This regularization method is based on the crack band theory originally proposed by Bazant and Oh [5] for smeared crack models of concrete.

At element level, a characteristic element length is introduced, and the stress-strain relationship is adjusted depending on this quantity. It is assumed that the characteristic element length (calculated internally based on the actual element geometry) represents the width of the localization zone.

In Mechanical APDL damage models, the damage-evolution law is modified. Instead of directly using an equivalent strain such as quantity , the damage parameter is defined as a function of a corresponding equivalent relative displacement . As a result, the spurious mesh dependency of the damage variable itself and the related nominal stresses is significantly reduced.

This approach does not limit the size of the localization zone; rather, localization of strains at element level is a fundamental assumption of this method. Consequently, the size of the localization zone decreases on mesh refinement. The corresponding strains remain mesh-dependent within the damaged region. This method nevertheless preserves the correct amount of energy dissipated during the damage process. Important results (such as damage, stresses, or deformation) are objective with respect to mesh refinement.

For more information about how the characteristic element length is calculated, see:

Use viscous regularization to overcome stability issues during the solution of a damaged structure. A rate effect is introduced in the damage formulation:

where:

= rate-independent damage variable defined by the standard damage-evolution law

= corresponding rate-dependent viscous damage variable

= a characteristic viscous relaxation time

The nominal stress is redefined in terms of the viscous damage variable:

This modification results in a general rate-dependency of the damage evolution. If the viscous effect is introduced for stabilization only, the characteristic viscous relaxation time must be significantly smaller than the time step of the load step:

In the context of cyclic loading where each load cycle comprises individual load segments, is the time step of such a segment. If this condition is not satisfied, a spurious rate-dependency of the results occurs.

The viscous regularization is an optional feature, applied in addition to the crack-band regularization.

For more information about viscous regularization, see:

Singularities caused by local strain-softening material models often lead to ill-posed differential equations, unconverged numerical results, and mesh sensitivity. The problem can be mitigated via implicit gradient regularization, a class of nonlocal methods, where local variables are enhanced by considering their nonlocal counterparts as extra degrees of freedom governed by Helmholtz-type equations.

The governing equations are therefore given by the linear momentum-balance equation and modified Helmholtz equations describing the nonlocal equivalent strain field :

where:

= Cauchy stress tensor

= body force

= nonlocal variable

= local variable to be enhanced

= length-scale parameter

= unit normal vector to the outer boundary of the nonlocal field

The gradient regularization enhances the continuum damage model, where is the local variable to be enhanced, and is the nonlocal counterpart.

The length-scale parameter controls the range of nonlocal interaction. For hints about identifying the length-scale parameter, see Identifying the Nonlocal Interaction-Range Parameter c.

With the homogeneous Neumann boundary conditions , no explicit definition of boundary conditions for the extra degrees of freedom is required.

In contrast to the crack-band regularization approach, this method introduces a localization limiter, represented by the length-scale parameter . Consequently, the size of the damage zone and the corresponding strains become insensitive to the mesh size.

The implicit gradient regularization method introduces additional degrees of freedom for the nonlocal variables. The following coupled pore-pressure-thermal mechanical solid elements support this regularization method: CPT212, CPT213, CPT215, CPT216, and CPT217. The nonlocal degree of freedom is activated via KEYOPT(18).

For more information, see:

CNVTOL,GFRS,1e-7,.001
CNVTOL,GFV,1e-4,1e-5

CUTCONTROL,DMGLIMIT,1e-3

This data table defines the criteria type for determining the onset of material damage under loading.

Specify the TBOPT value on the TB,DMGI command to correspond to failure criteria (TBOPT = 1 or FCRT).

The following table shows the coefficient values addressed for the TBOPT values:

To complete the material damage definition, it is also necessary to specify a compatible material damage-evolution law (TB,DMGE). Without a damage-evolution law, the damage-initiation criteria have no effect on the material. The following table summarizes the compatible damage-evolution laws with specific damage-initiation criteria:

This data table defines the material damage-evolution law (or the way a material degrades) following the initiation of damage (TB,DMGE).

Specify the TBOPT value on the TB,DMGE command to correspond to the instant stiffness reduction (TBOPT = 1 or MPDG).

The following table shows the coefficient values addressed for the available TBOPT values:

Table 4.38: Energies Dissipated per Unit Area

Energies dissipated per unit area Gc are specified individually for all damage modes (fiber tension, fiber compression, matrix tension, and matrix compression). For a specific damage mode, Gc is given by: For complex stress state, the equivalent stresses and strains are calculated based on Hashin failure criteria.

