The INIEVO criterion allows a two-step failure approach, divided into an initiation phase, in which damage has no effect on the stress computation, and a damage evolution phase, in which a stress softening can be generated.
Ductile material behavior can be modelled in Optistruct with Damage Initiation DMGINI, and Damage Evolution DMGEVO criteria via the MATF or the MATS1 material cards. It is supported for Explicit (NLEXPL) and Implicit Non-Linear Static and Transient (NLSTAT and NLTRAN) solutions for Solid and Shell elements.
Failure of materials is strongly influenced by the loading conditions and thus, the stress state and strain rate. Damage initiation is based on the notions of Stress Triaxiality, Strain Rate and optionally to the Lode parameter to describe the loading conditions (uniaxial tension, pure shear, plane strain etc.).
When the initiation criterion is reached, stress softening damage variable evolution is triggered. This evolution can be based on a plastic displacement at the time of failure or on a certain value of fracture energy. In addition, the shape can be chosen between a classic linear stress reduction or an exponential drop of the load-carrying capacity.
1. Material Failure Criterion
To describe the ductile failure criterion, the value Stress Triaxiality η, Strain Rate έ, and the Lode Parameter ξ, are needed.
1.1 Stress Triaxiality
Stress Triaxiality (η) is used to differentiate between compressive and tensile loadings and depends on the trace of the stress tensor. It can determine the position of the stress state on the hydrostatic axis. It is computed as follows:
Where, σ_{VM} is the equivalent Von Mises Stress.
Stress Triaxiality for some common Stress States:
1.2 Stain Rate
Strain rate is the time derivative of strain of a material during the experimental test. It is computed as follows:
Different strain rates lead to different Yield stress, Ultimate stress and Failure Strain for the same material.
1.3 Lode Angle
To describe 3D loading conditions, another important quantity is the Lode Angle (θ) given by:
Where:
J3 is the third deviatoric invariant.
The Lode Angle determines the position of the stress state in the deviatoric section.
Under plane stress hypothesis (for shell elements), the Lode Angle and the Stress Triaxiality are linked and thus one for them can be used to recover the other:
As it is much easier to deal with normalized value instead of radians, the lode angle is usually switched by the Lode Parameter denoted ξ, given by:
2. DUCTILE DAMAGE
Damage Mechanics
Considering an undamaged volume, the stress response is given by:
When the same volume is damaged, the effective area decreases and the imperfections area increases, where the damaged area is given by:
A damage variable is defined by the ratio between damaged and undamaged area:
The damage variable evolves between 0 and 1:
The stress response of the damaged volume is then given by:
Which after manipulation can be expressed as:
The curve below demonstrates the typical true stress-strain curve including damage:
Optistruct Ductile Damage: INIEVO criterion
The INIEVO failure criterion is a two-step failure criterion based on plastic strain and stress state. These two steps consist in the computation of two successive damage variables:
1st step: The damage initiation phase, in which an internal variable evolution denoted is computed. This variable is a purely internal value that has no influence on the stress computation. Once this initiation variable reaches the value 1.0, the second stage of the failure is triggered by the computation of the damage variable evolution denoted D.
2nd step: The damage variable D evolution is computed, generating a stress softening effect until the complete failure of the element and its deletion.
Damage Initiation
On ductile damage, the damage initiation occurs due to nucleation, growth and propagation of material imperfections. Damage initiation defines the point of initiation of degradation of stiffness. The damage initiation variable ω_{D}_{ }is computed incrementally for each time step as:
Where:
_{: }Plastic strain with respect to stress triaxiality, optionally with strain rate, and lode angle
P: Hydrostatic pressure −Tr(σ)/3
σ_{VM}: von Mises stress
When the damage initiation variable ω_{D}=1, the damage variable evolution D is triggered, generating stress softening.
2.2 Damage Evolution
Once the material is damaged, the stress progression at the integration point is significantly impacted. The material fails due to the gradual decrease in material stiffness and the stress-strain relationship can’t accurately predict the material’s behavior. At this stage, there is a significant strain localization, leading to a strong mesh dependency. To overcome this, the evolution of damage is formulated using the equivalent plastic displacement , or fracture energy dissipation , while also considering the element characteristic length .
Using Optistruct, the damage evolution can be expressed in terms of both Equivalent Plastic Displacement, or Fracture Energy:
Plastic displacement at failure (DISP):
According to the effective plastic displacement criterion, damage is characterized by the plastic displacement following the damage initiation. This displacement is not influenced by the size of the element.
The evolution of the damage variable with respect to plastic displacement can be expressed as linear or exponential form:
Linear Shape:
Exponential shape:
Where,α is the exponential shape parameter.
Dissipated fracture energy (ENER):
In this method of evolution, the fracture energy necessary for material failure is defined. This is calculated as the area under the stress-strain curve post-damage initiation.
