Generalized Kelvin-Voigt Model (LAW35)

This law uses a generalized viscoelastic Kelvin-Voigt model whereas the viscosity is based on the Navier equations.

The effect of the enclosed air is taken into account via a separate pressure versus compression function. For open cell foam, this function may be replaced by an equivalent "removed air pressure" function. The model takes into account the relaxation (zero strain rate), creep (zero stress rate), and unloading. It may be used for open cell foams, polymers, elastomers, seat cushions, dummy paddings, and so on. In Radioss the law is compatible with shell and solid meshes.

The simple schematic model in Figure 1 describes the generalized Kelvin-Voigt material model where a time-dependent spring working in parallel with a Navier dashpot is put in series with a nonlinear rate-dependent spring. If

σ_{m} = \frac{I_{1}}{3}

$σ_{m} = \frac{I_{1}}{3}$ is the mean stress, the deviatoric stresses

s_{i j}

$s_{i j}$ at steps

n

$n$ and

n + 1

$n + 1$ are computed by the expressions:(1)

s_{i j}^{n} = σ_{i j}^{n} - δ_{i j} σ_{m}^{n}

$s_{i j}^{n} = σ_{i j}^{n} - δ_{i j} σ_{m}^{n}$

for

i = j

$i = j$ else,

δ_{i j} = 0

$δ_{i j} = 0$

(2)

s_{i j}^{n + 1} = s_{i j}^{n} + {\dot{s}}_{i j} d t

$s_{i j}^{n + 1} = s_{i j}^{n} + {\dot{s}}_{i j} d t$

with:(3)

{\dot{s}}_{i j} = 2 G {\dot{e}}_{i j} - (\frac{G + G_{t}}{η_{0}} s_{i j} (t)) + \frac{2 G \cdot G_{t}}{η_{0}} e_{i j}

${\dot{s}}_{i j} = 2 G {\dot{e}}_{i j} - (\frac{G + G_{t}}{η_{0}} s_{i j} (t)) + \frac{2 G \cdot G_{t}}{η_{0}} e_{i j}$

for $i \neq j$ $i \neq j$

(4)

{\dot{s}}_{i j} = G {\dot{e}}_{i i} - (\frac{G + G_{t}}{η_{0}} s_{i i} (t)) + \frac{G \cdot G_{t}}{η_{0}} e_{i i}

${\dot{s}}_{i j} = G {\dot{e}}_{i i} - (\frac{G + G_{t}}{η_{0}} s_{i i} (t)) + \frac{G \cdot G_{t}}{η_{0}} e_{i i}$

for $i = j$ $i = j$

Where,

G

$G$ and

G_{t}

$G_{t}$ are defined as:(5)

G = \max (\frac{E}{2 (1 + ν)}, \frac{A \dot{e} + B}{2 (1 + ν)})

$G = \max (\frac{E}{2 (1 + ν)}, \frac{A \dot{e} + B}{2 (1 + ν)})$

(6)

G_{t} = \frac{E_{t}}{2 (1 + ν_{t})}

$G_{t} = \frac{E_{t}}{2 (1 + ν_{t})}$

In Equation 5 the coefficients

A

$A$ and

B

$B$ are defined for Young's modulus updates (

E = E_{1} \dot{ε} + E_{2}

$E = E_{1} \dot{ε} + E_{2}$ ).

The expressions used by default to compute the pressure is:(7)

\frac{d P}{d t} = C_{1} K {\dot{ε}}_{k k} - C_{2} [\frac{K + K_{t}}{3 λ + 2 η_{0}} σ_{k k}] + C_{3} [\frac{K K_{t}}{3 λ + 2 η_{0}} ε_{k k}]

$\frac{d P}{d t} = C_{1} K {\dot{ε}}_{k k} - C_{2} [\frac{K + K_{t}}{3 λ + 2 η_{0}} σ_{k k}] + C_{3} [\frac{K K_{t}}{3 λ + 2 η_{0}} ε_{k k}]$

Where,(8)

K = \frac{E}{3 (1 - 2 v)}

$K = \frac{E}{3 (1 - 2 v)}$

(9)

K_{t} = \frac{E_{t}}{3 (1 - 2 v_{t})}

$K_{t} = \frac{E_{t}}{3 (1 - 2 v_{t})}$

(10)

P = - \frac{1}{3} σ_{k k}

$P = - \frac{1}{3} σ_{k k}$

(11)

ε_{k k} = \ln (\frac{V}{V_{0}})

$ε_{k k} = \ln (\frac{V}{V_{0}})$

$λ$ $λ$ and $η_{0}$ $η_{0}$ are the Navier Stokes viscosity coefficients which can be compared to Lame constants in elasticity. $λ + \frac{2 η_{0}}{3}$ $λ + \frac{2 η_{0}}{3}$ is called the volumetric coefficient of viscosity. For incompressible model, $ε_{k k}^{v} = 0$ $ε_{k k}^{v} = 0$ and $λ \to \infty$ $λ \to \infty$ and $μ_{0} = \frac{μ}{3}$ $μ_{0} = \frac{μ}{3}$ . In Equation 11, C₁, C₂ and C₃ are Boolean multipliers used to define different responses. For example, C₁=1, C₂=C₃=0 refers to a linear bulk model. Similarly, C₁=C₂=C₃=1 corresponds to a visco-elastic bulk model.

For polyurethane foams with closed cells, the skeletal spherical stresses may be increased by:(12)

P_{a i r} = - \frac{P_{0} \cdot γ}{1 + γ - Φ}

$P_{a i r} = - \frac{P_{0} \cdot γ}{1 + γ - Φ}$

Where,

$γ$ $γ$: Volumetric strain
$Φ$ $Φ$: Porosity
$P_{0}$ $P_{0}$: Initial air pressure

In Radioss, the pressure may also be computed with the $P$ $P$ versus $μ = \frac{ρ}{ρ_{0}} - 1$ $μ = \frac{ρ}{ρ_{0}} - 1$ , by a user-defined function. Air pressure may be assumed as an "equivalent air pressure" versus $μ$ $μ$ . You can define this function used for open cell foams or for closed cell by defining a model identical to material LAW 33 (FOAM_PLAS).