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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
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Second-order work in barodesy.

Gertraud Medicus1, Barbara Schneider-Muntau1, Dimitrios Kolymbas1

  • 1Division of Geotechnical and Tunnel Engineering, University of Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria.

Acta Geotechnica
|July 21, 2020
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Summary
This summary is machine-generated.

Second-order work analyses reveal that material failure can precede the yield limit. In barodesy, vanishing second-order work in shear bands indicates imminent failure in boundary value problems.

Keywords:
BarodesyConstitutive modelFinite element simulationsSecond-order work

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Area of Science:

  • Geotechnical Engineering
  • Continuum Mechanics
  • Computational Mechanics

Background:

  • Second-order work analyses are crucial for predicting material failure, often indicating failure before the limit yield condition is met.
  • Existing models based on elasto-plasticity frequently employ these analyses.
  • Understanding the behavior of second-order work in different constitutive models is essential for accurate failure prediction.

Purpose of the Study:

  • To investigate second-order work behavior in barodesy, comparing it with elasto-plasticity and hypoplasticity.
  • To analyze standard element tests and finite element applications within the barodesy framework.
  • To identify critical stress states and failure precursors using second-order work indicators.

Main Methods:

  • Application of second-order work investigations to barodesy.
  • Analysis of standard element tests and finite element applications.
  • Examination of stress states relative to the critical limit surface.

Main Results:

  • Second-order work was shown to vanish at stress states within the critical limit surface in barodesy.
  • This finding is consistent with observations in hypoplasticity and elasto-plasticity.
  • For boundary value problems, a shear band with vanishing second-order work signifies imminent failure.

Conclusions:

  • Vanishing second-order work is a reliable indicator of impending failure in various material models, including barodesy.
  • The study confirms the utility of second-order work analyses for predicting failure in geotechnical and continuum mechanics.
  • Shear bands exhibiting vanishing second-order work are critical zones for failure in boundary value problems.