94
3 Continuum Mechanics and Nonlinear Elasticity
by P = f/A 0 , which is called the first Piola-Kirchhoff stress or engineering
stress. For small deformation, A ∼ = A 0 and the difference between σ and P is
relatively small. In this case, P can be used as a good approximation for true
stress, as is generally done for bone and other hard tissues. For soft tissue, however,
the distinction becomes significant as deformations grow large, and Cauchy stress
becomes the only physically meaningful stress at the continuum level.
Consider now a differential element of length dX and cross-sectional area dA 0 in
the unloaded bar (Fig. 3.12b). During loading, end forces df stretch the element to
a new length dx (stretch ratio λ = dx/dX), while the cross-sectional area changes
from dA 0 to dA. The Cauchy and first Piola-Kirchhoff stresses in the element are
σ = df/dA and P = df/dA 0 , respectively, which are related by P = σ dA/dA 0 .
For a given stretch, the change in cross-sectional area depends on the mechanical
properties of the material making up the bar (the so-called Poisson’s ratio effect). For
most materials, the cross-sectional area decreases with increasing λ, giving σ > P
for λ > 1 (tension) and |σ | < |P | for λ < 1 (compression; σ, P < 0). Consistent
with this observation, Eq. (3.79) yields P = σ dA/dA 0 = (J /λ)σ , where J is the
volume ratio.
You may be wondering whether there is a second Piola-Kirchhoff stress. Yes,
there is, and we denote it by S. For reasons that will become clear later, S is related
to the other stresses through the relation
σ = J
−1 λP = J
−1 λ
2 S
(3.99)
in one dimension. For small deformation (λ, J → 1), the values of all these stresses
are nearly identical.
3.4.2 Stress in 2D
Consider an unloaded body B in the XY -plane (Fig. 3.13a). Contact and body forces
deform B through the deformation gradient tensor F into the loaded configuration
b. Imagine that b is cut into two pieces, b 1 and b 2 , revealing contact forces
distributed over the cut surfaces (Fig. 3.13b). By Newton’s third law of motion
(action-reaction), these forces represent equal and opposite loads applied by b 2 on
b 1 and vice versa.
Focus now on an arbitrary element of area dA with outward unit normal n on the
cut surface of b 1 (Fig. 3.13c, right). By convention, the normal to a surface points
outward from the material bounded by the surface. In general, this surface can be
curved, but as dA → 0, the element becomes essentially flat. On this area element,
we define the Cauchy (true) stress vector
T =
df
dA
,
(3.100)
where df is the resultant force acting on dA.
3 Continuum Mechanics and Nonlinear Elasticity
by P = f/A 0 , which is called the first Piola-Kirchhoff stress or engineering
stress. For small deformation, A ∼ = A 0 and the difference between σ and P is
relatively small. In this case, P can be used as a good approximation for true
stress, as is generally done for bone and other hard tissues. For soft tissue, however,
the distinction becomes significant as deformations grow large, and Cauchy stress
becomes the only physically meaningful stress at the continuum level.
Consider now a differential element of length dX and cross-sectional area dA 0 in
the unloaded bar (Fig. 3.12b). During loading, end forces df stretch the element to
a new length dx (stretch ratio λ = dx/dX), while the cross-sectional area changes
from dA 0 to dA. The Cauchy and first Piola-Kirchhoff stresses in the element are
σ = df/dA and P = df/dA 0 , respectively, which are related by P = σ dA/dA 0 .
For a given stretch, the change in cross-sectional area depends on the mechanical
properties of the material making up the bar (the so-called Poisson’s ratio effect). For
most materials, the cross-sectional area decreases with increasing λ, giving σ > P
for λ > 1 (tension) and |σ | < |P | for λ < 1 (compression; σ, P < 0). Consistent
with this observation, Eq. (3.79) yields P = σ dA/dA 0 = (J /λ)σ , where J is the
volume ratio.
You may be wondering whether there is a second Piola-Kirchhoff stress. Yes,
there is, and we denote it by S. For reasons that will become clear later, S is related
to the other stresses through the relation
σ = J
−1 λP = J
−1 λ
2 S
(3.99)
in one dimension. For small deformation (λ, J → 1), the values of all these stresses
are nearly identical.
3.4.2 Stress in 2D
Consider an unloaded body B in the XY -plane (Fig. 3.13a). Contact and body forces
deform B through the deformation gradient tensor F into the loaded configuration
b. Imagine that b is cut into two pieces, b 1 and b 2 , revealing contact forces
distributed over the cut surfaces (Fig. 3.13b). By Newton’s third law of motion
(action-reaction), these forces represent equal and opposite loads applied by b 2 on
b 1 and vice versa.
Focus now on an arbitrary element of area dA with outward unit normal n on the
cut surface of b 1 (Fig. 3.13c, right). By convention, the normal to a surface points
outward from the material bounded by the surface. In general, this surface can be
curved, but as dA → 0, the element becomes essentially flat. On this area element,
we define the Cauchy (true) stress vector
T =
df
dA
,
(3.100)
where df is the resultant force acting on dA.