Nonlinear Stress-strain Curves Research Articles

Nonlinear stress-strain curves for solids containing closed cracks with friction

Solutions for the uniaxial stress-strain response of a body containing a distribution of non-interacting nonlinear cracks are derived. First, building on energy formalisms outlined by previous workers, general solutions are derived for the body containing cracks with dissipative tractions at their surfaces, in either tension or compression loading. The special case of a body in compression loading with sliding closed cracks governed by a general friction law is then considered as a case study. The friction law contains two shear resistance terms: a “friction coefficient” term proportional to the resolved normal compression stress across the crack plane; and a “cohesion” term representing the intrinsic shear resistance of the closed crack. Inclusion of the latter term is critical to the existence of a well-defined yield point in the stress-strain curve. It is assumed that the cracks do not extend at their ends during the loading-unloading-reloading cycle; they are, however, allowed to undergo reverse sliding during the unloading. Two crack distributions are considered: all cracks aligned, leading to linear expressions for both the elastic and quasi-plastic stressstrain regions; and cracks randomly oriented, with more complex (but nonetheless tractable) expressions for the quasi-plastic regions. The resultant nonlinear stress-strain curves exhibit cyclic hysteresis, to an extent dependent on friction and crack configuration parameters. Illustrative stress-strain curves are generated for selected ranges of these controlling parameters. An outcome of the analysis is the potential link to microstructural variables, via the crack configuration parameter, offering the prospect for predictions of damage accumulation in real microstructures. The model also offers the prospect of accounting for fatigue properties, via attrition of the frictional resistance at the sliding crack surfaces.

A constitutive equation for nonlinear stress–strain curves of crystalline polymers

We present a new method for the analysis of stress–strain curves for crystalline polymers such as polypropylene and polyethylene. A nonlinear constitutive equation that includes terms that cover the plastic deformation and anharmonicity of the spring is developed. In order to quantitatively characterize the nonlinear viscoelasticity using this equation, data on the transient moduli during elongation at a constant rate of strain are required. Hence, the simultaneous measurements of linear oscillatory viscoelastic moduli during a constant rate of elongation were investigated. It was found that the present method makes possible the evaluation of the plastic deformation fraction and the Gruneisen constant for crystalline polymers. © 1998 Chapman & Hall

Strength Characteristics of Giant Magnetostrictive Materials Fabricated by Power Metallurgy.

The strength and mechanical properties of giant magnetostrictive materials fabricated by powder metallurgy and Bridgman method were investigated. The effect of the shape and volume of test pieces on compressive strength and compressive fatigue properties were examined for the powder metallurgy specimens. Test results showed : (1) The compressive strength of the powder metallurgy material specimens (PM) was about 2/3 of the Bridgman material specimens (BM), while the strength scatter of PM was smaller than BM. (2) The radial compressive strength was about 1/20 of the compressive strength, and both kinds of specimens showed brittle fracture features. (3) Specimens of both materials showed nonlinear stress-strain curves, and the modulus of elasticity rapidly increased with increasing stress in the low stress region. (4) The compressive strength of PM was influenced by the shape of specimens : the compressive strength of prism type specimens was lower than that of rod type specimens. However, the strength was not influenced by the volume of specimens. (5) There was an apparent fatigue limit at about 60% of the compressive strength for PM.

Influence of elastic nonlinearity on arterial anastomotic compliance.

This study illustrates how the highly nonlinear elastic behavior of artery wall material can cause unusual structural characteristics that do not occur with a linear-elastic material. An example mathematical model of an end-to-end anastomosis successfully predicts the experimentally observed area of elevated elastic compliance, called the "Para-anastomotic Hypercompliant Zone" (PHZ). The elastic hypercompliance is shown to occur because the anastomosis locally restricts the arterial diameter, thus forcing the adjacent material to remain in a lower strain, and correspondingly a lower stiffness, part of its non-linear stress-strain curve. Elevated elastic compliance can be avoided by locally matching both the arterial diameter and the elastic compliance within the physiological pressure range.

