Wave speeds, shear bands and the second-order work for incrementally nonlinear constitutive models

Thermoplastic pipes are being used increasingly for water supply lines, storm sewers, and leachate collection systems in landfills. To facilitate limit states design for buried polymer pipes, nonlinear constitutive models have recently been developed to characterize the highly nonlinear and time-dependent material behavior of high-density polyethylene (HDPE). These models have been implemented in a finite element program to permit structural analysis for buried HDPE pipes and to provide information regarding performance limits of the structures. Predictions of HDPE pipe response under parallel plate loading and hoop compression in a soil cell are reported and compared with pipe response measured in laboratory tests. Effects on the structural performance of pipe material nonlinearity, geometrical nonlinearity, and backfill soil properties were investigated. Good correlations were found between the finite element predictions and the experimental measurements. The models can be used to predict pipe response under many different load histories (not just relaxation or creep). Work is ongoing to develop nonlinear constitutive models for polyvinylchloride and polypropylene to extend the predictive capability of the finite element model to these materials.

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Nonlinear constitutive models for pultruded FRP composites

Mechanics of Materials ◽

10.1016/s0167-6636(02)00207-7 ◽

2003 ◽

Vol 35 (8) ◽

pp. 791-801 ◽

Cited By ~ 23

Author(s):

Rami Haj-Ali ◽

Hakan Kilic

Keyword(s):

Constitutive Models ◽

Frp Composites ◽

Nonlinear Constitutive Models

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On the instability in second-order systems for acoustic VTI and TTI media

Geophysics ◽

10.1190/geo2011-0250.1 ◽

2012 ◽

Vol 77 (5) ◽

pp. T171-T186 ◽

Cited By ~ 25

Author(s):

Kenneth P. Bube ◽

Tamas Nemeth ◽

Joseph P. Stefani ◽

Ray Ergas ◽

Wei Liu ◽

...

Keyword(s):

Wave Equation ◽

Dispersion Relation ◽

Differential Equations ◽

Shear Wave ◽

Transversely Isotropic ◽

Second Order ◽

Wave Speeds ◽

Tti Media ◽

Second Order Systems ◽

Vti Media

We studied second-order wave propagation systems for vertical transversely isotropic (VTI) and tilted transversely isotropic (TTI) acoustic media with variable axes of symmetry that have their shear-wave speeds set to zero. Acoustic TTI systems are commonly used in reverse-time migration, but these second-order systems are susceptible to instablities appearing as nonphysical stationary noise growing linearly in time, particularly in variable-tilt TTI media. We found an explanation of the cause of this phenomenon. The instabilities are not caused only by the numerical schemes; they are inherent to the differential equations. These instabilities are present even in homogeneous VTI media. These instabilities are caused by zero wave speeds at a wide variety of wavenumbers — a direct consequence of setting the shear-wave speeds to zero — coupled with the second time derivative in these systems. Although the second-order isotropic wave equation allows smooth time-growing solutions, a larger class of time-growing solutions exists for the second-order acoustic TI systems, including nonsmooth solutions. Boundary conditions appear to be less effective in controlling these time-growing solutions than they are for the isotropic wave equation. These systems conserve an incomplete energy that does not prevent the instabilities. The corresponding steady-state systems are no longer elliptic differential equations and can have nonsmooth solutions that are related to the instabilities. We started initially with homogeneous VTI media, and then extended these results to heterogeneous variable-tilt TTI media. We also developed a second-order acoustic system for heterogeneous variable-tilt TTI media derived directly from the full-elastic system for heterogeneous variable-tilt TTI media. All second-order systems with a dispersion relation obtained by setting the shear-wave speeds to zero in the elastic dispersion relation allowed these nonphysical time-growing solutions; however, knowing the cause of these instabilities, it may be possible to prevent or control the activation of these solutions.

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On the Significance of Normal Stress Effects in the Flow of Glaciers

Journal of Glaciology ◽

10.1017/s0022143000008832 ◽

1987 ◽

Vol 33 (115) ◽

pp. 268-273 ◽

Cited By ~ 62

Author(s):

Chi-Sing Man ◽

Quan-Xin Sun

Keyword(s):

Normal Stress ◽

Creep Behaviour ◽

Laboratory Data ◽

Constitutive Models ◽

Second Order ◽

Material Parameters ◽

Stress Effects ◽

Polycrystalline Ice ◽

Glacier Ice ◽

Flow Law

AbstractMcTigue and others (1985) identified a possible problem in the type of constitutive equation usually used for modeling the creep behaviour of polycrystalline ice. They pointed out that Glen’s flow law necessarily excludes the consideration of normal stress effects, which are of great significance in other disciplines that consider non-Newtonian fluids. Using the second-order fluid (with material parameters evaluated from laboratory data) as a tentative model for ice, they reached the conclusion that normal stress effects may be discernible in natural glacier flow. But, as noted by McTigue and others, the second-order fluid “fails to represent the non-linear rate dependence of ice in shear”; therefore it is in fact not a suitable constitutive model for glacier ice in shearing flow. In this note, parallel to what McTigue and others did for the second-order fluid, we present a similar analysis for (I) the modified second-order fluid and (II) the power-law fluid of grade 2, both of which are constitutive models recently proposed by Man as a tentative generalization of Glen’s flow law. Both models (I) and (II) can represent normal stress effects, and both agree with Glen’s flow law in the prediction of the depth profile of velocity in the steady laminar flow of glaciers. For ease of comparison, the same creep data of McTigue and others are used in quantifying the material parameters in these two models. Both models (I) and (II) predict far less pronounced normal stress effects in glaciers than those estimated by McTigue and others (whose data analysis in fact suffered from inconsistencies and over-parameterization).

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Stick-slip and wear phenomena at the contact interface between an elastic beam and a rigid substrate

Mathematics and Mechanics of Solids ◽

10.1177/1081286520971671 ◽

2020 ◽

pp. 108128652097167

Author(s):

Francesco D’Annibale ◽

Arnaldo Casalotti ◽

Angelo Luongo

Keyword(s):

A Priori ◽

Constitutive Models ◽

Elastic Beam ◽

Longitudinal Direction ◽

Static Problem ◽

Rigid Substrate ◽

Equilibrium Equations ◽

Stick Slip ◽

Contact Points ◽

Nonlinear Constitutive Models

In this paper, the static behavior of an elastic beam resting on a rigid substrate is investigated. The structure lies on a rigid substrate and exchanges with it tangential forces, in correspondence with a finite number of contact points. These actions entail extension of the beam in the longitudinal direction together with a negligible bending, owing to the small eccentricity between the beam’s axis line and the rigid substrate. The beam obeys a linear elastic law, while, at the interface, different nonlinear constitutive models are considered to account for stick-slip phenomena due to friction, as well as wear due to abrasion. It is assumed that the contact points are a-priori known, thus entailing that the structural system can be treated as naturally discrete. The static problem is accordingly shown to be governed by a system of nonlinear ordinary differential equations in time, which rules, in incremental form, the equilibrium at the contact points in the longitudinal direction. A numerical solution for the equilibrium equations is carried out, under different imposed time histories of the longitudinal displacement assigned at the boundary. Numerical results are presented to compare and discuss the in-time evolution of the contact interactions between the beam and the substrate.

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