PSI - Issue 42

Tamás Fekete et al. / Procedia Structural Integrity 42 (2022) 1467–1474 Tamás Fekete, Éva Feketéné-Szakos / Structural Integrity Procedia 00 (2019) 000–000

1472

6

In NLFTFM –see Eq. (5)–, the constitutive behavior of a SM is determined by the HP , which has a strong effect on the stability of the crack-front –see Eq. (6)–. The aim of the qualitative study was to find a HP , which: (1) is adequate to model the characteristic of elastic-plastic materials that, after the elastic deformation El ε , behave hardening plastic up to 1 cr pl ε , then gradually soften up to 2 cr pl ε , followed by a progressive softening up to 3 cr pl ε , where fast fracture occurs; (2) after appropriate transformations, is suitable to describe brittle behavior, i.e., the small plastic response up to 1 cr pl ε is followed by sudden fast fracture. Such a potential is known to exist. To meet (1) this potential must be non-convex, such as the Ginzburg-Landau Potential ( GBP ), which has been used for decades in phase transitions dynamics. The GBP is of the form ( ) 2 4 V a b ε ε ε = − ⋅ + ⋅ –see Fig. 1–, but it is unstable at the origin. Since SM s are stable in an unloaded state, a modified GBP is used that is stable at the origin. The initial stored energy density of SM is denoted by 0 E ⌢ , the absorbed energy density is St E ⌢ , as described in Eq. (7). The maximum value of St E ⌢ is Crit E , reached at 2 cr pl ε . Beyond 2 cr pl ε the SM loses its stability, and changes from being an energy absorber to an energy emitter, leading to progressive softening that ends with fast fracture at 3 cr pl ε . Graph of St E ⌢ is shown in Fig. 1, where its critical points are also shown. HP is defined by H Crit St E E E = − ⌢ ⌢ –see also Eq. (7)–. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 4 0 ( , ˆ, ) 0 0 0 ˆ ˆ ( , , , ) ( ) ˆ ˆ ˆ ˆ ˆ ( ) ( ) ( ) ( , , ), where , , , ˆ ˆ ( , , , ) ( , , , ) St E p E E pl pl pl pl G k C H E p i i H E p Crit St E p E E a a b E E E e E G k k E E E τ τ τ τ ε ε τ τ τ τ ε τ τ ε τ τ ε τ τ τ τ τ ε ε τ τ τ τ τ ε ε τ τ ε ε τ τ ∆ − ∆ ⋅∆   = + ⋅ + ⋅ − ⋅   → + ∆ = + ∆ = ∆ = −  ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ (7) Aging is described by transformations that shrink the HP , with all its critical points ( ) 1 2 3 , , , cr cr cr El pl pl pl ε ε ε ε to the point ( ) ( ) 0 ˆ , Crit E ε τ on the global time scale. At ( ) ( ) 0 ˆ , Crit E ε τ –which may be non-physical–, the system has its most degenerated singularity, called the Organizing Center ( OC ) –see Stewart (1981)–. If the dissipative dynamics in Eq. (5) is conformally invariant, the sequence of conformal transformations –see Eq. (7), middle, –see also Ne’eman, Hehl, Mielke (1993)– shifts the system persistently towards the OC . Thus, the triplet of originally separate singular points ( ) 1 2 3 , , cr cr cr pl pl pl ε ε ε –of which ( ) 3 cr pl ε is initially degenerate–, is merged with ( ) 2 cr pl ε , which then becomes even more degenerate. The emergence of degenerate singularities in the mathematical picture means that the processes around the critical point suddenly accelerate in SM s. Note that the OC concept –Stewart (1981)– is based on a very profound result of modern mathematics, Malgrange's preparation theorem –Malgrange (1964)–. The novelty of OC lies in the fact that it is used not to examine the singular points of a system separately –as is usual in stability theory–, but to find the system of singularities in it, and help sort and regularize them. To illustrate the discussed points, for a hypothetical SM , the St E ⌢ and H E ⌢ were determined according to the system of Eq. (7) in a new condition and during a hypothetical ageing process. Then the constitutive functions Σ were constructed from H E ⌢ . St E ⌢ with its critical points is shown in Fig. 2, Σ with its critical points in Fig. 3, while the crx pl ε locations –as a function of the global time ˆ τ , placed on the y-axis– are presented in Fig. 4.

Fig. 1. Landau-Ginzburg vs. the Griffith and the modified Landau-Ginzburg potential.