Equivalent stress theory

Equivalent stress distribution which is shown in Figure 13 (loading below yield stress) is in good correlation with Hertzian contact theory. 

The new ASME Section VIII, Division 2, Part 5 utilizes the distortion energy theory to establish the equivalent stress in an elastic analysis where in the pre-2007 edition this was done with the maximum shear stress theory. 

The results of the equivalent stress calculation from the distortion energy theory indicate that reduced shell thickness values may be obtained if a more rigorous analysis is performed. This is the basis for part of the design-by-analysis section in Section Vm, Division 2. 

Based on the mathematical theory of plasticity, the plastic deformation behavior of the material can be described by the three components of the rate-independent plasticity model, namely yield criterion, flow rule, and hardening rule. The yield criterion determines the stress level at which yielding is initiated. This is represented by the equivalent stress Ueq, which is a function of the individual stress vector components a. Plastic strain is developed in the metal parts when the equivalent stress is equal to a material yield parameter ay finally, the flow rule determines the direction of plastic straining ... 

Where a number of stress components are at work in different planes, their values can be mathematically combined to produce a compound or equivalent stress, Cy. To determine the equivalent stress for a thick-layer elastic adhesive, normal stress theory may be used. This is commonly employed for components that are mechanically restrained from undergoing expansion. 

Voorhees analysis assumes that the creep-rupture life of a vessel under complex stressing is controlled by an equivalent stress, J, termed the shear-stress invariant. This average stress is also known as the octahedral shear stress, the effective stress, the intensity of stress, and the quadratic invariant. The theory for the biaxial-stress condition was developed by Von Mises (205), and this theory was further developed to apply to the triaxial-stress condition independently by Hencky (206, 207, 208) and by Huber (209). A derivation of the relationship between the equivalent stress, /, and the three principal stresses, /i, /2, and/s where /i > /2 > /s was given by Eichinger (210). The relationship between these stresses is ... 

Maximum distortion energy (or maximum octahedral shear stress) theory (von Mises) Failure occurs when the maximum distortion energy (or maximum octahedral shear stress) at an arbitrary point in a stressed medium reaches the value equivalent to the maximum distortion energy (or maximum octahedral shear stress) at failure (yield) in simple tension... 

Equation (9.17) defines the equivalent stress Mises theory. Under plain stress conditions (biaxial) Eq. (9.17) becomes... 

The present fatigue life model, expressed by Eqs. (5-7), is based on three existing fatigue life theories. The basic concept of fatigue integral is due to loannides and Harris [18]. The incorporation of fatigue limit stress, the mathematical form of the depth factor, and the use of the Mises stress as the equivalent stress are also part of the loannides-Harris theory [17-18]. 

In 1936, de Boer formulated his theory of a stressed bond which, despite its simplicity, still constitutes the basis for most models of chemical reactivity under stress [92], In order to fracture an unstressed bond which, in the absence of any vibration, is approximated by the Morse potential of Fig. 18, an energy D must be supplied. If, however, the bond is under tension due to a constant force feitt pulling on either end, the bond rupture activation energy will be decreased by an amount equivalent to the work performed by the mechanical force over the stretching distance from the equilibrium position. The bond potential energy in the presence of stress is given by ... 

Many physical and process constraints limit the cycle time, where cycle time was defined as the time to the maximum exotherm temperature. The obvious solution was to wind and heat the mold as fast and as hot as possible and to use the polymer formulation that cures most rapidly. Process constraints resulted in a maximum wind time of 3.8 minutes where wind time was defined as the time to wind the part plus the delay before the press. Process experiments revealed that inferior parts were produced if the part gelled before being pressed. Early gelation plus the 3.8 minute wind time constrained the maximum mold temperature. The last constraint was based upon reaction wave polymerization theory where part stress during the cure is minimized if the reaction waves are symmetric or in this case intersect in the center of the part (8). The epoxide to amine formulation was based upon satisfying physical properties constraints. This formulation was an molar equivalent amine to epoxide (A/E) ratio of 1.05. 

Cations can be seen as acting as ionic crosslinks between polyanion chains. Although this may appear a naive concept, crosslinking can be seen as equivalent to attractions between polyions resulting from the fluctuation of the counterion distribution (Section 4.2.13). Moreover, it relates to the classical theory of gelation associated with Flory (1953). Divalent cations (Zn and Ca +) have the potential to link two polyanion chains. Of course, unlike covalent crosslinks, ionic links are easily broken and re-formed under stress there could therefore be chain slipping and this may explain the plastic nature of zinc polycarboxylate cement. 

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

To explain the difference between the experimental results and theory, Doherty et al. (4J have given an empirical and a theoretical hypothesis. The theoretical hypothesis concerns the question of the meaning to be attached to the concept of the "equivalent random link" in the statistical theory of the randomly-jointed chain. According to Doherty et al., the assumption that the optical properties of the chain are describable by a randomly jointed model, using the same value of n, as for the description of stress has no strictly logical foundation. 

Transport phenomena modeling. This type of modeling is applicable when the process is well understood and quantification is possible using physical laws such as the heat, momentum, or diffusion transport equations or others. These cases can be analyzed with principles of transport phenomena and the laws governing the physicochemical changes of matter. Transport phenomena models apply to many cases of heat conduction or mass diffusion or to the flow of fluids under laminar flow conditions. Equivalent principles can be used for other problems, such as the mathematical theory of elasticity for the analysis of mechanical, thermal, or pressure stress and strain in beams, plates, or solids. 

It should be stressed that the relation between the SCRF and SAPT approaches is not obvious, as the former describes the solvation energetics in terms of the free energy of solvation at a finite temperature T, while in the latter one considers the interaction energy between the molecule of the solute and all molecules of the solvent at T = 0 K. One should also note that in the SCRF theory the solvent is modeled by a polarizable continuum, so the SCRF Hamiltonian is semiempirical. Still, by assuming a discrete equivalent of the SCRF Hamiltonian one can get approximate relations between SAPT and SCRF at T = 0 K. A SAPT analysis of the free energy of solvation AG within the SCRF method was reported in Ref. (234). It was shown that the free energy of solvation AG is given by,... 

This is because although 0 = (10), in general, cr(10) oQ (it will usually be less). In principle, the quantities we have defined, E(t), Dit), Gif), and J(i), provide a complete description of tensile and shear properties in creep and stress relaxation (and equivalent functions can be used to describe dynamic mechanical behavior). Obviously, we could fit individual sets of data to mathematical functions of various types, but what we would really like to do is develop a universal model that not only provides a good description of individual creep, stress relaxation and DMA experiments, but also allows us to relate modulus and compliance functions. It would also be nice to be able formulate this model in terms of parameters that could be related to molecular relaxation processes, to provide a link to molecular theories. 

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