Time-dependent Nonlinear Behavior Research Articles

Nonlinear time-dependent behavior of rheodictic polymers: A theoretical and experimental investigation

The present study examined the nonlinear time-dependent behavior of rheodictic polymers, a class of noncrosslinked materials that exhibit flow. Such behavior was addressed with extended Schapery's nonlinear viscoelastic model by introducing new physical quantities, i.e., the flow term Φflow and corresponding nonlinear shift parameter g2,flow. While Φflow portrays irrecoverable deformation, g2,flow depicts a nonlinear contribution to flow acceleration. This theory was accompanied by analytical and experimental methodologies for identifying all the parameters in the linear and nonlinear viscoelastic domains. Predictions of long-term time-dependent behavior (in shear) at various stress states show excellent agreement with the experimental data, i.e., within 5% error, obtained for polycarbonate at 130°C. Surprisingly, the newly introduced g2,flow indicates that flow retardation occurs with increasing stress, implying that a highly deformed entangled system hinders molecular reptation/disentanglement. Nevertheless, the proposed extension of Schapery's nonlinear viscoelastic model not only allows accurate predictions of the nonlinear time-dependent behavior of rheodictic polymers but also enables a detailed outlook on the underlying molecular mechanisms under severe environmental and loading conditions.

Multi-scale FE[formula omitted] investigation of non linear rate dependent 3D composite structures accounting for fiber–matrix damage

In the present paper, a two-scale FE2 technique based on periodic homogenization theory is investigated to predict the macroscopic non-linear behavior of polymer matrix composite structures.The computational technique accounts for the fiber/matrix interfacial damage, the matrix ductile damage and the effect of 3D periodic microstructure. The developed approach integrates the geometric description and the non-linear time-dependent local behavior of the different constituents (fibers, matrix and interface). For numerical calculations, advanced User Defined Material and User Element subroutines are developed at the two scales, simultaneously activated to solve macroscopic and microscopic problems through an incremental scheme in the finite element commercial code Abaqus/Implicit The computational efficiency of the developed multi-scale approach is demonstrated by predicting the overall response of 3D composite structures under complex loading paths. The composite structures consist of thermoplastic polymer matrix with elasto-viscoplastic behavior and ductile damage, reinforced by elastic aligned short fibers that are coated by a cohesive zone, which obeys the general unified potential. The numerical results obtained by FE2 simulation with and without accounting for interface effect are analyzed and compared for two examples: Meuwissen-like and 3D corner shape structures. The main benefits of the developed approach lie in accessing the microscopic strain fields, the distributions of the internal variables and the damage evolution in both polymer matrix and interface, as well as identifying their repercussions on the macroscopic response.

Study on Nonlinear Viscoelastic Constitutive Equations for Rock

Numerical simulation is one of the effective tools for the long-term stability assessment of underground structures. The reliability improvement of the numerical simulation requires precise constitutive equations to reproduce time-dependent behavior of rock such as loading-rate dependence, creep, and relaxation. These time-dependent behaviors of rock have nonlinearity between stress and strain rate and are influenced by surrounding environment. A number of constitutive equations have been proposed in previous studies; however, the equations based on the theory of linear viscoelasticity have difficulty reproducing accurately the time-dependent behavior of rock, and most of the equations based on the theory of viscoplasticity have been applied to the limited loading condition such as creep. In this article, nonlinear viscoelastic constitutive equations for rock proposed by the authors were introduced, and their exact solutions and application to experimental results were summarized. These equations can reproduce nonlinear time-dependent behavior and not only brittle but also ductile failures of rock and can be applied to the behavior under various loading conditions and experiment environment. In the equations, compliance or inelastic strain is assumed to be increased with time, but the functions and their combinations are of great variety. Hence, the differences of the solutions and applicability of the equations were detailed, and in addition future subjects were presented.

Nonlinear time-dependent mechanical behavior of mammalian collagen fibrils.

