Nonlinear Stress-strain Relationship Research Articles

Coupled purely nonlinear oscillators: normal modes and exact solutions for free and forced responses

This article is concerned first with free vibrations in a chain of two-mass oscillators with purely nonlinear springs whose power of nonlinearity can be any real number higher than unity. Similar normal modes are obtained by uncoupling the equations of motion. The corresponding time responses are determined in exact analytical forms in terms of Ateb and Jacobi elliptic functions. The external excitation is designed so as to uncouple the equations of motion again and to determine the exact solution for forced vibrations. Then, the case of a three-mass chain is investigated, and similar normal modes are found for the nonlinearity of interest. In addition, these novel results are extended to a continuous system—a rod exhibiting longitudinal vibrations when the material of the rod is characterized by a purely nonlinear stress–strain relationship.

Long-term performance of prestressed concrete bridges under the intertwined effects of concrete damage, static creep and traffic-induced cyclic creep

The current progress towards sustainable transportation demands for a prolonged lifespan, i.e., over 100years, for large-span bridges. To accomplish this goal, realistic prediction of the long-term performance of bridges plays an indispensable role in bridge engineering. In this study, the long-term deflection and the deflection-induced damage growth are investigated for the large-span prestressed concrete bridges carrying busy traffic flows. A unified constitutive model, taking into account the coupled effects of concrete viscoelastic behavior and concrete tensile cracking and plastic softening, is developed to estimate the bridge performance. Here the static creep of concrete is approximated by a series of Kelvin units. The elasto-plastic damage of concrete is described by the plasticity yield surfaces hinging on the flow rules and anisotropic damage laws anchored at the strain equivalence hypothesis. In addition to the static creep, the cyclic creep of concrete induced by vehicles is integrated in the nonlinear stress-strain incremental relation based on the fatigue growth of microcracks. With the aid of the 3D rate-type formulation, the proposed unified model is implemented in ABAQUS and a large-span prestressed concrete bridge carrying heavy traffic loads is analyzed. Compared to the inspection reports, the deflection history at the midspan, the deformed profile of the whole bridge and the distribution of concrete cracking are all adequately captured in the simulations. Further sensitivity investigation on bridge deflection finds that the cyclic load-induced damage accumulation process plays a non-negligible role and traffic management plans are needed for heavy trucks to mitigate the risk of excessive deflection and concrete degradation.

Effects of gas pressure on failure and deviatoric stress on permeability of reservoir rocks: initial studies on a Vosges sandstone

The geomechanical behaviour of a specific type of sandstone is investigated using tri-axial compression tests, a hydrostatic compression test to measure Biot’s coefficient, tri-axial tests with pressurised gas, and permeability tests under deviatoric stress. The tri-axial compression tests reveal a nonlinear stress–strain relationship at low confining pressures. Biot’s coefficient initially decreases with increasing confining pressure, and then gradually increases if the confining pressure is increased beyond a certain level. At constant confining pressure and deviatoric stress, the gas pressure loading path has a significant influence on the mechanical failure of the sandstone sample. If the gas pressure is increased by comparatively small increments, the critical pressure the sample can withstand is found to be close to the theoretical value, whereas if the gas pressure increments are comparatively large, the sample is found to fail at lower values of gas pressure. The injection and release of gas also accelerate the sample’s failure. At high confining pressures, there is a significant decrease in permeability with increasing deviatoric stress. However, at low confining pressures, the decrease in permeability is not as significant as that observed under high confining pressures. The reloading–unloading of deviatoric stress does not have a predictable effect on permeability.

Nonlinear Fracture of Functionally Graded Beams Under Mode II Loading Conditions

A theoretical study of Mode II fracture in functionally graded beams was carried out. A beam configuration with two symmetric longitudinal cracks was suggested in order to generate Mode II loading conditions. The beam mechanical behavior was described by nonlinear stress–strain relation. The fracture behavior was analyzed by applying the J-integral approach. Closed form analytical solutions were derived of the J-integral for two laws for variation of the modulus of elasticity along the beam height. Nonlinear analyses of the strain energy release rate were performed by considering the energy balance in order to verify the J-integral solutions. The results obtained can be applied to optimize the functionally graded beam structure with respect to the Mode II fracture performance. Also, the analytical solutions derived are very convenient for parametric studies of nonlinear fracture behavior of functionally graded beams. The present paper contributes towards the development of nonlinear fracture mechanics of functionally graded materials.

