastic stiffness matrix, which is evaluated by the stress and elastic coefficients, at each atom point during tension to evaluate the local stability. Contrary to the crystal, many atoms show negative value even in the initial equilibrium of amorphous metal. These “unstable” atoms turn out to be in the outer-shell of the local cluster such as (0, 0, 12, 0) icosahedron or in the boundary of the local clusters. On the other hand, the center atoms of the local clusters show high stability, resulting in the system stability of about 2×1012 GPa6. It is also demonstrated that the change in the local stability reveals the collapse or reconfiguration of local clusters.

Fibrous proteins and tensile elasticity of the human vocal fold

Viscoelastic response of the human vocal fold under tension has been reported previously, demonstrating nonlinearity and hysteresis of stress–strain curves. However, the importance of various matrix molecules for tensile elasticity of the vocal fold is not well understood. It is important to correlate the biomechanical behavior of the vocal fold with its histological microstructure, so as to examine the relative contributions of the matrix proteins such that cultured tissue constructs for repairing voice disorders may be engineered accordingly. This study attempted to test the hypothesis that collagen and elastin play a major role in determining the tensile elasticity of the vocal fold. Uniaxial tensile elastic properties of the vocal cover and vocal ligament were quantified using a servo-control lever system. The specimens were also studied histologically with elastin van Gieson stain. Digital image analysis of the vocal fold cover showed that a more profoundly nonlinear stress–strain curve and a higher elastic modulus (tangent Young’s modulus) were associated with higher densities of collagen and elastin in the specimens. These findings suggested that both collagen and elastin fibers likely contribute significantly to elasticity of the vocal fold under tension, thereby regulating vocal fold length changes and fundamental frequency control. [Work supported by NIH.]

Development of nonlinear cross-anisotropic model for the pre-failure deformation of geomaterials

Nonlinearity and anisotropy of a stress–strain response have been accepted as important factors influencing the pre-failure characteristics of soils. To predict this complex behavior of soils, a simplified soil model was developed using the property of cross-anisotropic elasticity to represent the anisotropic stiffness of soils and the Ramberg–Osgood model to simulate nonlinear stress–strain curves. A hypo-elastic formulation of the stress–strain relationship can describe the directional stiffness as a function of the current stress in a given direction. The model employs the separate formulations of the initial and equivalent elastic moduli for describing the stress-dependency of elastic response of soils. The reliability and usefulness of the model were verified by applications of hollow cylinder tests and triaxial tests on silts. Predictions of soil behavior using this model were compared with representative stress–strain data. The close agreement confirmed that the model reasonably well predicted the nonlinear response of anisotropic deformation at the pre-failure state of soils.

An Investigation of Non-linear Stress-strain Behavior of Thin Metal Films

ABSTACTNon-linear stress-strain curves of various films with thickness of one to four microns were determined from nanoindentation experiments with a Berkovich indenter. The measured load-depth data are analysed using a method based on neural networks. An inverse mapping of depth-dependent dimensionless hardness and stiffness data yields the material parameters describing a non-linear elastic-plastic stress-strain curve of the Armstrong-Frederick type. Suppositions for the application of this method are a sufficiently different hardness between film and substrate, and hardness and stiffness data available in a depth range of 10 to 200% of the film thickness. For all films considered a significant thickness dependence in strength has been observed. In addition, a remarkable influence of the substrate material on the strength of the film material was found and has been confirmed by focussed ion beam microscopy.

Computational analysis of the deformability of leukocytes modeled with viscous and elastic structural components

The objective of this work is to systematically include non-Newtonian effects in a previous Newtonian model of the leukocyte and to study the effects thereof on leukocyte rheology. The standard Newtonian-drop model of the cell is enhanced in three respects: (1) The cortical layer is treated as an elastic membrane with a nonlinear stress–strain curve to simulate unfolding of the excess surface area of the membrane. (2) A power-law shear thinning fluid is used for the cytoplasm. (3) A three-layer or compound cell model is used, which is comprised of the membrane cortex, cytoplasm and nucleus. Combinations of these aspects are also investigated. The governing equations for this multifluid system are solved in the Stokes limit. The immersed boundary technique is used to simulate the interaction of the elastic membrane with the flow field. Results indicate that each of these additional elements in the leukocyte model yield significant deviations from the Newtonian deformation and recovery behavior of the leukocyte. However, the added modeling sophistication does not appear to be sufficient to fully capture all the distinctive responses of the leukocytes under a wide variety of deformation and recovery protocols. It is shown that although a comprehensive model for the leukocyte remains elusive, the three-layer leukocyte model with cortical elasticity is a promising candidate.

