# Poster presentation: A new approach to the theory of stress and deformation based on the theory of potentials

This page contains the text of a poster which I have shown on several conferences. It contains the following sections
• a mathematical proof that the Cauchy stress tensor does not exist,
• a physical argument to the effect that the entire theoretical framework of continuum mechanics is unsuited to do justice to the nature of elasticity and stress as a physical phenomenon,
• an abbreviated outline of the approach I propose,
• some predictions made with the new approach.

 Cauchy stress theory at variance with Gauss Divergence Theorem The thermodynamic definition of pressure                P = ¶U/¶V                (eqn.1) or P = DU/DV in integrated form is an explicit statement of the fundamental requirement of potential theory: in a given state U, the ratio of mass in a system to its potential is scale-independent. The thermodynamic equilibrium condition is Psyst + Psurr = 0. Forces are acting on the system from the surrounding, and vice versa; that is: there are two independent groups of forces with opposite sign in the sense that they are directed either inside or outside.                 The two groups must balance. Thus the equilibrium condition translates into           ò fsyst × n dA + ò fsurr × n dA = 0, or fsyst + fsurr = 0         (eqn.2) where f is the source density. The Cauchy theory is based on the assumption that the ratio P = |f|/A ® const as A ® 0, whereas potential theory holds that the ratio P = U/V ® const as V ® 0. Can both conditions hold together? The source density f in the divergence theorem                 ò f × n dA = ò Ñ × f dV = f                (eqn.3) is proportional to mass. In a continuum of mass, f is therefore a linear function of the integration limits of the RHS, and Ñ × f = const. If r = |r| is the magnitude of the radius of the thermodynamic system, V µ r3 and A µ r2. Since LHS is a surface integral, but RHS is a volume integral, it follows that |f| in LHS must be a linear function of scale, i.e.                 |f|/|r| = const.                 (eqn.4) Thus if V ® 0, f vanishes with r: the ratio |f|/A ® ¥ as V ® 0, but the ratio U/V = P is constant. The Cauchy stress theory assumes that |f|/A approaches a finite value as V ® 0. Therefore the Cauchy theory is at variance with the divergence theorem. The Cauchy stress tensor does not exist. The Newtonian definition P = |f|/A is not not applicable in thermodynamics. The Newtonian definition applies to free planes whereas the divergence theorem and P = U/V consider a closed plane. Free planes divide space into top/bottom, left/right, front/back; only a closed plane divides space into inside/outside. Cauchy's approach to stress never distinguished system and surrounding, it is incompatible with the properties of a thermodynamic system. Newtonian theory unsuited for continuum mechanics and thermodynamics The basics of continuum mechanics were outlined by Euler in the mid-18th century. He adapted the concepts of Newtonian mechanics, the only one available at his time, which was conceived to understand celestial mechanics. Its main tools are the equation of motion f = ma, which is the Newtonian definition of force, the Newtonian definition of work = force times distance, w = f × x, Bernoulli's energy conservation law, Ukin + Upot = const. In physics, one cannot choose concepts at random. Choice of a particular concept implies a lot of context. Can Newton's tools be indiscriminately applied to continuum mechanics? Consider: The Newtonian force definition employs the inertial mass, measured in [kg]. It is thus implied that work is done against inertia, either by displacing a discrete body of mass m in empty space, or by accelerating it. Therefore f = ma refers to a velocity potential. Newtonian work is path-independent in Euclidean space, i.e. the 3D-space we live in. Bernoulli's law, the energy conservation law of classical mechanics states: the energetic state of a system undergoing a process in line with Newtonian physics is invariant by definition. Are there alternatives? Consider thermodynamics: Mass is dimensionless in thermodynamics and measured in Mol because the thermodynamic potential must be scaled per atom. Thermodynamic work is done not by acceleration against inertia but, e.g., by changing bond lengths in solids. Thermodynamic work is path-independent in PV-space, the energetic plane. Thermodynamics is the physics of changes of state, that is: the energetic state of a system is variable by definition. Its energy conservation law is the First Law: dU = dw + dq. What is the physical nature of an elastic deformation? The buildup of an elastic potential is a change of state: work is done on a system. Therefore, its proper theoretical