Linear Stress-strain Relationship Research Articles

A finite-element model relating regional myocardial contraction to left ventricular systolic function

Background: A finite-element model of the left ventricle (LV) is used to produce LV pressure and volume curves over the entire cardiac cycle under normal and ischemic conditions. The regional myocardial contractions defined by finite elements are then related to the time-varying LV elastance (instantaneous pressure over volume ratio). This model provides a computationally efficient way to assess how localized ischemic zones affect global LV systolic function. Method: A 3D model of the LV with an axisymmetric truncated-ellipsoidal geometry is represented by a 2D grid of long-axis cross-section. The initial end-diastolic grid is generated from a set of elliptic curves with user-defined parameters such as wall thickness, long and short axis lengths. The strain-stress relationship for each element is assumed to be linear but with a time-varying Young's modulus. Empirical curves of the Young's modulus for the finite elements are defined by the user to simulate normal and ischemic myocardium. To configure an ischemic zone of certain size and geometry, the contractile state of each finite element over a cardiac cycle can be individually assigned. Result: Simulations of normal and ischemic LV's were performed. Figures below show a comparison between control and the presence of an ischemic zone. The ejection fraction (EF) was reduced from 64% of control (A) to 43% of ischemia (B). The stroke volume (SV) was reduced from 70 ml to 50 ml. The LV pressure was set the same in both cases to show how the ischemic zone affects the LV volume curve over a cardiac cycle (C). The LV elastance curves were compared to the time-varying Young's modulus curves assigned to normal and ischemic finite elements (D). Conclusion: The model provides a computational method that relates regional myocardial impairment to LV systolic function. The drastic difference in wave shape between individual Young's modulus and global LV elastance suggests a built-in nonlinearity in the geometry of the finite-element model, even under the assumption of linear stress-strain relationships. The model should be useful for studying the hemodynamic consequences of different severity, size and location of regional myocardial impairments.

Evidence of power-law flow in the Mojave desert mantle.

Studies of the Earth's response to large earthquakes can be viewed as large rock deformation experiments in which sudden stress changes induce viscous flow in the lower crust and upper mantle that lead to observable postseismic surface deformation. Laboratory experiments suggest that viscous flow of deforming hot lithospheric rocks is characterized by a power law in which strain rate is proportional to stress raised to a power, n (refs 2, 3). Most geodynamic models of flow in the lower crust and upper mantle, however, resort to newtonian (linear) stress-strain rate relations. Here we show that a power-law model of viscous flow in the mantle with n = 3.5 successfully explains the spatial and temporal evolution of transient surface deformation following the 1992 Landers and 1999 Hector Mine earthquakes in southern California. A power-law rheology implies that viscosity varies spatially with stress causing localization of strain, and varies temporally as stress evolves, rendering newtonian models untenable. Our findings are consistent with laboratory-derived flow law parameters for hot and wet olivine--the most abundant mineral in the upper mantle--and support the contention that, at least beneath the Mojave desert, the upper mantle is weaker than the lower crust.

Early Age Deformation and Resultant Induced Stress in Expansive High Strength Concrete

Expansive additive is well known to be effective in compensating early-age shrinkage and the resultant induced stress in reinforced high-strength concrete (HSC) members. On the other hand, there have been few studies on numerical analysis methods for evaluating such early-age induced stress, which are vital to verify the risk of cracking. The present study formulates a 3-dimensional finite element method as well as a practical calculation method based on the beam theory, both of which consider the principle of superposition and linear stress-strain relationship of creep, in order to evaluate the early-age shrinkage/expansion-induced stress in reinforced members. The applicability of the proposed methods is evaluated by comparing computed values with experimental values on shrinkage/expansion-induced stress in RC beam specimens composed of various HSCs, using expansive additive and/or shrinkage reducing chemical agent and/or low-heat Portland cement. The results demonstrate that the proposed finite element method can accurately simulate induced stress in the reinforced concrete beams, even when expansive additive is used, and this indicates that the linear stress-strain relationship may be valid for expansive high strength concrete. Furthermore, there is a good agreement between the finite element method and a practical calculation method based on the beam theory, even in the case of RC beams with stirrups that cause a three-dimensional restraint condition in concrete.

Closed form analysis of wrinkled membranes with linear stress-strain relation

This paper considers the wrinkling phenomenon of membranes for the special case of a linear relationship between membrane forces and strains. Beginning from basic equations of regular wrinkling, the problem is treated analytically. An application of the theory to simple examples will be given.

Effect of atropine on the biomechanical properties of the oesophageal wall in humans.

