김경종
323 lines
25 KiB
Markdown
323 lines
25 KiB
Markdown
<!-- source-page: 981 -->
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The Virtual Crack Closure Technique (VCCT) criterion uses the principles of linear elastic fracture mechanics (LEFM) and, therefore, is appropriate for problems in which brittle crack propagation occurs along predefined surfaces.
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VCCT is based on the assumption that the strain energy released when a crack is extended by a certain amount is the same as the energy required to close the crack by the same amount. For example, Figure 11.4.3–4 illustrates the similarity between crack extension from i to j and crack closure at j.
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<details>
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<summary>text_image</summary>
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a δa
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crack closed
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i j
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a δa
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</details>
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Figure 11.4.3–4 Mode I: The energy released when a crack is extended by a certain amount is the same as the energy required to close the crack.
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In Figure 11.4.3–5 nodes 2 and 5 will start to release when
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<!-- source-page: 982 -->
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$$
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f = \frac {G _ {I}}{G _ {I C}} = \frac {1}{2} \left(\frac {v _ {1 , 6} F _ {v , 2 , 5}}{b d}\right) \frac {1}{G _ {I C}} \geq 1. 0,
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$$
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where $G _ { I }$ is the Mode I energy release rate, $G _ { I C }$ is the critical Mode I energy release rate, b is the width, d is the length of the elements at the crack front, $F _ { v , 2 , 5 }$ is the vertical force between nodes 2 and 5, and $v _ { 1 , 6 }$ is the vertical displacement between nodes 1 and 6. Assuming that the crack closure is governed by linear elastic behavior, the energy to close the crack (and, thus, the energy to open the crack) is calculated from the previous equation. Similar arguments and equations can be written in two dimensions for Mode II and for three-dimensional crack surfaces including Mode III.
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<details>
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<summary>text_image</summary>
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Load
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F_{v,2,5}
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6
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Load
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F_{v,2,5}
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5
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4
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2
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3
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1
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d
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y, v
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x, u
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Load
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F_{v,2,5}
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F_{v,2,5 crit}
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Area = G_{IC} db
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V_{2,5 crit}
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Displacement
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V_{2,5}
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</details>
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Figure 11.4.3–5 Pure Mode I modified.
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In the general case involving Mode I, II, and III the fracture criterion is defined as
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$$
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f = \frac {G _ {e q u i v}}{G _ {e q u i v C}} \geq 1. 0,
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$$
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where $G _ { e q u i v }$ is the equivalent strain energy release rate calculated at a node, and $G _ { e q u i v C }$ is the critical equivalent strain energy release rate calculated based on the user-specified mode-mix criterion and the bond strength of the interface. The crack-tip node will debond when the fracture criterion reaches the value of 1.0.
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Abaqus provides three common mode-mix formulae for computing $G _ { e q u i v C } \mathrm { : }$ the BK law, the power law, and the Reeder law models. The choice of model is not always clear in any given analysis; an appropriate model is best selected empirically.
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# BK law
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The BK law model is described in Benzeggagh (1996) by the following formula:
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$$
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G _ {e q u i v C} = G _ {I C} + \left(G _ {I I C} - G _ {I C}\right) \left(\frac {G _ {I I} + G _ {I I I}}{G _ {I} + G _ {I I} + G _ {I I I}}\right) ^ {\eta}.
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$$
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<!-- source-page: 983 -->
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To define this model, you must provide $G _ { I C } , G _ { I I C }$ and . This model provides a power law relationship combining energy release rates in Mode I, Mode II, and Mode III into a single scalar fracture criterion.
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Input File Usage: \*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=BK
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Abaqus/CAE Usage: Interaction module: Create Interaction Property: Contact, Mechanical→Fracture Criterion, Type: VCCT, Mixed mode behavior: BK
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# Power law
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The power law model is described in Wu (1965) by the following formula:
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$$
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\frac {G _ {\text {equiv}}}{G _ {\text {equiv} C}} = \left(\frac {G _ {I}}{G _ {I C}}\right) ^ {a _ {m}} + \left(\frac {G _ {I I}}{G _ {I I C}}\right) ^ {a _ {n}} + \left(\frac {G _ {I I I}}{G _ {I I I C}}\right) ^ {a _ {o}}.