The viscoelastic mechanical behavior of collagenous tissues has been studied extensively at the macroscale, yet a thorough quantitative understanding of the time-dependent mechanics of the basic building blocks of tissues, the collagen fibrils, is still missing. In order to address this knowledge gap, stress relaxation and creep tests at various stress (5-35 MPa) and strain (5-20%) levels were performed with individual collagen fibrils (average diameter of fully hydrated fibrils: 253±21 nm) in phosphate buffered saline (PBS). The experimental results showed that the time-dependent mechanical behavior of fully hydrated individual collagen fibrils reconstituted from Type I calf skin collagen, is described by strain-dependent stress relaxation and stress-dependent creep functions in both the heel-toe and the linear regimes of deformation in monotonic stress-strain curves. The adaptive quasilinear viscoelastic (QLV) model, originally developed to capture the nonlinear viscoelastic response of collagenous tissues, provided a very good description of the nonlinear stress relaxation and creep behavior of the collagen fibrils. On the other hand, the nonlinear superposition (NSP) model fitted well the creep but not the stress relaxation data. The time constants and rates extracted from the adaptive QLV and the NSP models, respectively, pointed to a faster rate for stress relaxation than creep. This nonlinear viscoelastic behavior of individual collagen fibrils agrees with prior studies of macroscale collagenous tissues, thus demonstrating consistent time-dependent behavior across length scales and tissue hierarchies. STATEMENT OF SIGNIFICANCE: Pure stress relaxation and creep experiments were conducted for the first time with fully hydrated individual collagen fibrils. It is shown that collagen nanofibrils have a nonlinear time-dependent behavior which agrees with prior studies on macroscale collagenous tissues, thus demonstrating consistent time-dependent behavior across length scales and tissue hierarchies. This new insight into the non-linear viscoelastic behavior of the building blocks of mammalian collagenous tissues may serve as the foundation for improved macroscale tissue models that capture the mechanical behavior across length scales.

Generic creep behavior and creep modeling of an aged surface support liner under tension

Polymer-based materials have been motivated to be an alternative support system element in the mining/tunneling industry due to their logistic and geotechnical benefits. Thin spray-on liner (TSL), a term to define the application of the material on the rock surface with a layer ranging from 2 mm to 10 mm in thickness, shows some promising results. TSLs are mainly composed of plastic, polymer, or cement-based ingredients to a certain proportion. This study intends to reveal the time-dependent response of TSL specimens, cured throughout 500 d, under four constant stress levels for stable laboratory conditions. The results were correlated using two interrelated equations to predict the material's service life (creep-rupture envelopes). The proposed correlations offered an insight into both the effective permanent support time and the strain amount at the liner failure. The time-dependent deformation of TSL, whose performance is highly responsive to creep behavior, was obtained so that the design engineers may use the findings to avoid the severe problems of material creep. Experimental data were also used to develop a Burgers (four-element) creep model. Since the liner has a nonlinear time-dependent behavior, creep models were built for each stress level separately. Subsequently, a generic equation was obtained using the nonlinear parametric dependencies. There is a good agreement between the proposed model and the experimental results. The proposed model can be used as a basis for future numerical studies related to the support behavior of aged surface support liners.

An extended gradient-enhanced damage-plasticity model for concrete considering nonlinear creep and failure due to creep

For high stress levels, the dependence of the creep strain rate of concrete on the acting stress is nonlinear. Furthermore, very high stress levels may lead to a critical growth of microcracks resulting in material failure. For the lifetime assessment of creep sensitive concrete structures by means of numerical simulations, the realistic description of nonlinear creep is crucial. However, many concrete models are formulated for either nonlinear time-independent material behavior, or time-dependent material behavior, restricted to linear creep. This is the motivation for extending a damage-plasticity model, aiming at a unified and computationally efficient approach for representing the highly nonlinear time-dependent material behavior of concrete. Well established approaches for modeling the evolution of material properties, inelastic deformation, damage, creep and shrinkage serve as basis for considering nonlinear creep and material failure due to high sustained acting stress. The extended material model for concrete is validated by comprehensive experimental results from the literature.

A master curve approach to model short- and long-term time-dependent nonlinear behavior of polyethylene

This paper presents a new approach for modeling the constitutive behavior of time-dependent materials by applying two phenomenological constitutive models to creep curves. Data obtained from short-term test are used for predicting the long-term behavior. The nonlinear viscoelastic response is modeled by using the multi-Kelvin model and the viscoplastic behavior is described by a power law. The methodology proposed herein for obtaining the constitutive formulation is based on developing master curves. Results obtained for polyethylene are discussed and compared using creep tests at different stress levels and has proved to be reliable based on the results analyzed in this work.

Analysis of Lightning-Related Stress in Transmission Lines Considering Ionization and Frequency-Dependent Properties of the Soil in Grounding Systems

This article analyzes the performance of grounding systems buried in the ionized soil during the lightning discharge. To this aim, at first, a method to simultaneously analyze the transient response of grounding system considering the time-dependent nonlinear behavior arising from soil ionization and frequency dependence of soil electrical parameters related to the frequency content of the lightning surge is presented. The proposed method is performed by a combination of the harmonic balance method and the method of moment, which is aimed at representing an equivalent circuit to model the grounding system during the lightning discharge. The proposed model is validated by comparing the associated results with the published measurement and the numerical data for different grounding systems. Finally, an application of the proposed method is presented and the generated overvoltages across the insulator string caused by lightning strokes, the outage rates of transmission lines, and the lifetime of surge arresters are discussed.

