2 S. Yoshida and T. Sasaki the field theory. In this study, we numerically solve the wave equation derived by the field theory and compare the resultant behavior of the displacement field with experimental results. 1.2 Theoretical Background Detailed description of the field theory can be found elsewhere [7]. In short, the theory derives a set of field equations that govern the elastoplastic dynamics of solid materials based on the physical principle known as local symmetry [8] and associated Lagrangian formalism. The wave equation can be put in the following form. ∂ 2−→v ∂t2 +σ ∇ · ∂−→v ∂t ∂−→v ∂t − G ρ ∇ 2−→v =− G ρ ∇ ∇ · −→v +α λ+2G ρ ∇ ∇ · −→v (1.1) Here −→v is the differential displacement vector, σ is the material constant that represents the energy dissipative nature, G is the shear modulus, α is a parameter that indicates the degree of elasticity in the elastoplastic regime (0 <α < 1), and λis Lamé’s first parameter. When wave Equation (1.1) is expressed in the principal coordinate system, the wave solution in each regime can be classified as shown in Table 1.1. In the linear elastic regime, the differential displacement field exhibits the well-known compression wave characteristics. In the principal coordinate system, the distortion tensor does not have shear components. In the elastoplastic regime, the distortion tensor components start to take non-zero values. This excites transverse waves in the differential displacement field. The irreversibility due to plasticity is reflected in the wave dynamics as the fact that the longitudinal and transverse waves decay. In the pre-fracture regime, the wave is concentrated in a certain location of the specimen and travels as a solitary wave. When the solitary wave becomes stationary, the fracture is in the final stage, and material discontinuity is generated at the point where the solitary wave becomes stationary. 1.3 Results and Discussions Figure 1.1 illustrates the experimental arrangement. A plate specimen is attached to a test machine for loadings. An in-plane sensitive dual-beam Electronic Speckle-Pattern Interferometer (ESPI) is configured to measure the differential displacement continuously as the test machine applies the load. At each time step, the current image is subtracted from the image taken in a previous time step. This generates the so-called subtraction fringe pattern as indicated in Fig. 1.1. The dark fringes in the fringe pattern represent the contour of the differential displacement occurring during the time interval between the two time steps involved in the image subtraction. Figure 1.2 shows typical fringe patterns obtained by the ESPI setup where a tensile load at a constant pulling rate is applied to the specimen (aluminum alloy A5083, 2%Mg). Labels (a)–(c) indicate the location on the stress-strain curve at which each group of fringe patterns are formed. The dark fringes represent the contour of the differential displacement vector component parallel to the applied load. The fringe patterns (a)–(c) can be characterized as follows. (a) It exhibits a uniformly distanced dark fringes running approximately perpendicular to the tensile axis. Since the dark fringes represent the differential displacement component parallel to the tensile axis, this pattern of the dark fringes indicate that the wave is longitudinal. At stage (b), the dark fringes become tilted to the tensile axis, indicating that the shear components become non-zero in the strain tensor. It also indicates that a transverse wave is being exited at this stage. The fringe pattern in stage (c) consists of curved dark fringes sandwiching parallel, slant linear fringes. These slant linear fringes can be interpreted as representing a solitary wave [7]. As the three fringe images indicate, the solitary wave travels along the specimen. The fact that the solitary wave is dynamic indicates that the specimen is not ready for the final fracture. Figure 1.3 shows fringe patterns from the same type of experiment as Fig. 1.2. In this case, both the displacement vector components parallel and perpendicular to the tensile axis were measured with an ESPI setup similar to Fig. 1.1 Table 1.1 Forms of differential displacement wave Regime Linear elastic Elastoplastic Pre-fracturing Wave Longitudinal waves Decaying longitudinal/transverse waves Solitary waves
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