Recently, we reported a novel ultrasound technique to assess biomechanical properties of the oesophagus in human subjects. In the present study, we use the technique, in combination with atropine, to determine the active and passive biomechanical properties of the oesophagus in normal healthy humans. A manometric catheter equipped with a high-compliance bag and a high-frequency intraluminal ultrasonography probe was used to record pressure and oesophageal geometry. Oesophageal distensions with either isovolumic (5-20 ml water) or with isobaric (10-60 mmHg) technique were performed. Intra-bag pressure and ultrasound images of the oesophagus were recorded simultaneously. Following injection of atropine (15 microg kg-1, I.V.), the oesophageal distensions were repeated. The oesophageal wall compliance, circumferential wall tension, stress, strain and elastic modulus were calculated. Atropine resulted in an increase in the oesophageal wall compliance during isobaric distension, but no change in compliance was observed during isovolumic distension. The stress-strain relationship was found to be linear during both types of distension, before as well as after atropine. The Young's modulus, which is the slope of a linear stress-strain relationship, was significantly higher after atropine in the isovolumic study but not in the isobaric study. The stress-strain relationship of the active component (muscle contraction) was different during isovolumic and isobaric distensions but the passive components were similar. The passive and active stress-strain relationships of the human oesophagus resemble those of other soft biological tissues. Furthermore, the method of oesophageal distension has significant influence on the active but not the passive biomechanical properties due to a strain-rate effect.

High temperature deformation behavior of the Zr 41.2Ti 13.8Cu 12.5Ni 10Be 22.5 bulk metallic glass

The high temperature deformation behavior of a Zr 41.2Ti 13.8Cu 12.5Ni 10Be 22.5 bulk amorphous alloy has been studied in the temperature range between 351 and 461 °C under compressive as well as tensile loading. Three types of nominal stress–strain curves under compressive loading have been identified depending on the strain rate and test temperature, viz., a linear stress–strain relationship with fracture at the maximum stress, large plastic deformation after stress overshoot, and steady state plastic flow without stress overshoot. The DSC analysis for the compressed specimens was also carried out to determine the change in the crystallization fraction under the various test conditions. Under tensile loading, superplastic-like deformation with a maximum elongation over 500% was observed at 431 °C under a relatively high strain rate of 2×10 −2/s. The tensile test specimens were, however, observed to exhibit a brittle fracture mode at temperatures below the glass transition temperature and above the crystallization temperature under the same initial strain rate. The superplastic-like properties characterized by the low flow stress and large elongation then appear to be very sensitive to test temperature and initial strain rate.

Numerical modelling of the Storfjorden (Svalbard) polynya development due to wind stress: role of the sea ice rheology and damping forces

Remote sensing of the ice cover in Storfjorden (Svalbard) revealed the persistence and evolution of latent heat polynyas during the winter of 1997/98. Latent heat polynyas open mechanically under wind stress or ocean currents that transport the ice cover away. In the present work we used mathematical modelling to simulate the Storfjorden polynya size and geometry caused by wind stress, measured at the meteorological station on the island of Hopen in winter 1997/98. The dependence of the polynya outlines on the wind velocity is presented. Two approaches were used: quasi-static and dynamic. Quasi-static simulations are based on a time-independent, linear ice stress-strain relationship valid for the low strain rates only. Time dependence of the ice cover fracture is joined with stress-strain nonlinearity caused by ice delayed-elastic recovery and viscosity. Results are compared to satellite observations from the synthetic aperture radar (SAR) of ERS-2. The simulation results show that a northern wind opens a larger polynya (ca. 30%) than does a north-eastern wind with the same speed. The results also indicate that the bathymetry and geometry of the fjord might have a stronger influence on the polynya opening and development than the location of individual islands and reefs.

Flow state in molecular-dynamics-simulated deformed amorphousNi0.5Zr0.5

Results are reported from molecular dynamics simulations about the deformation behavior of amorphous metallic ${\mathrm{Ni}}_{0.5}{\mathrm{Zr}}_{0.5}$ alloys near their glass transition temperature. Constant shear strain rate and constant shear stress deformations are analyzed. Stress-strain curves from constant shear strain rate deformations reveal different stages of deformation, first a regime with approximately linear stress-strain relation, then a transient regime with pronounced strain softening, and finally, for large strains, a nearly strain-independent flow stress region. The strong stress overshoot in the transient regime is ascribed to a delayed transition under steady-state deformations from the initial glassy state into a structurally and dynamically different one. This flow state shows a decreased viscosity, an increased potential energy per atom, and a loss of short-range order, the latter being deduced from the radial distribution functions. In the flow state, the self-part of the intermediate scattering function reveals an unchanged vibration spectrum of the system but a faster decay of density correlations compared to the starting structure. Creep curves from constant shear stress deformations show a nearly linearly increasing strain for moderate shear stresses and moderate deformations. The strain rate depends in a non-Newtonian manner on stress with an activation volume of $7--8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}29}{\mathrm{m}}^{3}.$ Deformation simulation with large constant shear stress leads to an instability, identified as the transition into the flow state.

