Elastic-Plastic Orthotropic Composite Solids

The material LAW14 (COMPSO) in Radioss allows to simulate orthotropic elasticity, Tsai-Wu plasticity with damage, brittle rupture and strain rate effects. The constitutive law applies to only one layer of lamina. Therefore, each layer needs to be modeled by a solid mesh. A layer is characterized by one direction of the fiber or material. The overall behavior is assumed to be elasto-plastic orthotropic.

Direction 1 is the fiber direction, defined with respect to the local reference frame

\vec{r}, \vec{s}, \vec{t}

as shown in 図 1.

For the case of unidirectional orthotropy (that is, $E_{33} = E_{22}$ and $G_{31} = G_{12}$ ) the material LAW53 in Radioss allows to simulate an orthotropic elastic-plastic behavior by using a modified Tsai-Wu criteria.

Linear Elasticity

When the lamina has a purely linear elastic behavior, the stress calculation algorithm:

Transform the lamina stress, $σ_{i j} (t)$ , and strain rate, $d_{i j}$ , from global reference frame to fiber reference frame.
Compute lamina stress at time $t + Δ t$ by explicit time integration:
(1) $σ_{i j} (t + Δ t) = σ_{i j} (t) + D_{i j k l} d_{k l} Δ t$
Transform the lamina stress, $σ_{i j} (t + Δ t)$ , back to global reference frame.

The elastic constitutive matrix

C

of the lamina relates the non-null components of the stress tensor to those of strain tensor:(2)

{σ} = [D] {ε}

The inverse relation is generally developed in term of the local material axes and nine independent elastic constants:(3)

{\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{23} \\ γ_{31} \end{matrix}} = [\begin{matrix} \frac{1}{E_{11}} & - \frac{ν_{21}}{E_{22}} & - \frac{ν_{31}}{E_{33}} & 0 & 0 & 0 \\ \frac{1}{E_{22}} & - \frac{ν_{32}}{E_{33}} & 0 & 0 & 0 \\ \frac{1}{E_{33}} & 0 & 0 & 0 \\ \frac{1}{2 G_{12}} & 0 & 0 \\ S y m m . & \frac{1}{2 G_{23}} & 0 \\ \frac{1}{2 G_{31}} \end{matrix}] {\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{23} \\ σ_{31} \end{matrix}}

Where,

$E_{i j}$: Young's modulus
$G_{i j}$: Shear modulus
$ν_{i j}$: Poisson's ratios
$γ_{i j}$: Strain components due to the distortion

Orthotropic Plasticity

Lamina yield surface defined by Tsai-Wu yield criteria is used for each layer:(4)

\begin{array}{l} F = f (W_{p}) = F_{1} σ_{1} + F_{2} σ_{2} + F_{3} σ_{3} + F_{11} σ_{1}^{2} + F_{22} σ_{2}^{2} + F_{33} σ_{3}^{2} + F_{44} σ_{12}^{2} \\ + F_{55} σ_{23}^{2} + F_{66} σ_{31}^{2} + 2 F_{12} σ_{1} σ_{2} + 2 F_{23} σ_{2} σ_{3} + 2 F_{13} σ_{1} σ_{3} \end{array}

with:

$F_{i} = - \frac{1}{σ_{i y}^{c}} + \frac{1}{σ_{i y}^{t}}$ ( $i$ =1,2,3);

$F_{11} = \frac{1}{σ_{1 y}^{c} σ_{1 y}^{t}}$ ; $F_{22} = \frac{1}{σ_{2 y}^{c} σ_{2 y}^{t}}$ ; $F_{33} = \frac{1}{σ_{3 y}^{c} σ_{3 y}^{t}}$ ;

$F_{44} = \frac{1}{σ_{12 y}^{c} σ_{12 y}^{t}}$ ; $F_{55} = \frac{1}{σ_{23 y}^{c} σ_{23 y}^{t}}$ ; $F_{66} = \frac{1}{σ_{31 y}^{c} σ_{31 y}^{t}}$ ;

$F_{12} = - \frac{1}{2} \sqrt{(F_{11} F_{22})}$ ; $F_{23} = - \frac{1}{2} F_{22}$

Where,

σ_{i}

is the yield stress in direction

i

c

and

t

denote respectively for compression and tension.

f (W_{p})

represents the yield envelope evolution during work hardening with respect to strain rate effects:(5)

f (W_{p}) = (1 + B \cdot W_{p}^{n}) (1 + c .1 n (\frac{\dot{ε}}{{\dot{ε}}_{0}}))

Where,

$W_{p}$: Plastic work
$B$: Hardening parameter
$n$: Hardening exponent
$c$: Strain rate coefficient

f (W_{p})

is limited by a maximum value

f_{\max}

:(6)

f (W_{p}) \leq f_{\max} = {(\frac{σ_{\max}}{σ_{y}})}^{2}

If the maximum value is reached the material is failed.

In 式 5, the strain rate effects on the evolution of yield envelope. However, it is also possible to take into account the strain rate

\dot{ε}

effects on the maximum stress

σ_{\max}

as shown in 図 3.

(a) Strain rate effect on $σ_{\max}$	(b) No strain rate effect on $σ_{\max}$

$σ = σ_{y} (1 + c .1 n (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$ $σ_{\max} = σ_{\max}^{0} (1 + c .1 n (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$ $f_{\max} = {(\frac{σ_{\max}}{σ_{y}})}^{2}$	$σ = σ_{y} (1 + c .1 n (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$ $σ_{\max} = σ_{\max}^{0}$

(a) Strain rate effect on

σ_{\max}

(b) No strain rate effect on

σ_{\max}

σ = σ_{y} (1 + c .1 n (\frac{\dot{ε}}{{\dot{ε}}_{0}}))

$σ_{\max} = σ_{\max}^{0} (1 + c .1 n (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$

$f_{\max} = {(\frac{σ_{\max}}{σ_{y}})}^{2}$

σ = σ_{y} (1 + c .1 n (\frac{\dot{ε}}{{\dot{ε}}_{0}}))

$σ_{\max} = σ_{\max}^{0}$

図 3. Strain Rate Dependency

Unidirectional Orthotropy

LAW 53 in Radioss provides a simple model for unidirectional orthotropic solids with plasticity. The unidirectional orthotropy condition implies:(7)

\begin{array}{l} E_{33} = E_{22} \\ G_{31} = G_{12} \end{array}

The orthotropic plasticity behavior is modeled by a modified Tsai-Wu criterion (Orthotropic Plasticity, 式 4) in which:(8)

F_{12} = \frac{2}{{(σ_{y}^{45 c})}^{2}} - \frac{1}{2} (F_{11} + F_{22} + F_{44}) + \frac{F_{1} + F_{2}}{σ_{y}^{45 c}}

Where,

σ^{45}_{y}^{c}

is yield stress in 45° unidirectional test. The yield stresses in direction 11, 22, 12, 13 and 45° are defined by independent curves obtained by unidirectional tests (図 4). The curves give the stress variation in function of a so-called strain

ε_{v}

:(9)

ε_{ν} = 1 - (T r a c e [ε])