58
3 Mechanical Aspects of Biosystems
3.6 Strain in Materials
The local ‘strain’ in a material is characterized as the fractional distortion of a
material element as a result of stress. For example, if a bone is compressed, the
change in length of the bone divided by the original length is the compressional
strain. If the bone were twice the length, we should expect twice the change in
length. Dividing the change in length by the original length produces an intrinsic
quantity. In fact, strain is measured by a unitless number. The local effects of strain
in uniform bone material can be described without having to know the size of the
bone.
Shearing stress can cause one layer of material to shift relative to an adjacent
layer. This relative shift is called a shearing strain. If your foot is attached to a
ski, and you twist your body, each element of material in your femur undergoes a
shearing strain. For a small material element rectangular in volume, a pure shear
between the top and bottom of the volume causes the top to shift relative to the
bottom, keeping the distance between top and bottom fixed. The shearing strain is
then the size of the displacement of the top relative to the bottom, divided by the
distance between the top and bottom. We will look at this case in more detail later.
We can define strain through the relative shift of material elements, but we should
be aware that pure rotation produces relative motion of material elements without
distortion. So, we can say that strain is that part of the relative shifts of material
elements which causes distortion. Mathematically, if ξ (x, y, z) represents the vector
displacement of a material element which started at position (x, y, z), then the
relative shift in the material in the ith direction due to a displacement in the j th
direction per unit length is given by ∂ξ j /∂x i which we will shorten to ∂ i ξ j . We can
always write: ∂ i ξ j ≡ (1/2)(∂ i ξ j + ∂ j ξ i ) + (1/2)(∂ i ξ j − ∂ j ξ i ). The second term can
be shown to be pure rotation. 32 The strain in the material in the ith direction due to
a displacement in the j th direction is defined as 33
σ ij ≡ (1/2)(∂ i ξ j + ∂ j ξ i ) .
(3.17)
There are six independent elements of the strain. Pure shearing strains are
represented in the three independent off-diagonal elements of the three-by-three
matrix formed from these terms, and the relative volume changes by the sum of
32 For a small rotation through an angle δθ, Eq. (3.3) or the Fig. 3.1 gives ξ = δθ × (r − r o ), where
r o locates a point on the axis of rotation. Taking the vector curl of both sides shows that half the
curl of the displacement field ξ (x, y, z, t) is a vector along the axis of rotation with size equal to
the angle of rotation: δθ = (1/2) ∇ × ξ . The components of this curl are just the antisymmetric
combination of the derivatives of the displacements we separated from the changes in displacement
across the material to define the strain tensor. Inversely, if the change of the displacements across
a small region of the body can be expressed as a curl, then this region is not being distorted, but
rather it is being rotated.
33 Caution: Engineers typically do not include the factor of (1/2) in their definition of strain.
3 Mechanical Aspects of Biosystems
3.6 Strain in Materials
The local ‘strain’ in a material is characterized as the fractional distortion of a
material element as a result of stress. For example, if a bone is compressed, the
change in length of the bone divided by the original length is the compressional
strain. If the bone were twice the length, we should expect twice the change in
length. Dividing the change in length by the original length produces an intrinsic
quantity. In fact, strain is measured by a unitless number. The local effects of strain
in uniform bone material can be described without having to know the size of the
bone.
Shearing stress can cause one layer of material to shift relative to an adjacent
layer. This relative shift is called a shearing strain. If your foot is attached to a
ski, and you twist your body, each element of material in your femur undergoes a
shearing strain. For a small material element rectangular in volume, a pure shear
between the top and bottom of the volume causes the top to shift relative to the
bottom, keeping the distance between top and bottom fixed. The shearing strain is
then the size of the displacement of the top relative to the bottom, divided by the
distance between the top and bottom. We will look at this case in more detail later.
We can define strain through the relative shift of material elements, but we should
be aware that pure rotation produces relative motion of material elements without
distortion. So, we can say that strain is that part of the relative shifts of material
elements which causes distortion. Mathematically, if ξ (x, y, z) represents the vector
displacement of a material element which started at position (x, y, z), then the
relative shift in the material in the ith direction due to a displacement in the j th
direction per unit length is given by ∂ξ j /∂x i which we will shorten to ∂ i ξ j . We can
always write: ∂ i ξ j ≡ (1/2)(∂ i ξ j + ∂ j ξ i ) + (1/2)(∂ i ξ j − ∂ j ξ i ). The second term can
be shown to be pure rotation. 32 The strain in the material in the ith direction due to
a displacement in the j th direction is defined as 33
σ ij ≡ (1/2)(∂ i ξ j + ∂ j ξ i ) .
(3.17)
There are six independent elements of the strain. Pure shearing strains are
represented in the three independent off-diagonal elements of the three-by-three
matrix formed from these terms, and the relative volume changes by the sum of
32 For a small rotation through an angle δθ, Eq. (3.3) or the Fig. 3.1 gives ξ = δθ × (r − r o ), where
r o locates a point on the axis of rotation. Taking the vector curl of both sides shows that half the
curl of the displacement field ξ (x, y, z, t) is a vector along the axis of rotation with size equal to
the angle of rotation: δθ = (1/2) ∇ × ξ . The components of this curl are just the antisymmetric
combination of the derivatives of the displacements we separated from the changes in displacement
across the material to define the strain tensor. Inversely, if the change of the displacements across
a small region of the body can be expressed as a curl, then this region is not being distorted, but
rather it is being rotated.
33 Caution: Engineers typically do not include the factor of (1/2) in their definition of strain.