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@@ -570,7 +587,7 @@ are no boundary forces, i.e.
, and that the deformation of the
body is driven by body forces
alone. It is also worth noting that when integrating by parts, we
would get terms of the form
, but that we replace it with the term involving the
@@ -598,14 +615,14 @@ symmetric gradient
instead of
. Due to
the symmetry of
The system at time step
, to be solved on the old domain
, has exactly the form of a stationary elastic
problem, and is therefore similar to what we have already implemented
@@ -639,7 +656,7 @@ continuous finite elements.
There are differences, however:
-- We have to move the mesh after each time step, in order to be
+
- We have to move (update) the mesh after each time step, in order to be
able to solve the next time step on a new domain;
@@ -649,7 +666,7 @@ There are differences, however:
-->
to compute the next incremental
displacement, i.e. we need to compute it at the end of the time step
@@ -672,12 +689,12 @@ Updating the stress variable
As indicated above, we need to have the stress variable
available
when computing time step
, and we can compute it using
@@ -685,7 +702,7 @@ when computing time step
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(18) | |
@@ -694,7 +711,7 @@ when computing time step
|
@@ -753,16 +770,16 @@ course replace this term by numerical quadrature:
where
are the quadrature weights and
the quadrature points on
cell
. This should make clear that what we really need is not the stress
in itself, but only the values of the stress in the quadrature
points on all cells. This, however, is a simpler task: we only have to provide
@@ -782,7 +799,7 @@ end of each time step we then only have to evaluate
, multiply it by the stress-strain tensor
at quadrature point
.
@@ -809,14 +826,14 @@ due to the fact that we compute
on the domain
,
and then use this displacement increment to both update the stress as well as
@@ -825,7 +842,7 @@ move the mesh nodes around to get to
on which the next increment
is computed. What we have to make sure, in this context, is that moving the
@@ -838,10 +855,11 @@ can be understood as follows: locally, the incremental deformation
can be decomposed into three parts, a linear translation (the constant part
-of the displacement field in the neighborhood of a point), a dilational
+of the displacement increment field in the neighborhood of a point), a
+dilational
component (that part of the gradient of the displacement field that has a
nonzero divergence), and a rotation. A linear translation of the material does
not affect the stresses that are frozen into it - the stress values are
@@ -853,14 +871,14 @@ situation where
, with which
). Nevertheless, if the the material was pre-stressed in a certain
direction, then this direction will be rotated along with the material. To
@@ -869,20 +887,20 @@ this end, we have to define a rotation matrix
that
describes, in each point the rotation due to the displacement increments. It
is not hard to see that the actual dependence of
on
can
only be through the curl of the displacement, rather than the displacement
@@ -890,7 +908,7 @@ itself or its full gradient (as mentioned above, the constant components of
the increment describe translations, its divergence the dilational modes, and
the curl the rotational modes). Since the exact form of
is cumbersome, we
only state it in the program code, and note that the correct updating formula
@@ -900,7 +918,7 @@ for the stress variable is then
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