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24.2 Damage and failure for ductile metals
• “Damage and failure for ductile metals: overview,” Section 24.2.1
• “Damage initiation for ductile metals,” Section 24.2.2
• “Damage evolution and element removal for ductile metals,” Section 24.2.3
24.2.1 DAMAGE AND FAILURE FOR DUCTILE METALS: OVERVIEW
Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
References
• “Progressive damage and failure,” Section 24.1.1
• “Damage initiation for ductile metals,” Section 24.2.2
• “Damage evolution and element removal for ductile metals,” Section 24.2.3
• *DAMAGE INITIATION
• *DAMAGE EVOLUTION
• “Defining damage,” Section 12.9.3 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
Overview
Abaqus/Standard and Abaqus/Explicit offer a general capability for predicting the onset of failure and a capability for modeling progressive damage and failure of ductile metals. In the most general case this requires the specification of the following:
• the undamaged elastic-plastic response of the material (“Classical metal plasticity,” Section 23.2.1);
• a damage initiation criterion (“Damage initiation for ductile metals,” Section 24.2.2); and
• a damage evolution response, including a choice of element removal (“Damage evolution and element removal for ductile metals,” Section 24.2.3).
A summary of the general framework for progressive damage and failure in Abaqus is given in “Progressive damage and failure,” Section 24.1.1. This section provides an overview of the damage initiation criteria and damage evolution law for ductile metals. In addition, Abaqus/Explicit offers dynamic failure models that are suitable for high-strain-rate dynamic problems (“Dynamic failure models,” Section 23.2.8).
Damage initiation criterion
Abaqus offers a variety of choices of damage initiation criteria for ductile metals, each associated with distinct types of material failure. They can be classified in the following categories:
• Damage initiation criteria for the fracture of metals, including ductile and shear criteria.
• Damage initiation criteria for the necking instability of sheet metal. These include forming limit diagrams (FLD, FLSD, and MSFLD) intended to assess the formability of sheet metal and the Marciniak-Kuczynski (M-K) criterion (available only in Abaqus/Explicit) to numerically predict necking instability in sheet metal taking into account the deformation history.
These criteria are discussed in “Damage initiation for ductile metals,” Section 24.2.2. Each damage initiation criterion has an associated output variable to indicate whether the criterion has been met during the analysis. A value of 1.0 or higher indicates that the initiation criterion has been met.
More than one damage initiation criterion can be specified for a given material. If multiple damage initiation criteria are specified for the same material, they are treated independently. Once a particular initiation criterion is satisfied, the material stiffness is degraded according to the specified damage evolution law for that criterion; in the absence of a damage evolution law, however, the material stiffness is not degraded. A failure mechanism for which no damage evolution response is specified is said to be inactive. Abaqus will evaluate the initiation criterion for an inactive mechanism for output purposes only, but the mechanism will have no effect on the material response.
Input File Usage: Use the following option to define each damage initiation criterion (repeat as needed to define multiple criteria):
*DAMAGE INITIATION, CRITERION=criterion 1
Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Ductile Metals→criterion
Damage evolution
The damage evolution law describes the rate of degradation of the material stiffness once the corresponding initiation criterion has been reached. For damage in ductile metals Abaqus assumes that the degradation of the stiffness associated with each active failure mechanism can be modeled using a scalar damage variable, d _ { i } ( i \in N _ { \mathrm { a c t } } ) , where N _ { \mathrm { a c t } } represents the set of active mechanisms. At any given time during the analysis the stress tensor in the material is given by the scalar damage equation
\boldsymbol {\sigma} = (1 - D) \bar {\boldsymbol {\sigma}},
where D is the overall damage variable and is the effective (or undamaged) stress tensor computed in the current increment. are the stresses that would exist in the material in the absence of damage. The material has lost its load-carrying capacity when . By default, an element is removed from the mesh if all of the section points at any one integration location have lost their load-carrying capacity.
The overall damage variable, D, captures the combined effect of all active mechanisms and is computed in terms of the individual damage variables, d _ { i } , according to a user-specified rule.