Theoretical Investigation of Nonlinear Time-Dependent Behavior of Two-Way High-Strength Concrete Walls

High-strength concrete (HSC) walls have been increasingly used in the past decades. However, the time-dependent behavior of HSC wall panels in two-way action was not investigated, and the time effect of creep is not included in the design codes in most countries. For this purpose, the nonlinear long-term behavior of two-way HSC wall is investigated in this paper. A theoretical model is developed using time-stepping analysis considering geometric nonlinearity and creep of concrete. A rheological material model that is based on the generalized Maxwell chain is adopted to model the concrete creep. Von Karman plate theory is used to derive the incremental governing equations. The equations are solved numerically at each time step based on a Fourier series expansion of the deformations and loads and numerical multiple shooting method. It shows that the model can effectively predict the time-dependent behavior of two-way HSC panels, where the out-of-plane deflection and internal bending moments increase with time due to the combined effects of creep and geometric nonlinearity, which may ultimately lead to creep buckling failures. A parametric study shows that the long-term behavior of the panel is very sensitive to the in-plane load level and eccentricity, slenderness ratio, aspect ratio, and edge support conditions.

Dynamic viscoelastic behavior of bovine periodontal ligament in compression.

As the main connector of teeth to the surrounding bone, the periodontal ligament (PDL) plays an important role in absorption and distribution of physiological and para-physiological loading. Despite the difference in the mechanical response of the PDL in tension and compression, little information is available on its viscoelastic behavior in compression. To explore the contribution of the fluid phase of the PDL, the aim of the present study was to measure its nonlinear time-dependent behavior in compression. The in vitro dynamic compressive tests were carried out over a wide range of frequencies at three different preloads. Also, a generalized Maxwell model was proposed based on the experimental data to develop a viscoelastic model for subsequent computational analysis of the PDL. The higher values of loss factor in compression found in the range of 0.03-0.4 than those of 0.04-0.08 reported in tension, implies the focal role of the fluid phase in compressive dynamic response. Furthermore, the model parameters predicted that with an increase in preload, the role of the viscous components decrease, whereas the role of the elastic component increases. The viscous effect of the PDL in compression is greater than that in tension. Also, the dependence of the relaxation modulus of the bovine PDL on the applied load indicates its nonlinear viscoelastic behavior in compression.

Comprehensive Modeling of Grounding Electrodes Buried in Ionized Soil Based on MoM-HBM Approach

This paper presents a new approach for modelling grounding electrodes buried in two layers soils considering frequency-dependent modeling of soil parameters together with the time-dependent nonlinear behavior arising from soil ionization caused by a lightning current discharge through the grounding system. To this aim, in the absence of soil ionization, a Norton's equivalent circuit is extracted in the frequency domain by means of the method of moment. Then, by adding the nonlinear load representing soil ionization to the Norton's equivalent circuit, the grounding system is changed into a nonlinear Norton's equivalent circuit and treated by using the harmonic balance method. The proposed model is validated by comparing the associated results with published measurement and the numerical data obtained with low- and high-magnitude impulse currents. The results prove that the proposed method is reliable and effective. Finally, an application of the proposed method in predicting the lightning overvoltages across the insulator string is presented and the performance of grounding system for different parameters is discussed.

On the Growth and Saturation of the Gyroresonant Streaming Instabilities

Abstract The self-regulation of cosmic-ray (CR) transport in the interstellar and intracluster media has long been viewed through the lenses of linear and quasi-linear kinetic plasma physics. Such theories are believed to capture the essence of CR behavior in the presence of self-generated turbulence but cannot describe potentially critical details arising from the nonlinearities of the problem. We utilize the particle-in-cell numerical method to study the time-dependent nonlinear behavior of the gyroresonant streaming instabilities, self-consistently following the combined evolution of particle distributions and self-generated wave spectra in one-dimensional periodic simulations. We demonstrate that the early growth of instability conforms to the predictions from linear physics, but that the late-time behavior can vary depending on the properties of the initial CR distribution. We emphasize that the nonlinear stages of instability depend strongly on the initial anisotropy of CRs—highly anisotropic CR distributions do not efficiently reduce to Alfvénic drift velocities, owing to reduced production of left-handed resonant modes. We derive estimates for the wave amplitudes at saturation and the timescales for nonlinear relaxation of the CR distribution and then demonstrate the applicability of these estimates to our simulations. Bulk flows of the background plasma due to the presence of resonant waves are observed in our simulations, confirming the microphysical basis of CR-driven winds.

