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Implementation of the extended finite element method for coupled dynamic thermoelastic fracture of a functionally graded cracked layer

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Abstract

The effect of thermal shock on a cracked functionally graded material (FGM) layer is considered using the extended finite element method. Classical coupled thermoelastic equations are used in the calculations. The coupled dynamical system of equations obtained from the extended finite element discretization is solved by the Newmark method in the time domain. Micromechanical models for conventional composites are used to estimate the material properties of functionally graded layer. The interaction integral is then employed to calculate the dynamic thermal stress intensity factors (SIFs) at each time step. The effects of initial crack angle and volume fraction profiles of FGMs on SIFs are studied. Also crack propagation phenomenon is investigated in this paper. We have used MATLAB software to do the different stages of simulation from mesh generation to numerical computation of SIFs. Some numerical examples are implemented to investigate the validity and accuracy of attained results.

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Abbreviations

A :

Element area, area (m2)

\( A^{*} \) :

Area associated with the domain J-integral, area (m2)

a :

Crack length, length (m)

a :

Vector of nodal unknowns associated with FE shape functions, length (m)

Bf :

Body force vector, force (N/m3)

B x and B y :

Body force vector components, force (N/m3)

b :

Vector of nodal unknowns associated with heaviside enriched shape functions, length (m)

c :

Vector of nodal unknowns associated with crack tip enriched shape functions, length (m)

C :

Damping matrix

C ijkl :

Material constitutive matrix components (N/m2)

c t :

Specific heat capacity [J/(kg K)]

D :

Material modulus matrix (N/m2)

E :

Young’s modulus (N/m2)

F m :

Crack tip enrichment functions (m0.5)

Fr :

Force vector (N)

H :

Heaviside enrichment function, dimensionless

J :

J integral (N/m)

K :

Stiffness matrix

K eq :

Equivalent dynamic stress intensity factor (N/m1.5)

K I :

Mode I stress intensity factor (N/m1.5)

K IC :

Fracture toughness (N/m1.5)

K ID :

Fracture toughness (N/m1.5)

K II :

Mode II stress intensity factor (N m1.5)

k :

Thermal conductivity [W/(m K)]

L :

Length of specimen, length (m)

M :

Mass matrix (kg)

MI :

M-integral (N/m)

m :

Number of crack tip enrichment functions (m)

N :

FEM shape function, dimensionless

n x and n y :

Unit vectors in x and y directions

p :

Power exponent determining the volume fraction profiles, dimensionless

q :

Smoothing weight function, dimensionless

q i :

Component of heat flux vector per unit area (W/m2)

R :

Generated heat per unit volume (W/m3)

r :

Polar coordinate system component, length (m)

S :

Vector of XFE shape functions

T :

Temperature (K)

Tf :

Traction force vector (N/m2)

t :

Time (s)

t D :

Dimensionless time, dimensionless

t n x :

Traction vector components at x direction (N/m2)

t n y :

Traction vector components at y direction (N/m2)

u :

Displacement vector (m)

V :

Volume (m3)

V i :

Volume fraction of inclusion, dimensionless

W :

Height of specimen, length (m)

X :

Global Cartesian coordinate system component

x :

Local Cartesian coordinate system component

α :

Coefficient of thermal expansion (1/K)

β :

Coupling term [N/(m2 K)]

β i :

Universal functions, dimensionless

γ :

Coefficient of Newmark method, dimensionless

ζ :

Coefficient of Newmark method, dimensionless

δ ij :

Kronecker delta, dimensionless

ε :

Strain tensor, dimensionless

ε aux :

Auxiliary strain tensor, dimensionless

ε m ij :

Mechanical strain component, dimensionless

θ :

Temperature change (K)

μ :

Lamé constant (N/m2)

λ :

Lamé constant (N/m2)

υ :

Poisson’s ratio, dimensionless

ρ :

Density (kg/m3)

φ :

Polar coordinate system component, dimensionless

σ :

Stress tensor (N/m2)

σ aux :

Auxiliary stress tensor (N/m2)

Φ:

Enrichment shape function, dimensionless

Ψ:

Enrichment shape function, dimensionless

ω :

The angle between local and global coordinate systems, dimensionless

ω c :

Crack propagation direction, dimensionless

Δ:

Nodal displacements and temperature changes vector

h :

Relative to nodes in an element

i :

Relative to components of Cartesian coordinate system

j :

Relative to components of Cartesian coordinate system

l :

Relative to the shape functions

m :

Relative to crack tip enrichment functions

n :

Relative to nodes, time step and component of coordinate system

ne :

Relative to nodes in element e

ns :

Relative to XFE shape functions

tip:

Relative to crack tip

1:

Relative to the horizontal axis of coordinate system

2:

Relative to the vertical axis of coordinate system

aux:

Relative to auxiliary field

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Authors and Affiliations

  1. Mechanical Engineering Department, Shahrood University of Technology, 3619995161, Shahrood, Semnan, Iran

    Masoud Mahdizadeh Rokhi & Mahmoud Shariati

Authors
  1. Masoud Mahdizadeh Rokhi
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  2. Mahmoud Shariati
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Correspondence to Masoud Mahdizadeh Rokhi.

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Technical Editor: Lavinia Borges.

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Rokhi, M.M., Shariati, M. Implementation of the extended finite element method for coupled dynamic thermoelastic fracture of a functionally graded cracked layer. J Braz. Soc. Mech. Sci. Eng. 35, 69–81 (2013). https://doi.org/10.1007/s40430-013-0015-0

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  • DOI: https://doi.org/10.1007/s40430-013-0015-0

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