Nonlinear Stress-strain Behavior Research Articles

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.

Micromechanical modeling of non-linear stress-strain behavior of polycrystalline microcracked materials under tension

The stress-strain behavior of microcracked polycrystalline materials (such as ceramics or rocks) under conditions of tensile, displacement-controlled, loading is discussed. Micromechanical explanation and modeling of the basic features, such as non-linearity and hysteresis in stress-strain curves, is developed, with stable microcrack propagation and “roughness” of intergranular cracks playing critical roles. Experiments involving complex loading histories were done on large- and medium grain size β-eucryptite ceramic. The model is shown to reproduce the basic features of the observed stress-strain curves.

Strength and deflection behaviour of cold-formed steel back-to-back channels

Cold-formed steel (CFS) construction can lead to more efficient designs compared to hot-rolled steel members as a consequence of its high strength, light weight, ease of fabrication, and flexibility in their cross-section profiles. However, CFS members are vulnerable to local, distortional and overall buckling modes. This paper develops a numerical model to investigate the flexural strength and failure modes of CFS back-to-back channel beams and verifies the efficiency of an optimisation framework previously proposed. The model incorporates non-linear stress-strain behaviour and enhanced corner properties obtained from coupon tests, as well as initial geometric imperfections measured in physical specimens. To simulate the behaviour of a bolt bearing against a steel plate in the back-to-back section, a connector model is used that takes into account both slippage and bearing deformations. The developed Finite Element (FE) models are verified against six four-point bending tests on CFS back-to-back channel beams, where excellent agreement is found between the experimental results and the FE predictions. The validated FE models are then used to assess the adequacy of the effective width method in EC3 and the Direct Strength Method (DSM) in estimating the design capacity of conventional and optimum design CFS channel beam sections. The results indicate that both EC3 and DSM provide accurate predictions for the bending capacity of lipped channel beam sections. A comparison between FE predictions and tested results show that, the geometric imperfections can change the FE predictions of ultimate capacity by 7%, while the strain-hardening of CFS material at the round corners has negligible effects. It is also shown that EC3 uses a reduced cross-sectional property to calculate deflections, which can reasonably predict deflections with a slight overestimation (6%) at the serviceability load level.

Enhancement of a meso-scale material model for nonlinear elastic finite element computations of plain-woven fabric membrane structures

Due to their structure of crossed yarns embedded in coating, woven fabric membranes are characterised by a highly nonlinear stress-strain behaviour. In order to determine an accurate structural response of membrane structures, a suitable description of the material behaviour is required. Typical phenomenological material models like linear-elastic orthotropic models only allow a limited determination of the real material behaviour. A more accurate approach becomes evident by focusing on the meso-scale, which reveals an inhomogeneous however periodic structure of woven fabrics. The present work focuses on an established meso-scale model. The novelty of this work is an enhancement of this model with regard to the coating stiffness. By performing an inverse process of parameter identification using a state-of-the-art Levenberg-Marquardt algorithm, a close fit w.r.t. measured data from a common biaxial test is shown and compared to results applying established models. Subsequently, the enhanced meso-scale model is processed into a multi-scale model and is implemented as a material law into a finite element program. Within finite element analyses of an exemplary full scale membrane structure by using the implemented material model as well as by using established material models, the results are compared and discussed.

Bilinear approach to tensile properties of flax composites in finite element analyses

Flax fibers show potential in bio-efficiency compared to conventional composite fibers with good mechanical properties. The tensile behavior of flax fibers shows a nonlinear stress–strain relation. Within this work, a bilinear elastic–plastic approach is described, which is based on the generalized Hill potential theory and easily applicable in ANSYS. The method was used to model the nonlinear stress–strain behavior under quasi-static tensile loading of flax fiber-reinforced laminates. Hashin failure mechanisms using the stress–strain data of the bilinear model are applied as well. The results were compared to experimental data, carried out using pre-impregnated flax fibers in three types of tensile test specimens. The layups of investigation were $$[0]_6$$ , $$[90]_6$$ and $$[\pm \, 45]_8$$ . The strain was evaluated using the non-contact, digital image correlation measurement ARAMIS. Good agreement with the test results was achieved, and the method of bilinear elastic–plastic modeling of flax fiber-reinforced structures was evaluated as simple but effective for quasi-static elongation under uniaxial tensile loading.

