Finite difference method for solving crack problems in a functionally graded material

SIMULATION ◽  
2018 ◽  
Vol 95 (10) ◽  
pp. 941-953 ◽  
Author(s):  
A Dorogoy
Keyword(s):  
Finite Difference ◽  
Functionally Graded Materials ◽  
Functionally Graded ◽  
Solution Technique ◽  
Test Problems ◽  
Two Dimensional ◽  
Graded Materials ◽  
Finite Difference Technique ◽  
Analytical Results ◽  
Crack Problems

A linear elastic two-dimensional formulation for functionally graded materials is presented. The two-dimensional equilibrium equations and boundary conditions in an orthogonal curvilinear coordinate system are written explicitly. The finite difference technique is used to solve the above formulation. The solution technique is verified by solving two test problems, in which the material is graded horizontally and vertically. The results are compared to analytical results and have very good agreement. The solution technique is then applied to solve a long layer containing an edge crack in which it is assumed that the Young’s modulus varies continuously along its width. The problem is solved for two loading conditions: tension and bending. The mode I stress intensity factor is extracted by applying three methods: J line and two versions of a modified conservative J integral for graded materials. All three methods provide similar results, which are in excellent agreement with the semi-analytical results in the literature. These results demonstrate the applicability of the finite difference technique for solving crack problems in functionally graded materials.

scholarly journals A fracture mechanics model for a crack problem of functionally graded materials with stochastic mechanical properties

2012 ◽  
Vol 468 (2146) ◽  
pp. 2939-2961 ◽  
Author(s):  
Licheng Guo ◽  
Zhihai Wang ◽  
Naotake Noda
Keyword(s):  
Mechanical Properties ◽  
Fracture Mechanics ◽  
Functionally Graded Materials ◽  
Small Sample ◽  
Functionally Graded ◽  
Phase Volume ◽  
Graded Materials ◽  
Volume Fractions ◽  
Mechanics Model ◽  
Crack Problems

This study aimed to develop a method to build a ‘bridge’ between the macro fracture mechanics model and stochastic micromechanics-based properties so that the macro fracture mechanics model can be expanded to the fracture mechanics problem of functionally graded materials (FGMs) with stochastic mechanical properties. An analytical fracture mechanics model is developed to predict the stress intensity factors (SIFs) in FGMs with stochastic uncertainties in phase volume fractions. Considering the stochastic description of the phase volume fractions, a micromechanics-based method is developed to derive the explicit probabilistic characteristics of the effective properties of the FGMs so that the stochastic mechanical properties can be combined with the macro fracture mechanics model. A thought for choosing the samples efficiently is proposed so that the stable probabilistic characteristic of SIFs can be obtained with a very small sample size. The probability density function of SIFs can be determined by developing a histogram from the generated samples. The present method may provide a thought to establish an analytical model for the crack problems of FGMs with stochastic properties.

TRANSIENT HEAT CONDUCTION IN FUNCTIONALLY GRADED MATERIALS

2007 ◽  
Vol 04 (04) ◽  
pp. 603-619 ◽  
Author(s):  
S. M. HAMZA-CHERIF ◽  
A. HOUMAT ◽  
A. HADJOUI
Keyword(s):  
Heat Conduction ◽  
Temperature Distribution ◽  
Functionally Graded Materials ◽  
Degrees Of Freedom ◽  
Functionally Graded ◽  
P Element ◽  
Transient Temperature ◽  
Test Problems ◽  
Shape Functions ◽  
Graded Materials

The h-p version of the finite element method (FEM) is considered to determine the transient temperature distribution in functionally graded materials (FGM). The h-p version may be regarded as the marriage of conventional h-version and p-version. The graded Fourier p-element is used to set up the two-dimensional heat conduction equations. The temperature is formulated in terms of linear shape functions used generally in FEM plus a variable number of trigonometric shape functions representing the internal degrees of freedom (DOF). In the graded Fourier p-element, the function of the thermal conductivity is computed exactly within the conductance matrix and thus overcomes the computational errors caused by the space discretization introduced by the FEM. Explicit and easily programmed trigonometric enriched capacitance, conductance matrices and heat load vectors are derived for plates and cylinders by using symbolic computation. The convergence properties of the h-p version proposed and the results of the numbers of test problems are in good agreement with the analytical solutions. Also, the effect of the non-homogeneity of the FGM on the temperature distribution is considered.

