3.7 Connection Between Stress and Strain
59
the diagonal elements (called the ‘trace’ of a matrix). 34 The last two independent
elements determine asymmetric strain.
Elastic and Inelastic Distortion
‘Elastic’ behavior is defined by the property that if the material is stressed, the
energy needed to strain the material is all stored mechanically, so that no heat is
produced. In contrast, ‘inelastic’ behavior is observed when some work done to
deform a material is lost to heat or other forms of dissipation.
3.7 Connection Between Stress and Strain
Under stress, a material may deform in a variety of complicated ways, depending on
the nature of the bonding between its molecules, the ordering of those molecules,
the initial internal stresses, and the duration of the external stress.
For small stresses, we should expect there to be a proportionality between stress
and strain. This is a form of Hooke’s Law for materials:
S = T : σ .
(3.18)
If this proportionality holds, we say the material is acting with linear elasticity.
Displacements of a material under stress do not have to be in the same direction
as the stress causing them. When you push on a bone, the displacement this push
causes will be preferentially in the direction that the atoms and molecules can more
easily shift. Thus, the proportionality factor in Eq. (3.18) is not simply one constant,
but rather a set of constants:
S ij =
3
k,l=1
T ij kl σ kl .
(3.19)
The set of coefficients T ij kl are called ‘tensor’ components. Each of the labels
on T ij kl , such as i, is called an index, taking the values 1, 2, or 3 = x, y, or z.
With four indices, the values T ij kl are said to make up the components of a fourth
‘rank’ tensor. Selecting a set of particular values for the indices (i, j, k, l), such as
34 The sum of the diagonal elements becomes ∇ · ξ . That this represents a relative volume change
can be seen as follows: Let V = dxdydz be a small volume element in the material before it is
stressed, and δ((V ) be the change in that small volume after the material is stressed. If all three
sides of the original volume are displaced in the direction of their normal, we will have δ((V ) =
i [ξ ii (x i +dx i )−ξ ii (x i )], which is, to first order in the small dx i , (
i ∂ i ξ i ) dxdydz = (∇·ξ ) )V .
59
the diagonal elements (called the ‘trace’ of a matrix). 34 The last two independent
elements determine asymmetric strain.
Elastic and Inelastic Distortion
‘Elastic’ behavior is defined by the property that if the material is stressed, the
energy needed to strain the material is all stored mechanically, so that no heat is
produced. In contrast, ‘inelastic’ behavior is observed when some work done to
deform a material is lost to heat or other forms of dissipation.
3.7 Connection Between Stress and Strain
Under stress, a material may deform in a variety of complicated ways, depending on
the nature of the bonding between its molecules, the ordering of those molecules,
the initial internal stresses, and the duration of the external stress.
For small stresses, we should expect there to be a proportionality between stress
and strain. This is a form of Hooke’s Law for materials:
S = T : σ .
(3.18)
If this proportionality holds, we say the material is acting with linear elasticity.
Displacements of a material under stress do not have to be in the same direction
as the stress causing them. When you push on a bone, the displacement this push
causes will be preferentially in the direction that the atoms and molecules can more
easily shift. Thus, the proportionality factor in Eq. (3.18) is not simply one constant,
but rather a set of constants:
S ij =
3
k,l=1
T ij kl σ kl .
(3.19)
The set of coefficients T ij kl are called ‘tensor’ components. Each of the labels
on T ij kl , such as i, is called an index, taking the values 1, 2, or 3 = x, y, or z.
With four indices, the values T ij kl are said to make up the components of a fourth
‘rank’ tensor. Selecting a set of particular values for the indices (i, j, k, l), such as
34 The sum of the diagonal elements becomes ∇ · ξ . That this represents a relative volume change
can be seen as follows: Let V = dxdydz be a small volume element in the material before it is
stressed, and δ((V ) be the change in that small volume after the material is stressed. If all three
sides of the original volume are displaced in the direction of their normal, we will have δ((V ) =
i [ξ ii (x i +dx i )−ξ ii (x i )], which is, to first order in the small dx i , (
i ∂ i ξ i ) dxdydz = (∇·ξ ) )V .