Table 4.39: Viscous Damping Coefficients

Viscous damping coefficients η are also specified respectively for all four damage modes. For a specific damage mode, the damage evolution is regularized as follows:

To complete the material damage definition, it is also necessary to specify a compatible material damage-initiation criterion (TB,DMGI). Without a damage-initiation criterion, the damage-evolution law has no effect on the material. The following table summarizes the compatible damage-initiation criteria with specific damage-evolution laws:

 

4.25.3.2.1. Predicting Post-Damage Degradation of Brittle Anisotropic Materials

Use the progressive damage model (TB,DMGE) to predict post-damage degradation of brittle anisotropic materials, typically in fiber-reinforced composites. The undamaged material must be linearly elastic. The onset of damage is determined via failure criteria (TB,DMGI). For more information, see Failure Criteria in the Mechanical APDL Theory Reference.

Following the onset of damage, material stiffness reduction occurs immediately. The constitutive relationship for a damage material is given as:

For a general orthotropic material, the damaged elastic matrix is defined as:

For a transversely isotropic material with plane stress state, primarily adopted for thin fiber-reinforced composite structures, the damaged elastic matrix can be expressed as:

Four damage modes (fiber tension [rupture], fiber compression [kinking], matrix tension [cracking], and matrix compression [crushing]) are accounted for. Four damage variables (one for each mode) are used to measure damage. The damage variables for calculating the damaged elasticity matrix are determined as follows:

When a damage mode is initiated, the damage progresses immediately, indicated by the increasing damage variable for the mode.

Two damage-evolution methods are available:

• Material property degradation method (TB,DMGE,,,,MPDG)

  The material stiffness is instantly reduced based on the damage variables, explicitly specified via the TB,DMGE command. Any physical failure criteria can be used to detect the onset of the damage.

• Continuum damage mechanics method (TB,DMGE,,,,CDM)

  Damage variables increase gradually based on the energy amounts dissipated for the various damage modes. To achieve an objective response, the dissipated energy for each damage mode is regularized as follows:

  

  The characteristic length L_e is calculated from the element area A via the following expressions:

  

  With this characteristic length, the constitutive relation is converted from stress-strain to stress-displacement relation. The damage-evolution function is derived from Hashin failure criteria; therefore, only Hashin failure criteria are allowed via TB,DMGI for detecting damage onset. For a plane stress state, the equivalent displacements and equivalent stresses are given below for all damage modes.

 

4.25.3.2.1.1. Damage Modes

Four damage modes can be accounted for via the progressive damage model:

Fiber Tension Damage Mode	To ensure the continuity near the free fiber stress state , the coupling factor α is chosen as follows:
Fiber Compression Damage Mode
Matrix Tension Damage Mode
Matrix Compression Damage Mode

In expressions above, is the McCauley operator, defined as .

The damage variable d for a given mode is given as follows:

Structural components are often exposed to significant cyclic thermal loading and complex force loading simultaneously, leading to fatigue failure due to damage initiation and propagation through the component materials. Following the onset of damage, an ill-posed partial differential equation system is a common problem in strain-softening and damage materials. The implicit gradient-regularized approach addresses the issue.

Because a wide range of material types are used in various engineering applications, the material-damage law makes no specific assumptions about any given material law. Instead, a general framework ensures that the stresses, material tangent, and other solution-history variables are determined in a consistent manner based on general material inelastic quantities.

Damage initiation and evolution are driven by the accumulation of inelasticity. A regularized fatigue variable is determined and then used to apply the influence of damage to the stress, material tangent, and inelastic dissipation. The thermal governing law is derived to account for material dissipation due to inelastic characteristics. The result is a multi-field problem consisting of the strain field, the nonlocal variable, and the temperature, leading to a fully coupled system.

The generalized-damage option uses the following assumptions for the material model:

The inelasticity model provides the general material inelastic quantities used to develop the generalized damage framework:

It is further assumed that the damage is isotropic in nature. After the damage is calculated in the manner shown in Degradation of Stress and Material Tangent, the effective stress can be degraded to obtain the nominal stress :

The generalized damage model supports the following materials:

The governing equation for the nonlocal equivalent strain field is shown in Implicit Gradient Regularization.

The local internal variable representing the local deformation is:[1]

, where

where:

= normalization constant of the equivalent inelastic strain rate

= power index representing the time dependence of the lifetime fatigue behavior:

= normalization constant of the von Mises equivalent effective (undamaged) stress (defined as = , where is the effective deviatoric stress)

= stress-dependency exponent

= magnitude of the inelastic strain rate increment , where the magnitude of the inelastic strain increment is defined as with the assumption of additive decomposition of the inelastic strain .

The damage exponential function calculates the damage when the nonlocal variable reaches the threshold :

where the damage rate coefficient describes the rate at which the damage grows. is the Macauley bracket.

The governing equation for heat-transfer from Equation 4–92. The equation is rewritten to include the inelastic dissipation source:

where the inelastic dissipation rate is a function of the nominal stress and is calculated as (where is the inelastic-dissipation-rate fraction coefficient).

For simplicity, other source and body-force terms have not been included here in the heat-transfer equation.

The following coupled pore-pressure-thermal mechanical solid elements support the generalized damage model: CPT212, CPT213, CPT215, CPT216, and CPT217.

For nonlocal damage, set KEYOPT(18) = 1 to activate the required extra degree of freedom (GFV1). The extra degree of freedom has no boundary-condition input.