Parametric Variational Principle for Bi-Modulus Materials and Its Application to Nacreous Bio-Composites

Bi-modulus materials have different moduli in tension and compression and the stress–strain relation depends on principal stress that is unknown before displacement is determined. Establishment of variational principle is important for mechanical analysis of materials. First, parametric variational principle (PVP) is proposed for static analysis of bi-modulus materials and structures. A parametric variable indicating state of principal stress is included in the potential energy formulation and the nonlinear stress–strain relation is evolved into a linear complementarity constraint. Convergence of finite element analysis is thus improved. Then the proposed variational principle is extended to a dynamic problem and the dynamic equation can be derived based on Hamilton’s principle. Finite element analysis of nacreous bio-composites is performed, in which a unilateral contact behavior between two hard mineral bricks is modeled using the bi-modulus stress–strain relation. Effective modulus of composites can be determined numerically and stress mechanism of “tension–shear chain” in nacre is revealed. A delayed effect on stress propagation is found around the “gaps” between mineral bricks, when a tension force is loaded to nacreous bio-composites dynamically.

A MeshFree Micromechanical Analysis for Twill Composite

This paper presents homogenization of twill woven composites. It includes presenting formulated geometrical unit cell models, establishing meshfree micromechanical and CDM models for twill woven composites. The results cover meshfree nodal distributions, meshfree yarn geometries, predicted mechanical properties, effectiveness of yarn waviness, non-linear stress-strain relation of unit cell, stress distributions inside unit cell volume, and necessary comparisons with existing literatures.

A new constitutive model for simulation of softening, plateau, and densification phenomena for trabecular bone under compression

A new type of constitutive model and its computational implementation procedure for the simulation of a trabecular bone are proposed in the present study. A yield surface-independent Frank–Brockman elasto-viscoplastic model is introduced to express the nonlinear material behavior such as softening beyond yield point, plateau, and densification under compressive loads. In particular, the hardening- and softening-dominant material functions are introduced and adopted in the plastic multiplier to describe each nonlinear material behavior separately. In addition, the elasto-viscoplastic model is transformed into an implicit type discrete model, and is programmed as a user-defined material subroutine in commercial finite element analysis code. In particular, the consistent tangent modulus method is proposed to improve the computational convergence and to save computational time during finite element analysis. Through the developed material library, the nonlinear stress–strain relationship is analyzed qualitatively and quantitatively, and the simulation results are compared with the results of compression test on the trabecular bone to validate the proposed constitutive model, computational method, and material library.

The effect of walking on inclined surfaces (uphill and downhill) on gait parameters of healthy subjects using self-paced treadmill in virtual environments

This research presents the finite element modeling (FEM) of human-specific computed tomography (CT) data to study the effect of bone prominences on contact stress in the shoulder for prevention of pressure ulcers. The 3D geometry of scapula, skin, and surrounding soft tissues in the shoulder was reconstructed based on the anonymous CT data of a human subject in a prone posture (without loading on the shoulder) for FEM analysis of the contact stress. FEM analysis results show that the maximum stress is located at the prominence of the scapula with sharp bone geometry. This demonstrates that stress concentration at the bone prominence is a significant factor to cause the high contact stress, which is a source for pressure ulcers. For experimental validation, a physical shoulder model manufactured by 3D printing of the bone geometry and the mold for molding of tissue-mimicking silicone was developed. Compression tests of the mattress foam and silicone were conducted to find the nonlinear stress–strain relations as inputs for FEM. Experiments of compressing the shoulder model against the foam were carried out. Three flexible force sensors were embedded inside the model to measure the contact forces and compared to the FEM predictions. Results show that the FEM predicted forces match well with the experimental measurements and demonstrate that FEM can accurately predict the stress distributions in the shoulder to study the effect of bone geometry on the inception of pressure ulcers.

Computational study of the elastic modulus of single size pebble beds for fusion applications

In this paper, the discrete element method (DEM) was used to report the mechanical behaviors and elastic moduli of single size pebble beds. The effect of cyclic pressure p and the subtle difference in particle sizes within the granular system were considered. In this study, we re-confirmed the nonlinear elastic stress-strain relation, the much lower elastic moduli of a granular system than that of solid materials and the faster growth of moduli than the p1/3 law predicted by the effective medium theory (EMT). We also observed that cyclic pressure (mechanical excitation) would stiffen the granular system, but this effect will become smaller as the cycle number increases. This indicates that the stability of system will become stronger under this effect. Besides, the subtle difference in grain sizes was observed to soften the system although it caused a little higher packing density. Furthermore, although the effective moduli of different granular materials Es are diverse, they were found to nearly collapse to the same distribution when both Es and p are non-dimensionalized by particle moduli Ep. This can actually also be derived from the EMT theory.

