6.6 General Theory for Growth in 3D
283
6.6.2 Governing Equations for 3D Growth
The fundamental equations of the RHM theory for growth are provided below. The
formulation extends the general tensor equations of Sect. 3.7.1 to include volumetric
growth.
Kinematic Relations Three kinematic tensors are central to RHM theory
(Fig. 6.10). The growth tensor G transforms B 0 into B G , the elastic deformation
gradient tensor F ∗ transforms B G into b, and the total deformation gradient tensor
F defines the mapping from the initial (reference) configuration B to the current
configuration b. No additional tensors are needed, because cutting the stress-free
body B causes no changes in geometry, and B R is just a special case of b without
external loads. These tensors are related by
F = F ∗ · G,
(6.48)
which is the tensor form of Eq. (6.3).
Equation (6.48) embodies the fundamental concept of RHM theory, whereby
the total deformation of a growing elastic body is decomposed into growth and
elastic deformation. The elastic deformation preserves continuity when the growth
field is geometrically incompatible. For this reason, F ∗ is sometimes called the
accommodation tensor. This relation also gives the total volume ratio J = det F =
det(F ∗ · G) = (det F ∗ )(det G) or
J = J
∗ J G
J = det F
J
∗
= det F
∗
J G = det G,
(6.49)
in agreement with Eq. (6.8).
To avoid excessive repetition, the rest of the governing equations are now
listed without discussion (compare with those in Sect. 3.7.1). An asterisk denotes a
quantity referred to the current ZSS B G . Unless noted otherwise, all other variables
and operators maintain their usual definitions relative to states B and b, e.g., ∇ and
∇ are gradient operators in B and b, respectively. For further explanation, please
see Sect. 6.3.2.
Kinematics
F = (∇r)
T
= I + (∇u)
T
= F
∗
· G
(6.50)
283
6.6.2 Governing Equations for 3D Growth
The fundamental equations of the RHM theory for growth are provided below. The
formulation extends the general tensor equations of Sect. 3.7.1 to include volumetric
growth.
Kinematic Relations Three kinematic tensors are central to RHM theory
(Fig. 6.10). The growth tensor G transforms B 0 into B G , the elastic deformation
gradient tensor F ∗ transforms B G into b, and the total deformation gradient tensor
F defines the mapping from the initial (reference) configuration B to the current
configuration b. No additional tensors are needed, because cutting the stress-free
body B causes no changes in geometry, and B R is just a special case of b without
external loads. These tensors are related by
F = F ∗ · G,
(6.48)
which is the tensor form of Eq. (6.3).
Equation (6.48) embodies the fundamental concept of RHM theory, whereby
the total deformation of a growing elastic body is decomposed into growth and
elastic deformation. The elastic deformation preserves continuity when the growth
field is geometrically incompatible. For this reason, F ∗ is sometimes called the
accommodation tensor. This relation also gives the total volume ratio J = det F =
det(F ∗ · G) = (det F ∗ )(det G) or
J = J
∗ J G
J = det F
J
∗
= det F
∗
J G = det G,
(6.49)
in agreement with Eq. (6.8).
To avoid excessive repetition, the rest of the governing equations are now
listed without discussion (compare with those in Sect. 3.7.1). An asterisk denotes a
quantity referred to the current ZSS B G . Unless noted otherwise, all other variables
and operators maintain their usual definitions relative to states B and b, e.g., ∇ and
∇ are gradient operators in B and b, respectively. For further explanation, please
see Sect. 6.3.2.
Kinematics
F = (∇r)
T
= I + (∇u)
T
= F
∗
· G
(6.50)