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2022 | Book

Spatial and Material Forces in Nonlinear Continuum Mechanics

A Dissipation-Consistent Approach

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This monograph details spatial and material vistas on non-linear continuum mechanics in a dissipation-consistent approach. Thereby, the spatial vista renders the common approach to nonlinear continuum mechanics and corresponding spatial forces, whereas the material vista elaborates on configurational mechanics and corresponding material or rather configurational forces. Fundamental to configurational mechanics is the concept of force. In analytical mechanics, force is a derived object that is power conjugate to changes of generalised coordinates. For a continuum body, these are typically the spatial positions of its continuum points. However, if in agreement with the second law, continuum points, e.g. on the boundary, may also change their material positions. Configurational forces are then power conjugate to these configurational changes. A paradigm is a crack tip, i.e. a singular part of the boundary changing its position during crack propagation, with the related configurational force, typically the J-integral, driving its evolution, thereby consuming power, typically expressed as the energy release rate. Taken together, configurational mechanics is an unconventional branch of continuum physics rationalising and unifying the tendency of a continuum body to change its material configuration. It is thus the ideal formulation to tackle sophisticated problems in continuum defect mechanics. Configurational mechanics is entirely free of restrictions regarding geometrical and constitutive nonlinearities and offers an accompanying versatile computational approach to continuum defect mechanics. In this monograph, I present a detailed summary account of my approach towards configurational mechanics, thereby fostering my view that configurational forces are indeed dissipation-consistent to configurational changes.

Table of Contents

Frontmatter
1. Introduction
Abstract
This chapter introduces the overarching topic of this monograph. After recalling the need for continuum mechanics, it contrasts the concepts of spatial and material forces and highlights why the latter are a necessary concept in various branches of defect mechanics. It then reviews various vistas on material forces before commenting on their computational implications.
Paul Steinmann
2. Kinematics in Bulk Volumes
Abstract
This chapter recalls the pertinent spatial and material continuum kinematics in bulk volumes, thereby focusing on the nonlinear deformation maps, their associated tangent, cotangent and measure maps, and expresses their compatibility conditions.
Paul Steinmann
3. Kinematics on Dimensionally Reduced Smooth Manifolds
Abstract
This chapter reviews continuum kinematics on dimensionally reduced smooth manifolds, i.e. on boundary surfaces and on boundary curves, with an emphasis on the nonlinear deformation maps and their associated tangent, cotangent, and measure maps.
Paul Steinmann
4. Kinematics at Singular Sets
Abstract
This chapter revisits the relevant continuum kinematics at singular sets, i.e. at singular surfaces and at singular curves and points, thereby elaborating on the jumps in the nonlinear deformation maps and their associated tangent, cotangent, and measure maps.
Paul Steinmann
5. Generic Balances
Abstract
This chapter represents the formulation of generic balances for generic volume as well as surface and curve extensive quantities, thereby highlighting their global and local formats and resorting in both cases to material and spatial control volumes as well as control surfaces and control curves.
Paul Steinmann
6. Kinematical ‘Balances’*
Abstract
This chapter applies the formats of the generic balances to the spatial and material tangent, cotangent, and measure maps to formulate what, for the sake of semantic unification, may be called kinematical ‘balances’.
Paul Steinmann
7. Mechanical Balances
Abstract
This chapter details the generic balances for the case of mechanical balances of mass, spatial momentum, and its vector and scalar moment, respectively, with emphasis on their global and local formats and the distinction between material and spatial control volumes.
Paul Steinmann
8. Consequences of Mechanical Balances
Abstract
This chapter explores the consequences of the mechanical balances by elaborating on local and global formats of the balance of kinetic energy and the balance of material momentum and its vector and scalar moments, thereby differentiating between spatial and material control volumes.
Paul Steinmann
9. Virtual Work
Abstract
This chapter capitalizes on the referential setting when introducing the notions of spatial and material virtual displacements and discussing the accompanying spatial and material virtual work principles.
Paul Steinmann
10. Variational Setting
Abstract
This chapter expands on the variational setting in terms of extended Hamilton and Dirichlet principles for conservative elasto-dynamic and elasto-static cases, respectively, and carefully analyzes the resulting spatial and material Euler-Lagrange equations.
Paul Steinmann
11. Thermodynamical Balances
Abstract
This chapter specifies the generic balances for the case of thermodynamical balances of energy and entropy, whereby the case of exterior energy is demarcated from the common case of (interior) energy by the formal incorporation of the external potential energy into the notion of internal energy.
Paul Steinmann
12. Consequences of Thermodynamical Balances
Abstract
This chapter exploits the consequences of the thermodynamical balances and the resulting formats of the dissipation power inequalities by identifying the forces driving material (configurational) changes on the boundary and at singular surfaces as the appropriate contributions to the balance of material momentum. It is the here advocated thermodynamical derivation of material forces that qualifies the current approach as being dissipation-consistent.
Paul Steinmann
13. Computational Setting
Abstract
This chapter sketches the consequences for computational mechanics by outlining the material force method based on finite element discretization of the material virtual work principle and highlights its applicability to geometrically nonlinear fracture mechanics by some computational examples.
Paul Steinmann
Metadata
Title
Spatial and Material Forces in Nonlinear Continuum Mechanics
Author
Prof. Dr. Paul Steinmann
Copyright Year
2022
Electronic ISBN
978-3-030-89070-4
Print ISBN
978-3-030-89069-8
DOI
https://doi.org/10.1007/978-3-030-89070-4

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