An explicit is solved by calculating results in small time increments or time steps. The size of the time step depends
on many factors but is automatically calculated by Radioss.
Hyperelastic materials are used to model materials that respond elastically under very large strains. These materials
normally show a nonlinear elastic, incompressible stress strain response which returns to its initial state when unloaded.
Composite materials consist of two or more materials combined each other. Most composites consist
of two materials, binder (matrix) and reinforcement. Reinforcements come in three forms, particulate,
discontinuous fiber, and continuous fiber.
Optimization in Radioss was introduced in version 13.0. It is implemented by invoking the optimization capabilities of
OptiStruct and simultaneously using the Radioss solver for analysis.
LAW24 uses a Drücker-Prager criteria with or without a cap in yield to model a
reinforced concrete material. This material law assumes that the two failure mechanisms of
the concrete material are tensile cracking and compressive crushing.
Concrete Tensile Behavior
In LAW24, the options Ht,
Dsup, and
can be used to describe tensile cracking and failure in
tension.
In the initial very small elastic phase, the material has an elastic modulus
Ec.
Once tensile strength, ft is
reached, the concrete starts to soften with the slope
Ht. The maximum damage
factor, Dsup, is significant
because it enables the modeling of residual stiffness during and after a crack.
The residual stiffness is computed as:(1)
When there is crack closure, the concrete becomes elastic again, and the damage
factor (for each direction) is conserved.
The bearing capacity of concrete in tensile is much lower than in compression. It is
normally considered elastic when in tension.
It is recommended to choose a Dsup
value close to 1 (default is 0.99999) in order to minimize the current stiffness at
the end of the damage and consequently avoid residual stress in tension, which can
become very high if the element is highly deformed due to tension. This will happen
if the force causing the damage remains.
It is possible to adjust the Dsup
(and Ht) in order to simulate and
fit the behavior of concrete reinforced by fibers. The concrete material fails once
it reaches the total failure strain .
Concrete Yield Surface in Compression
For concrete, the yield surface is the beginning of the plastic hardening zone which
is between the failure surface, , and the yield surface.
The yield surface is assumed to be the same as the failure surface in the tension
zone. In compression, the yield surface is a scaled down failure surface using the
factor . The yield in LAW24 for concrete is:(2)
For Icap
=0 or 1 (without a cap in yield) the
yield curve is:
The material is above yield and below the failure surface which
is the plastic hardening phase.
The input parameter is the hydrostatic failure pressure in a uniaxial
tension test and is the hydrostatic pressure by failure in a uniaxial
compression test.
The scale factor is a function of mean stress and can be described as:
When (in tension) the scale factor . In this case, the yield surface equals the
failure surface, .
In the tension-compression region, , then
with
The rest of the curve depends on the
Icap option and the
different scale factors used.
For Icap
=0 or 1 and (in compression), then
For Icap
=2 (with cap in yield) and (in compression), then
In (in cap zone)
with
The material constant should be . A higher value of results in a higher yield surface.
For example, if Icap
=2 (yield with cap), the difference of yield surface between and (Figure 10). The default value of in LAW24 is 0.5.
Concrete Plastic Flow Rule in Compression
A non-associated plastic flow rule is used in LAW24. The plastic flow rule
is:(3)
Where,
Plastic dilatancy.
Governs the volumetric plastic flow.
First stress invariant (hydrostatic pressure).
Experimentally, is a linear function of :(4)
If
then, which means the material is in
yield.
If
then, becomes negative is the cap region.
If
then, which means the material has
failed.
The values of are used to describe the material beyond yield, but
before failure. It is recommended to use -0.2 and -0.1 for in LAW24. If very small values of are used, there is no volumetric plasticity (no cap
region).
