22 C. A. Sciammarella et al. |d| = u2 +v2 = P AcE δ 2 v +ν 2 δ 2 u 1 2 =KP (3.8) In Eq. (3.8), Pis the load applied to the specimen, Ac is the cross section of the specimen, Eis the modulus of elasticity of the specimen, and ν is the Poisson’s ratio. The modulus of the vector is proportional to the applied load, where K is a constant for the whole field. The angle of inclination of the displacement vector is θd =arctg δv νδu (3.9) The observed patterns are affine transformations of the same configuration and will remain this way until the metal yields. Then, yielding is characterized by the configuration changes of moiré fringes of projected displacement. This conclusion is true for any type of fringe patterns that are linearly dependent on the applied loads. The above conclusion can be mathematically expressed by the following conditions. An affine transformation is defined in 2D as x1 (X1, X2, t) x2 (X1, X2, t) = F11(t) F12(t) F21(t) F22(t) X1 X2 + c1T(t) c2T(t) (3.10) In this equation, xi (i =1,2) represent the Eulerian coordinates of a 2D medium in the deformed position, Xi represent the Lagrangian coordinates of the point in the undeformed position, and “t” is a parameter that can be the time but is related to the load, by P=k×t (3.11) The meaning of Eq. (3.11) is that the increment of the load is the product of a constant k multiplied the parameter t. The functions ciT(t) are arbitrary translations. The functions Fij(t) are the configurations of projected displacements (i,j =1,2). Provided that the above conditions are satisfied in a metal subject to loading, the analyzed body remains elastic. In the moment that the functions that relate projected fringes and loads become function of the locationXij and the parameter t, that is Fij =F(Xij, t) or ciT =(Xij, t), the metallic specimen enters the plastic state of deformation. Hence, nonlinear changes of fringe patterns of the projected displacement are an indication of the onset of plastic deformation of the observed metallic component. The onset of the plastic deformations is the first step in a process that ends with the actual failure of a metallic component. The failure can manifest itself in two ways, plastic collapse of the part where the part becomes a mechanism in motion under the collapse load. The other possibility is that the onset of plasticity is followed by actual fracture of the component. 3.3 Scale Dependence of the Experimental Observations of the Transitions to Plasticity and Fracture Before dealing with the subject matter of the paper, it is necessary to make some consideration concerning the scale dependence of the observed experimental information and resolution dependence. One should understand that described phenomena, plasticity and fracture, depend on the observation scales. Getting quantitative information depends on the selection of correct kinematics and dynamic variables. Transition from linearized strain and stress tensors to the nonlinear ones poses important and significant differences in data analysis. Also spatial resolution plays a very important role: if the spatial resolution or displacement resolutions are both low or one of them is low, only average values will be detected. The representative volume element (RVE) is a link between the discontinuous nature of materials and Continuum Mechanics. Materials are defined by their mechanical properties. These properties represent in Continuum Mechanics averaged values at certain subscale. The averages are computed at an area or volume with a given shape that for convenience in 2D is a square and in 3D a cube. Conditions that correspond to a given RVE are selected on the basis of the Hill-Mandel [12, 13], homogenization principle. It states that for given σrve and εrve, stresses and strains of the RVE, the virtual work in the macroscale equals the virtual work in the subscale. This principle is of great significance. The failure of fulfilling it results in important errors in the values of computed quantities as pointed out in [14–19]. The measurement of local kinematic and dynamic variables for a given RVE with large deformations and rotations requires removing limitations due to linearized kinematics and dynamics variables. Displacement fields cannot be described in a
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