3 Experimental Observations on the Fracture of Metals 23 Fig. 3.3 Illustration of three scales of experimental observation of images captured in Experimental Mechanics unique geometric reference system. One has to select a given RVE scale and the corresponding coordinate system to define local variables, choosing either a Lagrangian or a Eulerian representation. To discuss this aspect, we recall Fig. 3.3 from [20]. Figure 3.3 introduces graphically the relationship of measurement of mechanical properties at three different scales. In applications, many different scales can be introduced, each one will provide different aspects of the kinematics and dynamics of the observed materials. The observed fields are scale dependent and also depend on the spatial resolution that can be achieved. The same field observed with different spatial resolution will provide different results depending on the gradients of selected variables in the field of interest. At a given scale, the behavior of a material may appear to be in the field of quasielastic behavior, while at a different scale one can observe local transition to plasticity or transition of plasticity to fracture. The following relationships between scales should be valid. σm ij (x) =<σ μ ij (x)>= 1 VR VR σ μ ij (xR) (3.12) εmij(x) =<ε μ ij (x)>= 1 VR VR ε μ ij (xR) (3.13) The meaning of Eqs. (3.12 and 3.13) is that the field average at a lower scale should give the values of the stresses and strains at the corresponding points of the upper scale. Another requirement needs to be satisfied. It is necessary to adopt a stress tensor and a strain tensor compatible with each other in the Mandel-Hill sense. In Experimental Mechanics, since images are obtained in the deformed state, it is more convenient to work with the Eulerian description. In this case, the selected strain tensor should be compatible with the Eulerian description, and when strain tensor is selected, the stress tensor should be compatible with the strain tensor in the Hill-Mandel sense. 3.4 Transitions to Plasticity In this section and following sections, we are dealing with space-temporal transitions of tensile specimens from elasticity to plasticity and the transition of plasticity to fracture. All these transitions are affected by the applied strain-rates to the specimen as well as by the interactions between testing machine and specimen. The effect of these two factors are outside the scope of this study. Examples will be analyzed that have been subjected to different strain rates chosen for research reasons other than these two mentioned variables. Indeed, these two factors may have influenced the analyzed images, but their effects are not discussed in the present work. In Sect. 3.2, it has been established that the elastic behavior of a metal is characterized by an affine transformation making the distribution of displacements proportional to applied load. From the point of view of the kinematic of the continuum, variables are linearized expressions of strains, the simplified expression of the rigid body rotations, and the Hooke’s law of that relates strains to stresses. The actual stress tensor utilized is the Cauchy stress tensor that is compatible with the Eulerian
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