Influence of matric suction on nonlinear time-dependent compression behavior of a granular fill material

The site formation for the construction of a new airport in a mountainous region is typically performed by cutting and filling a hill section. The fill materials are subjected to seasonal changes and large variations in water content. The water content change renders the fill material to be characterized as unsaturated or saturated. This study aims to investigate the influence of matric suction on the time-dependent compression behavior of one local soil as a fill material for the construction of a new runway of an airport in Chongqing city, a mountainous region in China. A series of unsaturated drained triaxial tests were conducted on this coarse-grained soil to obtain the relationship between the effective stress parameter, χ, and the matric suction. Subsequently, multistaged compression tests were performed on this soil using a newly designed suction-controlled oedometer apparatus. The influence of suction on the time-dependent compression behavior of the fill material is emphasized. The results indicate that the matric suction can increase the compression stiffness, and that the unloading–reloading index varies nonlinearly with suction. A linear relationship between the time-dependent compression coefficient and normalized effective vertical loading is established. The linear relationship is subsequently used to predict the time-dependent compression coefficient to describe the time-dependent behavior of the fill under unsaturated conditions. Further, a nonlinear function based on the work of Yin (Geotechnique 49(5):699–707, 1999) is adopted to describe the development of time-dependent compression. The results indicate that the prediction obtained from the two newly proposed methods are promising, and can predict the nonlinear time-dependent compression behavior of this coarse-grained soil.

Experimental and modelling study on nonlinear time-dependent behaviour of thin spray-on liner

Thin spray-on liners (TSLs) are fast-setting multi component polymeric materials applied on rock or coal surface with a thickness of 2–5 mm that have fairly high tensile strength, adhesion, and elongation capabilities. Compared to conventional surface support elements, TSLs with polymer content exhibit different material responses over time. As a matter of fact, the creep behaviour of TSLs under a constant load has a significance for the evaluation of the long term performances of TSLs. This paper investigates the nonlinear creep behavior of a polymer based TSL for different curing times. Experimental studies were completed to show the nonlinear time-dependent response of the liner under various constant load and curing time conditions. Dogbone shape test specimens were sustained under constant loads up to 2 months of testing time for different test sets. New equations were proposed for rupture time prediction (creep rupture envelopes) which may aid support design engineers to estimate the effective permanent support time and maximum allowable strain amount of the TSL. Test results were used to construct nonlinear viscoplastic models and viscoelastic models consisting of one independent spring and multi Kelvin–Voight elements. A good agreement was obtained between model predictions and experimental values. Generated constitutive models might aid to further numerical studies in TSL literature.

Time-Domain Modeling of Grounding Systems’ Impulse Response Incorporating Nonlinear and Frequency-Dependent Aspects

At lightning discharge, the grounding systems’ transient behavior is influenced by time-dependent nonlinear behavior and frequency-dependent effects. These effects are related to complex phenomena arising from the soil ionization surrounding the conductors at high current and to the fast risetime of the impulse current, respectively. Most of the performed investigations were either focused on soil ionization effects or frequency-dependent effects. This paper proposes a model that incorporates both aspects including the mutual coupling and air–soil interface effects, giving a more realistic representation of the grounding systems behavior. Based on the transmission line approach, the model can compute the grounding systems transient response buried in uniform and nonuniform soil. The model is applied to horizontal, vertical electrodes, and grounding grid subjected to impulse currents of variable magnitudes and shapes. The model results are validated against published measurement results obtained with low-current and high-current impulses. The time-domain transient potential and transient impedances show good agreement with experimental results. Other impulse impedance parameters used for the grounding systems analysis are reproduced quite accurately. As an application, the impulse impedance of a horizontal electrode and a grounding grid buried in two-layer soil is discussed when the upper layer depth and the resistivity of soil layers are varied.