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$$
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To define this model, you must provide $G _ { I C } , G _ { I I C } , G _ { I I I C } , a _ { m } , a _ { n }$ and $a _ { o }$
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Input File Usage: \*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=POWER
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Abaqus/CAE Usage: Interaction module: Create Interaction Property: Contact, Mechanical→Fracture Criterion, Type: VCCT, Mixed mode behavior: Power
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# Reeder law
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The Reeder law model is described in Reeder (2002) by the following formula:
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$$
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G _ {e q u i v C} = G _ {I C} + \left(G _ {I I C} - G _ {I C}\right) \left(\frac {G _ {I I} + G _ {I I I}}{G _ {I} + G _ {I I} + G _ {I I I}}\right) ^ {\eta} +
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$$
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$$
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(G _ {I I I C} - G _ {I I C}) \left(\frac {G _ {I I I}}{G _ {I I} + G _ {I I I}}\right) \left(\frac {G _ {I I} + G _ {I I I}}{G _ {I} + G _ {I I} + G _ {I I I}}\right) ^ {\eta}.
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$$
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To define this model, you must provide $G _ { I C } , G _ { I I C } , G _ { I I I C }$ and . The Reeder law is best applied when $G _ { I I C } \neq G _ { I I I C }$ . When $G _ { I I C } = G _ { I I I C }$ , the Reeder law reduces to the BK law. The Reeder law applies only to three-dimensional problems.
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Input File Usage: \*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=REEDER
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Abaqus/CAE Usage: Interaction module: Create Interaction Property: Contact, Mechanical→Fracture Criterion, Type: VCCT, Mixed mode behavior: Reeder
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<!-- source-page: 984 -->
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Releasing multiple nodes in one increment in Abaqus/Standard
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For an unstable crack growth problem, sometimes it is more efficient to allow multiple nodes at and ahead of a crack tip to debond in one increment without cutting back the increment size when the VCCT fracture criterion is satisfied. This capability is activated automatically if you specify an unstable growth tolerance, $f _ { t o l } ^ { u }$ . In this case if the fracture criterion, f, is within the given unstable growth tolerance:
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$$
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1 + f _ {t o l} \leq f \leq 1 + f _ {t o l} ^ {u},
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$$
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where $f _ { t o l }$ is the tolerance described earlier in this section, rather than cut back the increment size, more nodes at and ahead of the crack tip are allowed to debond in one increment until $f < 1$ for all the nodes ahead of the crack tip. The forces at those debonded nodes are completely released immediately during the following increment. If you do not specify a value for the unstable growth tolerance, the default value is infinity. In this case the fracture criterion, f, for unstable crack growth is not limited by any upper-bound value in the above equation.
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Input File Usage: \*FRACTURE CRITERION, TYPE=VCCT, UNSTABLE GROWTH TOLERANCE=
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Abaqus/CAE Usage: Interaction module: Create Interaction Property: Contact, Mechanical→Fracture Criterion, Type: VCCT, toggle on Specify tolerance for unstable crack propagation: specify value
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Defining variable critical energy release rates
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You can define a VCCT criterion with varying energy release rates by specifying the critical energy release rates at the nodes.
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If you indicate that the nodal critical energy rates will be specified, any constant critical energy release rates you specify are ignored, and the critical energy release rates are interpolated from the nodes. The critical energy release rates must be defined at all nodes on the slave surface.
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Input File Usage: Use both of the following options:
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\*FRACTURE CRITERION, TYPE=VCCT, NODAL ENERGY RATE \*NODAL ENERGY RATE
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Abaqus/CAE Usage: Defining variable critical energy release rates is not supported in Abaqus/CAE.
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# Enhanced VCCT criterion
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This criterion is available only in Abaqus/Standard.
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The enhanced VCCT criterion is very similar to the original VCCT criterion described above. As in the original VCCT criterion, the fracture criterion in the general case involving Mode I, II, and III is defined as
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$$
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f = \frac {G _ {e q u i v}}{G _ {e q u i v C}} \geq 1. 0.