Nonlinear Pile Foundation Analysis Using Florida-Pier

Florida-Pier is a nonlinear finite-element program designed for analyzing bridge pier structures composed of pier columns and cap supported on a pile cap and piles with nonlinear soil. The program was developed in conjunction with the Florida Department of Transportation (FDOT) structure division. In a recent addition, the piles were converted to include nonlinear behavior through the addition of a nonlinear discrete element. This element uses the true material stress-strain curves (steel and concrete) to develop its behavior and stiffness modeling. A description of the discrete-element model and a method to form its secant stiffness for any deformed position using material nonlinear stress strain curves is presented. A summary of the modeling capabilities of the Florida-Pier program are also given. Two examples showing the newly added nonlinear pile element are given. The first example shows the response of a linear pile foundation as compared to a nonlinear pile foundation. The second example show the capabilities of the program to model group pile behavior. The results for both examples are compared to COM624P (Wang and Reese 1993) the industry standard for pile analysis

Effect of Porosity on Dynamic Mechanical Properties of Rubber in Compression

Abstract Cylindrical samples of different levels of porosity were prepared from a model carbon black filled rubber and tested in compression. Dynamic properties were determined with dynamic loads and dynamic displacements superimposed on static loads. Nonlinear stress-strain curves of porous rubber blocks produced a dependence of dynamic properties on the mode of testing. Both storage and loss moduli as well as tan γ were found to depend on the levels of porosity. The loss modulus was found to increase with higher strain levels in porous samples, while the storage modulus changed because of reduced sample stiffness. The dynamic data was in agreement with the results of Goodrich flexometer tests on porous and solid samples. The results suggest that the introduction of porosity reduces service life of rubber parts by providing stress raisers for crack initiation, rather than through changing the dynamic properties.

Tensile Behavior of Microcracking SiC‐TiB2 Composites

The stress–strain behavior of microcracking SiC‐TiB2 composites was studied using uniaxial tensile tests. The nonlinear stress–strain behavior provided the critical stresses for microcracking (defined as the proportional limit) and the magnitude of residual strain release as a consequence of stress‐induced microcracking. The release of residual strain was confirmed using the loading–unloading process. Both the critical stress for the nonlinear stress–strain curve and the release of residual strain were used to establish the maximum principal stress as the stress criterion for microcracking.

Ferroelectric/ferroelastic interactions and a polarization switching model

Ferroelectric and ferroelastic switching cause ferroelectric ceramics to depolarize and deform when subjected to excessive electric field or stress. Switching is the source of the classic butterfly shaped strain vs electric field curves and the corresponding electric displacement vs electric field loops [1]. It is also the source of a stress—strain curve with linear elastic behavior at low stress, non-linear switching strain at intermediate stress, and linear elastic behavior at high stress [2, 3]. In this work, ceramic lead lanthanum zirconate titanate (PLZT) is polarized by loading with a strong electric field. The resulting strain and polarization hysteresis loops are recorded. The polarized sample is then loaded with compressive stress parallel to the polarization and the stress vs strain curve is recorded. The experimental results are modeled with a computer simulation of the ceramic microstructure. The polarization and strain for an individual grain are predicted from the imposed electric field and stress through a Preisach hysteresis model. The response of the bulk ceramic to applied loads is predicted by averaging the response of individual grains that are considered to be statistically random in orientation. The observed strain and electric displacement hysteresis loops and the nonlinear stress—strain curve for the polycrystalline ceramic are reproduced by the simulation.

Lifetime and young's modulus changes of glass/phenolic and glass/polyester composites under fatigue

Fatigue of composites causes a loss of stiffness and development of damage before ultimate failure. These effects were compared for phenolic and polyester resins reinforced by five layers of woven roving/chopped strand combination glass mat. Such polyester laminates are widely used, e.g. in marine applications such as high speed craft. The phenolic laminates are promising candidates for applications where fire resistance is important. S-N and ε-N curves were obtained. In addition, stress versus strain curves were measured continuously under reverse loading ( R = -1) up to 10 6 cycles. Analysis of these curves allows further insight into the fatigue process. Tensile moduli drop dramatically (25%) with increasing number of cycles, but remain fairly constant in the compressive region. This behaviour can be explained by effects of matrix cracking. It also indicates the need to define different moduli for non-linear stress-strain curves. Absorbed energy (damping) can also be determined from the stress-strain curve and can be used as an indicator of the onset of rapid degradation of the material leading to ultimate failure. All data are compared for polyester and phenolic matrix laminates.

