TY - JOUR

T1 - Phase field modeling of fracture in Functionally Graded Materials

T2 - Γ-convergence and mechanical insight on the effect of grading

AU - Asur Vijaya Kumar, P. K.

AU - Dean, A.

AU - Reinoso, J.

AU - Lenarda, P.

AU - Paggi, M.

N1 - Funding Information: JR is grateful to the Consejería de Economía y Conocimiento of the Junta de Andalucía (Spain) for financial support under the contract US-1265577 -Programa Operativo FEDER Andalucía 2014–2020. JR acknowledges Prof. Dominique Leguillon (CNRS - Sorbonne Universit?) for inspiring and fruitful discussions on this matter. MP would like to acknowledge the financial support of the Italian Ministry of Education, University and Research to the Research Project of National Interest (PRIN 2017) XFAST-SIMS “Extra fast and accurate simulation of complex structural systems”, CUP: D68D19001260001.

PY - 2021/2

Y1 - 2021/2

N2 - A phase field (PF) approximation of fracture for functionally graded materials (FGM) using a diffusive crack approach incorporating the characteristic length scale as a material parameter is herein proposed. A rule of mixture is employed to estimate the material properties, according to the volume fractions of the constituent materials, which have been varied according to given grading profiles. In addition to the previous aspects, the current formulation includes the internal length scale of the phase field approach variable from point to point, to model a spatial variation of the material strength. Based on the ideas stemming from the study of size-scale effects, Γ-convergence for the proposed model is proved when the internal length scale is either constant or a bounded function. In a comprehensive sensitivity analysis, the effects of various model parameters for different grading profiles are analyzed. We first prove that the fracture energy and the elastic energy of FGM is bounded by their homogeneous constituents. Constitutive examples of boundary value problems solved using the BFGS solver are provided to bolster this claim. Finally, crack propagation events in conjunction with the differences with respect to their homogeneous surrogates are discussed through several representative applications, providing equivalence relationships for size-scale effects and demonstrating the applicability of the current model for structural analysis of FGMs.

AB - A phase field (PF) approximation of fracture for functionally graded materials (FGM) using a diffusive crack approach incorporating the characteristic length scale as a material parameter is herein proposed. A rule of mixture is employed to estimate the material properties, according to the volume fractions of the constituent materials, which have been varied according to given grading profiles. In addition to the previous aspects, the current formulation includes the internal length scale of the phase field approach variable from point to point, to model a spatial variation of the material strength. Based on the ideas stemming from the study of size-scale effects, Γ-convergence for the proposed model is proved when the internal length scale is either constant or a bounded function. In a comprehensive sensitivity analysis, the effects of various model parameters for different grading profiles are analyzed. We first prove that the fracture energy and the elastic energy of FGM is bounded by their homogeneous constituents. Constitutive examples of boundary value problems solved using the BFGS solver are provided to bolster this claim. Finally, crack propagation events in conjunction with the differences with respect to their homogeneous surrogates are discussed through several representative applications, providing equivalence relationships for size-scale effects and demonstrating the applicability of the current model for structural analysis of FGMs.

KW - Finite element method

KW - Fracture mechanics

KW - Functionally Graded Materials

KW - Phase field

KW - γ-convergence

UR - http://www.scopus.com/inward/record.url?scp=85095572320&partnerID=8YFLogxK

U2 - 10.1016/j.tws.2020.107234

DO - 10.1016/j.tws.2020.107234

M3 - Article

AN - SCOPUS:85095572320

VL - 159

JO - Thin-walled structures

JF - Thin-walled structures

SN - 0263-8231

M1 - 107234

ER -