Nanotechnology, smartness and orthotropic nonhomogeneous elastic medium effects on buckling of piezoelectric pipes

The effects of nanotechnology and smartness on the buckling reduction of pipes are the main contributions of present work. For this ends, the pipe is simulated with classical piezoelectric polymeric cylindrical shell reinforced by armchair double walled boron nitride nanotubes (DWBNNTs), The structure is subjected to combined electro-thermo-mechanical loads. The surrounding elastic foundation is modeled with a novel model namely as orthotropic nonhomogeneous Pasternak medium. Using representative volume element (RVE) based on micromechanical modeling, mechanical, electrical and thermal characteristics of the equivalent composite are determined. Employing nonlinear strains-displacements and stress-strain relations as well as the charge equation for coupling of electrical and mechanical fields, the governing equations are derived based on Hamilton\'s principal. Based on differential quadrature method (DQM), the buckling load of pipe is calculated. The influences of electrical and thermal loads, geometrical parameters of shell, elastic foundation, orientation angle and volume percent of DWBNNTs in polymer are investigated on the buckling of pipe. Results showed that the generated O improved sensor and actuator applications in several process industries, because it increases the stability of structure. Furthermore, using nanotechnology in reinforcing the pipe, the buckling load of structure increases.

The effect of nonlinear elasticity on the large amplitude free vibration behavior of elastic plates at small scale

The effect of nonlinear elasticity on the free vibration behavior of elastic plates has been evaluated by employing continuum mechanics model. The second-order non-linear stress–strain relationship has been considered and the Kirchhoff’s hypothesis has been applied on the elastic plate. The large deformation during vibration has also been considered. By using the Hamilton principle, the governing equations of the free vibration of the plate under different boundary condition have been obtained. In order to get the explicit solutions of the governing equations, the Galerkin’s method and the harmonic balance method have been utilized. The relationship between the vibration frequency and the vibration amplitude has been discussed and the vibration frequencies of different shaped plate have been compared. It is perceived that the nonlinear elasticity has a distinct effect on the free vibration of the plate.

Application of artificial neural networks to predict the deflections of reinforced concrete beams

Abstract Nonlinear structural mechanics should be taken into account in the practical design of reinforced concrete structures. Cracking is one of the major sources of nonlinearity. Description of deflection of reinforced concrete elements is a computational problem, mainly because of the difficulties in modelling the nonlinear stress-strain relationship of concrete and steel. In design practise, in accordance with technical rules (e.g., Eurocode 2), a simplified approach for reinforced concrete is used, but the results of simplified calculations differ from the results of experimental studies. Artificial neural network is a versatile modelling tool capable of making predictions of values that are difficult to obtain in numerical analysis. This paper describes the creation and operation of a neural network for making predictions of deflections of reinforced concrete beams at different load levels. In order to obtain a database of results, that is necessary for training and testing the neural network, a research on measurement of deflections in reinforced concrete beams was conducted by the authors in the Certified Research Laboratory of the Building Engineering Institute at Wrocław University of Science and Technology. The use of artificial neural networks is an innovation and an alternative to traditional methods of solving the problem of calculating the deflections of reinforced concrete elements. The results show the effectiveness of using artificial neural network for predicting the deflection of reinforced concrete beams, compared with the results of calculations conducted in accordance with Eurocode 2. The neural network model presented in this paper can acquire new data and be used for further analysis, with availability of more research results.

