# Tensors in Newtonian Physics and the Foundations of Classical Continuum Mechanics

## Abstract

**:**

## 1. Introduction

- Tensor representation of various characteristics of mechanical processes and states, in the first place, stresses and strains;
- The structure of mappings of these tensors and equations linking them, including differential and integral connections used in constitutive relations of media;
- The basic hypothesis for the construction of constitutive relations of bodies at finite deformations, the general reduced forms of relations, bases of their classification (specialization for classes of bodies and processes), possibilities of their experimental verification and confirmation.

## 2. Newtonian Objectivity

#### 2.1. Objective Tensors and Diagrams

**Definition**

**1.**

**objective**, and the lower binary multi-index ${\mathbf{k}}_{m}$–

**type of objectivity**(for objective scalars the absence of (multi-) index is formally considered as an empty multi-index ( ), showing the zero rank of the scalar as a tensor).

#### 2.2. Mappings of Objective Tensors

**Definition**

**2.**

**independent of the frame of reference**if it is a morphism (concomitant) of the free action of the group G of changes of the frame-reference, that is, the mathematical identity is fulfilled:

**Definition**

**3.**

## 3. Development of Foundations of Classical Continuum Mechanics

#### 3.1. Generalized Theory of Strain and Stress Tensor Measures

#### 3.2. Constitutive Relations: Foundations of the General Theory

**Definition**

**4.**

#### 3.3. Notions of Process Image and Five-Dimensional Isotropy at Finite Strains

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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Brovko, G.L.
Tensors in Newtonian Physics and the Foundations of Classical Continuum Mechanics. *Math. Comput. Appl.* **2019**, *24*, 79.
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**AMA Style**

Brovko GL.
Tensors in Newtonian Physics and the Foundations of Classical Continuum Mechanics. *Mathematical and Computational Applications*. 2019; 24(3):79.
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**Chicago/Turabian Style**

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2019. "Tensors in Newtonian Physics and the Foundations of Classical Continuum Mechanics" *Mathematical and Computational Applications* 24, no. 3: 79.
https://doi.org/10.3390/mca24030079