Low-Cycle Fatigue Analysis Using the Direct Cyclic Approach
A low-cycle fatigue analysis:
is characterized by states of stress high enough for inelastic
deformation to occur in most cases;
is a quasi-static analysis on a structure subjected to sub-critical
cyclic loading;
can be associated with thermal as well as mechanical loading;
uses the direct cyclic approach to obtain the stabilized cyclic
response of the structure directly;
models progressive damage and failure in bulk ductile material based
on a continuum damage mechanics approach, in which case damage initiation and
evolution are characterized by the accumulated inelastic hysteresis strain
energy per stabilized cycle;
models propagation of a discrete crack along an arbitrary,
solution-dependent path without remeshing in the bulk brittle material based on
the principles of linear elastic fracture mechanics
(LEFM) with the extended finite element
method, in which case the onset and growth of fatigue crack are characterized
by the relative fracture energy release rate;
models progressive delamination growth along a predefined path at the
brittle material interfaces in laminated composites, in which case the onset
and growth of fatigue delamination at the interfaces are characterized by the
relative fracture energy release rate;
uses the damage extrapolation technique to accelerate the low-cycle
fatigue analysis; and
assumes geometrically linear behavior and fixed contact conditions
within each loading cycle.
In simulations where the bulk material deformation is inelastic, the
direct cyclic approach is the preferred method. It can be much more
computationally efficient at obtaining a stabilized response than a classical
transient analysis, which may require the application of many loading cycles to
obtain the same result. However, in the case of linear elastic response with
brittle materials, it may not be optimal, or even desirable, to use a Fourier
series to represent the displacement and residual fields. The preferred method
in this case is to use the classical incremental method (see
Linear Elastic Fatigue Crack Growth Analysis).
The traditional approach for determining the fatigue limit for a structure
is to establish the
curves (load versus number of cycles to failure) for the materials in the
structure. Such an approach is still used as a design tool in many cases to
predict fatigue resistance of engineering structures. However, this technique
is generally conservative, and it does not define a relationship between the
cycle number and the degree of damage or crack length.
One alternative approach is to predict the fatigue life by using a
crack/damage evolution law based on the inelastic strain/energy when the
structure's response is stabilized after many cycles. Because the computational
cost to simulate the slow progressive damage in a material over many load
cycles is prohibitively expensive for all but the simplest models, numerical
fatigue life studies usually involve modeling the response of the structure
subjected to a small fraction of the actual loading history. This response is
then extrapolated over many load cycles using empirical formulae such as the
Coffin-Manson relationship (see
Coffin,
1954, and
Manson,
1953) to predict the likelihood of crack initiation and propagation.
Since this approach is based on a constant crack/damage growth rate, it may not
realistically predict the evolution of the crack or damage.
Low-Cycle Fatigue Analysis in Abaqus/Standard
The direct cyclic analysis capability in
Abaqus/Standard
provides a computationally effective modeling technique to obtain the
stabilized response of a structure subjected to periodic loading and is ideally
suited to perform low-cycle fatigue calculations on a large structure. The
capability uses a combination of Fourier series and time integration of the
nonlinear material behavior to obtain the stabilized response of the structure
directly. The theory and algorithm to obtain a stabilized response using the
direct cyclic approach are described in detail in
Direct cyclic algorithm.
The direct cyclic low-cycle fatigue procedure models the progressive damage
and failure both in bulk materials (such as in solder joints in an electronic
chip packaging or intra-laminar crack growth in laminated composites) and at
material interfaces (such as delamination in laminated composites). The former
can be based on either a continuum damage mechanics approach or the principles
of linear elastic fracture mechanics with the extended finite element method.
The response is obtained by evaluating the behavior of the structure at
discrete points along the loading history (see
Figure 1).
The solution at each of these points is used to predict the degradation and
evolution of material properties that will take place during the next
increment, which spans a number of load cycles, .
The degraded material properties are then used to compute the solution at the
next increment in the load history. Therefore, the crack/damage growth rate is
updated continually throughout the analysis.
The elastic material stiffness at a material point remains constant and
contact conditions remain unchanged when the stabilized solution is computed at
a given point in the loading history. Each of the solutions along the loading
history represents the stabilized response of the structure subjected to the
applied period loads, with a level of material damage at each point in the
structure computed from the previous solution. This process is repeated up to a
point in the loading history at which a fatigue life assessment can be made.
