Elastic-plastic Orthotropic Composite Shells

Two kinds of composite shells may be considered in the modeling:

Composite shells with isotropic layers
Composite shells with at least one orthotropic layer

The first case can be modeled by an isotropic material where the composite property is defined in element property definition as explained in Element Library. However, in the case of composite shell with orthotropic layers the definition of a convenient material law is needed. Two dedicated material laws for composite orthotropic shells exist in Radioss:

Material law COMPSH (25) with orthotropic elasticity, two plasticity models and brittle tensile failure,
Material law CHANG (15) with orthotropic elasticity, fully coupled plasticity and failure models.

These laws are described here. The description of elastic-plastic orthotropic composite laws for solids is presented in the next section.

Tensile Behavior

The tensile behavior is shown in Figure 2. The behavior starts with an elastic phase. Then, reached to the yield state, the material may undergo an elastic-plastic work hardening with anisotropic Tsai-Wu yield criteria. It is possible to take into account the material damage. The failure can occur in the elastic stage or after plastification. It is started by a damage phase then conducted by the formation of a crack. The maximum damage factor will allow these two phases to separate. The unloading can happen during the elastic, elastic-plastic or damage phase. The damage factor

d

varies during deformation as in the case of isotropic material laws (LAW27). However, three damage factors are computed; two damage factors

d_{1}

and

d_{2}

for orthotropy directions and the other

d_{3}

for delamination:(1)

\begin{array}{l} σ_{11} = E_{11} (1 - d_{1}) ε_{11} \\ σ_{22} = E_{22} (1 - d_{2}) ε_{22} \\ σ_{12} = G_{12} (1 - d_{1}) (1 - d_{2}) γ_{12} \end{array}

Where,

d_{1}

and

d_{2}

are the tensile damages factors. The damage and failure behavior is defined by introduction of the following input parameters:

$ε_{t 1}$: Tensile failure strain in direction 1
$ε_{m 1}$: Maximum strain in direction 1
$ε_{t 2}$: Tensile failure strain in direction 2
$ε_{m 2}$: Maximum strain in direction 2
d_max: Maximum damage (residual stiffness after failure)

Delamination

The delamination equations are:(2)

σ_{31} = G_{31} (1 - d_{3}) γ_{31}

(3)

σ_{23} = G_{23} (1 - d_{3}) γ_{23}

Where, $d_{3}$ is the delamination damage factor. The damage evolution law is linear with respect to the shear strain.

Let

γ = \sqrt{γ_{31}^{2} + γ_{23}^{2}}

then:(4)

for $d_{3}$ ⇒ $γ = γ_{i n i}$

for $d_{3}$ =1 ⇒ $γ = γ_{\max}$

Plastic Behavior

The plasticity model is based on the Tsai-Wu criterion ¹ which enable to model the yield and failure phases. (5)

F (σ) = F_{1} σ_{1} + F_{2} σ_{2} + F_{11} σ_{1}^{2} + F_{22} σ_{2}^{2} + F_{44} σ_{12}^{2} + 2 F_{12} σ_{1} σ_{2}

Where,(6)

F_{1} = - \frac{1}{σ_{1 y}^{c}} + \frac{1}{σ_{1 y}^{t}}

(7)

F_{22} = \frac{1}{σ_{2 y}^{c} σ_{2 y}^{t}}

(8)

F_{2} = - \frac{1}{σ_{2 y}^{c}} + \frac{1}{σ_{2 y}^{t}}

(9)

F_{44} = \frac{1}{σ_{12 y}^{c} σ_{12 y}^{t}}

(10)

F_{11} = \frac{1}{σ_{1 y}^{c} σ_{1 y}^{t}}

(11)

F_{12} = - \frac{α}{2} \sqrt{F_{11} F_{22}}

Where,

α

is the reduction factor. The six other parameters are the yield stresses in tension and compression for the orthotropy directions which can be obtained uniaxial loading tests:

