used here for validation, the tube is seamless, cold-drawn,
low-carbon steel-type ASTM 179 and the tubesheet mate-
rial is carbon steel-type ASTM A5 16 G70 with average
yield stress of 248 MPa and modulus of elasticity of
207 GPa (Al-Aboodi et al. 2008; Shuaib et al. 2003).
Because of expected large plastic strains, the Bilinear
Isotropic Hardening (BISO) option was used in the 3-D FE
model. The curve in the plastic region was approximated
by a linear relationship. The slope of the approximated line
in the plastic region of the true stress–strain diagram
defines the tangent modulus of plasticity (Ett). An elastic-
perfectly plastic material is that having zero tangent
modulus. The average value of Ett for the tube material
used in the experimental study was found to be 733 MPa
(Al-Aboodi et al. 2008). In order to investigate the effect of
tube material strain hardening on the residual stresses, Ett
values ranging from 0 to 1.2 GPa were considered
(Table 1). These values were chosen to cover the plastic
behavior of most of the steel materials used in similar
applications. Table 1 contains the input parameters to FE
code of the analysis. They consist of the range of clear-
ances (c), percentage wall reductions (%WR), and tube and
tubesheet material constants. The friction coefficient
between the tube and tubesheet is assumed to be zero
(Merah et al. 2003; Al-Zayer 2001).Rolling is performed by the rigid rollers that stay in contact
with the inner surface of the tube being expanded. During
expansion the rollers are modeled as rigid lines that
undergo outward radial displacement and simultaneous
circumferential motion that simulate tube expansion kine-
matics. According to the industrial rolling practice speci-
fied by a major roller expansion equipment manufacturer
(Cooper Power Tools 2005), the amount of the radial dis-
placement ðurÞ of the rigid rollers against the tube inner
surface during the tube expansion phase is obtained from
the values of the required tube percent wall reduction
(%WR = 5%), the tube thickness (t), and initial tube–
tubesheet clearance (c = 0–4.5 mm) as follows:
ur ¼ð%WRÞt þ c: ð1Þ
Once the expansion process reaches this limit of radial
deformation, the inner tube surface will be released. The
rolling process is simulated by three rigid rollers, 120 apart,
moving circumferentially by increments of 6. Four
revolutions are required to reach the maximum radial
displacement determined by Eq. 1. The rolling process is
described schematically in Fig. 2a. Graphical representation
of the displacement profile along the circumference of the
inner tube is shown in Fig. 2b. Figure 3 shows that for each
roller, it is assumed that a total of 11 nodes are effectively
deformed. Five nodes on each side of the roller are partially
displaced plus the middle node along the generating line.
Equation 2 is used to define the deformation profile yielding
maximum displacement at the mid node. It is a sinusoidal
function that represents the shape of the inner tube surface
displaced by the radial movement of the roller.
where ui is the nodal displacement at the ith node, us is the
step displacement and bi is a normalized nodal factor
taking the values 0, 0.25, 0.5, 0.75, 1.0, and 1.25; bi = 0
corresponds to the mid point.
The boundary condition in this model was applied by
constraining the primary side of tube from moving axially
(Fig. 1c) and the outer side of the tube at its secondary side
from moving tangentially. The model used in the current
investigation did not require that the analysis includes the
effects of torsional moments, because the tube was con- 胀管换热器英文文献和中文翻译(3):http://www.751com.cn/fanyi/lunwen_16315.html