Project in Continuum Mechanics: - Division of Solid Mechanics

Prof. Larsgunnar Nilsson 2011-02-22
Project in Continuum Mechanics:
Large axial and shear deformation of a rubber material.
Introduction
A circular cylindrical rubber specimen, see Figure 1, is tested in a combined loading
sequence:
1. The specimen is subjected to an axial compression, followed by (without unloading), see
Figure 2
2. The specimen is subjected to a shear deformation, see Figure 3
Fig 1. Rubber specimen before test
Fig 2. After full axial loading
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Prof. Larsgunnar Nilsson 2011-02-22
Fig. 3. After full axial and shear loading
The Finite Element representation
Your task is to evaluate certain measures in one finite element. The specific element is
individual to you. Each element is a hexahedron element, but you are supposed to construct a
tetrahedron element based sequentially on the first three node numbers plus the fifth node
number of that element. The sequence of four node numbers defines a four nodes tetrahedron
element, see Figure 4, which is your specific element.
The project documents can be found at
http://www.solid.iei.liu.se/Education/TMHL41/tmhl41.html
The nodal coordinates belonging to the hexahedron elements are listed in the file
ELEMENTS.
The coordinates of the nodes in the initial configuration is given the file UNDEFORMED.
The coordinates of the nodes in the final configuration is given the file DEFORMED.
Note: Length unit is mm.
Element theory
The current coordinates of the element is given by
x ( X,
t)
N ( X)
x ( t)
, i=1,3 I=1,2,3,4
i
I
iI
where N I (X) are the shape functions, which are functions of the Lagrangian (or material)
coordinates, and x iI =(x I ,y I ,z I ) are the nodal coordinates of node I.
The shape functions of a four node tetrahedron element can be obtained as follow:
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Prof. Larsgunnar Nilsson 2011-02-22
Define the matrix A
1
X
1
Y1
Z1
1 X
2
Y2
Z
2
A
1
X
3
Y3
Z
3
1
X
4
Y4
Z
4
where (X I ,Y I , Z I ) are the nodal coordinates (I=1,4) and the local node numbering is defined by
choosing the first node number and then numbering the remaining three nodes in a counter
clockwise direction as seen from the first. The shape functions are given by
N ( X , Y,
Z)
m
m
I
IJ
1
( 1)
6V
1I
( I J
)
1
m
Aˆ
IJ
2I
X
m
3I
Y m
4I
Z
4
1
2
3
1
Fig 4. Four node tetrahedron element and its nodal numbering
And where  IJ are the minors of A, i.e. the determinant of the matrix obtained by deleting
column I and row J of A. The element volume is given by V=detA/6.
Note that the deformation gradient is given by
1
1
F
ij
x
X
i
j
N
X
I
j
x
iI
Task
Considering your specific element at its final deformed configuration, where you are
supposed to evaluate, at the centroid of the element, the following items:
1. The deformation gradient (F)
2. The Green Lagrange strain (E)
3. The principal Green Lagrange strains and the corresponding directions
4. The right stretch tensor (U), the principal stretches and the corresponding principal
directions
5. The logarithmic strain tensor
6. The rotation tensor (R)
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Prof. Larsgunnar Nilsson 2011-02-22
7. Compare the resulting principal directions of E and U from 3 and 4. Prove that they
are the same!
8. Discuss the principal stretches and the principal directions with regard to the analysed
rubber material test. Do these directions correspond to what you expected?
The results should be presented in a typed report written in MS-Word (or similar program),
clearly stating the starting values and each step of the derivation including a discussion of all
results. Text from the Matlab program is not allowed in the report.
Credits
A correct solution of all tasks gives 3 credit points, which can be summed to the credit points
of the written examine. These credits can be used just once at the examine occasion on
March 19, 2011.
Dead line
The report must be delivered to the undersigned not later than March 11, at 15.00.
Larsgunnar Nilsson
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