Chapter 2
Preliminaries
The material of music is sound and silence. Integrating these is
composing.
— John Cage, 1912–1992 —
For the formulation, the algorithmic treatment, and the analysis of the computational
material models to be considered in this treatise, a number of modelling, computational, and mathematical tools are needed.
Modelling tools are concerned with continuum mechanics in one dimension and
dissipation consistent material modelling based on concepts from convex analysis.
Computational tools are concerned with constitutive integrators for either strain
or stress control and—in order to give some context—the finite element method in
one dimension.
Mathematical tools are concerned with Laplace transformation, complex representations, Legendre transformation, and constrained optimization.
All these tools will be outlined only briefly as preliminaries in the sequel of this
chapter.
2.1 Modelling Tools
2.1.1 Continuum Mechanics
Let the one-dimensional Euclidean space E
1 be parameterized by the coordinate x. A
one-dimensional continuum body B consists of a continuous set of physical points p,
which are embedded into the Euclidean space by a mapping p → x ∈ E
1 (Fig. 2.1).
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. Steinmann and K. Runesson, The Catalogue of Computational Material Models,
https://doi.org/10.1007/978-3-030-63684-5_2
23
Preliminaries
The material of music is sound and silence. Integrating these is
composing.
— John Cage, 1912–1992 —
For the formulation, the algorithmic treatment, and the analysis of the computational
material models to be considered in this treatise, a number of modelling, computational, and mathematical tools are needed.
Modelling tools are concerned with continuum mechanics in one dimension and
dissipation consistent material modelling based on concepts from convex analysis.
Computational tools are concerned with constitutive integrators for either strain
or stress control and—in order to give some context—the finite element method in
one dimension.
Mathematical tools are concerned with Laplace transformation, complex representations, Legendre transformation, and constrained optimization.
All these tools will be outlined only briefly as preliminaries in the sequel of this
chapter.
2.1 Modelling Tools
2.1.1 Continuum Mechanics
Let the one-dimensional Euclidean space E
1 be parameterized by the coordinate x. A
one-dimensional continuum body B consists of a continuous set of physical points p,
which are embedded into the Euclidean space by a mapping p → x ∈ E
1 (Fig. 2.1).
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. Steinmann and K. Runesson, The Catalogue of Computational Material Models,
https://doi.org/10.1007/978-3-030-63684-5_2
23