Cohesive fracture modeling of elastic–plastic crack growth in functionally graded materials

Abstract

This work investigates elastic–plastic crack growth in ceramic/metal functionally graded materials (FGMs). The study employs a phenomenological, cohesive zone model proposed by the authors and simulates crack growth by the gradual degradation of cohesive surfaces ahead of the crack front. The cohesive zone model uses six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) to describe the constitutive response of the material in the cohesive zone. A volume fraction based, elastic–plastic model (extension of the original Tamura–Tomota–Ozawa model) describes the elastic–plastic response of the bulk background material. The numerical analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. Numerical values of volume fractions for the constituents specified at nodes of the finite element model set the spatial gradation of material properties with isoparametric interpolations inside interface elements and background solid elements to define pointwise material property values. The paper describes applications of the cohesive zone model and the computational scheme to analyze crack growth in a single-edge notch bend, SE(B), specimen made of a TiB/Ti FGM. Cohesive parameters are calibrated using the experimentally measured load versus average crack extension (across the thickness) responses of both Ti metal and TiB/Ti FGM SE(B) specimens. The numerical results show that with the calibrated cohesive gradation parameters for the TiB/Ti system, the load to cause crack extension in the FGM is much smaller than that for the metal. However, the crack initiation load for the TiB/Ti FGM with reduced cohesive gradation parameters (which may be achieved under different manufacturing conditions) could compare to that for the metal. Crack growth responses vary strongly with values of the exponent describing the volume fraction profile for the metal. The investigation also shows significant crack tunneling in the Ti metal SE(B) specimen. For the TiB/Ti FGM system, however, crack tunneling is pronounced only for a metal-rich specimen with relatively smaller cohesive gradation parameter for the metal.

Introduction

Advances in material synthesis technologies have spurred the development of a new class of materials, called functionally graded materials (FGMs), with promising applications in aerospace, transportation, energy, electronics and biomedical engineering [1], [2], [3]. An FGM comprises a multi-phase material with volume fractions of the constituents varying gradually in a pre-determined (designed) profile, thus yielding a nonuniform microstructure in the material with continuously graded properties. In applications involving severe thermal gradients (e.g. thermal protection systems), FGMs exploit the heat, oxidation and corrosion resistance typical of ceramics, and the strength, ductility and toughness typical of metals. Damage tolerance and defect assessments for structural integrity of FGM components require knowledge of the fracture behavior of FGMs. For ceramic/metal FGMs, cracks generally nucleate near the ceramic surface exposed to the environment and then grow towards the metal side. When a crack extends into the metal rich region, the substantial plastic deformation in the background FGM invalidates simple crack growth models [4], [5] based on linear-elastic crack tip analysis [6], [7].

This work describes an investigation of crack growth in ceramic/metal FGMs undergoing plastic deformation in the background (bulk) material. The present study focuses on three-dimensional (3-D) numerical modeling of elastic–plastic crack growth and utilizes results of recent experimental studies on fracture in ceramic/metal FGMs [8] for parameter calibration. The plasticity formulation follows a composite model proposed by Tamura et al. [9] (referred to as the TTO model henceforth), which has been employed in the study of plastic deformation of FGMs in [8], [10], [11]. The simulation of crack growth involves the gradual degradation of surfaces along a cohesive zone ahead of the crack. The cohesive zone approach proves to be a convenient and effective method to simulate and analyze crack growth in ductile and quasi-brittle materials. In a cohesive zone model, a narrow-band termed a cohesive zone, or process zone, exists ahead of the crack front. Material behavior in the cohesive zone follows a nonlinear cohesive constitutive law which relates the cohesive traction to the relative displacements of the adjacent cohesive surfaces. Material separation and thus crack growth occurs as the progressive decay of the cohesive tensile and shear tractions across the cohesive surfaces. Dugdale [12] first proposed a cohesive type model to study ductile fracture in a thin sheet of mild steel. Cohesive zone models have been extended to study fracture processes in quasi-brittle materials such as concrete (see, e.g., [13], [14]), ductile metals (see, e.g., [15], [16]), and metal matrix composites [17]. In a recent work, Jin et al. [18] proposed a new phenomenological cohesive zone model for two phase FGMs, and used the model to investigate crack growth in compact tension, C(T), and single-edge notched bend, SE(B), specimens made of a titanium/titanium monoboride (Ti/TiB) FGM without considering plastic deformation in the background material.

While two-dimensional (2-D) models approximate the behavior of “very thick” (plane strain) or “very thin” (plane stress) cracked structures, 3-D models describe more realistically the elastic–plastic stress and deformation states in cracked test specimens and structural components. Moreover, 3-D analyses enable modeling of crack tunneling phenomenon which becomes significant in common laboratory specimens and in components with surface breaking defects. Here we apply a computational framework of 3-D solid and interface-cohesive elements to analyze elastic–plastic crack growth in ceramic/metal FGMs using the new cohesive zone model for FGMs of Ref. [18].

The paper is organized as follows. Section 2 reviews the elastic–plastic model for two-phase composites, including FGMs, proposed by Tamura et al. [9] (TTO model). We describe an extension of this model to incorporate more realistic power-law hardening behavior of the metal and metal-rich background material. Section 3 summarizes a new phenomenological cohesive zone model for FGMs, which was recently presented by the authors [18]. Section 4 describes the 3-D finite element formulation with graded solid and interface-cohesive elements tailored for applications to FGMs. Section 5 describes the procedures to calibrate the cohesive parameters and presents results of a parametric study of elastic–plastic crack growth analyses for an SE(B) specimen made of a TiB/Ti FGM system. Finally, Section 6 provides some concluding remarks.