Harmonic analysis in rheological property measurement

Sudden change in the linear stress-strain relationship has long been known to produce interesting discontinuities in the flow properties of polymer melts, concentrated particle suspensions, and gels as the shear rate amplitude is increased. The non-linear effect produces stress harmonics, which contain information about the material properties that is otherwise difficult to obtain. This note describes a generalized procedure for extracting yield stress material functions from the discrete Fourier transform of the stress signal.

On modelling phase propagation in SMAs by a Maxwellian thermo-viscoelastic approach

In this paper we propose a Maxwellian thermo-viscoelastic approach to the problem of phase transformation in shape memory alloys. An explicit temperature-dependent non-monotone piecewise linear stress–strain relation is considered and the corresponding free energy function is used to establish the heat propagation equation. The heat exchange between the material and its environment is also taken into account. The numerical simulations of three end-displacement rates lead to serrated hysteresis loops and result in inhomogeneous deformations and exothermic/endothermic behavior during loading/unloading tests. It is shown that the influence of the strain rate on the size and shape of the hysteresis loop is due in fact to the heat exchange between the bar and its environment. The predictions are compared qualitatively with experimental results.

Hencky's elasticity model and linear stress-strain relations in isotropic finite hyperelasticity

Hencky's elasticity model is an isotropic finite elasticity model assuming a linear relation between the Kirchhoff stress tensor and the Hencky or logarithmic strain tensor. It is a direct generalization of the classical Hooke's law for isotropic infinitesimal elasticity by replacing the Cauchy stress tensor and the infinitesmal strain tensor with the foregoing stress and strain tensors. A simple, straightforward proof is presented to show that Hencky's elasticity model is exactly a hyperelasticity model, derivable from a quadratic potential function of the Hencky strain tensor. Generally, Hill's isotropic linear hyperelastic relation between any given Doyle-Ericksen or Seth-Hill strain tensor and its work-conjugate stress tensor is studied. A straightforward, explicit expression of this general relation is derived in terms of the Kirchhoff stress and left Cauchy-Green strain tensors. Certain remarkable properties of Hencky's model are indicated from both theorectical and experimental points of view.

An efficient beam–column formulation for 3D reinforced concrete frames

This paper presents a new beam–column formulation which can be used for the accurate, yet efficient, modelling of 3D reinforced concrete (R/C) frames. The formulation is intended for modelling the nonlinear elastic behaviour of a whole R/C beam–column with only one element, which is an essential ingredient of adaptive elasto-plastic analysis. On the longitudinal axis level, quartic shape functions are used to represent the two transverse displacements. A constant axial force criterion is employed instead of shape functions for the axial displacement, which is largely responsible for the accuracy of the proposed formulation. For concrete, the formulation assumes a nonlinear compressive stress–strain relationship and no tensile resistance; whereas for steel, a linear stress–strain relationship is utilised. On the cross-sectional level, the formulation is capable of modelling the interaction between the axial force and the biaxial moments for a general R/C cross-section, with explicit expressions obtained using a novel approach based on integration over triangular subdomains. The paper provides the details of the proposed formulation, and presents several verification examples to demonstrate the accuracy of this formulation and its ability to model the nonlinear elastic response of reinforced concrete beam–columns with only one element per member.

Poromechanics: from Linear to Nonlinear Poroelasticity and Poroviscoelasticity

Due to the impact on productivity and oil in place estimates, reliable modeling of rock behavior is essential in reservoir engineering. This paper examines several aspects of rock poroelastic behavior within the framework of Biot's mechanics of fluid saturated porous solids. Constitutive laws of linear and nonlinear poroelasticity are first determined from a fundamental stress decomposition, which allows to clearly connect linear and nonlinear models. Concept of effective stress and rock compressibility are considered. Linear incremental stress-strain relations are derived from the proposed nonlinear constitutive law by defining tangent elastic properties. These characteristics are naturally functions of strains and pore pressure, but explicit expressions as functions of stresses and pore pressure are established herein. Experiments performed on a reservoir sandstone illustrate these points. A constitutive law of poroviscoelasticity is finally presented and applied to experimental data obtained on clay.

Linear stress-strain relations in nonlinear elasticity

Nonlinear strain measures which are compatible with the existence of an elastic potential and lead to a linear stress-strain relation are obtained. An unusual property of these measures is their dependence on material parameters. Standard strain measures commonly used in nonlinear elasticity are shown to be consistent with linear stress-strain relations only for particular cases of Poisson's ratio. Corresponding potentials for these cases are presented.

