Failure Models

In addition to the possibility to define user's material failure models, Radioss integrates several failure models. These models use generally a global notion of cumulative damage to compute failure. They are mostly independent to constitutive law and the hardening model and can be linked to several available material laws. A compatibility table is given in the Radioss Reference Guide. 表 1 provides a brief description of available models.

表 1. Failure Model Description
Failure Model	Type	Description
BIQUAD	Strain failure model	Direct input on effective plastic strain to failure
CHANG	Chang-Chang model	Failure criteria for composites
CONNECT	Failure	Normal and Tangential failure model
EMC	Extended Mohr Coulomb failure model	Failure dependent on effective plastic strain
ENERGY	Energy isotrop	Energy density
FABRIC	Traction	Strain failure
FLD	Forming limit diagram	Introduction of the experimental failure data in the simulation
HASHIN ³ ⁴	Composite model	Hashin model
HC_DSSE	Extended Mohr Coulomb failure model	Strain based Ductile Failure Model: Hosford-Coulomb with Domain of Shell-to-Solid Equivalence
JOHNSON	Ductile failure model	Cumulative damage law based on the plastic strain accumulation
LAD_DAMA	Composite delamination	Ladeveze delamination model
NXT	NXT failure criteria	Similar to FLD, but based on stresses
PUCK	Composite model	Puck model
SNCONNECT	Failure	Failure criteria for plastic strain
SPALLING	Ductile + Spalling	Johnson-Cook failure model with Spalling effect
TAB1	Strain failure model	Based on damage accumulation using user-defined functions
TBUTCHER	Failure due to fatigue	Fracture appears when time integration of a stress expression becomes true
TENSSTRAIN	Traction	Strain failure
WIERZBICKI	Ductile material	3D failure model for solid and shells
WILKINS	Ductile Failure model	Cumulative damage law

Johnson-Cook Failure Model

High-rate tests in both compression and tension using the Hopkinson bar generally demonstrate the stress-strain response is highly isotropic for a large scale of metallic materials. The Johnson-Cook model is very popular as it includes a simple form of the constitutive equations. In addition, it also has a cumulative damage law that can be accesses failure:(1)

D = \sum \frac{Δ ε}{ε_{f}}

with:(2)

ε_{f} = [D_{1} + D_{2} \exp (D_{3} σ^{*})] [1 + D_{4} \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}})] [1 + D_{5} T^{*}]

Where $Δ ε$ is the increment of plastic strain during a loading increment, $σ^{*} = \frac{σ_{m}}{σ_{V M}}$ the normalized mean stress and the parameters $D_{i}$ the material constants. Failure is assumed to occur when $D$ =1.

Wilkins Failure Criteria

An early continuum model for void nucleation is presented in ¹. The model proposes that the decohesion (failure) stress

σ_{c}

is a critical combination of the hydrostatic stress

σ_{m}

and the equivalent von Mises stress

σ_{V M}

:(3)

σ_{c} = σ_{m} + σ_{V M}

In a similar approach, a failure criteria based on a cumulative equivalent plastic strain was proposed by Wilkins. Two weight functions are introduced to control the combination of damage due to the hydrostatic and deviatoric loading components. The failure is assumed when the cumulative reaches a critical value

D_{c}

. The cumulative damage is obtained by:(4)

D_{c} = \int W_{1} W_{2} d {\bar{ε}}_{p} \approx \sum_{i = 1}^{n} W_{1} W_{2} Δ {\bar{ε}}_{p}^{i}

Where,

$W_{1} = {(\frac{1}{1 + a P})}^{α}$
$P = \frac{1}{3} \sum_{j = 1}^{3} σ_{j j}$
$W_{2} = {(2 - A)}^{β}$
$A = \max (\frac{s_{2}}{s_{3}}, \frac{s_{2}}{s_{1}})$
$s_{1} \geq s_{2} \geq s_{3}$

Where,

$Δ {\bar{ε}}_{p}$: An increment of the equivalent plastic strain
$W_{1}$: Hydrostatic pressure weighting factor
$W_{2}$: Deviatoric weighting factor
$s_{i}$: Deviatoric principal stresses
a, $α$ and $β$: The material constants

Tuler-Butcher Failure Criteria

A solid may break owning to fatigue due to Tuler-Butcher criteria: ²

(5)

D = \int_{0}^{t} \max (0, {(σ - σ_{r})}^{λ}) d t

Where,

$σ_{r}$: Fracture stress
$σ$: Maximum principal stress
$λ$: Material constant
$t$: Time when solid cracks
$D$: Another material constant, called damage integral

Forming Limit Diagram for Failure (FLD)

In this method the failure zone is defined in the plane of principal strains (図 1). The method usable for shell elements allows introducing the experimental results in the simulation.