Abaqus supports different models of damage evolution in ductile metals and provides controls associated with element deletion due to material failure, as described in “Damage evolution and element removal for ductile metals,” Section 24.2.3. All of the available models use a formulation intended to alleviate the strong mesh dependency of the results that can arise from strain localization effects during progressive damage.
Input File Usage: Use the following option immediately after the corresponding *DAMAGE INITIATION option to specify the damage evolution behavior:
*DAMAGE EVOLUTION
Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Ductile Metals→criterion: Suboptions→Damage Evolution
Elements
The failure modeling capability for ductile metals can be used with any elements in Abaqus that include mechanical behavior (elements that have displacement degrees of freedom).
For coupled temperature-displacement elements the thermal properties of the material are not affected by the progressive damage of the material stiffness until the condition for element deletion is reached; at this point the thermal contribution of the element is also removed.
The damage initiation criteria for sheet metal necking instability (FLD, FLSD, MSFLD, and M-K) are available only for elements that include mechanical behavior and use a plane stress formulation (i.e., plane stress, shell, continuum shell, and membrane elements).
24.2.2 DAMAGE INITIATION FOR DUCTILE METALS
Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
References
• “Progressive damage and failure,” Section 24.1.1
• *DAMAGE INITIATION
• “Defining damage,” Section 12.9.3 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
Overview
The material damage initiation capability for ductile metals:
• is intended as a general capability for predicting initiation of damage in metals, including sheet, extrusion, and cast metals as well as other materials;
• can be used in combination with the damage evolution models for ductile metals described in “Damage evolution and element removal for ductile metals,” Section 24.2.3;
• allows the specification of more than one damage initiation criterion;
• includes ductile, shear, forming limit diagram (FLD), forming limit stress diagram (FLSD) and Müschenborn-Sonne forming limit diagram (MSFLD) criteria for damage initiation;
• includes in Abaqus/Explicit the Marciniak-Kuczynski (M-K) and Johnson-Cook criteria for damage initiation;
• can be used in Abaqus/Standard in conjunction with Mises, Johnson-Cook, Hill, and Drucker-Prager plasticity (ductile, shear, FLD, FLSD, and MSFLD criteria); and
• can be used in Abaqus/Explicit in conjunction with Mises and Johnson-Cook plasticity (ductile, shear, FLD, FLSD, MSFLD, Johnson-Cook, and MK criteria) and in conjunction with Hill and Drucker-Prager plasticity (ductile, shear, FLD, FLSD, MSFLD, and Johnson-Cook criteria).
Damage initiation criteria for fracture of metals
Two main mechanisms can cause the fracture of a ductile metal: ductile fracture due to the nucleation, growth, and coalescence of voids; and shear fracture due to shear band localization. Based on phenomenological observations, these two mechanisms call for different forms of the criteria for the onset of damage (Hooputra et al., 2004). The functional forms provided by Abaqus for these criteria are discussed below. These criteria can be used in combination with the damage evolution models for ductile metals discussed in “Damage evolution and element removal for ductile metals,” Section 24.2.3, to model fracture of a ductile metal. (See “Progressive failure analysis of thin-wall aluminum extrusion under quasi-static and dynamic loads,” Section 2.1.16 of the Abaqus Example Problems Guide, for an example.)
Ductile criterion
The ductile criterion is a phenomenological model for predicting the onset of damage due to nucleation, growth, and coalescence of voids. The model assumes that the equivalent plastic strain at the onset of damage, \bar { \varepsilon } _ { \mathrm { D } } ^ { p l } , is a function of stress triaxiality and strain rate:
\bar {\varepsilon} _ {\mathrm{D}} ^ {p l} (\eta , \dot {\bar {\varepsilon}} ^ {p l}),
where \eta = - p / q is the stress triaxiality, p is the pressure stress, q is the Mises equivalent stress, and \dot { \bar { \varepsilon } } ^ { p l } is the equivalent plastic strain rate. The criterion for damage initiation is met when the following condition is satisfied:
\omega_ {\mathrm{D}} = \int \frac {\mathrm{d} \bar {\varepsilon} ^ {p l}}{\bar {\varepsilon} _ {\mathrm{D}} ^ {p l} (\eta , \dot {\bar {\varepsilon}} ^ {p l})} = 1,
where \omega _ { \mathrm { D } } is a state variable that increases monotonically with plastic deformation. At each increment during the analysis the incremental increase in \omega _ { \mathrm { D } } is computed as
\Delta \omega_ {\mathrm{D}} = \frac {\Delta \bar {\varepsilon} ^ {p l}}{\bar {\varepsilon} _ {\mathrm{D}} ^ {p l} (\eta , \dot {\bar {\varepsilon}} ^ {p l})} \geq 0.
