Cohesive fracture modeling of elastic–plastic crack growth in functionally graded materials
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.
Section snippets
Stress–strain curves for ceramic/metal FGMs
While the classical Hooke’s law describes the linear-elastic response of FGMs with the elastic properties evaluated approximately by micromechanics models for conventional composites, determination of the elastic–plastic behavior of FGMs remains a challenging task. Previous studies [8], [10], [11] have adopted the J2 flow theory for ceramic/metal FGMs and evaluated the material properties (yield stress and tangent modulus) using the volume fraction based model proposed by Tamura et al. [9] (TTO
Cohesive zone model for FGMs
While the cohesive zone approach has proven a convenient and effective method to simulate and analyze crack growth in homogeneous materials, generalization of the cohesive zone concept to model fracture in FGMs represents a challenging task because of the complicated microstructures and the related failure mechanisms in FGMs. Jin et al. [18] proposed a volume fraction based, phenomenological cohesive fracture model suitable for engineering scale applications. Such volume fraction based formulae
Graded solid and interface elements formulation
This section describes the small-displacement formulation of both the 3-D solid element and interface-cohesive element with graded material properties (graded elements). Previous studies [16], [26] of crack growth in thin aluminum panels using 3-D cohesive elements show that a small-displacement scheme yields a slightly lower calibrated peak cohesive traction than the value obtained in a finite-deformation framework due to the thickness reduction effect predicted in the finite-deformation
Specimen geometries, materials, and finite element models
We performed 3-D numerical analyses of elastic–plastic crack growth for both Ti metal and TiB/Ti FGM SE(B) specimens containing an initially sharp, straight crack front over the thickness. Fig. 7 shows the geometry of the SE(B) specimens used in the crack growth study. A layered TiB/Ti FGM SE(B) specimen has been recently tested as described in [8] and a Ti metal only SE(B) specimen tested as described in [19]. The company CERCOM Inc. developed the TiB/Ti FGM system in a layered structural form
Concluding remarks
This study employs a new phenomenological cohesive zone model [18] and extends the TTO model [9] within a 3-D computational framework that includes graded solid and interface-cohesive elements to investigate elastic–plastic crack growth in Ti metal and TiB/Ti FGM SE(B) specimens. The cohesive zone model involving 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)
Acknowledgements
This work was supported by the NASA-Ames “Engineering for Complex System Program”, and the NASA-Ames Chief Engineer’s Office (Dr. Tina Panontin) through grant NAG 2-1424. Additional support for this work was provided by the National Science Foundation (NSF) under grant CMS-0115954 (Mechanics & Materials Program).
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