Mesoscopic characterization of magnetoelastic hybrid materials: magnetic gels and elastomers, their particle-scale description, and scale-bridging links

Magnetic hybrid materials in the form of magnetic gels and elastomers, that is, magnetic or magnetizable colloidal particles embedded in an elastic polymer matrix, are fascinating substances. By addressing and adjusting the magnetic interactions between the particles through external magnetic fields, their overall material properties can be tuned reversibly while in operation. A central goal is to understand how these features can be optimized and which structural properties of the materials determine their overall behavior and its tunability. Mesoscopic theories and modeling are necessary for these purposes, resolving the arrangement of the embedded particles and linking it to the macroscopic scale of the overall material behavior. Here, we overview such recent developments of mesoscopic approaches. Particularly, we address coarse-grained but efficient dipole-spring models, explicit analytical calculations using linear elasticity theory, numerical approaches that allow to characterize nonlinear effects, or density functional theory. In this way, various properties and types of behavior of these materials are revealed, for instance, their reversible tunability of static and dynamic mechanical moduli by magnetic fields, elastic interactions between the embedded particles mediated through the polymeric matrix, or a pronounced and reversibly tunable nonlinear stress–strain behavior. Links from the mesoscopic to the micro- and macroscopic level are outlined. We mention combined efforts of theoretical descriptions, modeling, numerical simulations, and experimental investigations. It becomes evident from our treatment that an integrated approach of theory, simulations, and experiments will significantly increase our further understanding of these materials in the future and will draw possible applications into sight.

Wave-induced dynamics of marine pipelines in liquefiable seabed

This paper presents a simple but workable modeling method to simulate the wave-induced liquefaction scenarios around a marine pipeline within the framework of the Biot's theory, incorporating the main features such as relation for the consolidation describing the pore-volume reduction, hysteretic stress-strain behavior of soil skeleton and soil-pipe contact effect. In this context, special attention is paid to the implementation of a well-calibrated cyclic soil model for hysteretic and nonlinear stress-strain behavior (i.e. strain softening and cyclic degradation), associated with a semi-empirical shear-volume coupling equation for capturing the accumulative volumetric change, which links the increment of volumetric strain per cycle of wave with the shear strain occurring during that particular cycle. The proposed modeling framework is then incorporated into an explicit time matching finite difference analysis procedure, allowing a full non-linear dynamic analysis of the intensive interactions between the pipeline and the seabed undergoing buildup of pore pressure and residual liquefaction. Retrospective simulation of the wave flume test performed by Sumer et al. (2006c) using the proposed model shows good agreement, calibrating the reliability of the modeling method for the prediction of wave-induced liquefaction of sandy seabed and failure process of the buried pipelines. Finally, the liquefaction mechanism around a buried pipeline under a nonlinear wave loading is investigated by numerical examples. The obtained results interpret the cause of liquefaction and the resulting consequence for pipeline stability in wave environment.

Design rules for stainless steel welded I-columns based on experimental and numerical studies

Stainless steel is characterised by its nonlinear stress-strain behaviour with significant strain hardening, although current design codes treat it as an elastic, perfectly plastic material like carbon steel. The continuous strength method (CSM) is a newly developed strain based design approach which was proposed for nonlinear metallic materials. With recent developments, CSM can be used to predict the cross-section resistance for stocky and slender sections, and CSM design rules have recently been proposed for predicting the buckling resistance of cold-formed RHS and SHS columns. Welded sections, however, could behave differently from cold-formed sections due to the presence of residual stresses. Despite offering more economic options in many design cases, research on stainless steel welded sections is very limited to date. In this study, the behaviour of stainless steel welded I-sections was investigated through a test program, and the investigation was complemented by finite element (FE) modelling. The test program covered tensile coupon tests, residual stress and initial geometric imperfection measurements, stub column tests and flexural buckling tests of pin-ended long columns. FE models were developed for both major and minor axis buckling based on test results, and the verified FE modelling technique was used to investigate the effects of cross-section slenderness λp, section height-to-width ratio H/Band the ratio of flange thickness-to-web thickness tf/tw on column curves of welded I-sections. Buckling formulas for welded I-columns were eventually proposed following the same philosophy recently adopted by the authors for cold-formed hollow section columns. The imperfection parameter was recalibrated appropriately to incorporate special features of welded I-sections. Two sets of equations were proposed to tackle the observed variation in buckling behaviour against major and minor axis buckling. Buckling resistance predictions obtained from the proposed method were deemed reliable showing good accuracy and consistency with test and FE results.