$σ_{1}_{y}^{t}$: Tension in direction 1 of orthotropy
$σ_{2}_{y}^{t}$: Tension in direction 2 of orthotropy
$σ_{1}_{y}^{c}$: Compression in direction 1 of orthotropy
$σ_{2}_{y}^{c}$: Compression in direction 2 of orthotropy
$σ_{12}_{y}^{c}$: Compression in direction 12 of orthotropy
$σ_{12}_{y}^{t}$: Tension in direction 12 of orthotropy

The Tsai-Wu criteria are used to determine the material behavior: (12)

$F (σ) < 1$ : elastic state
$F (σ) = 1$ : plastic admissible state
$F (σ) > 1$ : plastically inadmissible stresses

For

F (σ) = 1

the cross-sections of Tsai-Wu function with the planes of stresses in orthotropic directions is shown in Figure 3.

F (σ) > 1

, the stresses must be projected on the yield surface to satisfy the flow rule.

F (σ)

is compared to a maximum value

F (W_{p})

varying in function of the plastic work

W_{p}

during work hardening phase:(13)

F (σ) = F (W_{p}) = 1 + b W_{p}^{n}

Where,

$b$: Hardening parameter
$n$: Hardening exponent

Therefore, the plasticity hardening is isotropic as illustrated in Figure 4.

Failure Behavior

The Tsai-Wu flow surface is also used to estimate the material rupture by means of two variables:

plastic work limit $W_{p}^{\max}$ ,
maximum value of yield function $F_{\max}$

If one of the two conditions is satisfied, the material is ruptured. The evolution of yield surface during work hardening of the material is shown in Figure 5.

The model will allow the simulation of the brittle failure by formation of cracks. The cracks can either be oriented parallel or perpendicular to the orthotropic reference frame (or fiber direction), as shown in Figure 6. For plastic failure, if the plastic work

W_{p}

is larger than the maximum value

W_{p}^{\max}

for a given element, then the element is considered to be ruptured. However, for a multi-layer shell, several criteria may be considered to model a total failure. The failure may happen:

If $W_{p} > W_{p}^{\max}$ for one layer,
If $W_{p} > W_{p}^{\max}$ for all layers,
If $W_{p} > W_{p}^{\max}$ or tensile failure in direction 1 for each layer,
If $W_{p} > W_{p}^{\max}$ or tensile failure in direction 2 for each layer,
If $W_{p} > W_{p}^{\max}$ or tensile failure in directions 1 and 2 for each layer,
If $W_{p} > W_{p}^{\max}$ or tensile failure in direction 1 for all layers,
If $W_{p} > W_{p}^{\max}$ or tensile failure in direction 2 for all layers,
If $W_{p} > W_{p}^{\max}$ or tensile failure in directions 1 and 2 for each layer.

The last two cases are the most physical behaviors; but the use of failure criteria depends, at first, to the analyst’s choice. In Radioss the flag I_off defines the used failure criteria in the computation.

In practice, the use of brittle failure model allows to estimate correctly the physical behavior of a large rang of composites. But on the other hand, some numerical oscillations may be generated due to the high sensibility of the model. In this case, the introduction of an artificial material viscosity is recommended to stabilize results. In addition, in brittle failure model, only tension stresses are considered in cracking procedure.

The ductile failure model allows plasticity to absorb energy during a large deformation phase. Therefore, the model is numerically more stable. This is represented by CRASURV model in Radioss. The model makes also possible to take into account the failure in tension, compression and shear directions.

Strain Rate Effect

The strain rate is taken into account within the modification of Plastic Behavior, Equation 13 which acts through a scale factor:(14)

F (W_{p}, \dot{ε}) = (1 + b {(\frac{W_{p}}{W_{p}^{r e f}})}^{n}) * (1 + c \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))

Where,

$W_{p}$: Plastic work
$W_{p}^{r e f}$: Reference plastic work
$b$: Plastic hardening parameter
$n$: Plastic hardening exponent
$c$: Strain rate coefficient (equal to zero for static loading)

The last equation implies the growing of the Tsai-Wu yield surface when the dynamic effects are increasing. The effects of strain rate are illustrated in Figure 7.