Spalling with Johnson-Cook Failure Model

In this model, the Johnson-Cook failure model is combined to a Spalling model where we take into account the spall of the material when the pressure achieves a minimum value P_min. The deviatoric stresses are set to zero for compressive pressure. If the hydrostatic tension is computed, then the pressure is set to zero. The failure equations are the same as in Johnson-Cook model.

Bao-Xue Wierzbicki Failure Model

Bao-Xue-Wierzbicki model ⁵ represents a 3D fracture criterion which can be expressed by:(6)

{\bar{ε}}_{f}^{n} = {\bar{ε}}_{\max}^{n} - [{\bar{ε}}_{\max}^{n} - {\bar{ε}}_{\min}^{n}] {(1 - {\bar{ξ}}^{m})}^{1 / m}

(7)

{\bar{ε}}_{\max} = C_{1} e^{- C_{2} \bar{η}}

(8)

{\bar{ε}}_{\min} = C_{3} e^{- C_{4} \bar{η}}

Where,

C_{1}

C_{2}

C_{3}

C_{4}

γ

and

m

are the material constants,

n

is the hardening parameter and

\bar{η}

and

\bar{ξ}

are defined as:

for solids:
If I_moy=0:

$\bar{η} = \frac{σ_{m}}{σ_{V M}}$ ; $\bar{ξ} = \frac{27}{2} \frac{J_{3}}{σ_{V M}^{3}}$

If I_moy=1:
$\bar{η} = \frac{\int_{0}^{ε_{p}} \frac{σ_{m}}{σ_{V M}} d ε_{p}}{ε_{p}}$ $\bar{ξ} = \frac{\int_{0}^{ε_{p}} \frac{27 J_{3}}{2 σ_{V M}^{3}} d ε_{p}}{ε_{p}}$
for shells:
$\bar{η} = \frac{σ_{m}}{σ_{V M}}$ ; $\bar{ξ} = - \frac{27}{2} \bar{η} ({\bar{η}}^{2} - \frac{1}{3})$

Where,

$σ_{m}$: Hydrostatic stress
$σ_{V M}$: The von Mises stress
$J_{3} = s_{1} s_{2} s_{3}$: Third invariant of principal deviatoric stresses

Strain Failure Model

This failure model can be compared to the damage model in LAW27. When the principal tension strain

ε_{1}

reaches

ε_{t 1}

, a damage factor

D

is applied to reduce the stress, as shown in 図 3. The element is ruptured when

D

=1. In addition, the maximum strains

ε_{t 1}

and

ε_{t 2}

may depend on the strain rate by defining a scale function.

Energy Density Failure Model

When the energy per unit volume achieves the value

E_{1}

, then the damage factor

D

is introduced to reduce the stress. For the limit value

E_{2}

, the element is ruptured. In addition, the energy values

E_{1}

and

E_{2}

may depend on the strain rate by defining a scale function.

XFEM Crack Initialization Failure Model

This failure model is available for Shell only.

In /FAIL/TBUTCHER, the failure mode criteria are written as:

For ductile materials, the cumulative damage parameter is:(9)

D = \int_{0}^{t} \max (0, {(σ - σ_{r})}^{λ}) d t

Where,

$σ_{r}$: Fracture stress
$σ$: Maximum principal stress
$λ$: Material constant
$t$: Time when shell cracks for initiation of a new crack within the structure
$D$: Another material constant called damage integral

For brittle materials, the damage parameter is:(10)

\dot{D} = \frac{1}{K} {(σ - σ_{r})}^{a}

(11)

σ_{r} = σ_{0} {(1 - D)}^{b}

(12)

D = D + \dot{D} Δ t

¹ Argon A.S., J. Im, and Safoglu R., 「Cavity formation from inclusions in ductile fracture」, Metallurgical Transactions, Vol. 6A, pp. 825-837, 1975.

² Tuler F.R. and Butcher B.M., 「A criteria for time dependence of dynamic fracture」, International Journal of Fracture Mechanics, Vol. 4, N°4, 1968.

³ Hashin, Z. and Rotem, A., 「A Fatigue Criterion for Fiber Reinforced Materials」, Journal of Composite Materials, Vol. 7, 1973, pp. 448-464.9.

⁴ Hashin, Z., 「Failure Criteria for Unidirectional Fiber Composites」, Journal of Applied Mechanics, Vol. 47, 1980, pp. 329-334.

⁵ Wierzbicki T., 「From crash worthiness to fracture; Ten years of research at MIT」, International Radioss User's Conference, Nice, June 2006.