Development and verification of large deformation model considering stiffness deterioration and shear dilation effect in FLAC3D

The existing constitutive models of rock with strain softening cannot successfully reflect the damage-induced anisotropy and nonlinearity of the post-peak failure behavior under progressive loading. In order to better reflect the nonlinear stress-strain behavior of rock, especially for the post-peak failure behavior, with expected stiffness degradation and large deformation, a modified constitutive model of rock considering stiffness degradation and dilatation behavior at large deformation was proposed in this study. This study analyzes and discusses various attenuation parameters in the proposed nonlinear plastic constitutive model using FLAC3D software. The excavation-induced stress in rocks was calculated by FLAC3D using the Mohr-Coulomb model, conventional strain model, and the proposed modified model. The obtained results of these models were analyzed and compared with field data. This study shows that the simulated results using the proposed new constitutive model matched much more closely with the measured field data, with an average error less than 5%. This new model can successfully reflect the damage-induced stiffness degradation at large deformation, and can provide a theoretical basis for stability design and evaluation of underground excavation.

Elastic and fracture characteristics of graphene-silicon nanosheet composites using nonlinear finite element method

A simple and efficient methodology is proposed for computing the nonlinear stress-strain behavior and fracture strength under tensile loading of graphene-silicon nanosheet composites. A nonlinear finite-element model (FEM) is developed to obtain the stress-strain relationships, which in general are computed using the computationally expensive procedure of molecular dynamics (MD) simulations. The modelling method, based on the interconnection of continuum micro-mechanics approaches, incorporates bonded contact at the graphene-silicon interface with the graphene being modelled as the multilinear elastic and the silicon assumed as an isotropic material is fed into an atomistic progressive fracture model. Using this model, we get the nonlinear behaviour of graphene, silicon and its composites, as a nonlinear stress-strain curve with their critical stress at inflection point leading to failure. The obtained results of the stress-strain curves, the elastic modulus and the critical stresses of single layer graphene (SLG), silicon nanosheet and their composites (GSNC) with different thickness of silicon nanosheet are in good agreement with the MD results available in the literature. These results elucidate that the tensile strength and Young’s modulus of the silicon nanosheet increase enormously by putting the graphene layer on the top and bottom surface of the silicon nanosheet. This model is then employed to study the non-linear stress behaviour for the different orientations of silicon with varying thickness and their corresponding composites with graphene and the effect of various chirality of SLGs are also considered. Thus, we try to establish FEM as a reliable numerical method to obtain the mechanical behaviour of graphene-silicon nanosheet composites which for long-time has been solved using the classical approach of MD. The parametric study which is done using the developed FEM model shows that the GSNC helps to enhance the mechanical properties of silicon nanosheet, which in turn is helpful for the silicon-based semiconductor industry. This fact correlates well with the other methods, well-established in the literature.