CRASURV Model

The CRASURV model is an improved version of the former law based on the standard Tsai-Wu criteria. The main changes concern the expression of the yield surface before plastification and during work hardening. First, in CRASURV model the coefficient

F_{44}

in Plastic Behavior, Equation 5 depends only on one input parameter:(15)

F_{44} (W_{p}^{*}, \dot{ε}) = \frac{1}{σ_{12}^{} (W_{p}^{*}, \dot{ε}) \cdot σ_{12}^{} (W_{p}^{*}, \dot{ε})}

With $σ_{12}^{} (W_{p}^{*}, \dot{ε}) = σ_{12 y}^{} (1 + b_{12}^{} {(W_{p}^{*})}^{n_{12}^{}}) (1 + c_{12}^{} \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$

Another modification concerns the parameters

F_{i j}

in Plastic Behavior, Equation 5 which are expressed now in function of plastic work and plastic work rate as in Strain Rate Effect, Equation 14:(16)

\begin{array}{l} F (W_{p}^{*}) = F_{1} (W_{p}^{*}, \dot{ε}) σ_{1} + F_{2} (W_{p}^{*}, \dot{ε}) σ_{2} + F_{11} (W_{p}^{*}, \dot{ε}) σ_{1}^{2} + F_{22} (W_{p}^{*}, \dot{ε}) σ_{2}^{2} \\ + 2 F_{12} (W_{p}^{*}, \dot{ε}) σ_{1} σ_{2} + F_{44} (W_{p}^{*}, \dot{ε}) σ_{12}^{2} \end{array}

With

$F_{i} (W_{p}^{*}, \dot{ε}) = - \frac{1}{σ_{i}^{c} (W_{p}^{*}, \dot{ε})} + \frac{1}{σ_{i}^{t} (W_{p}^{*}, \dot{ε})}$ ( $i$ =12)

$F_{i i} (W_{p}^{*}, \dot{ε}) = \frac{1}{σ_{i}^{c} (W_{p}^{*}, \dot{ε}) \cdot σ_{i}^{t} (W_{p}^{*}, \dot{ε})}$ ( $i$ =12)

$F_{12} (W_{p}^{*}, \dot{ε}) = - \frac{α}{2} \sqrt{F_{11} (W_{p}^{*}, \dot{ε}) F_{22} (W_{p}^{*}, \dot{ε})}$

$F_{44} (W_{p}^{*}, \dot{ε}) = \frac{1}{σ_{12}^{} (W_{p}^{*}, \dot{ε}) \cdot σ_{12}^{} (W_{p}^{*}, \dot{ε})}$

And(17)

σ_{i}^{t} (W_{p}^{*}, \dot{ε}) = σ_{i y}^{t} (1 + b_{i}^{t} {(W_{p}^{*})}^{n_{i}^{t}}) (1 + c_{i}^{t} \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))

(18)

σ_{i}^{c} (W_{p}^{*}, \dot{ε}) = σ_{i y}^{c} (1 + b_{i}^{c} {(W_{p}^{*})}^{n_{i}^{c}}) (1 + c_{i}^{c} \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))

(19)

σ_{12}^{} (W_{p}^{*}, \dot{ε}) = σ_{12 y}^{} (1 + b_{12}^{} {(W_{p}^{*})}^{n_{12}^{}}) (1 + c_{12}^{} \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))

And $W_{p}^{*} = \frac{W_{p}}{W_{p}^{r e f}}$

Where the five sets of coefficients

b

n

and

c

should be obtained by experience. The work hardening is shown in Figure 8.

