**Authors:** Ehsan Azadi

In this article, we give a general exact mathematical framework that all of the fundamental relations and conservation equations in continuum mechanics can be derived based on it. We consider a general integral equation contains the parameters that act on the volume and the surface of the integral's domain. The idea is to determine how many local relations can be derived from this general integral equation and what these local relations are? Thus, we first derive the general Cauchy lemma and then by a new general exact tetrahedron argument derive two other local relations. So, there are three local relations that can be derived from the general integral equation. Then we show that all of the fundamental laws in continuum mechanics, include the conservation of mass, linear momentum, angular momentum, energy, and the entropy law, can be shown and considered in this general framework. So, we derive the integral form of these fundamental laws in this framework and applying the general three local relations lead to exactly derivation of the mass flow, continuity equation, Cauchy lemma for traction vectors, existence of stress tensor, general equation of motion, symmetry of stress tensor, existence of heat flux vector, differential energy equation, and differential form of the Clausius-Duhem inequality for entropy law. The general exact tetrahedron argument is an exact proof that removes all of the challenges on the derivation of fundamental relations in continuum mechanics. During this proof, there is no approximating or limiting process and the parameters are exact point-base functions. Also, it gives a new understanding and a deep insight into the origins and the physics and mathematics of the fundamental relations and conservation equations in continuum mechanics. This general mathematical framework can be used in many branches of continuum physics and the other sciences.

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[v1] 2017-07-07 10:28:47

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