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$$
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<!-- source-page: 985 -->
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The crack-tip node debonds when the fracture criterion reaches the value of 1.0. However, unlike the original VCCT criterion, you can specify two different critical fracture energy release rates: $G _ { C }$ for the onset of a crack and $G _ { C } ^ { P }$ for the growth of a crack. When the enhanced VCCT criterion is used in the general case involving Mode I, II, and III fracture, the amount of energy dissipated associated with the release of the debondpropagate the crack, $G _ { e q u i v C } ^ { \bar { P } }$ is controlled by the critical equivalent strain energy release rate required to, rather than by the critical equivalent strain energy release rate required to initiate the crack, for different mix $G _ { e q u i v C }$ The formulae for calculating fracture criteria. $G _ { e q u i v C } ^ { P }$ are identical to those used for $G _ { e q u i v C }$
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Input File Usage: \*FRACTURE CRITERION, TYPE=ENHANCED VCCT
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Abaqus/CAE Usage: Interaction module: Create Interaction Property: Contact, Mechanical→Fracture Criterion, Type: Enhanced VCCT
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# Low-cycle fatigue criterion
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This criterion is available only in Abaqus/Standard.
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If you specify the low-cycle fatigue criterion, progressive delamination growth at the interfaces in laminated composites subjected to sub-critical cyclic loadings can be simulated. This criterion can be used only in a low-cycle fatigue analysis using the direct cyclic approach (“Low-cycle fatigue analysis using the direct cyclic approach,” Section 6.2.7). The onset and delamination growth are characterized by using the Paris law, which relates the relative fracture energy release rate to crack growth rates as illustrated in Figure 11.4.3–6. The fracture energy release rates at the crack tips in the interface elements are calculated based on the above mentioned VCCT technique.
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The Paris regime is bounded by the energy release rate threshold, $G _ { t h r e s h }$ , below which there is no consideration of fatigue crack initiation or growth, and the energy release rate upper limit, $G _ { p l }$ , above which the fatigue crack will grow at an accelerated rate. $G _ { C }$ is the critical equivalent strain energy release rate calculated based on the user-specified mode-mix criterion and the bond strength of the interface. The formulae for calculating $G _ { C }$ have been provided above for different mixed mode fracture criteria. You can specify the ratio of $G _ { t h r e s h }$ over $G _ { C }$ and the ratio of $G _ { p l }$ over $G _ { C }$ . The default values are $\begin{array} { r } { \frac { G _ { t h r e s h } } { G _ { C } } = 0 . 0 1 } \end{array}$ Gc and $\begin{array} { r } { \frac { G _ { p l } } { G _ { C } } = 0 . 8 5 } \end{array}$ Gpl Gc .
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Input File Usage: \*FRACTURE CRITERION, TYPE=FATIGUE
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Abaqus/CAE Usage: The low-cycle fatigue criterion is not supported in Abaqus/CAE.
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# Onset of delamination growth
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The onset of delamination growth refers to the beginning of fatigue crack growth at the crack tip along the interface. In a low-cycle fatigue analysis the onset of the fatigue crack growth criterion is characterized by $\Delta G$ , which is the relative fracture energy release rate when the structure is loaded between its maximum and minimum values. The fatigue crack growth initiation criterion is defined as
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$$
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f = \frac {N}{c _ {1} \Delta G ^ {c _ {2}}} \geq 1. 0,
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$$
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where $c _ { 1 }$ and $c _ { 2 }$ are material constants and is the cycle number. The interface elements at the crack tips will not be released unless the above equation is satisfied and the maximum fracture energy release
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<!-- source-page: 986 -->
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<details>
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<summary>line</summary>
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| G | da/dN |
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| -------- | ------ |
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| G_thresh | Low |
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| G_pl | High |
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| G_C | High |
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</details>
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Figure 11.4.3–6 Fatigue crack growth govern by Paris law.
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rate, $G _ { m a x ; \ l }$ , which corresponds to the cyclic energy release rate when the structure is loaded up to its maximum value, is greater than $G _ { t h r e s h }$ .
