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Finite element analysis of cracked elastic bodies under unsteady translation and rotation

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Summary

The broad objective of the investigations described in this report is the accurate finite element computation of stress intensity factors in cracked elastic bodies under steady or unsteady translational and rotational loads. Applications of interest include fatigue under adverse environments, such as stall in turbomachinery and critical speeds in elastic mechanisms. The enriched element formulation of Gifford and Hilton [1] has been extended to include inertial, centrifugal, Coriolis and angular acceleration effects, and to include Mode III (tearing), which is coupled to Modes I and II under unsteady rotation. The extension is based on the fact, which does not appear to be widely appreciated, that inertial effects near the crack tip are completely accounted for by the time dependence of the stress intensity factors. For Mode III, the Mindlin formulation for a plate in bending has been adopted in order to extend the enriched element formulation to torsion and bending effects. The centrifugal and angular acceleration effects are fully coupled to the displacements. A finite element code by Gifford and Hilton [2] for static problems in Modes I and II has been thoroughly rewritten and expanded to implement the dynamic features and Mode III. The code incorporates matrices and vectors representing translational, Coriolis, centrifugal and angular acceleration effects. It implements the Newmark method for time integration, and a non-symmetric ‘wave front’ solver. The elements are validated by comparison with several benchmark problems.

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Abbreviations

A :

element area

A *,B *,C *,D *,E * :

shape functions of singular displacements

B :

elastic body

B′:

undeformed elastic body

B :

strain-displacement matrix

D :

equivalent viscous damping matrix

D :

material stiffness matrix

E :

elastic modulus

f b :

balancing force

f c :

consistent force

f ω :

force due to centrifugal acceleration

f α :

force due to angular acceleration

H :

centrifugal matrix

h :

thickness of element

J :

torsion constant

K :

stiffness matrix

K I,K II,K III :

stress intensity factor in Mode I, II and III

M :

consistent mass matrix

N :

composite shape function

N i :

shape function

r, θ:

polar coordinates

S :

surface

t, τ:

time

t :

surface traction vector

u i ,v i ,w i :

x i ,y i andz i displacement in element

u n :

nonsingular part of displacement field

u s :

singular part of displacement field

u, v, w :

cartesian displacement components

V :

volume

x, y, z :

Cartesian coordinates

x i ,y i ,z i :

X, Y, Z coordinate fori th node in each element

ζ, ξ:

natural coordinates

α:

angular acceleration

β:

crack angle

\(\dot u\) :

time derivative ofu

δr :

variation ofr

ɛ:

strain tensor

Γ:

Coriolis matrix

γ:

vector of nodal + nodeless unknowns

Λ:

angular acceleration matrix

μ:

shear modulus

ν:

Poisson's ratio

ω:

angular velocity

ϱ:

density of material

σ:

stress tensor

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Author notes

  1. C. -Y. Gau

    Present address: Research Associate, Design and Manufacturing Institute, Dept. of Mechanical Engineering, Stevens Institute of Technology, 07030, Hoboken, NJ, USA

  2. D. W. Nicholson

    Present address: Dept. of Mechanical and Aerospace Engineering, University of Central Florida, 32816, Orlando, FL, USA

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    1. C. -Y. Gau
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    2. D. W. Nicholson
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    Gau, C.Y., Nicholson, D.W. Finite element analysis of cracked elastic bodies under unsteady translation and rotation. Acta Mechanica 101, 111–137 (1993). https://doi.org/10.1007/BF01175601

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