24 C. A. Sciammarella et al. Fig. 3.4 Load vs. displacement plot for a tensile specimen description of the continuum [20]. However, in the linearized Continuum Mechanics, the distinction between the Lagrangian and Eulerian description is dropped. Hence, in dealing with plasticity, one must be careful in the selection of variables to discuss the transition to plasticity. The transition implies a change in the description of the continuum and consequently of the variables that define a material between the two different behaviors. This is an important point to be considered if one wants to define the transition of state using displacement patterns. A representation of force vs. displacement, Fig. 3.4, provides the classical method to detect transition from the elastic regime to the plastic regime. With a certain precision a line with the slope Es (N/mm) is drawn, the elastic regimen is established and separated from the plastic regimen that at each point can be characterized by the local slopes. As explained in Sect. 3.2, for a given displacement δ (mm), the corresponding force is P(N) =Es N mm×δmm (3.14) In Eq. (3.14), Es is the slope of the plot. A similar criterion can be applied for a 2D displacement distribution utilizing properties of the isothetic lines (moiré fringes). In [7], it was introduced the family of fringes of equal projected displacements calling them isotachis fringes. Also, families of the derivatives of the isotachis fringes were introduced calling them isotachis of the partial derivatives. In this paper, we will call these fringes iso-derivatives fringes. Calling U(x,P) and V(x,P) the projected displacements with respect to the coordinates, the symbol boldxrepresents the vector x=xi +yj. The notations U(x,P) andV(x,P,) will be utilized to indicate families of fringes, u(x,P) andv(x,P) for individual values. As the applied load or displacement on a body is increased, the displacement field is modified. This change results in a change of the spatial velocity of the displacement field at a given point. In the linear case, for small deformation and rotation theory, these changes are components of the strain tensor. In the general case, these derivatives are no longer strains, but we call them spatial velocities or changes of the projected displacements per unit of length. We introduce the notation for spatial derivatives ∂u(x, P) ∂x =εu (x, P) (3.15) ∂v(x, P) ∂y =εv (x, P) (3.16) Hence, it is important to make a distinction between iso-derivatives, or loci of the points with the same spatial velocity and iso-strain lines, since only for small rotations and small deformations the strain components are linear functions of the derivatives of the displacements. The moiré method provides the means to generate iso-derivative fringes, and since the common way in moiré method is to project displacements, there will be two families of iso-derivative fringes. Iso-derivatives fringes are important tools to define the transition from elasticity to plasticity in 2D or 3D.
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