The CRASURV model will allow the simulation of the ductile failure of orthotropic shells. The plastic and failure behaviors are different in tension and in compression. The stress softening may also be introduced in the model to take into account the residual Tsai-Wu stresses. The evolution of CRASURV criteria with hardening and softening works is illustrated in Figure 9.

Chang Chang Model

Chang-Chang law ², ³ incorporated in Radioss is a combination of the standard Tsai-Wu elastic-plastic law and a modified Chang-Chang failure criteria. ⁴ The affects of damage are taken into account by decreasing stress components using a relaxation technique to avoid numerical instabilities.

Six material parameters are used in the failure criteria:

$S_{1}$: Longitudinal tensile strength
$S_{2}$: Transverse tensile strength
$S_{12}$: Shear strength
$C_{1}$: Longitudinal compressive strength
$C_{2}$: Transverse compressive strength
$β$: Shear scaling factor

Where,

$1$: Fiber direction

The failure criterion for fiber breakage is written as:

Tensile fiber mode: $σ_{11} > 0$
(20) $e_{f}^{2} = {(\frac{σ_{11}}{S_{1}})}^{2} + β {(\frac{σ_{12}}{S_{12}})}^{2} - 1.0 {\begin{cases} \geq 0 f a i l e d \\ < 0 e l a s t i c - p l a s t i c \end{cases}$
Compressive fiber mode: $σ_{11} < 0$
(21) $e_{c}^{2} = {(\frac{σ_{11}}{C_{1}})}^{2} - 1.0 {\begin{cases} \geq 0 f a i l e d \\ < 0 e l a s t i c - p l a s t i c \end{cases}$

For matrix cracking, the failure criterion is:

Tensile matrix mode: $σ_{22} > 0$
(22) $e_{m}^{2} = {(\frac{σ_{22}}{S_{2}})}^{2} + β (\frac{σ_{12}}{S_{12}}) - 1.0 {\begin{cases} \geq 0 f a i l e d \\ < 0 e l a s t i c - p l a s t i c \end{cases}$
Compressive matrix mode: $σ_{22} < 0$
(23) $e_{d}^{2} = {(\frac{σ_{22}}{2 S_{12}})}^{2} + [{(\frac{C_{2}}{2 S_{12}})}^{2} - 1] \frac{σ_{22}}{C_{2} +} {(\frac{σ_{12}}{S_{12}})}^{2} - 1.0 {\begin{cases} \geq 0 f a i l e d \\ < 0 e l a s t i c - p l a s t i c \end{cases}$

If the damage parameter is equal to or greater than 1.0, the stresses are decreased by using an exponential function to avoid numerical instabilities. A relaxation technique is used by gradually decreasing the stress:(24)

[σ (t)] = f (t) * [σ_{d} (t_{r})]

With: $f (t) = \exp (\frac{t - t_{r}}{T})$ and $t \geq t_{4}$

Where,

$t$: Time
$t_{r}$: Start time of relaxation when the damage criteria are assumed
$τ$: Time of dynamic relaxation

$[σ_{d} (t_{r})]$ is the stress components at the beginning of damage (for matrix cracking $[σ_{d} (t_{r})] = (\begin{array}{l} σ_{d 22} \\ σ_{d 12} \end{array})$ ).

¹ Tsai S.W. and Wu E.M., “A general theory of strength for anisotropic materials”, Journal of Composite Materials, 58-80, 1971.

² Chang, F.K. and Chang, K.Y., “A Progressive Damage Model For Laminated Composites Containing Stress Concentrations”, Journal of Composite Materials, Vol 21, 834-855, 1987.

³ Chang, F.K. and Chang K.Y., “Post-Failure Analysis of Bolted Composites Joints in Tension or Shear-Out Mode Failure”, Journal of Composite Materials, Vol 21, 809-833, 1987.

⁴ Matzenmiller A. and Schweizerhof K., “Crashworthiness Simulation of composite Structures-a first step with explicit time integration”, Nonlinear Computational Mechanics-State of the Art Ed. p. Wriggers and W. Wagner, 1991.