Investigation of Transient Creep in Thick-walled Tubes under Axially Symmetric Loading

  • J. F. Besseling
Conference paper
Part of the IUTAM Symposia book series (IUTAM)

Summary

The feasibility of a numerical analysis of nonstationary creep pro blems is investigated for thick-walled tubes under axially symmetric loading. It is shown how approximate solutions may be obtained with the aid of automatic digital computers by means of a numerical method ; based upon the application of an extremum principle for the rate of deformation. The primary creep phase can be included in the analysis if creep equations based upon the concept of microscopic inhomogeneity of the material give an adequate description of this phenomenon.

As an illustration of the method here presented the exact integrodifferential equation for the transition from the elastic to the steady state creep solution in the case of an incompressible material, showing only secondary creep governed by a power law in the stress, has been solved numerically for four combinations of the parameters.

Keywords

Dinate 

Nomenclature

A

matrix of coefficients of unknowns

C

bulk modulus of elasticity

G

modulus of rigidity

I2

second invariant of elastic strain tensor I 2 = 1/2 (e α β-ē α β) ( α -ēβ α)

J2

second invariant of strain rate tensor J 2 = 1/2 ė α β ėβ α

N

number of subelements of element of volume

T

absolute temperature

Tc

characteristic temperature of creepprocess

V

volume

ai

deformation parameter

b, c, d

column vectors

eij

strain-deviation tensor, e ij = ε ij - ε δ ij, e αα = 0

ēij

inelastic strain tensor

f(I2)

function determining stress dependence of creep rate

g(T)

function determining temperature dependence of creep rate

h (J2, T)

function determining dependence of I2 on J2 and T

p

constant appearing in f(I 2)

p1

internal pressure on tube

p2

external pressure on tube

pa

pressure determining total axial force in tube by p a r 1 2 /2

q

exponent of power law in the stress for creep rate

q′

temperature corrected exponent

r

radial coordinate

r1

inner radius of tube

r2

outer radius of tube

sij

stress-deviation tensor, s ij = σ ij - σ δ ij , s αα = 0

t

time

ui

displacement field

u

radial displacement

v′, w

row vectors

xi

rectangular cartesian coordinates

x

radial component of elastic strain deviator, x = e r - ē r

y

tangential component of elastic strain deviator, y = e t - ē t

α

coefficient of cubic thermal expansion

δij

Kronecker delta, δ ij = 1 if i = j, δ ij = 0 if ij

εij

strain tensor

ε

isotropic strain, 3 ε = ε αα

x

ratio of bulk modulus of elasticity and modulus of rigidity, x = C/G

λ

ratio of outer and inner radio of tube, λ - r 2/r 1

ϱ

non-dimensional radial coordinate, ϱ = r/r 1

σij

stress tensor

τ

isotropic stress, 3 σ = σ αα

σ

temperature corrected non-dimensional time parameter

ψk

portion of the volume occupied by subelements of class k

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References

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Copyright information

© Springer-Verlag OHG., Berlin/Göttingen/Heidelberg 1962

Authors and Affiliations

  • J. F. Besseling
    • 1
  1. 1.Technical UniversityDelftHolland

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