In bulk material, there are two approaches to modeling the progressive
damage and failure. One approach is based on continuum damage mechanics. This
approach is more appropriate for ductile material, in which the cyclic loading
leads to stress reversals and the accumulation of plastic strains, which in
turn cause the initiation and propagation of cracks. The damage initiation and
evolution are characterized by the stabilized accumulated inelastic hysteresis
strain energy per cycle as illustrated in
Figure 2.
The other approach is based on the principles of linear elastic fracture
mechanics with the extended finite element method. This approach is more
appropriate for brittle material or material with small scale yielding, in
which the cyclic loading leads to material strength degradation causing fatigue
crack growth along an arbitrary path. The onset and growth of the crack are
characterized by the relative fracture energy release rate at the crack tip
based on the Paris law (Paris,
1961).
At interfaces of laminated composites the cyclic loading leads to interface
strength degradation causing fatigue delamination growth. The onset and growth
of delamination are also characterized by the relative fracture energy release
rate at the crack tip based on the Paris law (Paris,
1961).
Both the progressive damage mechanism in the bulk material and the
progressive delamination growth mechanism at interfaces can be considered
simultaneously, with the failure occurring first at the weakest link in a
model.
Defining a low-cycle fatigue analysis using the direct cyclic approach is
similar to defining a direct cyclic analysis. See
Direct Cyclic Analysis
for details on how to specify the number of Fourier terms, number of
iterations, and the increment sizes. You specify the maximum numbers of cycles,
,
when you define the low-cycle fatigue analysis step.
Determining Whether to Use the Fourier Coefficients from the Previous Step
A low-cycle fatigue step using the direct cyclic approach can be the only
step in an analysis, can follow a general or linear perturbation step, or can
be followed by a general or linear perturbation step. Multiple low-cycle
fatigue analysis steps can be included in a single analysis. In such a case the
Fourier series coefficients obtained in the previous step can be used as
starting values in the current step. By default, the Fourier coefficients are
reset to zero, thus allowing application of cyclic loading conditions that are
very different from those defined in the previous low-cycle fatigue step.
As in a direct cyclic analysis, you can specify that a low-cycle fatigue
step in a restart analysis should use the Fourier coefficients from the
previous step, thus allowing continuation of an analysis to simulate more
loading cycles. In a low-cycle fatigue analysis a restart file is written at
the end of the stabilized cycle. Consequently, a restart analysis that is a
continuation of a previous low-cycle fatigue analysis will start with a new
loading cycle at
(see
Restarting an Analysis).
Progressive Damage and Damage Extrapolation in Bulk Ductile Material Based on Continuum Damage Mechanics Approach
Low-cycle fatigue analysis in
Abaqus/Standard
allows modeling of progressive damage and failure for ductile materials in any
elements whose response is defined in terms of a continuum-based constitutive
model (About the Material Library).
This includes cohesive elements modeled using a continuum approach (Modeling of an Adhesive Layer of Finite Thickness).
The inelastic definition in a material point must be used in conjunction with
the linear elastic material model (Linear Elastic Behavior),
the porous elastic material model (Elastic Behavior of Porous Materials),
or the hypoelastic material model (Hypoelastic Behavior).
After damage initiation the elastic material stiffness is degraded
progressively in each cycle (as shown in
Figure 1)
based on the accumulated stabilized inelastic hysteresis energy. It is
impractical and computationally expensive to perform a cycle-by-cycle
simulation for a low-cycle fatigue analysis; Instead, to accelerate the
low-cycle fatigue analysis, each increment extrapolates the current damaged
state in the bulk material forward over many cycles to a new damaged state
after the current loading cycle is stabilized.
Damage Initiation and Evolution
Damage initiation refers to the beginning of degradation of the response of
a material point. In a low-cycle fatigue analysis the damage initiation
criterion is characterized by the accumulated inelastic hysteresis energy per
cycle, .
and material constants are used to determine the number of the cycle in which
damage is initiated, .
At the end of a stabilized loading cycle, ,
Abaqus/Standard
checks to see if the damage initiation criterion
is satisfied in any material point; material stiffness at a material point will
not be degraded unless this criterion is satisfied. The calculations and output
associated with damage initiation are discussed in detail in
Damage Initiation for Ductile Materials in Low-Cycle Fatigue.
