60
3 Mechanical Aspects of Biosystems
1223, picks ‘a component’, in this case T 1223 , of T. Each component of T is simply a
number, with units of force over area. We will refer to T as the ‘stress-strain’ tensor.
This kind of object, T, is of general interest to mathematicians as well as
physicists and engineers. In fact, the name tensor historically came from the relation
expressed by Eq. (3.19)! These days, any object which behaves like a vector in each
of its n indices is called a tensor of rank n. 35 (See Appendix G.) We have seen
one good example already: The vector cross-product contains the operation which
converts a vector into a vector. The entity T is more general because it relates one
second rank tensor to another.
Symmetry reduces the number of independent components of T. Balancing
torques on a cubical piece of material requires the stress tensor to be symmetric:
S ij = S ji . The strain tensor is symmetric because it cannot have any part which is a
pure rotation. These two conditions mean that the stress-strain tensor is symmetric
in both its first and its second pair of indices. Thus T has no more than 6 × 6 = 36
independent components for any material. (That’s still a lot of numbers to measure
for a material!)
If the structure of the material looks the same in a mirror, then the stress-strain
will have only 6 independent components left. But mirror symmetry will not apply
to many biological materials, because amino acids and sugars have a left or right
‘handedness’. Material with structural handedness may show more resistance to a
left-handed twist than a right-handed twist. Twisting DNA to a tighter helix is harder
than twisting it to a looser helix.
If the material is a cubic crystal (such as table salt), only 3 components of T are
independent. And if the material is isotropic (the same in any direction), there are
only 2 left! 36 Typically, amorphous materials and fluids have such isotropy. The
two independent components can be determined by the linear stretch and shear
properties of the material, which are rather easily measured and tabulated.
For an isotropic and homogeneous material, the relationship between elastic
stress and strain is often measured by how much the material stretches under stress
and contracts at right angles to the stretch:
F
A
= Y
l
l
,
(3.20)
w
w
=
h
h
= −n P
l
l
,
(3.21)
35 Tensors are any entity which transform under coordinate changes x = a · x with the product of
coordinate transformations and/or their inverses: T =a ···a : T : a −1 ···a −1 .
36 There are only two kinds of tensors in three dimensions which are unchanged by rotations. They
are proportional to (1) The Kronecker delta:
δ ij
, which is 1 for i = j and zero otherwise; and
(2) the completely antisymmetric tensor
ij k
in a three dimensional space, which is 1 when
ij k = 123 or any even permutation, −1 for odd permutation, and 0 otherwise. To be invariant
under rotation and symmetric in the first and the second pair of indices, the tensor T ij kl must be of
the form a δ ij δ kl + b (δ ik δ jl + δ jk δ il ).
3 Mechanical Aspects of Biosystems
1223, picks ‘a component’, in this case T 1223 , of T. Each component of T is simply a
number, with units of force over area. We will refer to T as the ‘stress-strain’ tensor.
This kind of object, T, is of general interest to mathematicians as well as
physicists and engineers. In fact, the name tensor historically came from the relation
expressed by Eq. (3.19)! These days, any object which behaves like a vector in each
of its n indices is called a tensor of rank n. 35 (See Appendix G.) We have seen
one good example already: The vector cross-product contains the operation which
converts a vector into a vector. The entity T is more general because it relates one
second rank tensor to another.
Symmetry reduces the number of independent components of T. Balancing
torques on a cubical piece of material requires the stress tensor to be symmetric:
S ij = S ji . The strain tensor is symmetric because it cannot have any part which is a
pure rotation. These two conditions mean that the stress-strain tensor is symmetric
in both its first and its second pair of indices. Thus T has no more than 6 × 6 = 36
independent components for any material. (That’s still a lot of numbers to measure
for a material!)
If the structure of the material looks the same in a mirror, then the stress-strain
will have only 6 independent components left. But mirror symmetry will not apply
to many biological materials, because amino acids and sugars have a left or right
‘handedness’. Material with structural handedness may show more resistance to a
left-handed twist than a right-handed twist. Twisting DNA to a tighter helix is harder
than twisting it to a looser helix.
If the material is a cubic crystal (such as table salt), only 3 components of T are
independent. And if the material is isotropic (the same in any direction), there are
only 2 left! 36 Typically, amorphous materials and fluids have such isotropy. The
two independent components can be determined by the linear stretch and shear
properties of the material, which are rather easily measured and tabulated.
For an isotropic and homogeneous material, the relationship between elastic
stress and strain is often measured by how much the material stretches under stress
and contracts at right angles to the stretch:
F
A
= Y
l
l
,
(3.20)
w
w
=
h
h
= −n P
l
l
,
(3.21)
35 Tensors are any entity which transform under coordinate changes x = a · x with the product of
coordinate transformations and/or their inverses: T =a ···a : T : a −1 ···a −1 .
36 There are only two kinds of tensors in three dimensions which are unchanged by rotations. They
are proportional to (1) The Kronecker delta:
δ ij
, which is 1 for i = j and zero otherwise; and
(2) the completely antisymmetric tensor
ij k
in a three dimensional space, which is 1 when
ij k = 123 or any even permutation, −1 for odd permutation, and 0 otherwise. To be invariant
under rotation and symmetric in the first and the second pair of indices, the tensor T ij kl must be of
the form a δ ij δ kl + b (δ ik δ jl + δ jk δ il ).