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Piecewise defined functions are mathematical models where different expressions define a function over distinct intervals of the domain. These functions are useful for representing systems with varying behaviors depending on input values.For example, the function:  uses a linear rule for inputs less than or equal to –1 and a quadratic rule for values greater than –1. Although it has two formulas, it still defines a single function.Another common type is the absolute value...
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The piecewise parabolic method for elastic-plastic flow in solids.

Wei Zhang1,2, Cheng Chen3, Kun Liu4

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This study introduces a new numerical method for simulating elastic-plastic flow, accurately capturing material behavior under high-speed impacts. The technique precisely models large deformations and material interfaces for solid mechanics applications.

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Area of Science:

  • Solid Mechanics
  • Computational Physics
  • Materials Science

Background:

  • Elastic-plastic flow simulations are crucial for understanding material behavior under extreme conditions.
  • Existing numerical methods face challenges in accurately modeling large deformations and high strain rates.
  • Capturing material interfaces in multi-material simulations remains a significant hurdle.

Purpose of the Study:

  • To develop a high-order numerical technique for simulating elastic-plastic flow.
  • To accurately model the constitutive relations of materials under strong impact.
  • To enable precise simulation of elastic-plastic waves and multi-material interfaces.

Main Methods:

  • A high-order piecewise parabolic method combined with the HLLD Riemann solver was developed.
  • Plasticity was introduced by decomposing the deformation gradient tensor and adding a plastic source term.
  • A temporal splitting strategy and semi-implicit scheme were employed to solve the resulting equations.
  • The level-set algorithm was integrated for accurate interface tracking in multi-material scenarios.

Main Results:

  • The numerical model successfully simulated large deformations and high strain rates.
  • The method accurately captured the elastic-plastic behavior and waves in solid materials.
  • Multi-material interfaces were tracked sharply and correctly in high-speed impact problems.

Conclusions:

  • The developed numerical method provides an accurate and robust tool for simulating elastic-plastic flow.
  • This technique is effective for analyzing material responses under extreme impact conditions.
  • The approach enhances the simulation of complex phenomena like elastic-plastic waves and multi-material interactions.