Once the damage initiation criterion is satisfied at a material point, the
damage state is calculated and updated based on the inelastic hysteresis energy
for the stabilized cycle.
Abaqus/Standard
assumes that the degradation of the elastic stiffness can be modeled using the
scalar damage variable, .
The rate of the damage in a material point per cycle, ,
is calculated based on the accumulated inelastic hysteresis energy, the
characteristic length associated with an integration point, and material
constants. For details, see
Damage Evolution for Ductile Materials in Low-Cycle Fatigue.
Typically, a material has completely lost its load-carrying capacity when
.
You can remove an element from the mesh if all of the section points at all
integration locations of the element have lost their load-carrying capability.
Damage Extrapolation Technique in the Bulk Material
If the damage initiation criterion is satisfied in any material point at the
end of a stabilized cycle, ,
Abaqus/Standard
extrapolates the damage variable
from the current cycle forward to the next increment over a number of cycles,
.
The new damage state, ,
is given by
You specify the minimum ()
and maximum ()
number of cycles over which the damage is extrapolated forward in any given
increment. The default values are 100 and 1000, respectively.
Discrete Crack Propagation along an Arbitrary Path Based on the Principles of Linear Elastic Fracture Mechanics with the Extended Finite Element Method
Low-cycle fatigue analysis in
Abaqus/Standard
allows the modeling of discrete crack growth along an arbitrary path based on
the principles of linear elastic fracture mechanics with the extended finite
element method. You complete the definition of the crack propagation capability
by defining a fracture-based surface behavior and specifying the fracture
criterion in enriched elements. The fracture energy release rates at the crack
tips in enriched elements are calculated based on the modified virtual crack
closure technique (VCCT).
VCCT
uses the principles of linear elastic fracture mechanics. Therefore,
VCCT
is appropriate for problems in which brittle fatigue crack growth occurs,
although nonlinear material deformations can occur somewhere else in the bulk
materials. For more information about defining fracture criteria and
VCCT
in enriched elements, see
Modeling Discontinuities as an Enriched Feature Using the Extended Finite Element Method.
To accelerate the low-cycle fatigue analysis, the damage extrapolation
technique is used, which advances the crack by at least one element length
after each stabilized cycle.
Onset and Growth of Fatigue Crack
The onset and growth of fatigue crack at an enriched element are
characterized by using the Paris law, which relates the relative fracture
energy release rate, ,
to crack growth rates. Two criteria must be met to initiate fatigue crack
growth: one criterion is based on material constants, ,
and the current cycle number, ;
the other criterion is based on the maximum fracture energy release rate,
,
which corresponds to the cyclic energy release rate when the structure is
loaded up to its maximum value. Once the onset of fatigue crack growth
criterion is satisfied at the enriched elements, the crack growth rate,
,
is a piecewise function based on material constants and
(the Paris law). The criteria for fatigue crack onset and growth are discussed
in detail in
Modeling Discontinuities as an Enriched Feature Using the Extended Finite Element Method.
Damage Extrapolation Technique
If the onset of crack growth criterion is satisfied at any crack tip in the
enriched element at the end of a stabilized cycle, ,
Abaqus/Standard
extends the crack length, ,
from the current cycle forward over a number of cycles,
,
to
by fracturing at least one enriched element ahead of the crack tips. Given the
material constants
and
(as defined in
Modeling Discontinuities as an Enriched Feature Using the Extended Finite Element Method),
combined with the known element length and likely propagation direction
at the enriched elements ahead of the crack tips, the number of cycles
necessary to fail each enriched element ahead of the crack tip can be
calculated as ,
where
represents the enriched element ahead of the th
crack tip. The analysis is set up to advance the crack by at least one enriched
element per increment after the loading cycle is stabilized. The element with
the fewest cycles is identified to be fractured, and its
is represented as the number of cycles to grow the crack equal to its element
length, .
