Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Notes on rate equations in nonlinear continuum mechanics

Daniel Aubram

TL;DR

This work provides a geometric treatment of rate equations in nonlinear continuum mechanics, emphasizing objectivity, frame invariance, and the role of corotational rates. It systematically develops rate constitutive models (hypoelasticity, hypoelasto-plasticity, and hypoplasticity), derives rate forms of virtual power, and details objective time integration algorithms (notably Hughes and Winget) for large deformations. The hypoelastic simple shear example shows how different objective rates (Zaremba-Jaumann vs Green-Naghdi) yield markedly different stress responses, highlighting the need for careful rate selection and robust numerical schemes. Overall, the paper integrates differential-geometry foundations with finite-element-oriented time-integration techniques to enable consistent, incremental-objective simulations of nonlinear inelastic solids.

Abstract

The paper gives an introduction to rate equations in nonlinear continuum mechanics which should obey specific transformation rules. Emphasis is placed on the geometrical nature of the operations involved in order to clarify the different concepts. The paper is particularly concerned with common classes of constitutive equations based on corotational stress rates and their proper implementation in time for solving initial boundary value problems. Hypoelastic simple shear is considered as an example application for the derived theory and algorithms.

Notes on rate equations in nonlinear continuum mechanics

TL;DR

This work provides a geometric treatment of rate equations in nonlinear continuum mechanics, emphasizing objectivity, frame invariance, and the role of corotational rates. It systematically develops rate constitutive models (hypoelasticity, hypoelasto-plasticity, and hypoplasticity), derives rate forms of virtual power, and details objective time integration algorithms (notably Hughes and Winget) for large deformations. The hypoelastic simple shear example shows how different objective rates (Zaremba-Jaumann vs Green-Naghdi) yield markedly different stress responses, highlighting the need for careful rate selection and robust numerical schemes. Overall, the paper integrates differential-geometry foundations with finite-element-oriented time-integration techniques to enable consistent, incremental-objective simulations of nonlinear inelastic solids.

Abstract

The paper gives an introduction to rate equations in nonlinear continuum mechanics which should obey specific transformation rules. Emphasis is placed on the geometrical nature of the operations involved in order to clarify the different concepts. The paper is particularly concerned with common classes of constitutive equations based on corotational stress rates and their proper implementation in time for solving initial boundary value problems. Hypoelastic simple shear is considered as an example application for the derived theory and algorithms.

Paper Structure

This paper contains 30 sections, 60 theorems, 287 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Continuum Mechanics
  3. Motion of a Body
  4. Deformation Gradient and Strain
  5. Stress and Balance of Momentum
  6. Constitutive Theory and Frame Invariance
  7. Rates of Tensor Fields
  8. Fundamentals
  9. Corotational Rates
  10. Rate Constitutive Equations
  11. Hypoelasticity
  12. Hypoelasto-Plasticity
  13. Hypoplasticity
  14. Rate Forms of Virtual Power
  15. Initial Boundary Value Problem and Principle of Virtual Power
  16. ...and 15 more sections

Key Result

Proposition 2.1

For a regular $C^1$-motion, the Lie derivative of an arbitrary, possibly time-dependent, spatial tensor field $\boldsymbol{t}_t\in\mathfrak{T}^p_q(\mathcal{S})$ along the spatial velocity $\boldsymbol{v}$ can be expressed by

Figures (4)

  • Figure 1: Simple shear and schematic diagram of associated right polar decomposition $\boldsymbol{F}=\boldsymbol{R}\boldsymbol{U}$.
  • Figure 2: Comparison of the shear stress in hypoelastic simple shear using the Zaremba-Jaumann and Green-Naghdi stress rates.
  • Figure 3: Relative shear stress error $\left|(\sigma^{12}_{\mathrm{num}}-\sigma^{12}_{\mathrm{exact}})/\sigma^{12}_{\mathrm{exact}}\right|$ in simple shear for $k=1.0$ ($\beta=\uppi/4$) using the Zaremba-Jaumann stress rate and Green-Naghdi stress rate.
  • Figure 4: Lie derivative of a time-in-de-pen-dent vector field $\boldsymbol{v}$ along a time-de-pen-dent vector field $\boldsymbol{u}$; reprint from Aub2009

Theorems & Definitions (273)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.1
  • proof
  • ...and 263 more