Observations of inelastic deformation during high-temperature fatigue and failure of a polycrystalline silicon nitride

Abstract The effects of rate-dependent inelastic deformation were observed during tensile static, dynamic, and cyclic fatigue testing of a hot-isotatically pressed, monolithic, polycrystalline silicon nitride in ambient air at temperatures of 1150, 1260 and 1370°C. Constant stresses in static fatigue ranged from 50 to 300 MPa. Stress rates in dynamic fatigue ranged from 10−4 to 101 MPa/s. Waveforms in cyclic fatigue included trapezoid, triangle and sine at a stress ratio, R, of 0.1, frequencies of 0.1 and 10 Hz, and maximum stresses ranging from 75 to 325 MPa. At 1150°C, all fatigue results showed a similar slow crack growth failure mechanism with no separate cyclic fatigue mechanism. However, at 1260 and 1370°C the failure mechanism was multi-faceted. For both temperatures, the failure in static fatigue was dominated by the accumulation of diffusion-controlled creep cavities. In dynamic fatigue, the inelastic deformation exhibited by non-linear stress-strain curves supported the conclusion of failure by slow crack growth alone at high stress rates and by a mixture of slow crack growth and creep damage at low stress rates. Under cyclic loading, ‘enhanced’ times to failure, attributed to viscous rate effects, occurred regardless of waveform or frequency. The stress rate was related to the stress at the onset to non-linearity, σ0, indicating a quasi-endurance limit below which cyclic fatigue had little ‘enhancing’ effect. Creep compliance relations indicated non-linear viscoelastic (or viscoplastic) behavior, leading to speculation on the role of this behavior in producing the ‘enhanced’ cyclic fatigue resistance.

A predictive model for the mechanical behavior of particulate composites. Part I: Model derivation

AbstractA method for predicting the highly nonlinear stress‐strain behavior and dilatation induced by cavitation of highly filled particulate composites from constituent properties has been developed. The approach presented uses a variation of linear elasticity throughout and has no adjustable parameters, unlike the methods currently used, which require large numbers of fitting factors and complicated nonlinear analyses. An energy balance derived from the first law of thermodynamics calculates critical strain values at which filler particles will debond when subjected to deformation. Repeated calculations of critical strain values using re‐evaluated material properties accounting for the damage caused by debonding give very nonlinear stress‐strain and dilatation curves. Experimentally observed dependencies on particle size, filler concentration, adhesion, and matrix and filler properties are correctly predicted. The method can be generalized for any state of stress or particle shape. Comparisons of experimental data with the model results give good agreement.

Tensile Tests of a Fluid-Filled Poroelastic Core Sample : Theoretical Consideration

The literature shows that nonlinear stress-strain curves have been obtained in tensile tests of an epoxy-coated hourglass-shaped rock specimen under confining pressures and concludes that this nonlinearity is caused in part by plasticity and mainly by nonlinear elastic behavior. However, in addition to these causes, some contribution of pore-fluid diffusion in a specimen is expected to occur. The hourglass-shaped specimen was examined in a previous paper. For a noncoated cylindrical specimen (core sample), the present paper analyzes the contribution of radial pore-fluid diffusion to nonlinearity and yields the same conclusion as that in the previous paper; that is, this apparent nonlinearity is small but not negligible. A formula is also given to estimate the bend of the stress-strain curve due to diffusion. Furthermore, stress-strain hysteresis curves are given for the same purpose.

Non-linear compression stress-strain curve of pitch-based high modulus carbon fibre composites and structural responses

The longitudinal compression behaviour of unidirectional composites is studied to understand the role of the fibre compressive property in deformation and failure by systematically varying the tensile modulus of reinforcing high modulus carbon fibre. As the composites deform, their softness increases with increasing compressive strain, and the loading path is traced back when the load is removed. The intensity of softening is correlated to the fibre's tensile modulus and possible softening mechanisms are discussed in conjunction with fibre and matrix properties. Further, it is investigated how the non-linear stress-strain relation affects the stress and strain distributions and deformation when plates fabricated from these fibres are tested by the three-point bending test.

Tensile Test of Fluid-Filled Poroelastic Specimen. Theoretical Consideration-Supplement.

The literature shows that nonlinear stress-strain curves have been obtained in tensile tests of an epoxy-coated hourglass-shaped rock specimen under confining pressures and concludes that this nonlinearity is caused in part by plasticity and mainly by nonlinear elastic behavior. However, in addition to these causes, some contribution of the pore-fluid diffusion is expected. The abovementioned specimen was examined in the previous paper. The present analysis for a noncoated cylindrical specimen (core sample) shows the contribution of the radial pore-fluid diffusion to the nonlinearity and gives the same conclusion as that in the previous paper ; that is, this apparent nonlinearity is small but not negligible. A formula is given to estimate the bend of the stress-strain curve due to the diffusion. Furthermore, sress-strain hysteresis curves are given for the same purpose.