Engineering biologically extensible hydrogels using photolithographic printing

Biomaterials for tissue engineering that recapitulate the mechanical response and biological function of native tissue are highly sought after to lessen the burden of damaged or diseased tissue. Poly(ethylene glycol) diacrylate (PEGDA) hydrogels are a popular candidate because of their favorable bioactive properties. However, their mechanical behavior is very dissimilar to that of biological tissue, which behaves in a mechanically anisotropic, nonlinear, and viscoelastic fashion. It has been previously shown that PEGDA hydrogels can be patterned in alternating linear strips of different stiffnesses to generate anisotropic behavior, but these constructs still have a linear stress-strain response. In this study, we imparted nonlinear mechanical properties to PEGDA hydrogels by fabricating composite hydrogel constructs consisting of a stiff sinusoidal reinforcement embedded into a softer base matrix. This was achieved by polymerizing low molecular weight (MW) PEGDA hydrogel precursor into a stiff sinusoidal shape and then polymerizing this construct into a high MW precursor. Samples were generated with different relative stiffness between the two components and a range of sinusoid periodicities to assess the tunability of the resulting stress-strain curve. Tensile testing indicates that the sinusoidal patterning gives rise to nonlinear stress-strain behavior. Varying the relative stiffness was shown to tune the slope of the linear region of the stress-strain curve, and varying periodicity was shown to affect the length of the toe region of this curve. We conclude that composite hydrogels with stiff sinusoidally-patterned reinforcements display mechanical properties more similar to those of biological tissue than uniform or linearly-patterned hydrogels. Statement of SignificanceHydrogel biomaterials are a popular candidate for engineering constructs that can mimic the properties of native tissue for disease modeling and tissue-engineering applications. Studies have shown that poly(ethylene) glycol diacrylate (PEGDA) hydrogels can be fabricated to display many biological aspects of native tissue. However, they are unable to recapitulate fundamental mechanical properties of such tissue, such as anisotropy and nonlinearity. Photolithographic techniques have been employed to generate anisotropic linear PEGDA hydrogels via patterned reinforcement. The present study indicates that such techniques can be modified to generate PEGDA constructs with a sinusoidal reinforcement that display a strongly nonlinear response to tensile loading. This work sets the stage for more intricate patterning for providing increased control over hydrogel mechanical response.

Strain rate-dependent mechanical behavior of quasi-unidirectional E-glass fabric reinforced polypropylene composites under off-axis tensile loading

To make a better use of quasi-unidirectional fabrics in structural application, the strain rate-dependent tensile behavior of quasi-unidirectional E-glass fabric reinforced polypropylene composites was studied. Monotonic tensile tests were firstly performed on 10°, 20° and 90° off-axis samples of composites. The experimental results show that both mechanical properties and nonlinear stress-strain behavior of composites are dependent of strain rate. A regression function is adopted to describe the variation in overall mechanical properties with the imposed strain rate. A strain rate-dependent coupled damage-plasticity model is built for quasi-unidirectional composites. The damage variables and plastic parameters in the proposed model are determined through cyclic loading-unloading tensile tests and can be represented as a function of the imposed strain rate. In addition, the damage mechanism of composites is observed to evolve with strain rate. Increasing strain rate can hinder the initiation and evolution of damage in composites.

Effects of material nonlinearity on the global analysis and stability of stainless steel frames

In structural frames, second order effects refer to the internal forces and moments that arise as a result of deformations under load (i.e. geometrical nonlinearity). EN 1993-1-1 states that global second order effects may be neglected if the critical load factor of the frame αcr is greater than or equal to 10 for an elastic analysis, or greater than or equal to 15 when a plastic global analysis is used. No specific guidance is provided in EN 1993-1-4 for the design of stainless steel frames, for which the nonlinear stress-strain behaviour of the material will result in greater deformations as the material loses its stiffness. A study of the effects of material nonlinearity on the stability of stainless steel frames is presented herein. A series of different frame geometries and loading conditions are considered. Based on the findings, proposals for the treatment of the influence of material nonlinearity on the global analysis and design of stainless steel frames are presented.

