Civil Engineering Reference
In-Depth Information
Therefore, as there are a fixed number of elements (
Elements
test
) in the test
region, the displacement of each element can be determined by
s
Elements
test
s
Elements
=
(6.18)
The strain in the elements in the length direction can be calculated using
s
+
L
initial,test
L
initial,test
ε
L
=
ln
(6.19)
where
L
initial,test
is the initial length of the entire test region and when assuming
isotropy, the width and thickness strain is
ε
w
= ε
t
=−
0.5
ε
L
(6.20)
The instantaneous size of the elements is determined using
x
uniform
=
s
Elements
+
x
(6.21)
t
uniform
=
t
e
ε
t
(6.22)
w
uniform
=
w
test
e
ε
w
(6.23)
where
x
uniform
is the length of the element,
t
uniform
is the thickness of the sheet
under uniform deformation,
t
is the initial thickness of the sheet,
w
uniform
is the
width of the sheet under uniform deformation, and
w
test
is the initial width of the
sheet in the testing region. As a result of the uniform deformation assumption, all
the elements have equal lengths, widths, and thicknesses. Additionally, the con-
duction areas and convection areas are determined from the new element sizes by
x
x
uniform
A
11_uniform
=
A
11
(6.24)
A
22_uniform
=
x
uniform
w
uniform
(6.25)
where
A
11_uniform
is the new conduction area in the test region of the sheet and
A
22_uniform
is the new convection area in the sheet for each element.
Last, the heat generated from the applied current now varies as a function of
deformation as the resistance of the sheet increases with elongation. The heat gen-
erated can now be given as follows:
I
2
R
uniform
t
uniform
w
uniform
e
gen_uniform
=
(6.26)
s
+
L
initial,test
where
s
+
L
initial,test
R
uniform
=
ρ
e
(6.27)
t
uniform
w
uniform