treatment requires an equation of state, not an equation of motion, and work done is related to PdV-work, it is not Newtonian work. None of this is reflected in any way in present-day continuum mechanics where the elastic potential for a volume-constant deformation is zero by definition. This is wrong by nature. Thus, the theoretical framework of classical continuum mechanics is unsuited to properly deal with the physical reality of an elastic deformation which represents a change of state. Two profoundly different energy conservation laws are thoroughly mixed up. Euler passed on in 1783. In the following year, Lagrange discovered vector fields and founded field theory. The nature of changes of state was only understood in 1842 when the golden age of thermodynamics began. Continuum mechanics, however, the physics of anisotropic changes of state, remained solidly stuck in the philosophy of the 18th century. It is obsolete by 200 years. A new approach to continuum mechanics A new approach to deformation theory must consider the nature of elastic deformation as a change of state. Thus an equation of state is required. 1. Solids have a much smaller molar volume than gases because of internal bonds. Thus they are said to have an internal pressure. An increase in external pressure must therefore be scaled to the internal pressure of the solid. An equation of state is proposed                 PkV = z                 (eqn.5a) where                 k = ln Vmolsolid/ ln Vmolideal gas                (eqn.5b) and z is solved by the Birch-Murnaghan equation. Eqn.5 is material- independent. For real materials, k is readily calculated; for modelling purposes it suffices to continue with the ideal gas law. The approach taken here is similar to that of Grüneisen 1908 (see historical paper).           A material-independent equation of state for solids: Bridgman (1958) compressed the alkali metals to a confining pressure (external pressure) of 100 kbar.           If Bridgman's data are scaled to the internal pressure of the respective substances, his four independent graphs (above) all converge into one single graph. Thus the material-independent equation of state for solids (eqn.5) is supported by experiments. Bridgman PW (1958) The physics of high pressure. Bell & Sons Ltd, London  2. Boyle's law is written in scalar terms; it must be transformed into vector form:                 PV = const ® fr = const                (eqn.6) where r = |r| is the radius of the thermodynamic system, and f = |f| is the force acting on the system surface (either fsyst from within, or fsurr from without). 3. Equilibrium can always be assumed to exist in the elastic state because system and surrounding are physically connected through chemical bonds in the solid; these bonding forces need to be taken into account. Disequilibrium is therefore impossible. 4. Both fsyst and fsurr are understood as classical force fields which are derived from potentials (E = material enthalpy, U = some externally controlled potential):                 fsyst = ei¶Esyst/¶xi         fsurr = ei¶Usurr/¶xi                 (eqn.7) with the field property tensors                 ¶2E/¶x2 = Fsyst          ¶2U/¶x2 = Fsurr                (eqn.8) such that f = ò Fdr, or simply Fr = f. Fsurr is the field property tensor for the external force field, it represents the boundary conditions. Fsyst represents the material properties. The two independent fields are then related to one another by means of the equilibrium condition. The result is the stress field. 5. Once the stress field is derived, the equation of state will provide the displacement field: just as PV = const translates into ln (V1/V0) = - DP/P0, the equation of state in vector form yields                 fr = const ® ln (r1/r0) = - Df/f0                (eqn.9) Thus, stress is treated as an anisotropic change of state. Stress field and displacement field have exactly identical geometric properties as a function of two independent sets of boundary conditions: the externally controlled set and the material properties. Results (1) Prediction of fabric property orientations in simple shear zones The eigendirections (stable directions) of the simple shear stress/displacement field are non-orthogonal. They are correlated with the observed fabric elements: c         contracting eigendirection: parallel to dilational joints that open           upon unloading, but controlled by residual stress e         extending eigendirection: parallel to S-plane in S-C fabrics R         bisector of small angle: Riedel plane, or C-plane in S-C           fabrics P         bisector of large angle: P-plane                 The figure shows the predicted properties of a dextral simple shear. These theoretically derived results compare very favorably with natural fabrics.