Cohesive crack tip modelling: size-scale transition from ductile to brittle failure

Carpinteri, Alberto

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 1 (1990) no. 1, pp. 59-79.

Abstract (VO)

The nature of the crack and the structure behaviour can range from ductile to brittle depending on tensile strength and fracture toughness of the material, as well as on the size-scale of the solid body. Strength and toughness present in fact different physical dimensions and any consistent fracture criterion must consider energy dissipation both per unit of volume and per unit of crack area. A cohesive crack model is proposed aiming at describing the size effects of fracture mechanics, i.e., the transition from ductile to brittle global behaviour caused by increasing the size-scale and keeping the geometrical shape unchanged. For extremely brittle cases (e.g., initially uncracked specimens, large and/or slender structures, low fracture toughness, high tensile strength, etc.) a snap-back instability in the load-deflection path occurs. If the loading process is deflection-controlled, the loading capacity presents a discontinuity with a negative jump. It is proved that such a catastrophical event tends to reproduce the classical LEFM-instability (\( K_{I} = K_{IC} \)).

Riassunto

Il comportamento di una fessura e del solido che la contiene può variare da duttile a fragile, in funzione della resistenza alla trazione e della tenacità alla frattura del materiale, così come delle dimensioni del solido stesso. Resistenza e tenacità presentano infatti dimensioni fisiche diverse, e un criterio di rottura coerente dovrebbe prevedere dissipazione energetica sia nell'unità di volume del solido che sull'unità di superficie della fessura che si forma. Nella presente nota si propone un modello matematico della fessura dotato di forze coesive, le quali simulano gli effetti plastici e permettono di descrivere la transizione duttile-fragile che si verifica all'aumentare delle dimensioni strutturali, pur mantenendo invariata la forma geometrica del solido. Per casi di estrema fragilità (es., solidi cristallini, strutture grandi e/o snelle, basse tenacità alla frattura, alte resistenze alla trazione), si evidenzia un fenomeno di instabilità catastrofica nel diagramma forza-spostamento. Se il processo di caricamento avviene imponendo una crescita monotona allo spostamento, la capacità portante in funzione dello spostamento mostrerà una discontinuità con un salto negativo. Si verifica che tale evento tende a riprodurre la ormai classica instabilità della Meccanica della Frattura Elastica Lineare (\( K_{I} = K_{IC} \)).

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