BME510
1
Soft & Hard Tissue Mechanics - Small Deformation Elasticity
z
y
x
i
F
F
F
F
Einstein indicial notation:
i
=1, 2, 3
dealing with quantities beyond scalars and vectors, including
stress, strain and constitutive coefficients
tensors

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BME510
2
The stress tensor:
The stress at any point is described by nine components
zz
zy
zx
yz
yy
yx
xz
xy
xx
Where only 6 are independent components:
ji
ij
, thus:
(tensor symmetry)
yx
xy
zy
yz
xz
zx
Normal Stresses:
xx
11
(
xx normal stress
)
yy
22
(
yy normal stress
)
zz
33
(
zz normal stress
)
Shear Stresses:
xy
12
(
xy shear stress
)
xz
13
(
xz shear stress
)
yx
21
(
yx shear stress
)
yz
23
(
yz shear stress
)
zx
31
(
zx shear stress
)
zy
32
(
zy shear stress
)
i.e.,
σ
yx
is stress acting in the x direction on an area (which is defined by the y normal vector) we have:
33
32
31
23
22
21
13
12
11
zz
zy
zx
yz
yy
yx
xz
xy
xx
j
i
Cauchy stress tensor
Each index refers to an axis in the same way as the
coordinate system designation, i.e.
x
=1,
y
=2,
z
=3
may be replaces with
σ
for all stresses
2nd order stress tensor

BME510
3
•
For small deformation elasticity (<5%) we can use the small deformation linear strain
tensor and the
Cauchy stress tensor
.
•
Any repeated indices represent a sum of terms containing the repeated indices (aka,
dummy indices).
For example, hydrostatic pressure is defined as one-third of the sum of normal
components of stress:
)
(
3
1
33
22
11
p
)
(
3
1
3
1
33
22
11
ii
p
the indices
ii
are repeated - the index
i
is NOT independent.
Therefore, the number of non-repeated
(or independent) indices is zero and the quantity is a
scalar
(tensor of 0th order). Accordingly, it is
non-directional (unlike vectors).
A traction force boundary condition for an elasticity problem ─ the product
between a 2nd order tensor and a vector (a tensor of 1
st
order):
j
ij
i
n
t
3
,
2
,
1
,
3
3
,
2
,
1
,
2
3
,
2
,
1
,
1
3
33
2
32
1
31
3
3
23
2
22
1
21
2
3
13
2
12
1
11
1
j
i
n
n
n
n
j
i
n
n
n
n
j
i
n
n
n
n
j
j
j
j
j
j
σ
is the 2
nd
order stress tensor and
n
is the vector normal to the boundary
i
is the independent index that is transferred on the resulted traction quantity
t
(
j
is a repeated index while
i
is a non-repeated or independent index - for each
i
we
sum over the
j
indices)

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