Yield stress and stress-strain curve of fiber reinforced matrix with plasticity

A model for local plastic yielding is proposed to predict the yield stress and stress-strain curves of the short fiber reinforced matrix with plasticity. A fiber named P-fiber is defined as a short fiber enclosed by the locally yielded area. This P-fiber is embedded into the matrix, producing the stiffness and macroscopic non-linear stress-strain curve. A modified Eshelby's equivalent inclusion method is used for two processes. One is the process for the stiffness of an anisotropic elastic P-fiber and the other is for the macroscopic behaviors of short fiber reinforced composite materials. The present model is applied to composites with both low and high Young's modulus ratios such as fiber reinforced metals (FRMs) and plastics (FRPs).

Mechanical Characterization of SCS-6/Ti-6-4 Metal Matrix Composite

Off-axis tension tests were performed on SCS-6/Ti-6-4 metal matrix com posite. A one-parameter plasticity model was used to characterize the elastic-plastic prop erties. In addition, a micromechanical model was developed assuming elastic fiber and elastic-plastic matrix properties. This model was employed to relate the apparent yielding with the fiber/matrix separation in the MMC. From the micromechanical model, the fiber/matrix interfacial bond strength was estimated, and with the aid of a damage model, the nonlinear off-axis stress-strain curves were accurately predicted.

A predictive model for the effects of thermal history on the mechanical behavior of amorphous polymers

AbstractThe effects of thermal history during formation on the mechanical behavior of amorphous polymer solids must be part of a constitutive description for polymers in order to enable efficient design of manufacturing processes. A nonlinear, thermoviscoelastic constitutive equation for amorphous polymers has been previously derived via the rational thermodynamics framework, where the dissipation is assumed to occur on a material timescale that is controlled by the non‐equilibrium entropy, In this paper we report on the predictions of this constitutive equation for poly(vlnyl acetate). The material properties required by the constitutive model were determined for poly(vinyl acetate) from PVT and linear viscoelastic shear and bulk moduli measurements. Predictions of specific volumes during isobarfc cooling from the rubber and nonlinear uniaxial stress‐strain curves just below Tg are in reasonable agreement with experimental data. The model predicts that samples cooled below Tg and then isothermally annealed for specified times will exhibit yield stresses that increase with Increasing annealing time, as well as the familiar effects of changes in temperature and strain rate on the uniaxial stress‐strain and yield behavior. Shortcomings and potential improvements in the model are discussed.

Tensile test of fluid-filled poroelastic specimen. Theoretical consideration.

The literature shows that nonlinear stress-strain curves have been obtained in tensile tests of epoxy-coated hourglass-shaped rock specimen under confining pressures and concludes that this nonlinearity is caused in part by plasticity and mainly by nonlinear elastic behavior. However, in addition to these causes, some contribution of the pore-fluid diffusion in specimen is expected. The present theoretical analysis based on the linear theory of poroelasticity shows how much the pore-fluid diffusion contributes to the nonlinearity and concludes that this apparent non-linearity is small but not negligible. A formula is given to estimate how much a stress-strain curve bends due to the diffusion.

Failure of Carbon/Epoxy Lamina Under Combined Stress

It is well known that matrix failure in carbon/epoxy composites is influenced by multiaxial states of stress. However most of the experimental work to measure this inter action has not focused exclusively on matrix failure, and a general multiaxial stress criterion for matrix failure has not been established. To examine this question experimen tally we have carried out a series of tests involving torsional shear combined with axial tension or compression of unidirectional hoop wound cylinders, using AS4/55A carbon/ epoxy lamina. The matrix failure stresses are seen to be well correlated with the Tsai-Wu quadratic polynomial. There is also a strong interaction between the strains at failure, with small amounts of transverse tension giving a significant reduction in the shear strain at fail ure. The effect of σ2 on the nonlinear shear stress-strain curves is also presented.

Tensile Behavior of Glass/Ceramic Composite Materials at Elevated Temperatures

This paper describes the tensile behavior of high-temperature composite materials containing continuous Nicalon ceramic fiber reinforcement and glass and glass/ceramic matrices. The longitudinal properties of these materials can approach theoretical expectations for brittle matrix composites, failing at a strength and ultimate strain level consistent with those of the fibers. The brittle, high-modulus matrices result in a nonlinear stress-strain curve due to the onset of stable matrix cracking at 10 to 30 percent of the fiber strain to failure, and at strains below this range in off-axis plies. Current fibers and matrices can provide attractive properties well above 1000°C, but composites experience embrittlement in oxidizing atmospheres at 800 to 1000°C due to oxidation of a carbon interface reaction layer. The oxidation effect greatly increases the interface bond strength, causing composite embrittlement.