### 22.6.3. Failure Models

The discussions in the sections on strength models explain how plastic yielding arises from the inability of real materials to support arbitrarily large shear stresses. The separation of the flow into separate elastic and plastic flow regimes is described, together with the criteria for recognizing the switch between one regime and the other.

In a similar manner real materials are not able to withstand tensile stresses which exceed the material’s local tensile strength.

The computation of the dynamic motion of materials assuming that they always remain continuous, even if the predicted local stresses reach very large negative values, will lead to unphysical solutions.

Some model has to be constructed to recognize when tensile limits are reached, to modify the computation to deal with this and to describe the properties of the material after this formulation has been applied.

Several different modes of failure have been implemented in Autodyn to provide the most suitable form of failure criterion for different types of materials. They can be divided into three categories of failure criteria:

Bulk (isotropic) failure

  These models consider the material behavior in an overall (isotropic) manner and allow for failure when some predefined flow variable reaches a critical value.

Directional failure

  These models can be used to model failure initiation which is directionally dependent. They are therefore very appropriate for different failure modes such as spalling, plugging, delamination, petalling, and discing. They are only implemented currently for Lagrange, ALE, Shell, and SPH solvers. They cannot be used for Euler since it is not possible to accurately track principal directions in Euler cells.

Cumulative Damage, Johnson-Holmquist Damage

  These failure models have been developed to describe the macroscopic inelastic behavior of materials such as ceramics and concrete where the strength of the material can be significantly impaired by crushing. The models can sometimes also be used to follow the behavior of metals which are subjected to tensile stress levels below the ultimate tensile limit but for periods of time long enough to initiate incipient spall.

Post-failure response of materials is isotropic for all of the above failure models.

Following failure, the failed material in that cell can no longer sustain any shear stress or any negative pressures except in the case of the hydrodynamic tensile limit model, where it is assumed that the material "reheals", i.e. while the material pressure is set to zero in the cell for that time step it is assumed that the material can sustain negative pressures down to the hydrodynamic tensile limits in subsequent flow (it may of course reach that limit again and the failure process will be repeated).

In Euler Parts, material(s) in a cell which has failed have their volumes adjusted and void created to satisfy the zero pressure criterion. These materials may be transported into neighboring cells in the following time-step. They will not carry with them any knowledge of the failure in the previous time-step and in the donor cell (other than the values of pressure, volume etc. arising from the failure). The criterion for failure in the recipient cells will test on the updated pressure etc. within those cells without knowledge of past histories.

While some of the different models can be used with all types of solvers, others are implemented in only specific types of solvers. The range of applicability is indicated below:

| Lagrange | ALE | Euler | Shell
--- | --- | --- | --- | ---
Hydrodynamic tensile limit | Yes | Yes | Yes | No
Bulk strain | Yes | Yes | Yes | Yes
Directional failure | Yes | Yes | No | Yes
Cumulative damage | Yes | Yes | Yes | No
Johnson-Holmquist damage | Yes | Yes | Yes | No

#### 22.6.3.1. Material Stress Failure

In this option, and the subsequent two material failure options, the specified directions, referred to as "principal" directions, are defined by the dominant geometric characteristics of the material.

These models are useful for materials which are likely to fail along pre-defined material planes, for example where delamination failure occurs between layers in a composite plate.

Failure will be initiated if any of the principal material stresses or strains exceed their respective specified failure levels.

Because the principal directions are defined by the material directions, there can be a finite, non-zero, shear stress and strain on the planes defined by these directions. Also the maximum direct stresses and strains can lie on planes which are not coincident with the principal material planes, and the maximum shear stress and shear strain may not be on a plane at 45 degrees to the principal material plane.

You must specify:

1. Maximum tensile failure stress 11

2. Maximum tensile failure stress 22

3. Maximum tensile failure stress 33

4. Maximum shear stress 12

5. Material axes option

6. Rotation angle (degrees)

7. X-origin (Polar space)

8. Y- origin (Polar space)

You define the initial principal material direction through the input parameters (5) to (8) as shown in the following figure:

Failure is initiated if any of the principal material stresses exceed their respective specified tensile failure stress limits. For shear failures the shear stress on planes parallel to the principal directions are checked against the maximum shear stress 12.

#### 22.6.3.2. Material Strain Failure

You must specify:

1. Maximum tensile failure strain 11

2. Maximum tensile failure strain 22

3. Maximum tensile failure strain 33

4. Maximum shear strain 12

5. Material axes option

6. Rotation angle (degrees)

7. X-origin (Polar space)

8. Y- origin (Polar space)

You define the initial principal material direction through the input parameters (5) to (8) as detailed in the previous figure.

Failure is initiated if any of the principal material strains exceeds the respective specified limit. For shear failures the shear strain on planes parallel to the principal directions are checked against the maximum shear strain 12.

#### 22.6.3.3. Material Stress/Strain Failure

You must specify:

1. Maximum tensile failure stress 11

2. Maximum tensile failure stress 22

3. Maximum tensile failure stress 33

4. Maximum shear stress 12

5. Maximum tensile failure strain 11

6. Maximum tensile failure strain 22

7. Maximum tensile failure strain 33

8. Maximum shear strain 12

9. Material axes option

10. Rotation angle (degrees)

11. X-origin (Polar space)

12. Y- origin (Polar space)

You define the initial principal material direction through the input parameters (9) to (12) as detailed in the previous figure.