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# Fatigue delamination growth using the Paris law
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Once the onset of delamination growth criterion is satisfied at the interface, the delamination growth rate, $d a / d N$ , can be calculated based on the relative fracture energy release rate, $\Delta G$ . The rate of the delamination growth per cycle is given by the Paris law if $G _ { t h r e s h } < G _ { m a x } < G _ { p l }$
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$$
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\frac {d a}{d N} = c _ {3} \Delta G ^ {c _ {4}},
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$$
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where $c _ { 3 }$ and $c _ { 4 }$ are material constants.
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At the end of cycle , Abaqus/Standard extends the crack length, $a _ { N }$ , from the current cycle forward over an incremental number of cycles, $\Delta N$ to $a _ { N + \Delta N }$ by releasing at least one element at the interface. Given the material constants $c _ { 3 }$ and $c _ { 4 }$ , combined with the known node spacing $\Delta a _ { N j } =$ $a _ { N + \Delta N } - a _ { N }$ at the interface elements at the crack tips, the number of cycles necessary to fail each
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<!-- source-page: 987 -->
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interface element at the crack tip can be calculated as $\Delta N _ { j }$ , where j represents the node at the jthe crack tip. The analysis is set up to release at least one interface element after the loading cycle is stabilized. The element with the fewest cycles is identified to be released, and its $\Delta N _ { m i n } = m i n ( \Delta N _ { j } )$ is represented as the number of cycles to grow the crack equal to its element length, $\Delta a _ { N m i n } = m i n ( \Delta a _ { N j } )$ . 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.
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If $G _ { m a x } > G _ { p l }$ , the interface elements at the crack tips will be released by increasing the cycle number count, , by one only.
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# Specifying how a debonding force is released after a fracture criterion is met in Abaqus/Standard
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After debonding, the traction between two surfaces is initially carried as equal and opposite forces at the slave node and the corresponding point on the master surface. The debonding force is released as the crack opens and advances. Once complete debonding has occurred at a point, the bond surfaces act like standard contact surfaces with associated interface characteristics. There are two different ways to release the debonding force, depending on the fracture criterion that you specify.
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# Specifying a debonding amplitude curve
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When you use the critical stress, critical crack opening displacement, or crack length versus time fracture criteria, you can define how this force is to be reduced to zero with time after debonding starts at a particular node on the bonded surface. You specify a relative amplitude, ${ \pmb a } ,$ as a function of time after debonding starts at a node. Thus, suppose the force transmitted between the surfaces at slave node N is $\mathbf { T } ^ { N } | _ { 0 }$ when that node starts to debond, which occurs at time $t ^ { N } | _ { 0 }$ . Then, for any time $t > t ^ { N } | _ { 0 }$ the force transmitted between the surfaces at node N is $a ( t - t ^ { N } | _ { 0 } ) \mathbf { T } ^ { N } | _ { 0 }$ . The relative amplitude must be 1.0 at the relative time 0.0 and must reduce to 0.0 at the last relative time point given.
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The best choice of the amplitude curve depends on the material properties, specified loading, and the crack propagation criterion. If the stresses are removed too rapidly, the resulting large changes in the strains near the crack tip can cause convergence difficulties. For large-strain problems severe mesh distortion can also occur. For problems with rate-independent materials a linear amplitude curve is normally adequate. For problems with rate-dependent materials the stresses should be ramped off more slowly at the beginning of debonding to avoid convergence and mesh distortion difficulties. To reduce the likelihood of convergence and mesh distortion difficulties, you can reduce the value of the debond stress by 25% in 50% of the time to debond. The solution should not be strongly influenced by the details of the unloading procedure; if it is, this usually indicates that the mesh should be refined in the debond region.
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Input File Usage: $* { \mathrm { D E B O N D } } , { \mathrm { S L A V E } } { = } s l a \nu e , { \mathrm { M A S T E R } } { = } m a s t e r$
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$D a t a \ l i n e s \ t o \ d e f i n e \ d e b o n d i n g \ a m p l i t u d e \ c u r \nu e$
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Abaqus/CAE Usage: Specifying a debonding amplitude curve is not supported in Abaqus/CAE.