A fine mesh is recommended, particularly at probable damage-prone regions. To observe mesh-independent results, the mesh size may need to be less than half the square root of the length-scale parameter .[2]

Define the inelastic material (TB). For example:

Define the damage parameters (TB,CDM,,,,GDMG), then specify the constants (TBDATA) for the CDM material data table:

Constant	Meaning	Property	Unit	Range
C1		Nonlocal parameter threshold	---
C2		Damage rate coefficient	---
C3		Stress dependency exponent	---
C4		Power index representing the time dependence of the lifetime fatigue behavior	---
C5		Normalization constant of the equivalent inelastic strain rate	Time-1
C6		Normalization constant of the equivalent effective stress	Force/Length2
C7		Length-scale parameter	Length2

Define Thermal Properties (if Needed)

The damage model is valid without the temperature degree of freedom active. If you wish to apply the effect of thermal evolution, activate the required extra degree of freedom (TEMP) via KEYOPT(11) = 1 and define the heat-transfer properties.

Optionally, define the inelastic dissipation rate fraction coefficient. If not explicitly defined, the default coefficient is 1. To change it, specify the new coefficient (TB,THERM,,,,DISS), then specify the constant (TBDATA) for the THERM material data table:

Constant	Meaning	Property	Unit	Range
C1		Inelastic dissipation rate fraction coefficient	---	Default:

/prep7
!Define element
ET,1,215
KEYO,1,18,1  ! nonlocal dof
KEYO,1,11,1  ! thermal dof

! parameters
rh=7860                  ! material density
cc=5.02e-4               ! heat capacity
k0=50                    ! thermal conductivity
alpha=10.8e-6            ! thermal expansion
temp_0=20.0              ! reference temperature

Em=2.35e5
nu=0.3
! Kinematic hardening
s0=400
gam1=2e2
gam2=1e2
Ck1=10
Ck2=5
Kr=1200
nn=1.5
br=6
Qr=1100

! Damage profile
c_eta=2.6
eta_cr=0
zz=0.06
! Fatigue damage
m0=1.0
n0=0.008
p0=0.6
a0=1.15e5

tb,elas,1
tbdata,1,Em,nu

tb,dens,1
tbdata,1,rh

tb,cte,1
tbdata,1,alpha

tref,temp_0

tb,therm,1,,,spht
tbdata,1,cc

tb,therm,1
tbdata,1,k0

tb,cdm,1,,,gdmg
tbdata,1,eta_cr,zz,m0,n0,p0,a0
tbdata,7,c_eta

TB,CHAB,1
TBDATA,1,s0,Ck1,gam1,Ck2,gam2

TB,RATE,1,,,EVH
TBDATA,1,s0,0,Qr,br,1/nn,Kr

These are the element quantities available for postprocessing (/POST1, /POST26), followed by their postprocessing labels:

According to Hooputra et al. [4], ductile damage initiates when the accumulated equivalent plastic strain exceeds a critical limit . The limit is not constant but depends on the effective stress triaxiality .

For a general nonlinear loading path, damage initiates when the ductile damage-initiation criterion is met:

The effective stress triaxiality is defined as the ratio between the effective hydrostatic stress and the effective von Mises stress :

where:

= first effective stress invariant

= second deviatoric effective stress invariant

The following table summarizes the stress triaxiality for various loading conditions:

Loading Condition	Effective Stress Triaxiality
Hydrostatic compression
Biaxial compression
Uniaxial compression
Shear
Uniaxial tension
Biaxial tension
Hydrostatic tension

The Mechanical APDL ductile-damage option introduces three additional history variables:

The parameter is a material constant provided as tabular input for different stress triaxiality values. (See Defining the Ductile-Damage Model.)

The evolution of the ductile-damage variable is based on the amount of plastic strain after damage has been initiated. To avoid mesh-dependency of the damage variable, the crack-band regularization method is applied. Consequently, the damage-evolution law is expressed in terms of a relative equivalent plastic displacement:

Mechanical APDL determines the characteristic element length based on the average integration-point volume, area, or length. A good approximation of the characteristic element length is given by:

where:

= element volume

= element area

= element length

= number of integration points associated with this geometrical entity

For shell elements, corresponds to the number of in-plane integration points. Mechanical APDL does not consider the integration points distributed over the shell thickness when calculating the characteristic element length.

 

4.25.6.2.1. Linear Damage Evolution

The linear damage-evolution law is defined as:

where is a material constant representing the relative equivalent plastic displacement at full degradation of the material. (See Defining the Ductile-Damage Model.)

Figure 4.68: Linear Damage-Evolution Law

 

4.25.6.2.2. Exponential Damage Evolution

The exponential damage-evolution law is defined as:

where is a material constant representing the initial slope of this function. (See Defining the Ductile-Damage Model.)

Figure 4.69: Exponential Damage-Evolution Law

The damage parameter is available for postprocessing (/POST1 and /POST26). The assigned label is GDMG.