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.
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Abbreviations
- A :
-
matrix of coefficients of unknowns
- C :
-
bulk modulus of elasticity
- G :
-
modulus of rigidity
- I 2 :
-
second invariant of elastic strain tensor I 2 = 1/2 (e α β-ē α β) (eβ α -ēβ α)
- J 2 :
-
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
- T c :
-
characteristic temperature of creepprocess
- V :
-
volume
- a i :
-
deformation parameter
- b, c, d :
-
column vectors
- e ij :
-
strain-deviation tensor, e ij = ε ij - ε δ ij, e αα = 0
- ē ij :
-
inelastic strain tensor
- f(I 2):
-
function determining stress dependence of creep rate
- g(T):
-
function determining temperature dependence of creep rate
- h (J 2, T):
-
function determining dependence of I2 on J2 and T
- p :
-
constant appearing in f(I 2)
- p 1 :
-
internal pressure on tube
- p 2 :
-
external pressure on tube
- p a :
-
pressure determining total axial force in tube by p a r 21 /2
- q :
-
exponent of power law in the stress for creep rate
- q′:
-
temperature corrected exponent
- r :
-
radial coordinate
- r 1 :
-
inner radius of tube
- r 2 :
-
outer radius of tube
- s ij :
-
stress-deviation tensor, s ij = σ ij - σ δ ij , s αα = 0
- t :
-
time
- u i :
-
displacement field
- u :
-
radial displacement
- v′, w′:
-
row vectors
- x i :
-
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
- δ ij :
-
Kronecker delta, δ ij = 1 if i = j, δ ij = 0 if i ≠ j
- ε ij :
-
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
- σ ij :
-
stress tensor
- τ :
-
isotropic stress, 3 σ = σ αα
- σ:
-
temperature corrected non-dimensional time parameter
- ψ k :
-
portion of the volume occupied by subelements of class k
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© 1962 Springer-Verlag OHG., Berlin/Göttingen/Heidelberg
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Besseling, J.F. (1962). Investigation of Transient Creep in Thick-walled Tubes under Axially Symmetric Loading. In: Hoff, N.J. (eds) Creep in Structures. IUTAM Symposia. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-86014-0_10
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DOI: https://doi.org/10.1007/978-3-642-86014-0_10
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