In Abaqus/Standard the ductile criterion can be used in conjunction with the Mises, Johnson-Cook, Hill, and Drucker-Prager plasticity models and in Abaqus/Explicit in conjunction with the Mises, Johnson-Cook, Hill, and Drucker-Prager plasticity models, including equation of state.
Input File Usage: Use the following option to specify the equivalent plastic strain at the onset of damage as a tabular function of stress triaxality, strain rate, and, optionally, temperature and predefined field variables:
*DAMAGE INITIATION, CRITERION=DUCTILE, DEPENDENCIES=n
Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Ductile Metals→Ductile Damage
Defining dependency of ductile criterion on Lode angle
Recent experimental results for aluminum alloys and other metals (Bai and Wierzbicki, 2008) reveal that, in addition to stress triaxility and strain rate, ductile fracture can also depend on the third invariant of deviatoric stress, which is related to the Lode angle (or deviatoric polar angle). Both Abaqus/Explicit and Abaqus/Standard allow the definition of the equivalent plastic strain at the onset of ductile damage, \bar { \varepsilon } _ { \mathrm { D } } ^ { p l } , as a function of the Lode angle, , by way of the functional form
\bar {\varepsilon} _ {\mathrm{D}} ^ {p l} (\eta , \xi (\Theta), \dot {\bar {\varepsilon}} ^ {p l}),
where
\xi (\Theta) = \cos (3 \Theta) = \left(\frac {r}{q}\right) ^ {3},
q is the Mises equivalent stress, and r is the third invariant of deviatoric stress, r = \left( \frac { 9 } { 2 } { \bf S } \cdot { \bf S } : { \bf S } \right) ^ { \frac { 1 } { 3 } } . The function can take values from \xi = - 1 , for stress states on the compressive meridian, to \xi = 1 , for stress states on the tensile meridian.
Input File Usage: Use the following option to indicate that the equivalent plastic strain at the onset of ductile damage is a function of the Lode angle:
*DAMAGE INITIATION, CRITERION=DUCTILE, LODE DEPENDENT
Abaqus/CAE Usage: Defining dependency of ductile criterion on Lode angle is not supported in Abaqus/CAE.
Johnson-Cook criterion
The Johnson-Cook criterion (available only in Abaqus/Explicit) is a special case of the ductile criterion in which the equivalent plastic strain at the onset of damage, \bar { \varepsilon } _ { \mathrm { D } } ^ { p l } , is assumed to be of the form
\bar {\varepsilon} _ {\mathrm{D}} ^ {p l} = \left[ d _ {1} + d _ {2} \mathrm{exp} \left(- d _ {3} \eta\right) \right] \left[ 1 + d _ {4} \mathrm{ln} \left(\frac {\dot {\bar {\varepsilon}} ^ {p l}}{\dot {\varepsilon} _ {0}}\right) \right] \left(1 + d _ {5} \hat {\theta}\right),
where d _ { 1 } { - } d _ { 5 } are failure parameters and \dot { \varepsilon } _ { 0 } is the reference strain rate. This expression differs from the original formula published by Johnson and Cook (1985) in the sign of the parameter d _ { 3 } . This difference is motivated by the fact that most materials experience a decrease in \bar { \varepsilon } _ { \mathrm { D } } ^ { p l } with increasing stress triaxiality; therefore, d _ { 3 } in the above expression will usually take positive values. \hat { \theta } is the nondimensional temperature defined as
\hat {\theta} \equiv \left\{ \begin{array}{c c c} 0 & \mathrm{for} & \theta < \theta_ {\mathrm{transition}} \\ (\theta - \theta_ {\mathrm{transition}}) / (\theta_ {\mathrm{melt}} - \theta_ {\mathrm{transition}}) & \mathrm{for} & \theta_ {\mathrm{transition}} \leq \theta \leq \theta_ {\mathrm{melt}} \\ 1 & \mathrm{for} & \theta > \theta_ {\mathrm{melt}} \end{array} \right.,
where is the current temperature, \theta _ { \mathrm { m e l t } } is the melting temperature, and \theta _ { \mathrm { t r a n s i t i o n } } is the transition temperature defined as the one at or below which there is no temperature dependence on the expression of the damage strain \bar { \varepsilon } _ { \mathrm { D } } ^ { p l } . The material parameters must be measured at or below the transition temperature.
