Abstract—This scientific research proposes a fundamental application of constructal theory developed by prof. Adrian BEJAN of Duke University in order to prove in a mathematical sense that the mechanical maximum work principle used by the theory of continuum media plasticity can be regarded as a solution of a general variational optimization problem. According to the first and second thermodynamics laws, the constructal principle search to complete the natural tendency for all finite-size system to raise the entropy obtaining specific optimal system design or material flow configurations. In accord with the constructal theory “all system searches to flow more and more easily over time using specific distribution of imperfections in order to maximize entropy and to minimize the losses”. In this sense, concerning the field of forming processes, all material flows under specified boundaries, loading and processing conditions are those which minimize the sum of dissipated deformation and friction power. Thus all the corresponding mechanical variables (velocities, stress, strain, strain rate) of the real mechanical state as those that minimizes the total dissipated power. It can be then obtained a variational constrained minimization problem. Equivalent form of the maximum work principle is proved also for the friction stresses together with the convexity properties of plastic or friction potential. An application in the case of a cylindrical upsetting shows the feasibility of the proposed minimization problem formulation to find analytical solution. To valid this theory, comparisons are made using the classical analytical analysis based on upper and lower bound theorems, slices method and numerical Finite Element Modelling (FEM). Index Terms—Constructal theory, continuum media plasticity, maximum work principle, variational optimization. I. INTRODUCTION Theoretical and applied scientific researches concerning the field of thermodynamic optimization theory for complex systems based on maximization of entropy, minimization of local or global flow resistance and corresponding maximization of system’s speed to reach a stable equilibrium state have been developed by Prof. Adrian BEJAN of Duke University. As can be mentioned in previous scientific works [1-2] the named constructal principle of Prof. A. BEJAN can be regarded as a general constrained optimization problem taking into account the principle that “for a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it”. Generally all systems search to optimize imperfections distribution to facilitate the flow and to Manuscript received Mai 20, 2018; revised June 20, 2018. A. Gavrus is with the National Institute of Applied Sciences of Rennes, Rennes, F-35708 France (e-mail: adinel.gavrus@insa-rennes.fr). minimize the local resistance or applied powers. A lot of studies [2] concerning examples from thermal problems, fluid flow or porous media properties, nature phenomena observations, economic or societal behavior confirm this principle. Starting from a recent scientific work of the author this paper proposes to consolidate the general proof of the mechanical maximum work principle [3]. This one is used by the plasticity theory [4] and is generally applied to obtain the governing equations of material flow or strength analysis and to make analytical or numerical computations particularly during forming processes. Previous works [5-6] have search to explain this principle only in a phenomenological way and only for metallic materials starting from local slips of atomic planes. Using the well-known mechanical virtual power principle and based on above mentioned constructal theory the maximum work principle can been obtained from a general variational optimization problem which search to minimize the material flow-resistance or the system dissipations. After detailed theoretical backgrounds together with their useful consequences proving convexity and normal rule of rheological or tribologic potential behaviour, lower-bound and upper-bound theorems, an application is presented concerning a cylindrical quasi-static crushing taking into account a Tresca friction. Starting from theoretical considerations, a validation of proposed techniques to find analytical solutions is also developed using comparisons between the obtained analytical estimations of the loads and the numerical FEM results. II. THEORETICAL BACKGROUNDS A. Mathematical Proof of Maximum Work Principle According to the continuum media theory considering a material flow on a defined body  the Newtonian local mechanical equilibrium balance can be written using the equivalent virtual power works principle (PPW):   * * d d * * ' '' : dV v dS dv * T v dS' T v dS'' f v dV v* dV dt                                     (1) The PPW principle is available for all virtual admissible velocities field * v following boundaries kinematic conditions, where    is the Cauchy stress tensor, *      is the virtual strain rate tensor defined by       T 1 grad( v*) grad( v*) 2  , Application of Constructal Theory to Write Mechanical Maximum Work Principle and Equilibrium State of Continuum Media Flow as a Solution of a Variational Optimization Problem A. Gavrus International Journal of Modeling and Optimization, Vol. 8, No. 4, August 2018 227 DOI: 10.7763/IJMO.2018.V8.655