Failure is initiated if any of the principal material stresses or strains exceed their respective failure levels. For shear failures the values of the shear stress and strain on planes parallel to the principal directions are checked against the inputted maximum shear stress and strain.

#### 22.6.3.4. Cumulative Damage Model

This model can be used to describe the macroscopic inelastic behavior of material such as ceramics and concrete where the strength of the material can be significantly degraded by crushing.

The model can be used only with the Linear equation of state but can be used in conjunction with any strength model (except Johnson-Holmquist which has its own associated cumulative damage model).

However, since experiments indicate that ceramics show a marked increase in compressive strength as the hydrostatic pressure is increased, it is most likely that this model will be used in conjunction with the Mohr-Coulomb model which uses a yield strength that is a function of the local hydrostatic pressure.

To model the progressive crushing and subsequent weakening of ceramic materials the model computes an r;damage factor which is usually related to the amount of straining the material is subjected to.

This damage factor is used to reduce the elastic moduli and yield strength of the material as the calculation proceeds. In the standard model damage is represented by a parameter D which is zero for all plastic deformation for which the effective plastic strain is less than a value EPS1. When the strain reaches a value of EPS1 the damage parameter D increases linearly with strain up to a maximum value Dmax (<1) at a value of the effective plastic strain EPS2, as shown in the following figure.

Thus

| (22–33)

If a different damage function is required you can program this by means of the user subroutine EXDAM.

To describe the progressive crushing of a material the damage function is used to reduce the material’s strength. Fully damaged material has some residual strength in compression but none in tension. The current value of the damage factor D is used to modify the bulk modulus, shear modulus and yield strength of the material.

The yield strength is reduced as follows:

- If the hydrostatic pressure is positive

  | (22–34)

  (providing some residual strength when D reaches its maximum Dmax)

- If the hydrostatic pressure is negative

  | (22–35)

These are illustrated graphically below.

Yield Stress as a Function of Cumulative Damage

The bulk modulus and shear modulus are unaffected in compression, while in tension they are progressively reduced to zero when damage is complete. In tension therefore they are both reduced by the factor (1 - D/Dmax) as shown graphically in the figure below.

Bulk and Shear Modulus as Functions of Cumulative Damage

For a more detailed account of the use of this type of model see Persson (1990 [13]).

#### 22.6.3.5. Orthotropic Softening

Advanced composite strength and damage models are available in Autodyn. The Orthotropic Softening Model forms part of this capability and is usually used in combination with an orthotropic equation of state and an orthotropic strength model.

The basic characteristics of these models are described in the figures below.

Outline of new orthotropic yield and damage models for composite materials

Full details of the model can be found in the composite modeling guidance document that is available on the ANSYS Customer Portal under Product Documentation, on the Explicit Dynamics page. This provides further details of the modeling techniques available and how to obtain/fit material data to the models.

Material data for a Kevlar-epoxy composite for this advanced model is provided in the standard Autodyn material library.

For full details of the original model development and characterization work, please refer to [12].

#### 22.6.3.6. Tsai - Hoffman - Hill Model

The Tsai-Wu, Hoffman and Tsai-Hill criteria can be used with the 3D composite shell element. These models require both compressive and tensile failure strengths, FTij and FCij, to be defined in the plane of the shell which are combined to give a bulk failure criteria as follows;

| (22–36)

where the constants Fij and Fi are defined for each of the failure models as follows;

Tsai-Wu | Hoffman | Tsai-Hil
--- | --- | ---

#### 22.6.3.7. MO Granular Failure Model

The MO Granular failure model allows a granular material to regain strength once is has failed. Therefore, the failure model can only be used with the MO Granular strength model.

The yield stress is made up of two components, one dependent on the density and one dependent of the pressure,

| (22–37)

where σY , σP , and σp denote the total yield stress, the pressure yield stress and the density yield stress respectively. F is initially supplied by the user as the Initial Fraction of Yield. Failure occurs if the Von Mises stress is greater than both the total yield stress σY and FSP where FS is the user defined variable Slope and P is pressure. After material failure, if the current density is greater than the density at which the material failed, F is determined by the following equation,

| (22–38)

where

| (22–39)

and ρF and ρ are the failure density and the current density respectively. HEXP is the user defined quantity Heal Exponential and ρF is initially supplied by the user as the Initial Failure Density. Therefore F is allowed to increase and the material may reheal.

Below is an example of a simulation which uses the MO Granular failure model:

The applied stress is time dependent as shown below:

This produces the following pressure density variation:

The model undergoes simple compression followed by uni-axial compression forcing the material to fail and then reheal. The initial values used to govern the failure nature of the material are: F = 1, FS = 3, HEXP = 40, and ρF = 10. Variations of the governing variables are shown below. The failure density is initially set to 10 and when failure occurs the failure density is set to the current density. Subsequently, as the density increases past this failure density then the material will begin to reheal.

The fraction of yield, F, increases as the density increases until the strength of the material is returned to its initial state:

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