The Johnson-Cook criterion can be used in conjunction with the Mises, Johnson-Cook, Hill, and Drucker-Prager plasticity models, including equation of state. When used in conjunction with the Johnson-Cook plasticity model, the specified values of the melting and transition temperatures should be consistent with the values specified in the plasticity definition. The Johnson-Cook damage initiation criterion can also be specified together with any other initiation criteria, including the ductile criteria; each initiation criterion is treated independently.
Input File Usage: Use the following option to specify the parameters for the Johnson-Cook initiation criterion:
*DAMAGE INITIATION, CRITERION=JOHNSON COOK
Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Ductile Metals→Johnson-Cook Damage
Shear criterion
The shear criterion is a phenomenological model for predicting the onset of damage due to shear band localization. The model assumes that the equivalent plastic strain at the onset of damage, \bar { \varepsilon } _ { \mathrm { S } } ^ { p l } , is a function of the shear stress ratio and strain rate:
\bar {\varepsilon} _ {\mathrm{S}} ^ {p l} (\theta_ {s}, \dot {\bar {\varepsilon}} ^ {p l}).
Here \theta _ { s } ~ = ~ ( q + k _ { s } p ) / \tau _ { \mathrm { m a x } } is the shear stress ratio, \tau _ { \mathrm { m a x } } is the maximum shear stress, and k _ { s } is a material parameter. A typical value of k _ { s } for aluminum is k _ { s } = 0 . 3 (Hooputra et al., 2004). The criterion for damage initiation is met when the following condition is satisfied:
\omega_ {S} = \int \frac {\mathrm{d} \overline {{\varepsilon}} ^ {p l}}{\overline {{\varepsilon}} _ {S} ^ {p l} (\theta_ {s} , \dot {\overline {{\varepsilon}}} ^ {p l})} = 1,
where \omega _ { \mathrm { S } } is a state variable that increases monotonically with plastic deformation proportional to the incremental change in equivalent plastic strain. At each increment during the analysis the incremental increase in \omega _ { \mathrm { { S } } } is computed as
\Delta \omega_ {\mathrm{S}} = \frac {\Delta \bar {\varepsilon} ^ {p l}}{\bar {\varepsilon} _ {\mathrm{S}} ^ {p l} (\theta_ {s} , \dot {\bar {\varepsilon}} ^ {p l})} \geq 0.
In Abaqus/Explicit the shear criterion can be used in conjunction with the Mises, Johnson-Cook, Hill, and Drucker-Prager plasticity models, including equation of state. In Abaqus/Standard it can be used with the Mises, Johnson-Cook, Hill, and Drucker-Prager models.
Input File Usage:
Use the following option to specify k _ { s } and to specify the equivalent plastic strain at the onset of damage as a tabular function of the shear stress ratio, strain rate, and, optionally, temperature and predefined field variables:
*DAMAGE INITIATION, CRITERION=SHEAR, KS= ,DEPENDENCIES=n
Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Ductile Metals→Shear Damage
Initial conditions
Optionally, you can specify the initial work hardened state of the material by providing the initial equivalent plastic strain values (see “Defining initial values of state variables for plastic hardening” in “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1) and, if residual stresses are also present, the initial stress values (see “Defining initial stresses” in “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1). Abaqus uses this information to initialize the values of the ductile and shear damage initiation criteria, and \omega _ { \mathrm { { S } } } , assuming constant values of stress triaxiality and shear shear ratio (linear stress path).