Tensile Strength and Deformation of a Two‐Dimensional Carbon–Carbon Composite at Elevated Temperatures

Tensile strength and creep behavior of a two‐dimensional (2D) laminate carbon–carbon composite (C/C) were examined from room temperature to 2773 K in an inert atmosphere. The tensile strength of the C/C was monotonically enhanced with increasing test temperatures. In particular, significant improvement was observed at temperatures higher than 1773 K. In this temperature range, nonlinear stress–strain curves were observed at low deformation rates, but with increasing test speed, the stress–strain curves became linear until total fracture. The source of the apparent nonlinearity was thus concluded to be creep deformation, which appeared from 1773 K. Two ruling mechanisms for the strength enhancement of the C/C at elevated temperatures were identified. The first source was degassing of absorbed water, which had a dominant influence on the strength enhancement up to 1773 K. The second was creep deformation. This phenomenon was notable at temperatures higher than 1773 K, and produced much larger enhancement than the degassing.

Pragmatic multiscale modelling of bone as a natural hybrid nanocomposite

A multiscale modelling strategy is proposed to predict the mechanical properties of polymer-inorganic hybrid nanocomposites. In particular, high volume fractions of oriented mineral plates in a ductile polymer matrix offer the potential of an attractive balance of stiffness and toughness. Bone is used here as a natural example of such a material, since it has a hierarchy of structures at all scales of dimension, and is an excellent subject for multiscale materials modelling. The key step in the model is to predict the individual properties of the component materials by a self-consistent mean-field method from their molecular structure. This allows the synergistic interaction of mineral and polymer at the large number of interfaces in a nanocomposite to be modelled as an energy sharing process, whose main effect is upon the polymer modulus. The interface-modified mechanical properties are then used in composite models at higher scales to predict the nonlinear stress–strain curve to failure as a function of the main variables of mineral volume fraction (MVF) and particle aspect ratio.

Strength Curves for Metal Plates in Compression

The paper describes a generalized strength formulation for metal plates. The focus is on metallic alloys with nonlinear stress strain curves, such as aluminum and stainless steel. The formulation is based on a generalized Winter-curve featuring two material-dependent parameters. Analytic expressions are derived for these parameters so that they can be determined for given material properties assumed to be expressed in terms of the Ramberg-Osgood parameters (E0,σ0.2,n). Thus, the generalized formulation allows the plate strength equation to be determined for a given alloy, requiring only the Ramberg-Osgood parameters. The strength curves obtained using the generalized formulation are shown to compare well with finite element results. The strength equations apply to uniformly compressed plates simply supported along all four edges. The validity ranges for the material properties are 3⩽n⩽100 and 0.001⩽e⩽0.003 where e=σ0.2/E0. The plates are assumed to be free of residual stress.

Constitutive characterization of vocal fold viscoelasticity based on a modified Arruda-Boyce eight-chain model

Previous studies have shown that vocal fold tissues exhibit nonlinear viscoelastic behavior under different loading conditions. Hysteresis and strain rate dependence of stress–strain curves have been observed for vocal fold ligament and muscle tissues when subjected to sinusoidal tensile loading. Nonlinear viscoelastic response and tissue failure have also been described for vocal fold mucosa subjected to constant strain-rate tests under large-strain shear. These findings cannot be adequately described by the traditional constitutive formulations of linear and quasilinear viscoelasticity. This study attempts to characterize some nonlinear behavior of vocal fold tissues under tensile loading based on a modified version of the Arruda-Boyce (Bergström-Boyce) hyperelastic model for polymers, which has been shown to adequately predict the rate-dependent behavior of some elastomers and biological tissues. Results indicated that the model was only capable of describing the relatively linear portion of the nonlinear stress–strain curves of the vocal fold muscle (at strain smaller than 20%), while failing to predict the exponential increase of stress at higher strain. However, the model was capable of predicting the dependence of stress on strain rate reasonably well. This finding was consistent with the model’s assumptions on the constitutive behavior of the two constituent polymer networks.