Transient Thermal Stresses in a Superelastic Shape Memory Alloy Strip.

The austenitic Shape Memory Alloy (SMA) can be loaded up to a large strain without any plastic deformation and behaves superelastically (pseudelastically) above its characteristic transition temperature. When loaded, the superelastic SMA undergoes a stress-induced martensitic transformation that will result in a large recoverable strain up to 8%. On unloading, the superelastic SMA experiences a large hysteresis loop that makes the alloy an useful candidate for strain energy absorption. By taking advantage of this effect, SMA is used in various fields. The SMA can be used to improve the impact damage tolerance of composite materials and to induce the relaxation of stress distribution. This investigation deals with the transient thermoelastic problem for a superelastic SMA strip under the heat exchange. In order to simplify the SMA model, a linearization of one dimensional constitutive equations is made. The piecewise linear stress-strain relation is separated into two parts : before and after the stress-induced martensitic phase transformation. Another assumption is the identical SMA stress-strain relation in tension and in compression. Results for the temperature and stresses distributions are obtained numerically. We discuss about the variation of the martensitic phase and the thermal stress relaxation.

Viscoelastic properties of the human temporomandibular joint disc in patients with internal derangement

Purpose: This study investigated the viscoelastic properties of human temporomandibular joint (TMJ) discs in patients with severe internal derangement (ID). Patients and Methods: TMJ discs obtained from 5 patients with severe TMJ internal derangement were analyzed. Normal discs derived from 2 fresh cadavers and 4 patients without ID served as the controls. The viscoelastic responses of the discs to tensile forces were evaluated by means of stress-strain analyses. Results: The discs in both groups exhibited a nonlinear stress-strain relationship that was represented by a power function of the strain. However, after stress relaxation, the ID discs were likely to exhibit a linear stress-strain relationship. The instantaneous elastic moduli were almost the same in both discs, but the relaxed elastic modulus of the ID discs was significantly greater than that of the controls in lower strain range of less than 2%. Conclusions: ID discs are more rigid than normal discs. These findings suggest that the changes in viscoelastic property of the discs in patients with ID are somewhat different from those that occur with aging. © 2000 American Association of Oral and Maxillofacial Surgeons

Membranes of revolution under partially vanishing normal load

This paper investigates the existence and uniqueness of solutions of some boundary value problems modeling the deformation of a membrane of revolution under a partially vanishing normal load. The frame of the membrane models we deal with is the Reissner theory of thin shells of revolution, in which strain-displacement relations are nonlinear, although it assumes a linear stress-strain relation.

Nonlinear stress-strain relationships in tissue and their effect on the contrast-to-noise ratio in elastograms

The practice of elastography is generally limited to small applied compressions (typically 1%), under the assumption of a linear stress-strain relationship in biological tissue. However, the recent reports of larger applied compressions and precompression levels to increase the strain contrast violate the above assumption. The nonlinear stress-strain relationships in different breast tissue types significantly alter the contrast in elastography, especially for large applied compression. The moduli of normal fibrous and glandular breast tissue (along with cancerous lesions) are strain-dependent, with tissue stiffness increasing with applied compression. In this paper, we illustrate that the strain-dependence of the modulus has a significant impact on the elastographic contrast and on the contrast-to–noise ratio, and may even cause a reversal of the contrast in certain situations. This paper also emphasizes the effect of the precompression strain level on the strain contrast.

A class of inverse problems for viscoelastic material with dominating Newtonian viscosity

Memory kernels in linear stress-strain relations involving a Newtonian viscosity are identified by solving a class of inverse problems. The inverse problems are reduced to nonlinear Volterra integral equations of the first kind which in turn lead to corresponding Volterra equations of the second kind by differentiation. Applying the contraction principle with weighted norms we derive global (in time) existence, uniqueness and stability of the solution to the inverse problems under similar assumptions as for related inverse problems in heat flow.

Non-linear eddy-viscosity modelling of transitional boundary layers pertinent to turbomachine aerodynamics

Abstract The predictive performance of three low-Reynolds-number turbulence models, one based on the linear (Boussinesq) stress-strain relationship and two adopting cubic forms, is examined by reference to transitional flat-plate boundary layers subjected to streamwise pressure gradient and moderate free-stream turbulence, the latter causing bypass transition. One of the cubic models involves the solution of two transport equations, while the other involves three equations, one of which relates to the second invariant of turbulence anisotropy. The investigation demonstrates that, for the conditions examined, the non-linear models return a more credible representation of transition than the linear variant, although none of the models may be said to be entirely satisfactory. Thus, while non-linear modelling shows promise and offers inherent advantages through the use of a more general stress-strain linkage and elaborate calibration, further refinement is needed in the present cubic variants for transitional flows.