Importance of test parameters, specimen type and use configuration on the identification of Sn/Ag solder behaviour laws

In this paper, solder bump interconnections based on a Tin brazing process are studied under the angle of their capacity to resist to repeated thermomechanical loading. Indeed, understanding and modelling nonlinear behaviour is one of the most important milestones in the process of assessing solder reliability. However, the identification of soldering behavioural laws is highly dependent on the context and in particular on the nature of the specimen, the processes used to manufacture it as well as its utilization. This gave rise to a large number of models for the description of the creep in solder materials. Within this framework, the article aims first at proposing a state of the art on the most relevant models describing the nonlinear time dependent behaviour of this kind of solder alloy and, in particular, the different works that have addressed the modelling of the behaviour of Sn/Ag specimens. Due to the diversity of existing models, a methodology is then proposed to match analytical models and experimental data. It is shown that the results depend on the experimental setup and on the temperature used for the tests. Shear relaxation tests are used to identify primary and secondary creep. The results are discussed with respect to models and range of temperature. Based on the previous developments, an adaptation of the laws found in the literature is proposed to describe the behaviour of a Sn/Ag3.5 wire bond solder used in an IGBT rail traction module. The outcomes are experimentally validated.

Modeling short- and long-term time-dependent nonlinear behavior of polyethylene

ABSTRACTA new methodology for developing macromechanical constitutive formulations for time-dependent materials is presented in this article. In particular, two phenomenological constitutive models for polymer materials are illustrated, describing time-dependent and nonlinear mechanical behavior. In this new approach, short-term creep test data are used for modeling both short-term and long-term responses. The differential form of a model is used to simulate typical nonlinear viscoelastic polymeric behavior using a combination of springs and dashpots. Unified plasticity theory is then used to develop the second model, which is a nonlinear viscoplastic one. Least squares fitting is applied for the determination of material parameters for both models, based on experimental results. Due to practical constraints, experimental data are usually available for short-term time-frames. In the presented proposed formulation, the material parameters determined from short-term testing are used to obtain material parameter relationships for predicting the long-term material response. This is done by extending short-term information for longer time frames. Finally, theoretical and experimental results of tensile tests on polyethylene subjected to various load levels and test times are compared and discussed. Very good agreement of the modeling results with experimental data shows that the developed formulation provides a flexible and reliable framework for predicting load responses of polymers.

Bend stiffener nonlinear viscoelastic time domain formulation

Bend stiffeners are conical polyurethane structures used in the offshore industry to ensure a smooth transition in the upper connection of flexible risers with the floating production unit. The polyurethane employed for bend stiffeners present a nonlinear viscoelastic response that is highly dependent on the loading rate and temperature. This may lead to different flexible riser response when compared to elastic or hyperelastic material modeling. In order to quantify this effect, tensile and relaxation tests are carried out to characterize the nonlinear time dependent mechanical behavior of the polyurethane. A constitutive equation based on the modified superposition method is then presented and a procedure for material identification proposed. A numerical scheme using the state variable approach is formulated for the nonlinear viscoelastic constitutive equation finite element implementation. The top connection system, consisting of a bend stiffener and a flexible pipe segment, is represented by a large displacement beam model accounting for geometrical and material nonlinearities. A case study is carried out to compare the system curvature response calculated for the nonlinear viscoelastic and hyperelastic (Marlow strain energy potential) bend stiffener subjected to harmonic loading conditions at different frequencies. The results show the importance of bend stiffener strain rate dependency on the riser curvature response.

Creep-recovery-creep tests to determine the yield stress of fluid gels containing gellan gum and Na+

In this work, combined creep-recovery-creep tests were designed to determine the yield stress and the inception of non-linear time-dependent rheological behaviour for fluid gels containing 0.2wt% gellan gum and 0.22M Na+ as promoting cation of the gel formation. Experiments were performed from 3Pa up to 6Pa and samples were allowed to relax for the required equilibration time to avoid the effects of shear history. The response of these tests at the smaller stresses studied (below critical shear stress of 3.6Pa) was controlled by non-linear viscoelasticity. Subsequently, a slight increase in shear stress above the critical shear stress by just 0.1Pa resulted in a dramatic change in the creep flow properties. This made it possible to determine a practical yield stress and the onset of thixotropic properties. In addition, it was possible to determine the shear rate dependence of viscosity from the creep-recovery-creep tests, which illustrated the very shear thinning behaviour of these fluid gels.

Nonlinear time dependent behaviour of epoxy resins

The nonlinear behaviour of epoxy resins is studied on standard tensile tests. A strain field measurement system is applied (Aramis) in order to monitor local strains. The residual strain is measured by recovering the specimens for up to 68 hours after unloading. The time span the specimen is exposed to load has a large influence on the creeping process and the residual strain after recovering. This is studied by comparison of instantaneous unloading with keeping the specimen under permanent load for thirty minutes. It is shown that moderate differences in the initial strain can lead to large differences in the creep behaviour as well as in the residual strain.