The fracture energy is the area below Stress – Plastic Displacement curve, given by:
The evolution of the damage variable with respect to fracture energy can be expressed as linear or exponential form:
Linear Shape:
Where σ_{y}: Yield stress at damage evolution triggering
Exponential shape:
3. Optistruct Set up
The damage initiation and evolution failure criteria can be defined in Optistruct using two methods:
1. Using the DAMAGE continuation line in the MATS1 Bulk Data Entry. This method is supported both for Implicit and Explicit Dynamic Analysis.
MATS1 Bulk Data Entry
INIEVO, implicit supports isotropic hardening, HR=1
More information of the material card can be found in Optistruct’s Documentation:
https://help.altair.com/hwsolvers/os/topics/solvers/os/mats1_bulk_r.htm
2. Using CRI=INIEVO in the MATF Bulk Data Entry. This method is supported only for Explicit Dynamic Analysis.
MATF Bulk Data Entry
More information of the material card can be found in Optistruct’s Documentation:
https://help.altair.com/hwsolvers/os/topics/solvers/os/matf_bulk_r.htm
DMGINI can be very versatile, i.e., it can define the damage onset curves/surfaces (equivalent plastic strain) as a function of stress triaxiality, strain rate, and Lode parameter, the last being valid only for solid elements.
Damage evolution, DMGEVO, supports two forms: equivalent plastic displacement (DISP) and energy (ENERGY). Moreover, damage evolution can follow linear (LIN) and exponential (EXP) shapes.
3.1 DMGINI: Defines the damage initiation criteria for Ductile Damage.
For CRI (Damage initiation criteria) =DUCTILE, the parameters denote:
More information for the card can be found in Optistruct’s Documentation:
https://help.altair.com/hwsolvers/os/topics/solvers/os/dmgini_bulk_r.htm?zoom_highlight=dmgini
3.2 DMGEVO: Defines the damage evolution criteria for Ductile Damage.
The parameters denote:
TYPE |
SHAPE |
W1 |
Equivalent Plastic Displacement (DISP) or Dissipated Fracture Energy (ENER) |
Linear (LIN) or Exponential (EXP) |
Corresponding Value |
More information for the card can be found in Optistruct’s Documentation:
https://help.altair.com/hwsolvers/os/topics/solvers/os/dmgevo_bulk_r.htm
3.3 DAMAGE OUTPUT
Damage Initiation Index (DII) and Damage Index (DI) results are automatically written.
•DII becomes non-zero when plasticity starts and reaches 1.0 when damage (softening) begins, whereas DI becomes 1.0 when the element is fully damaged.
•Reported results are element average results, and therefore, DII can be <1.0 and DI > 0.0.
•Damage onsets only after the equivalent plastic strain (DMGINI) is exceeded.
•Element erosion is based on the following rules
4. Ductile Damage Example in Optistruct
This example simulates a tensile test on a dogbone specimen with ductile damage.
Download Models:
To model the elastoplastic material, we use the MAT1 and MATS1 cards.
In MAT1 bulk data entry card we define the elastic properties of the material by giving the modulus of Elasticity E, Poisson’s Ratio v and Density RHO.
In MATS1 bulk data entry card we define the plastic properties of the material by adding the Plastic Material curve.
! In MATS1 the true stress vs plastic strain must be given for large displacement analysis.
A standard test produces Engineering or nominal stress and strain. Which are given from the following equations: σ_{eng}=F/Ao and ε_{eng}=DL/L0
True Stress-Strain can be computed from Engineering Stress-Strain by the following equations: σ_{t}= σ_{eng}(ε_{eng}+1) and ε_{t}= ln(ε_{eng}+1)
Plastic strain then can be computed by the following equation: ε_{p}= ε_{t} –(σ_{t}/E)
Implicit:
In the MATS1 card we add the DAMAGE continuation line referencing the initiation DMGINI and evolution DMGEVO card IDs.
We add DMGINI with the following values:
Plastic Strain at Failure |
Stress Triaxiality |
Strain Rate |
0.25 |
1.35 |
1 |
We add DMGEVO with the following values:
TYPE |
Shape |
Plastic displacement |
DISP |
LIN |
0.65 |
Optistruct Deck:
The DAMAGE continuation line has to be added manually in the current version of Hypermesh (2024.1) and it will get implemented in a future release.
Results: We can observe necking in the specimen before it fractures.
From the damage onset to final damage, element eq. plastic Displacement
𝑢𝑓=𝐿_{ref}(𝜀𝑓_{𝑝𝑙}−𝜀𝐷_{𝑝𝑙}) increases 0.65.
Where,
For Element with ID=5, we plot Von Mises stress vs Equivalent Plastic Strain:
In the specimen damage onsets around t=0.185. When damage index (element average result) becomes >0.0.
Element’s ID= 5 volume is 6.82 and the reference length is 1.896.
Plastic strain at full damage is 0.5928 = [Plastic strain at damage initiation: 0.25] + [plastic displacement (from DMGEVO: 0.65)] / [reference length (1.896)].
Explicit:
Material Set-up is similar with implicit but instead of DAMAGE continuation line in MATS1 we define a MATF material card with INIEVO.
Implicit/Explicit Results Comparison: