Reply to the discussion by G. Qu on “Modelling the combined effect of strain rate and temperature on one-dimensional compression of soils”Appears in the Canadian Geotechnical Journal, 46(9): 1100–1101.

AbstractThe failure characteristics of rock subjected to impact disturbance under one-dimensional static axial compression are helpful for studying the problems of pillar instability and rock burst in deep, high geostress surrounding rock under blasting disturbances. Improved split Hopkinson pressure bar equipment was used for one-dimensional dynamic–static combined impact tests of deep-seated dolomite specimens under axial compression levels of 0, 12, 24, and 36 MPa. The experimental results demonstrate that the dolomite specimens exhibit strong brittleness. The dynamic strength always maintains a strong positive correlation with the strain rate when the axial compression is fixed; when the strain rate is close, the dynamic elasticity modulus and peak strength of the specimens first increase and then decrease with the increase in axial compression, and the peak value appears at 24 MPa. The impact resistance of specimens can be enhanced when the axial compression is 12 or 24 MPa, but when it increases to 36 MPa, the damage inside the specimen begins to cause damage to the dynamic rock strength. Prior to the rock macroscopic failure, the axial static load changes the rock structure state, and it can store strain energy or cause irreversible damage.

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Transient Solutions for One-Dimensional Problems With Strain Softening

Journal of Applied Mechanics ◽

10.1115/1.3173062 ◽

1987 ◽

Vol 54 (3) ◽

pp. 513-518 ◽

Cited By ~ 22

Author(s):

T. Belytschko ◽

Xiao-Jun Wang ◽

Z. P. Bazant ◽

Y. Hyun

Keyword(s):

Strain Rate ◽

Closed Form ◽

Transient Response ◽

Softening Point ◽

Strain Softening ◽

Stress Strain ◽

One Dimensional ◽

Closed Form Solutions ◽

Transient Solutions

Closed-form solutions are presented for the transient response of rods in which strain softening occurs and the stress-strain laws exhibit nonvanishing stresses after the strain-softening regime. It is found that the appearance of any strain softening results in an infinite strain rate if the material is inviscid. For a stress-strain law with a monotonically decreasing stress the strains are infinite also. If the stress increases after the strain-softening portion, the strains remain finite and the strain-softening point moves through the rod.

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VERIFICATION OF THE SHB TEST RESULTS BASED ON THE ONE-DIMENSIONAL STRESS WAVE THEORY

International Journal of Modern Physics B ◽

10.1142/s0217979208046335 ◽

2008 ◽

Vol 22 (09n11) ◽

pp. 1068-1073

Author(s):

TOMOKAZU MASUDA ◽

KENJI SAITO ◽

IZUMI MORITA ◽

SHUSHI IKEDA ◽

KOICHI MAKII ◽

...

Keyword(s):

Strain Rate ◽

Stress Wave ◽

High Speed ◽

Wave Theory ◽

High Speed Photography ◽

Strain Rates ◽

One Dimensional ◽

Strain And Strain Rate ◽

Deformation Behaviors ◽

The One

In order to evaluate dynamic deformation behaviors under high strain rates, Kobe Steel has developed and applied a Split-Hopkinson Bar (SHB) apparatus. This paper discusses the validity of the strain measurements and strain rates measured by this SHB apparatus. The strain waves that propagated in the incident and transmitted bars and the specimen are captured using a high-resolution type high-speed photography in detail. The strain wave propagated many times in the incident and transmitted bars and the specimen when the specimen was not broken. The amount of the deformation of the specimen decreases with the propagation frequency of the incident wave. On the other hand, to improve accuracy at the strain and strain rate calculated by the one-dimensional stress wave theory, Young's modulus, the longitudinal wave speed, and the density were accurately determined. It was understood that the calculation value showed the strain and strain rate captured with the high-speed photography are a good agreement. As a result, the validity of the measurement accuracy of this SHB could be shown.