The most critical element is completely fractured with a zero constraint and a
zero stiffness at the cracked surfaces at the end of the stabilized cycle. As
the enriched element is fractured, the load is redistributed, and a new
relative fracture energy release rate must be calculated for the enriched
elements ahead of the crack tips for the next cycle. This capability allows at
least one enriched element ahead of the crack tips to be fractured after each
stabilized cycle and precisely accounts for the number of cycles needed to
cause fatigue crack growth over that length.
Progressive Delamination Growth along a Pre-Defined Path at Interfaces
Low-cycle fatigue analysis in
Abaqus/Standard
also allows the modeling of progressive delamination growth at the interfaces
in laminated composites. The interface along which the delamination (or crack)
propagates must be indicated in the model using a fracture criterion
definition. The fracture energy release rates at the crack tips in the
interface elements are calculated based on the virtual crack closure technique
(VCCT).
VCCT
uses the principles of linear elastic fracture mechanics. Therefore,
VCCT
is appropriate for problems in which brittle fatigue delamination growth occurs
along predefined surfaces, although nonlinear material deformations can occur
in the bulk materials. For more information about defining fracture criteria
and VCCT,
see
Crack Propagation Analysis.
To accelerate the low-cycle fatigue analysis, the damage extrapolation
technique is used, which releases at least one element length at the crack tip
along the interface after each stabilized cycle. When both brittle fatigue
delamination at interfaces and ductile damage or discrete crack growth in bulk
materials are considered in an analysis, failure occurs first at the weakest
link.
Onset and Growth of Fatigue Delamination
The onset and growth of fatigue delamination at a defined crack interface
are characterized by using the Paris law, which relates the relative fracture
energy release rate, ,
to crack growth rates. Two criteria must be met to initiate fatigue
delamination growth: one criterion is based on material constants,
,
and the current cycle number, ;
the other criterion is based on the maximum fracture energy release rate,
,
which corresponds to the cyclic energy release rate when the structure is
loaded up to its maximum value. Once the onset of delamination growth criterion
is satisfied at the interface, the delamination growth rate,
,
is a piecewise function based on material constants and
(the Paris law). The criteria for fatigue delamination onset and growth are
discussed in detail in
Fatigue Crack Growth Criterion.
Damage Extrapolation Technique at the Interface Elements
If the onset of delamination growth criterion is satisfied at any crack tip
in the interface at the end of a stabilized cycle, ,
Abaqus/Standard
extends the crack length, ,
from the current cycle forward over a number of cycles,
,
to
by releasing at least one element at the interface. Given the material
constants
and
(as defined in
Fatigue Crack Growth Criterion),
combined with the known node spacing
at the interface elements at the crack tips, the number of cycles necessary to
fail each interface element at the crack tip can be calculated as
,
where j represents the node at the
jth crack tip. The analysis is set up to release at least
one interface element per increment after the loading cycle is stabilized. The
element with the fewest cycles is identified to be released, and its
is represented as the number of cycles to grow the crack equal to its element
length, .
The most critical element is completely released with a zero constraint and a
zero stiffness at the end of the stabilized cycle. As the interface element is
released, the load is redistributed, and a new relative fracture energy release
rate must be calculated for the interface elements at the crack tips for the
next cycle. This capability allows at least one interface element at the crack
tips to be released after each stabilized cycle and precisely accounts for the
number of cycles needed to cause fatigue crack growth over that length.
Controlling the Solution Accuracy
Low-cycle fatigue analysis utilizes the direct cyclic approach to obtain the
stabilized cyclic solution iteratively by combining a Fourier series
approximation with time integration of the nonlinear material behavior using a
modified Newton method. The accuracy of the algorithm depends on the number of
Fourier terms used, the number of iterations taken to obtain the stabilized
solution, and the number of time points within the load period at which the
material response and residual vector are evaluated. Some methods for
controlling the solution accuracy in a direct cyclic analysis are described in
detail in
Direct Cyclic Analysis.
They all remain valid in a low-cycle fatigue analysis using the direct cyclic
approach. In addition, the accuracy of a low-cycle fatigue analysis depends on
the number of cycles over which the damage is extrapolated forward, as
described below.
Controlling the Accuracy of Damage Extrapolation in the Bulk Material When Using the Continuum Damage Mechanics Approach
To accelerate the low-cycle fatigue analysis, the damage extrapolation
technique is used at the end of a stabilized cycle. In addition to specifying
the minimum and maximum number of cycles over which the damage is extrapolated
(see
Damage Extrapolation Technique in the Bulk Material
above), you can specify the damage extrapolation tolerance,
,
to control the accuracy of damage extrapolation in the bulk material. The
default is .