Local-flexural interactive buckling of standard and optimised cold-formed steel columns

This paper aims to study the interaction of local and overall flexural buckling in cold-formed steel (CFS) channels under axial compression. Detailed nonlinear FE models were developed and validated against a total of 36 axial compression tests on CFS plain and lipped channel columns with pin-ended boundary conditions. The numerical models incorporated the non-linear stress-strain behaviour of CFS material and enhanced properties of cold-worked corner regions obtained from coupon tests. The effects of initial geometric imperfections of the specimens measured by a specially designed set-up with laser displacement transducers were also taken into account. The developed FE models produced excellent predictions of the ultimate strength of the specimens obtained from experimental tests. The validated FE models and experimental results were then used to assess the adequacy of the effective width method in Eurocode 3 (EC3) and Direct Strength Method (DSM) in estimating the design capacity of a wide range of conventional and optimised design CFS channel column sections. The results indicate that Eurocode 3 provides conservative predictions (on average 21% deviation) for the compressive capacity of plain and lipped channel sections, while in general DSM predictions are more accurate for lipped channels. A comparison between FE predictions and tested results show that geometric imperfections can change the FE predictions by up to 20% and 40%, respectively, for lipped and plain channel columns, while the strain hardening effect at the rounded corner regions of the cross-sections is negligible. The results also confirmed that the proposed numerical model is able to provide a consistent and reliable prediction on the efficiency of a previously proposed optimisation methodology.

Compressive response and deformation mechanisms of vertically aligned helical carbon nanotube forests

We study the dynamic compressive response of vertically aligned helical carbon nanotube forests using a mesoscale model. To describe the compressive response, the model includes the helical geometry of the constituent coils, the entanglement between neighboring coils, and the sideway interactions among coils. Coarse-grained simulations show forest densification and stress localization, which are caused by different deformation mechanisms such as coil packing, buckling, and crushing. We find that these mechanisms depend on the initial overlap between coils and lead to a nonlinear stress-strain behavior that agrees with recent impact experiments. The nonlinear stress-strain behavior was shown to be composed of an initial linear increase of stress in strain followed by an exponential growth. These regimes are an outcome of the characteristics of both the individual coils and the entangled morphology of the forests.

Investigation of tensile properties of RTV Silicone based Isotropic Magnetorheological Elastomers.

Magnetorheological elastomer (MRE) consists of an elastomer matrix and a Ferro-magnetic ingredient. The mechanical properties of MR elastomers can be reversibly controlled by applying a magnetic field of suitable intensity. The current work focusses on the enhancement of tensile property of RTV (Room Temperature Vulcanization) silicone based elastomer. The influence of Carbonyl iron powder (CIP) content and magnetic field were experimentally investigated. Addition of CIP increases the tensile modulus but it reduces the percentage elongation and tensile strength making it brittle. Under the influence of magnetic field, the enhancement of tensile properties up to 20% content was linear. The behavior above 20% is observed to be non-linear. The onset of non-linear stress-strain behavior is investigated. Regression equation is developed from the experimental data relating percentage content with the mechanical properties of MRE. The developed equation predicted the behavior of 27% MRE with an error of less than 8%. Hyperelastic model developed by Yeoh was fitted to the stress-strain behavior of MRE with minimal error.

Tearing energy and path-dependent J-integral evaluation considering stress softening for carbon black reinforced elastomers

For carbon black reinforced Ethylene-Propylene-Diene Monomer Rubber (EPDM) fracture mechanics investigations by using a Single Edge Notched Tension sample (SENT) are presented. Filled elastomers under cyclic loading typically display inelastic effects like permanent set, strong nonlinear S-shaped stress-strain behaviour and stress induced softening (discontinuous damage) as well as cyclic softening (continuous damage) better known as Mullins effect. These unique properties give this material class an interesting and a complex aspect regarding the experimental observations and numerical modelling of fracture mechanics. In simulations with homogenous material, the J-integral given as the evaluation of the Eshelby stress tensor along a path integral around the crack shows no dependency on the chosen integration path. In contrast, dissipative materials show strong path dependency of the J-integral. When the chosen path is near to the crack tip, this effect is pronounced, due to the inelastic effects, which are very high at the crack tip compared to the rest of the sample. In the process zone, the J-integral value depends strongly on the chosen path around the crack tip. This phenomenon occurs due to the intensity of inelastic effects near the crack tip. The path-dependency of J-integral using the variational formulation of a crack problem in a dissipative material has been given in [16]. Stumpf and Le have proposed a new formulation of the boundary value problem for elastoplastic cracked body using variational formulation, where the finite deformation was considered. For cyclic displacement controlled loading, the elastomers show typical stress softening behaviour, i.e. the force-displacement curve of each cycle is lower than the previous cycle and higher than the following cycle. This occurs due to the continuous damage, whereby the bondings between polymer chains and the filler particles are partially damaged. At small external amplitudes, the stress softening effect is significant only in the vicinity of the crack tip, because of the hysteresis which is large compared to the rest of the sample. The evaluation of the J-integral for an SENT sample under cyclic displacement controlled loading shows clearly that the resulting energy flux at the crack tip decreases from the virgin curve until the last repetition. The values found are in good agreement with experimentally evaluated tearing energy. Hence J-integral evaluation while considering a stress softening reflecting material model can be a promising approach used for estimation of the crack propagation or for lifetime prediction.