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One-Dimensional Dynamic Compressive Behavior of EPDM Rubber

Journal of Engineering Materials and Technology ◽

10.1115/1.1584492 ◽

2003 ◽

Vol 125 (3) ◽

pp. 294-301 ◽

Cited By ~ 55

Author(s):

B. Song ◽

W. Chen

Keyword(s):

Strain Rate ◽

High Speed ◽

Dynamic Stress ◽

Split Hopkinson Pressure Bar ◽

Constant Strain Rate ◽

Strain Rates ◽

Hopkinson Pressure Bar ◽

Stress Strain ◽

Epdm Rubber ◽

One Dimensional

Dynamic compressive stress-strain curves at various strain rates of an Ethylene-Propylene-Diene Monomer Copolymer (EPDM) rubber have been determined with a modified split Hopkinson pressure bar (SHPB). The use of a pulse-shaping technique ensures that the specimen deforms at a nearly constant strain rate under dynamically equilibrated stress. The validity of the experiments was monitored by a high-speed digital camera for specimen edge deformation, and by piezoelectric force transducers for dynamic stress equilibrium. The resulting dynamic stress-strain curves for the EPDM indicate that the material is sensitive to strain rates and that the strain-rate sensitivity depends on the value of strain. Based on a strain energy function theory, a one-dimensional dynamic constitutive equation for this rubber was modified to describe the high strain-rate experimental results within the ranges of strain and strain rates presented in this paper.

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Constitutive Model of Shape Memory Alloys: One-Dimensional Phase Transformation Model

Advances in Science and Technology ◽

10.4028/www.scientific.net/ast.56.84 ◽

2008 ◽

Vol 56 ◽

pp. 84-91

Author(s):

Tadashige Ikeda

Keyword(s):

Phase Transformation ◽

Constitutive Model ◽

Shape Memory Alloys ◽

Strain Rate ◽

Shape Memory ◽

Strain Rate Effect ◽

Rate Effect ◽

Reverse Transformation ◽

Stress Strain ◽

One Dimensional

A simple yet accurate macroscopic constitutive model of shape memory alloys has been developed. The features of this model are (1) energy-based phase transformation criterion, (2) one-dimensional phase transformation rule based on a micromechanical viewpoint, (3) dissipated energy with a form of a sum of two exponential functions, (4) duplication of the strain rate effect, and (5) adaptability to multi-phase transformation. This model is further improved to be able to express stress-strain relationships such that the reverse transformation starts at a higher stress than the martensitic transformation starts. Here, the ideal reversible transformation temperature is empirically described by a function of the martensite volume fraction. In this paper, an outline of our model is given, where the improvement is introduced. Then, it is shown that the model can quantitatively duplicate the major and minor hysteresis loops, strain rate effect, and asymmetry in tension and compression on the stress-strain relationship. And that it can also duplicate the stress-strain relationships having the reverse transformation start stress higher than the forward one.

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ONE-DIMENSIONAL CHAOS IN IMPULSED LINEAR OSCILLATING CIRCUITS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493000787 ◽

1993 ◽

Vol 03 (04) ◽

pp. 921-941 ◽

Cited By ~ 2

Author(s):

LAURA GARDINI ◽

RENZO LUPINI

Keyword(s):

Asymptotic Behavior ◽

Fixed Points ◽

Critical Point ◽

Critical Points ◽

Phase Plane ◽

Global Bifurcation ◽

Homoclinic Bifurcation ◽

Combined Effect ◽

Chaotic Oscillations ◽

One Dimensional

The dynamics of a damped linear oscillating circuit subject to impulses is represented by a one-dimensional endomorphism (or noninvertible map) π: ℝ → ℝ. The asymptotic behavior of orbits in the phase-plane is characterized in terms of critical points and point singularities of π (fixed points or cycles). Their combined effect, that is, the merging of a critical point into a repelling cycle, causes a global bifurcation or a homoclinic bifurcation, with transition to chaotic oscillations.

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