Improved crack analysis of two-dimensional functionally graded materials by enriched natural element method

2020 ◽  
Vol 234 (10) ◽  
pp. 2012-2023
Author(s):  
Jin-Rae Cho
Keyword(s):  
Stress Intensity ◽  
Functionally Graded Materials ◽  
Displacement Field ◽  
Stress Intensity Factors ◽  
Functionally Graded ◽  
Two Dimensional ◽  
Intensity Factors ◽  
Graded Materials ◽  
Natural Element Method ◽  
Natural Element

The numerical calculation of stress intensity factors of two-dimensional functionally graded materials is introduced by an enriched Petrov–Galerkin natural element method (enriched PG-NEM). The overall trial displacement field is basically approximated in terms of Laplace interpolation functions and it is enriched by the near-tip asymptotic displacement field. The overall strain and stress fields which were approximated by PG-NEM were smoothened and enhanced by the patch recovery. The modified interaction integral [Formula: see text] is used to evaluate the stress intensity factors of functionally graded materials with the spatially varying elastic modulus. The validity of present method is justified through the evaluation of crack-tip stress distributions and the stress intensity factors of four numerical examples. It has been found that the proposed method effectively and successfully captures the near-tip stress singularity with a remarkably improved accuracy, even with the remarkably coarse grid, when compared with an extremely fine grid and the analytical and numerical reference solutions.

Evaluation of Fracture Toughness of Ceramic-Metal Functionally Graded Materials

2010 ◽  
Vol 123-125 ◽  
pp. 971-974
Author(s):  
Sheikh Md. Rasel ◽  
Foisal Ahmed Mirza ◽  
Ali Md. Afsar ◽  
Jung I. Song
Keyword(s):  
Compressive Load ◽  
Functionally Graded ◽  
Correlation Technique ◽  
Two Dimensional ◽  
Graded Materials ◽  
Three Point Bending ◽  
Initiation And Propagation ◽  
Crack Problems ◽  
Growth Initiation ◽  
Model Crack

The main objective of this study is to examine the two dimensional surface crack problems in a system with an interface between two elastic-plastic solids of different yield strength subjected to mode I mechanical loading. The surface cracks growth is considered to occure along the interface direction of bimaterials which is perfectly bonded to each others. A two dimensional finite elementmethod is used to solve the structural problem. Solid 183-node elements are utilized to simulate the strain singularity around the crack front. The crack surface is subjected to a compressive load by three point bending. The stress intesity factors are computed by using the displacement correlation technique. The primary goal is to develop a model crack tip stresses and strains in a manner that is useful for crack growth initiation and propagation in a FGM.

Thermoelastic Fracture Mechanics for Nonhomogeneous Material Subjected to Unsteady Thermal Load

1999 ◽  
Vol 67 (1) ◽  
pp. 87-95 ◽  
Author(s):  
B. L. Wang ◽  
J. C. Han ◽  
S. Y. Du
Keyword(s):  
Functionally Graded Materials ◽  
Material Properties ◽  
Crack Problem ◽  
Laminated Composite ◽  
Functionally Graded ◽  
Comprehensive Treatment ◽  
Numerical Illustration ◽  
Graded Materials ◽  
Nonhomogeneous Material ◽  
Crack Problems

This article provides a comprehensive treatment of cracks in nonhomogeneous structural materials such as functionally graded materials. It is assumed that the material properties depend only on the coordinate perpendicular to the crack surfaces and vary continuously along the crack faces. By using a laminated composite plate model to simulate the material nonhomogeneity, we present an algorithm for solving the system based on the Laplace transform and Fourier transform techniques. Unlike earlier studies that considered certain assumed property distributions and a single crack problem, the current investigation studies multiple crack problems in the functionally graded materials with arbitrarily varying material properties. The algorithm can be applied to steady state or transient thermoelastic fracture problem with the inertial terms taken into account. As a numerical illustration, transient thermal stress intensity factors for a metal-ceramic joint specimen with a functionally graded interlayer subjected to sudden heating on its boundary are presented. The results obtained demonstrate that the present model is an efficient tool in the fracture analysis of nonhomogeneous material with properties varying in the thickness direction. [S0021-8936(00)01601-9]

Sign in / Sign up

Export Citation Format

Share Document