Improving the sensitivity to map nonlinear parameters for hyperelastic problems

The mechanical properties of tissues are important indicators of tissue “health”. Diseased tissues due to cancer and other causes tend to stiffen with increase in strain, exhibiting a nonlinear stress–strain behavior. The hyperelastic mechanical behavior of soft tissues can be characterized by an exponential model with two material parameters, namely the shear modulus μ and a nonlinearity parameter γ. A variety of methods and techniques have been developed to solve inverse problems in nonlinear elasticity to determine these properties given the mechanical response of the tissues. Reconstruction of the nonlinear parameter from noisy measurements of displacement response is a difficult problem, and obtaining a well-recovered solution is challenging. This article is directed towards the improvement in the reconstruction of the nonlinear parameter, γ, by introducing a new parameter, which is a combination of γ and the first invariant of the Cauchy–Green deformation tensor. Comparative study is carried out between reconstructions of γ directly from previously existing formulations and the reconstruction of γ from the new parameter, for 2D problems. The new method is compared with previous methods using numerical experiments in terms of shape of the stiff regions, the contrast in γ and robustness under different loading conditions. We find that the new method is a considerable improvement to previous methods and could be a valuable tool in biomedical applications.

Constitutive modelling of elastomer/graphene platelet nanocomposites

Elastomers are used in a wide variety of structural and engineering applications. They exhibit a nonlinear elastic stress-strain behaviour known as hyperelasticity which is generally described by hyperelastic strain energy functions. The question raised in the current study was; which model can accurately describe and predict the actual behaviour of the elastomer nanocomposites. The tensile data were used to fit the various elastomeric material models available in MSC.MARC finite element analysis package. The relative percentage error was calculated to determine the goodness of fit in order to select the best model. Numerical results showed that the third order deformation model was the best among the various material models since giving a maximum relative error of fit was 2.7% at small and large strains. To verify the effectiveness of third order deformation model, FE simulations for tensile test was carried out. The results showed that the third order model is sufficiently enough to regenerate the experimental data for uniaxial test and efficiently capture the hyperelastic behavior as good as the experiments.

Parameter identification methods for visco- and hyperelastic material models

Many materials in modern civil engineering applications, such as interlayers for laminated safety glass, are polymer-based. These materials are showing distinct viscoelastic (strain-rate) and temperature dependent behaviour. In literature, different mathematical representations of these phenomena exist. A common one is the ‘Prony-series’ representation, which is implemented in many state-of-the-art Finite-Element-Analysis-Software to incorporate linear viscoelastic material behaviour. The Prony-parameters at a certain reference temperature can either be determined by relaxation or retardation experiments in the time domain or with a steady state oscillation in the frequency domain in the so called ‘Dynamic Mechanical Thermal Analysis’ followed by a ‘Time–Temperature-Superposition-Principle’. However, present research shows that polymeric materials also may need to have constitutive equations which include hyperelasticity (nonlinear stress–strain behaviour in a quasi-static condition) when undergoing large deformations, so that the material model should be expanded for a more realistic representation in numerical simulations. A novel method for the whole identification process for a numerical material model in terms of a linear Generalized Maxwell Model (Prony-series) based on experimental data will be presented. Furthermore, material parameters for different hyperelastic material models based on experimental investigations will be shown and compared. Future research activities as well as extensions of the presented novel method are also highlighted within this paper.