Concrete Crushing Failure in Compression
For concrete, the compression failure curve can be defined with a strength of:
Uniaxial tension (triaxiality is 1/3)
Uniaxial compression (triaxiality is -1/3)
Biaxial compression (triaxiality is -2/3)
Confined compression strength (tri-axial test)
Under confined pressure
The best way to fully determine the 3D failure envelope is to get experimental data
for all of these values, which are schematically illustrated in Figure 12.
It is possible to plot most of the 3D failure envelope characteristic points on the
plane stress surface. The compression strength values is a failure point out of this
plane.
Even if the compression strength point is located out of this plane, its position
shapes the envelope on the plane stress surface (Figure 14).
Also notice from these plots that the failure envelope is not a convex surface. Figure 15 shows this behavior with other
parameters.
In this particular case, the compressive strength is changing but all other ratios
are fixed, . This leads to an envelope scaling (Figure 18).
All other ratios are fixed and the ratios in the space (used to define the concrete failure) are:
Where the failure curve is defined using and is the mean stress (pressure), then and are the first and second stress invariants.
The material fails once it reaches the failure curve
.
Concrete
Reinforcement
In Radioss there are two
different ways to simulate the reinforcement in concrete.
One way is to use beam or truss elements and connect them to the concrete
with kinematic conditions.
Another way is to use the parameters in LAW24 along with the orthotropic
solid property /PROP/TYPE6 to define the reinforced
direction. Parameters in LAW24 are used to define the
reinforcement cross-section area ratio to the whole concrete section area in
direction 1, 2, 3.(5)
Where, is the yield stress of the reinforcement. If steel
is used as a reinforcement, then is the yield stress of steel and is the modulus of steel in the plastic phase.
Concrete Material (/MAT/LAW81)
LAW81 can be used to model rock or concrete materials.
Drücker-Prager Yield Criteria
LAW81 uses a Drücker–Prager yield criterion where the yield surface and the failure
surface are the same. The yield criteria is:(6)
Where,
von Mises stress with
Pressure is defined as
The yield surface can be described in two parts:
The linear part (), where the scale function is which leads to the von Mises stress being
linearly proportional to pressure:(7)
Where,
Cohesive and is the intercept of yield envelope with the
shear strength.
If , the material has no
strength under tension.
Angle of internal friction, which defines the slope of the
yield envelope.
and are also used to define the Mohr-Coulomb yield
surface. The Drücker-Prage yield surface is a smooth version of the
Mohr-Coulomb yield surface.
The second part () of the yield surface simulates a cap limit.
An increase of pressure in a rock or concrete material will increase the
yield of the material; but, if pressure increases enough, then the rock or
concrete material will be crushed. The Drücker-Prager model with the cap
limit can be used to model this behavior. The cap limit defined in part
and uses the scale
function:(8)
The von Mises stress is:(9)Where,
Curve is defined using the
fct_IDPb
input
Computed by Radioss using the
input ratio value.
with .
Where, is the maximum point of yield curve,
where
If , then and the yield function is then,
which means the material is
crushed.
The input parameters need to determine for the Drücker–Prager yield
surface. At least four tests are needed to fit these parameters. In the simplest
case, uniaxial tension and uniaxial compression can be used to determine the linear
part, . To determine biaxial compression tests and
compression/compression tests are needed (refer to CC00 and CC01 in RD-E: 4700 Concrete Validation).
For most materials such as metal, the plastic strain increment could be considered
normal to yield surface. However, if the plastic strain increment normal to yield
surface is used for rock or concrete materials, the plastic volume expansion is
overestimated. Therefore, a non-associated plastic flow rule is used in these
materials. In LAW81 the plastic flow function defined as:
if
if
if
Since the pressure is , the yield function and plastic flow function are the same and the following condition is
fulfilled:(10)(11)
The pressure can be calculated using the yield surface where . With defined as (12)
The parameter can be determined using the von Mises stress at pressure, in the function.
1 Han, D. J.,
and Wai-Fah Chen. "A nonuniform hardening plasticity model for concrete
materials." Mechanics of materials 4, no. 3-4 (1985):
283-302