THE ANALYSIS BEHAVIOUR OF ARCHED STRIP FOUNDATIONS

The construction of bearing walls structures does not necessarily require for using plane strip reinforce concrete footings. These structures can construct by using present simple arched plain and reinforced concrete strip footings on silty clay and silty sand soil using finite element. This paper aims at analyzing plain and reinforced concrete arched strip footings, as foundation system of bearing walls structures, as an alternative solution to the construction of buildings. The effect of soil type, arched strip footing's height and the bearing walls vertical load on the dimensions and capacity of arched strip footings study in this paper. A numerical model for the non-linear analysis of arched strip footing-soil interaction problem based on the finite and infinite element implement. A computer program develops to model the arched strip footing-soil installation. The material and geometrical non-linearity of the concrete strip footing takes into account the non-linear stress-strain relation of concrete and presence for cracking also considers. In addition, Duncan-Mohr-Coulomb Modified model uses to simulate soil non-linearity. The obtained numerical results were compared with the traditional method in designing of strip footings commonly used by structural engineers. Design charts propose and presented for structural designers in order to calculate arched P.C & R.C strip footing dimensions according to type of soil and vertical load such strip footings which consider the cost less than traditional bearing walls construction system. The thickness of strip footing expresses in a non-dimensional ratio (t/B), where (B) is the breadth of the strip footing, with three ratios of 0.1, 0.2, and 0.3 respectively. The height of the arched strip footing was expressed in a non-dimensional form (h/B) with three ratios of 0.1, 0.2 and 0.3 respectively. Four different values of vertical load (P = 20t/m', 30t/m', 40t/m' & 50t/m') investigate in the analysis. Two types of soil; silty clay, and silty sand consider in this study to represent the cases of weak and stiff soil. This result leads to exceptionally low cost up to 30% and safe structures than in case of plane strip footing. The present investigation shows some results that the minimum etc.&nbsp

Characterization of the nonlinear elastic behavior of chinchilla tympanic membrane using micro-fringe projection

The mechanical properties of an intact, full tympanic membrane (TM) inside the bulla of a fresh chinchilla were measured under quasi-static pressure from −1.0 kPa to 1.0 kPa applied on the TM lateral side. Images of the fringes projected onto the TM were acquired by a digital camera connected to a surgical microscope and analyzed using a phase-shift method to reconstruct the surface topography. The relationship between the applied pressure and the resulting volume displacement was determined and analyzed using a finite element model implementing a hyperelastic 2nd-order Ogden model. Through an inverse method, the best-fit model parameters for the TM were determined to allow the simulation results to agree with the experimental data. The nonlinear stress-strain relationship for the TM of a chinchilla was determined up to an equibiaxial tensile strain of 31% experienced by the TM in the experiments. The average Young's modulus of the chinchilla TM from ten bullas was determined as approximately 19 MPa.

Modeling of Nonlinear Mechanical Behavior for 3D Needled C/C-SiC Composites Under Tensile Load

This paper established a macroscopic constitutive model to describe the nonlinear stress–strain behavior of 3D needled C/C-SiC composites under tensile load. Extensive on- and off-axis tensile tests were performed to investigate the macroscopic mechanical behavior and damage characteristics of the composites. The nonlinear mechanical behavior of the material was mainly induced by matrix tensile cracking and fiber/matrix debonding. Permanent deformations and secant modulus degradation were observed in cyclic loading-unloading tests. The nonlinear stress–strain relationship of the material could be described macroscopically by plasticity deformation and stiffness degradation. In the proposed model, we employed a plasticity theory with associated plastic flow rule to describe the evolution of plastic strains. A novel damage variable was also introduced to characterize the stiffness degradation of the material. The damage evolution law was derived from the statistical distribution of material strength. Parameters of the proposed model can be determined from off-axis tensile tests. Stress–strain curves predicted by this model showed reasonable agreement with experimental results.

A Hydrogen-Induced Decohesion Model for Treating Cold Dwell Fatigue in Titanium-Based Alloys

Cold dwell fatigue in near-alpha Ti alloys is a time-dependent fracture process at ambient temperature that involves fatigue in the presence of creep to produce cracking on low-energy fracture (e.g., cleavage) facets in hard alpha grains. In this article, cold dwell fatigue is treated as a hydrogen-induced decohesion process by using a nonlinear cohesive stress-strain relation to describe the decrease in the cohesive strength with increasing local hydrogen contents. It is postulated that during cold dwell fatigue, time-dependent deformation occurs by 〈a〉 slip that results in dislocation pileups in soft alpha grains. The stress and dilatational fields of the dislocation pileups assist the transport of internal hydrogen atoms from soft grains to neighboring hard grains. The accumulation of internal hydrogen atoms at the trap sites leads to decohesion along crystallographic planes, which can be slip or hydride habit planes. The decohesion model is applied to treat cold dwell fatigue in Ti-6Al-4V with a basal-transverse texture by modeling the effects of hydrogen-induced decohesion on the stress-fatigue life (S-N f) response, the time-dependent crack growth response (da/dt), and the fracture toughness (K c) as functions of grain orientation. A probabilistic time-dependent fatigue crack growth analysis is then performed to assess the influence of microtexture on the dwell fatigue life of a Ti-6Al-4V ring disk subjected to a long-duration hold at the peak stress of the loading cycle. The results of the probabilistic life computations indicate that dwell fatigue resistance in Ti-6Al-4V may be improved and the risk of disk fracture may be reduced significantly by controlling the microtexture or reducing the size and volume fraction of hard alpha grains in the microstructure.