Determining the Increment over Which Damage Is Extrapolated Forward
Abaqus/Standard
uses an adaptive algorithm to determine the number of cycles over which the
damage is extrapolated forward in each increment. By default,
Abaqus/Standard
starts with 500 cycles (half of the default value of maximum increment in
number of cycles) and determines the maximum damage increment at any material
points based on
If the maximum damage increment, ,
is greater than the damage extrapolation tolerance that you specify, the number
of cycles over which the damage is extrapolated forward is reduced accordingly
to ensure the maximum damage increment is less than the damage extrapolation
tolerance. On the other hand, if the maximum damage increment at all material
points is less than half of the damage extrapolation tolerance that you
specify, the number of cycles is increased accordingly to ensure the maximum
damage increment is equal to the damage extrapolation tolerance.
Controlling Element Fracture
In addition to elements forecast to be fully or almost fully damaged after , additional elements are allowed to fracture if they are within the
tolerances described below in the current cycle. This approach avoids a jagged (not
smooth) crack front. The traction is removed immediately upon fracture or ramped down
gradually (see Specifying How a Debonding Force Is Released after a Fracture Criterion Is Met in Abaqus/Standard).
Two criteria are available to control additional fracture of elements ahead of the
current crack front: a cycle-based criterion (with a tolerance ) and a damage-based criterion (with a tolerance ). If both tolerances are specified, the damage-based tolerance takes
precedence.
Elements that satisfy the following expression fracture if the cycle-based criterion is in
effect:
Elements that satisfy the following expression fracture if the damage-based criterion is
in effect:
where and are the scalar damage variables at the end of cycles and , respectively.
Initial Conditions
Initial values of stresses, temperatures, field variables,
solution-dependent state variables, etc. can be specified (see
Initial Conditions).
Boundary Conditions
Boundary conditions can be applied to any of the displacement or rotation
degrees of freedom. During the analysis, prescribed boundary conditions must
have an amplitude definition that is cyclic over the step: the start value must
be equal to the end value (see
Amplitude Curves).
If the analysis consists of several steps, the usual rules apply (see
Boundary Conditions).
At each new step the boundary condition can either be modified or completely
defined. All boundary conditions defined in previous steps remain unchanged
unless they are redefined.
Loads
The following loads can be prescribed in a low-cycle fatigue analysis using
the direct cyclic approach:
Concentrated nodal forces can be applied to the displacement degrees of
freedom (1–6); see
Concentrated Loads.
Distributed pressure forces or body forces can be applied; see
Distributed Loads.
The distributed load types available with particular elements are described in
Abaqus Elements Guide.
During the analysis each load must have an amplitude definition that is
cyclic over the step where the start value must be equal to the end value (see
Amplitude Curves).
If the analysis consists of several steps, the usual rules apply (see
About Loads).
At each new step the loading can either be modified or completely defined. All
loads defined in previous steps remain unchanged unless they are redefined.
Predefined Fields
The following predefined fields can be specified in a low-cycle fatigue
analysis using the direct cyclic approach, as described in
Predefined Fields:
Temperature is not a degree of freedom in a low-cycle fatigue analysis
using the direct cyclic approach, but nodal temperatures can be specified as a
predefined field. The temperature values specified must be cyclic over the
step: the start value must be equal to the end value (see
Amplitude Curves).
If the temperatures are read from the results file, you should specify initial
temperature conditions equal to the temperature values at the end of the step
(see
Initial Conditions).
Alternatively, you can ramp the temperatures back to their initial condition
values, as described in
Predefined Fields.
Any difference between the applied and initial temperatures will cause thermal
strain if a thermal expansion coefficient is given for the material (Thermal Expansion).
The specified temperature also affects temperature-dependent material
properties, if any.
The values of user-defined field variables can be specified. These
values affect only field-variable-dependent material properties, if any. The
field variable values specified must be cyclic over the step.
Material Options
Most ductile material models that describe mechanical behavior are available
for use in a low-cycle fatigue analysis. The inelastic definition in a material
point must be used in conjunction with the linear elastic material model (Linear Elastic Behavior),
the porous elastic material model (Elastic Behavior of Porous Materials),
or the hypoelastic material model (Hypoelastic Behavior).
