20 C. A. Sciammarella et al. the J-integral formulation. In both elastic and elastoplastic approaches, instability of the crack propagation is based on the balance of the elastic energy stored in the material and the energy required to form new crack surfaces. In the approach of the atomistic-quasi-continuum mechanics, transitions from plasticity to fracture are also formulated on the basis of the energy analysis. There is an alternative approach to the formulation of the onset of plasticity and of the transition from plasticity to fracture based on the gauge theoretical formalism of physical mesomechanics. Based on an initial work of V.E. Panin [1], it has been further developed and experimentally supported by the work of S. Yoshida and his collaborators. In this approach, the relationship between materials science and continuum kinematics is based on the utilization of Maxwell equations to formulate the relationship of the microstructure with the kinematics of the continuum. A very interesting yield of this method is the experimental tool utilized to connect theoretical predictions with experimental observations [2–6]. This tool is termed as cine-speckle interferometry and is an independent discovery of a family of fringes initially introduced by A.J. Durelli and V.J. Parks [7]. These fringes are space derivatives of the isothetic lines (moiré fringes). Additional applications of this family of fringes can be found in [8–11]. The present study attempts to reformulate this approach fostering a closer connection between materials sciences and continuum kinematics. The final goal is to develop a practical tool to experimentally determine the transitions to plasticity and from plasticity to fracture in metals. These transitions are dynamic events that can be observed via optical methods of Experimental Mechanics. This approach opens the possibility of establishing the connection of microstructure and continuum mechanics predictions on quantitative basis. The developments presented in this paper are restricted to metals and concentrate on tensile tests results. 3.2 Analysis of the Displacement Field of a Tensile Specimen Let us review some fundamental aspects of the kinematics of tensile specimens. One of the manifestations of the onset of plasticity in metals from the Experimental Mechanics point of view is the change in shape of projected displacements fringe patterns, isothetic lines, or moiré fringes. In the case of materials like crystalline metals, the linear relationship between applied loads and fringe patterns implies that the geometry of the isothetic lines is preserved in the following sense. The distribution of displacements and associated loads remains proportional, or also one can say displacements and loads are linearly correlated. Increasing the load, the displacements increase in the same proportion. As a consequence of this proportionality, the ratios of displacements for successive loadings at a given point remain constant throughout the entire body, and this ratio will be in the same proportion of the ratio of the applied loadings but different for different points. Since isothetic lines represent loci of equal projected displacements and the gradients reflect changes in these projected displacements, the resulting configuration of fringes by increasing the load will reflect this constant ratio for the entire region under observation. A simple example, a tensile specimen of constant section subjected to axial load in the elastic range, will be analyzed in what follows (Fig. 3.1). According to the basic law of isothetic lines, the displacement vectors uand v have moduli, |u| =p (3.1) |v| =p (3.2) In Eqs. (3.1) and (3.2), p is the pitch of the printed cross-grating on the surface of a tensile specimen. The modulus of the gradient in the x-direction can be computed by the approximate expression, εx = ∂u ∂x ≈ p δu (3.3) And in the y-direction, εy = ∂v ∂y ≈ p δv (3.4) The displacement vector is d=u i +vj (3.5)
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