Mechanical behavior of jute hybrid bio-composites

Abstract Hybrid bio-composites are environmentally friendly and can provide a sustainable alternative to existing engineering materials in several applications. While these hybrid composites have relatively low modulus, their material consistency lend them to be used in many structural applications. Many low modulus natural fibers exhibit nonlinear axial stress strain relations. Orthotropic material nonlinearity is typically analyzed for composites in the transverse and shear directions, and very few computational models consider axial nonlinearity. In this manuscript two new macro and one micromechanical constitutive models are developed to characterize the nonlinear orthotropic behavior of these material systems in the axial, transverse and shear directions. These models are then implemented within finite element (FE) code. A hybrid bio-composite in the form of pultruded layers manufactured with jute bio-fibers, combined with unidirectional roving E-glass, and embedded in a polymeric matrix was chosen for this study. Stress strain curves are generated for these dually reinforced systems in transverse, axial and shear modes to calibrate the nonlinear parameters for computational models. Photomicroscopy was also used to characterize the microsctructure to calibrate the micromechanical constitutive model. All three models are then validated under a multi-axial state of stress by full-field stress/strain analysis via Digital Image Correlation (DIC) and Thermoelastic Stress Analysis (TSA) of open-hole specimens. The results show that all of the models match the full-field TSA and DIC results under a multi-axial state of stress; however, the Anisotropic Potential Theory (APT) model based on the work of Hahn Tsai showed more response at stress concentrations than the Anisotropic Deformation Theory (ADT) model based on the work of Hashin. Differences may have resulted from the correction scheme implemented in the APT model. Also the Nine-Cell micromechanical model in this study based on the work of Haj-Ali et al. was developed for comparison with the APT and ADT macromodels. The macromodels and micromodel presented here were implemented in ABAQUS user material subroutines, and are beneficial for analysis and design of structures with soft fiber constituents that have a nonlinear axial response.

The application of J integral to measure cohesive laws under large-scale yielding

A method is developed to obtain the mode I cohesive law of elastic–plastic materials using a Double Cantilever Beam sandwich specimen loaded with pure bending moments. The approach is based on the validity of the J integral for materials having a non-linear stress–strain relationship without unloading of any material point. This assumption is not met exactly as there is a small where the material unloads. To examine the error of the method, a numerical parameter study is performed for a wide range of material and specimen parameters. The error of the method is below 16% and thus the method can be used to measure cohesive laws including their shape.

Analysis of Beams on Heterogeneous and Nonlinear Soil

AbstractA new method for nonlinear analysis of Euler-Bernoulli beams resting on heterogeneous, multilayered soil is presented. The governing differential equations for beam and soil displacements are obtained using the principle of virtual work, and these equations are solved using one-dimensional finite-element and finite-difference methods. Using the analysis, beam displacements can be obtained as a nonlinear function of applied loads if the beam and soil geometry and properties are known. The distinct feature of the analysis is that beam responses with accuracy comparable with those obtained from equivalent two-dimensional finite-element analysis are obtained within seconds. Examples illustrate how the proposed method can be applied in practice to obtain beam displacements in soil deposits that are characterized with nonlinear stress-strain relationships, have multiple layers, and have a modulus varying spatially within each soil layer. Parametric studies are performed that highlight the effects of soi...

応力反転後の非線形弾塑性挙動の考慮によるスプリングバック予測精度向上

Abstract Prediction of springback during press forming has become more important due to the recent trend of application of high strength steels in automotive white body parts. The accuracy of springback prediction depends on the estimation of recovery strain after die release. As it is well known that unloading behavior after pre-strain shows a nonlinear stress-strain relationship, a material model which considers nonlinear elastoplastic behavior is necessary for accurate springback prediction. In this study, the nonlinear behaviors of high strength steel sheets under uniaxial and biaxial stress states were measured. A new material model that considers nonlinear behavior was proposed and implemented in finite element method (FEM) software. Press-forming tests of curved hat-shaped parts and the corresponding springback analyses were performed, and the springback angles were evaluated. The springback predictions made using the proposed model showed good agreement with the experimental results.