The following material properties are not active during a low-cycle fatigue analysis: acoustic
properties, thermal properties (except for thermal expansion), mass diffusion properties,
electrical conductivity properties, piezoelectric properties, and pore fluid flow
properties.
Different types of output are available for postprocessing and for
monitoring a low-cycle fatigue analysis using the direct cyclic approach.
Message File Information
As in a direct cyclic analysis, low-cycle fatigue analysis using the direct
cyclic approach in
Abaqus/Standard
prints the residual force, time average force, and a flag to indicate if
equilibrium was satisfied in the message (.msg) file at
different time increments for each iteration in each loading cycle. You can
control the frequency in increments at which information is printed to the
message file, and you can suppress the output; the default is to print output
every 10 increments (see
The Abaqus/Standard Message File
for more information).
Abaqus/Standard
also prints the number of Fourier terms used, the maximum residual coefficient,
the maximum correction to displacement coefficients, and the maximum
displacement coefficient in the Fourier series in the message file at the end
of each iteration in each cycle. An example of the output is shown below:
CYCLE 5 STARTS
ITERATION 26 STARTS
INC TIME STEP LARG. RESI. TIME AVG. FORCE
INC TIME FORCE FORCE EQUV.
10 0.250 2.50 1.008E+01 50.9 N
20 0.250 5.00 1.622E+01 76.8 N
30 0.250 7.50 4.622E-02 99.8 Y
ITERATION 26 SUMMARY
NUMBER OF FOURIER TERMS USED 40, TOTAL NUMBER OF INCREMENTS 120
CYCLE/STEP TIME 30.0, TOTAL TIME COMPLETED 31.0
AVERAGE FORCE 21.2 TIME AVG. FORCE 25.7
MAX. COEFFICIENT OF DISP. 0.142 AT NODE 24 DOF 2
MAX. COEFF. OF RESI. FORCE ON CONST. TERM 31.7 AT NODE 44 DOF 1
MAX. COEFF. OF RESI. FORCE ON PERI. TERMS 0.82 AT NODE 6 DOF 3
MAX. CORR. TO COEFF. OF DISP. ON CONST. TERM 0.002 AT NODE 50 DOF 3
MAX. CORR. TO COEFF. OF DISP. ON PERI. TERMS 0.015 AT NODE 50 DOF 3
Results Output
Element and nodal output are written only when the stabilized cycle is
reached. If a stabilized cycle has not been reached at the end of a cycle,
output is written for the last iteration of the cycle. All standard output
variables in
Abaqus/Standard
(Abaqus/Standard Output Variable Identifiers)
are available. In addition, the following variables are available for
progressive damage in bulk ductile material based on the continuum damage
mechanics approach:
STATUS
Status of element (the status of an element is 1.0 if the element is active,
0.0 if the element is not).
SDEG
Scalar stiffness degradation, D.
CYCLEINI
Number of cycles to initialize the damage at the material point.
The following variables are available for discrete crack propagation along
an arbitrary path based on the principles of linear elastic fracture mechanics
with the extended finite element method:
STATUSXFEM
Status of the enriched element. (The status of an enriched element is 1.0 if
the element is completely cracked, 0.0 if the element is not. If the element is
partially cracked, the value lies between 1.0 and 0.0.)
CYCLEINIXFEM
Number of cycles to initialize the crack at the enriched element.
ENRRTXFEM
All components of strain energy release rate range.
Recovering Additional Results for a Stabilized Cycle
Output at exact times is not supported for low-cycle fatigue analysis. If
output at exact times is requested,
Abaqus
will issue a warning message and change the output to an output at approximate
times.
Limitations
A low-cycle fatigue analysis using the direct cyclic approach is subject to
the following limitations:
Contact conditions cannot change during a given cycle when direct cyclic
analysis is used iteratively to obtain a stabilized solution.
The analysis may not perform well when there is compressive load on the
crack surface during a loading cycle because the global stiffness is formed
only one time at the beginning of each given loading cycle.
Geometric nonlinearity can be included only in any general step prior to
a direct cyclic step; however, only small displacements and strains will be
considered during the cyclic step.
