Small Strain Formulation

Radioss uses two different methods to calculate stress and strain. The method used depends on the type of simulation. The two types are Large strain and Small strain.

The large strain formulation has been discussed before and is used by default. Small strain analysis is best used when the deformation is known to be small, for example, linear elastic problems.

Large strain is better suited to nonlinear, elastoplastic behavior where large deformation is known to occur. However, large mesh deformation and distortion can create problems with the time step. If an element is deformed excessively, the time step will decrease too much, increasing the CPU time. If the element reaches a negative volume, the computation will stop or the element will have to be removed. Using small strain can eliminate these problems.

Using a small strain formulation for part of a large deformation process introduces of course errors. These errors depend on the specific case, but they can provide a better solution than element deletion.

On the other side, materials like honeycomb, which have no Poisson's effect, can have the small strain limitations corrected by using adjusted stress-strain curves.

A small strain, small displacement formulation can thus be specified for some specific material behavior, like honeycomb, or can be implemented when the time step with a large strain formulation reaches a minimum value that is defined by the user. This allows the computation to proceed at an acceptable rate.

The small displacement formulation is, however, not recommended for some simulations, e.g. crash analysis.

Small Strain Option

Assuming a constant Jacobian matrix during time and also a constant volume, previous equations degenerate into a small strain and small displacement formulation. All spatial variables are then values defined at time t=0 (or at the time the small strain formulation is initiated).

Time step then becomes constant:(1)

Δ t = \frac{l_{0}}{c}

and the effective negative volume has no effect on the computation (only the initial volume is used).

The Jacobian matrix time transformation is dependent upon element deformation and element rigid body rotation (Kinematic Description, Equation 24 and Equation 25). On the other hand, rigid body translation has no effect on the Jacobian matrix.

A small strain formulation is achieved if the element deformation is not taken into account. Likewise, a small displacement formulation is obtained if the element rigid body rotation is ignored.

From a practical point of view, small strain formulation will be obtained if, instead of recomputing the Jacobian matrix at each cycle, the initial matrix is updated taking into account element rigid body rotation:(2)

F (t + δ t) = F (t) Ω

Where,

$Ω$: Rigid body angular velocity

An alternative solution that accounts for element rigid body rotation consists in computing the internal forces in a local reference frame attached to the element. This solution is used for shell elements and convected brick elements.

Unlike the large strain formulation, the small strain formulation uses values based on the initial configuration. This is either at the beginning of the simulation or at the beginning of the small strain implementation.

Hence, the strain rate is calculated using:(3)

{\dot{ε}}_{i j} = {(\frac{\partial Φ_{I}}{\partial x_{j}})}_{t = 0} v_{i I}

with $Φ_{I}$ the interpolating shape functions and $v_{i I}$ the components of velocity at node $I$ .

The strain in an arbitrary

x

direction is calculated by:(4)

ε_{x} = \sum {\dot{ε}}_{x} d t = \sum (\frac{Δ δ x}{x_{0}}) = \frac{Δ x}{x_{0}}

Thus, the strain is the engineering strain.

The stress is calculated using the strain rate and the material law provided by the user. The later is integrated over the element volume to produce the internal force vector, which is summed over the elements to obtain the overall force vector:(5)

f_{i I}^{int} = {\int_{Ω} σ_{i j} (\frac{\partial φ_{I}}{\partial x_{j}})}_{t = 0} d Ω

The stress is the engineering stress.

The volumetric strain using the small strain formulation is independent of density. For one dimensional deformation, one has:(6)

μ = - (ε_{x x} + ε_{y y} + ε_{z z}) = - \frac{δ l}{l_{0}}

The small strain formulation for solid elements was developed for specific material, like honeycomb. In the crushing direction, honeycomb has no Poisson's effect and stress integration over the initial surface is acceptable. The effect on strain is small during elastic deformation and can be corrected in the plastic phase by using a modified engineering stress-engineering strain material curve.

For materials like crushable foam, with a small Poisson's ratio, this formulation can be applied successfully in certain situations. However, for other materials, this formulation has to be used very carefully.

Shell elements have fewer limitations than solid elements. For crash applications, the main shell deformation is bending. The small strain formulation has no effect on the bending description if membrane deformation is small.

The small strain formulation can be applied to some elements for which the time step is reaching a user specified value.

If the critical time step is small, compared to the initial one, this formulation gives acceptable results and is more accurate than removing the deformed elements.

Large Strain Option

By default Radioss uses a large strain large displacement formulation with explicit time integration. By computing the derivative of shape functions at each cycle, large displacement formulation is obtained. The large strain formulation results from incremental strain computation. Stresses and strains are therefore true stresses and true strains.

The spatial derivatives of isoparametric brick shape functions are given by:(7)

\frac{\partial Φ_{I}}{\partial x_{j}} = F {(t)}^{- 1} \frac{\partial Φ_{I}}{\partial r}

Where,

$F (t)$: Jacobian matrix

For each element the internal forces are integrated over the volume with one integration point:(8)

f_{i I}^{int} = \int_{Ω} (σ_{i j} {(\frac{\partial Φ_{I}}{\partial x_{i}})}_{t = 0} d Ω = σ_{i j} \frac{\partial Φ_{I}}{\partial x_{i}} Ω)

Time integration of Cauchy stress (true stress):(9)

σ_{i j} (t + δ t) = σ_{i j} (t) + \frac{d σ_{i j} (t)}{d t} d t

uses objective stress rate, meaning that the stress tensor follows the rigid body rotation of the material. Stress rate is a function of element average rigid body rotation and of strain rate. Strain rate is obtained from spatial velocity derivative:(10)

\frac{d ε_{i j}}{d t} = \frac{1}{2} [\frac{\partial v_{i}}{\partial x_{j}} + \frac{\partial v_{j}}{\partial x_{i}}]

Where,(11)

\frac{\partial v_{i}}{\partial x_{j}} = \frac{\partial Φ_{I}}{\partial x_{j} v_{i}}

Stability of explicit scheme is given by the Courant condition:(12)

Δ t < \frac{l}{c}

with

$l$: Element characteristic length
$c$: Sound speed

The time step is computed at each cycle.

Large element deformation can give a large time step decrease. For overly large deformations a negative volume can be reached and it then becomes impossible to invert the Jacobian matrix and to integrate the stresses over the volume.

Stress and Strain Definition

With large strain formulation, stresses are true stresses and strains are true strains:(13)

ε = \sum Δ Δ l / l \equiv \ln \frac{l}{l_{0}}

(14)

σ = \frac{F}{S}

With small strain formulation stresses become engineering stresses and strains engineering strains:(15)

ε = \sum Δ Δ l / l_{0} \equiv \frac{Δ l}{l_{0}}

(16)

σ = \frac{F}{S_{0}}

The definition of volumic strain is also modified. For large strain Radioss uses a volumic strain computed from density:(17)

μ = (\frac{ρ}{ρ_{0}} - 1) = \frac{Δ V}{V} = \frac{Δ l}{l}

For small strain you have:(18)

μ = - (ε_{x} + ε_{y} + ε_{z}) = - \frac{Δ l}{l_{0}}