Using Voigt convention, the strain rate of
Strain Rate,
Equation 5 can be written as:
(1)
With,
It is useful to take the Belytschko-Bachrach's mix form
1 of the shape functions written
by:
(2)
Where,
The derivation of the shape functions is given by:
(3)
It is decomposed by a constant part which is directly formulated with the Cartesian
coordinates, and a non-constant part which is to be approached separately. For the strain
rate, only the non-constant part is modified by the assumed strain. You can see in the
following that the non-constant part or the high order part is just the hourglass terms.
You now have the decomposition of the strain rate:
(4)
with:
;
Belvtschko and Bindeman
2 ASQBI assumed strain is used:
(5)
with
Where,
;
; and
.
To avoid shear locking, some hourglass modes are eliminated in the terms associated with
shear so that no shear strain occurs during pure bending. That is,
in
terms and all fourth hourglass modes in shear terms are also
removed since this mode is non-physical and is stabilized by other terms in
.
The terms with Poisson coefficient are added to obtain an isochoric assumed strain field
when the nodal velocity is equivoluminal. This avoids volumetric locking as
. In addition, these terms enable the element to capture
transverse strains which occurs in a beam or plate in bending. The plane strain expressions
are used since this prevents incompatibility of the velocity associated with the assumed
strains.
Incompressible or
Quasi-incompressible Cases
Flag for new solid element: Icpre =1,2,3
For incompressible or quasi- incompressible materials, the new solid elements have no
volume locking problem due to the assumed strain. Another way to deal with this problem is
to decompose the stress field into the spherical part and the deviatory part and use reduced
integration for spherical part so that the pressure is constant. This method has the
advantage on the computation time, especially for the full integrated element. For some
materials which the incompressibility can be changed during computation (for example:
elastoplastic material, which becomes incompressible as the growth of plasticity), the
treatment is more complicated. Since the elastoplastic material with large strain is the
most frequently used, the constant pressure method has been chosen for
Radioss usual solid elements. The flag
Icpre has been introduced for new solid
elements.
- Icpre
- =0
- Assumed strain with
terms is used.
- =1
- Assumed strain without
terms and with a constant pressure method is used. The
method is recommended for incompressible (initial) materials.
- =2
- Assumed strain with
terms is used, where
is variable in function of the plasticity state. The
formulation is recommended for elastoplastic materials.
- =3
- Assumed strain with
terms is used.