Input File Template
The following is an example of modeling progressive damage and failure in
the bulk material based on the continuum damage mechanics approach and
progressive delamination growth at the interface:
HEADING
…
BOUNDARYData lines to specify zero-valued boundary conditionsINITIAL CONDITIONSData lines to specify initial conditionsAMPLITUDEData lines to define amplitude variations
**
MATERIALOptions to define material propertiesDAMAGE INITIATION, CRITERION=HYSTERESIS ENERGYData lines to define material constants for bulk ductile material damage initiationDAMAGE EVOLUTION, TYPE=HYSTERESIS ENERGYData lines to define material constants for bulk ductile material damage evolution
**
SURFACE, NAME=secondaryData lines to define secondary surface at delamination interfaceSURFACE, NAME=mainData lines to define main surface at delamination interfaceCONTACT PAIRsecondary, main
**
STEP (,INC=)
Set INC equal to the maximum number of increments in a single loading cycleDIRECT CYCLIC, FATIGUEData line to define time increment, cycle time, initial number of Fourier terms,
maximum number of Fourier terms, increment in number of Fourier terms,
and maximum number of iterationsData line to define minimum increment in number of cycles,
maximum increment in number of cycles, total number of cycles,
and damage extrapolation toleranceDEBOND, SECONDARY=secondary, MAIN=mainFRACTURE CRITERION, TYPE=FATIGUEData lines to define material constants used in Paris law and fracture criterion
**
BOUNDARY, AMPLITUDE=
Data lines to prescribe zero-valued or nonzero boundary conditionsCLOAD and/or DLOAD, AMPLITUDE=
Data lines to specify loadsTEMPERATURE and/or FIELD, AMPLITUDE=
Data lines to specify values of predefined fields
**
END STEP
The following is an example of modeling discrete crack growth
in the bulk material based on the principles of linear elastic fracture
mechanics with the extended finite element method and progressive delamination
growth at the interface:
HEADING
…
ENRICHMENT, TYPE=PROPAGATION CRACK, INTERACTION=INTERACTION,
ELSET=ENRICHED
BOUNDARYData lines to specify zero-valued boundary conditionsINITIAL CONDITIONSData lines to specify initial conditionsAMPLITUDEData lines to define amplitude variations
**
MATERIALOptions to define material propertiesSURFACE, INTERACTION=INTERACTIONSURFACE BEHAVIORFRACTURE CRITERION, TYPE=FATIGUEData lines to define material constants used in the Paris law and fracture criterion in the bulk
material for enriched elements
**
SURFACE, NAME=secondaryData lines to define secondary surface at delamination interfaceSURFACE, NAME=mainData lines to define main surface at delamination interfaceCONTACT PAIRsecondary, main
**
STEP (,INC=)
Set INC equal to the maximum number of increments in a single loading cycleDIRECT CYCLIC, FATIGUEData line to define time increment, cycle time, initial number of Fourier terms,
maximum number of Fourier terms, increment in number of Fourier terms,
and maximum number of iterationsData line to define minimum increment in number of cycles,
maximum increment in number of cycles, total number of cycles,
and damage extrapolation toleranceDEBOND, SECONDARY=secondary, MAIN=mainFRACTURE CRITERION, TYPE=FATIGUEData lines to define material constants used in the Paris law and fracture criterion at the interface
**
BOUNDARY, AMPLITUDE=
Data lines to prescribe zero-valued or nonzero boundary conditionsCLOAD and/or DLOAD, AMPLITUDE=
Data lines to specify loadsTEMPERATURE and/or FIELD, AMPLITUDE=
Data lines to specify values of predefined fields
**
END STEP
References
Coffin, L., “A
Study of the Effects of Cyclic Thermal Stresses on a Ductile
Metal,” Transactions of the American Society
of Mechanical
Engineering, vol. 76, pp. 931–951, 1954.
Manson, S., “Behavior
of Materials under Condition of Thermal
Stress,” Heat Transfer Symposium, University
of Michigan Engineering Research Institute, Ann Arbor,
MI, pp. 9–75, 1953.
Paris, P., M. Gomaz, and W. Anderson, “A
Rational Analytic Theory of Fatigue,” The
Trend in
Engineering, vol. 15, 1961.