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<!-- source-page: 988 -->
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# Ramping down debonding force for the VCCT and the enhanced VCCT criteria
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For the VCCT and the enhanced VCCT criteria, when the energy release rate exceeds the critical value at a crack tip, you can either release the traction between the two surfaces at the crack tip immediately during the following increment or release the traction gradually during succeeding increments with the reduction of the magnitude of the debonding force being governed by the critical fracture energy release rate. The latter approach is sometimes recommended to avoid sudden loss of stability when the crack tip is advanced. The enhanced VCCT criterion is meaningful only when used in conjunction with the latter approach. When the former approach is used, the results obtained by using the enhanced VCCT criterion are identical to those obtained by using the original VCCT criterion.
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Input File Usage: Use the following option to release the traction immediately:
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\*DEBOND, SLAVE=slave, MASTER=master, DEBONDING FORCE=STEP
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Use the following option to release the traction gradually:
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\*DEBOND, SLAVE=slave, MASTER=master, DEBONDING FORCE=RAMP
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Abaqus/CAE Usage: Interaction module: Special→Crack→Create: Name: crack name, Type: Debond using VCCT, select the step and the surface to surface (Standard) interaction, Debonding force: Step or Ramp
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# Procedures
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Crack propagation analysis can be performed for static or dynamic overloadings using the following procedures:
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• “Static stress analysis,” Section 6.2.2
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• “Quasi-static analysis,” Section 6.2.5
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• “Implicit dynamic analysis using direct integration,” Section 6.3.2
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• “Explicit dynamic analysis,” Section 6.3.3
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• “Fully coupled thermal-stress analysis,” Section 6.5.3
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It can also be performed for sub-critical cyclic fatigue loadings using the following procedure:
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• “Low-cycle fatigue analysis using the direct cyclic approach,” Section 6.2.7
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# Controlling time incrementation during debonding in Abaqus/Standard
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When automatic incrementation is used for any criteria other than VCCT, enhanced VCCT, or low-cycle fatigue, you can specify the size of the time increment used just after debonding starts. By default, the time increment is equal to the last relative time specified. However, if a fracture criterion is met at the beginning of an increment, the size of the time increment used just after debonding starts will be set equal to the minimum time increment allowed in this step.
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<!-- source-page: 989 -->
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For fixed time incrementation the specified time increment value will be used as the time increment size after debonding starts if Abaqus/Standard finds it needs a smaller time increment than the fixed time increment size. The time increment size will be modified as required until debonding is complete.
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Input File Usage: \*DEBOND, SLAVE=slave, MASTER=master, TIME INCREMENT=t
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Abaqus/CAE Usage: Controlling time incrementation during debonding is not supported in Abaqus/CAE.
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# Viscous regularization for VCCT in Abaqus/Standard
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The simulation of structures with unstable propagating cracks is challenging and difficult. Nonconvergent behavior may occur from time to time. While the usual stabilization techniques (such as contact pair stabilization and static stabilization) can be used to overcome some convergence difficulties, localized damping is included for VCCT or enhanced VCCT by using the viscous regularization technique. Viscous regularization damping causes the tangent stiffness matrix of the softening material to be positive for sufficiently small time increments.
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Input File Usage: Use one of following options:
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$\scriptstyle * \mathrm { F R A C T U R E } \mathrm { C R I T E R I O N } , \mathrm { T Y P E } = \mathrm { V C C T } , \mathrm { V I S C O S I T Y } = \mu$
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$* { \mathrm { F R A C T U R E ~ C R I T E R I O N } } , { \mathrm { T Y P E } } { \mathrm { = } } { \mathrm { E N H A N C E D ~ V C C T , ~ V I S C O S I T Y } } { \mathrm { = } } \mu$
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Abaqus/CAE Usage: Interaction module: Create Interaction Property: Contact, Mechanical→Fracture Criterion, Type: VCCT or Enhanced VCCT, Viscosity
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# Linear scaling to accelerate convergence for VCCT in Abaqus/Standard
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For most crack propagation simulations using VCCT or the enhanced VCCT criterion, the deformation can be nearly linear up to the point of the onset of crack growth; past this point the analysis becomes very nonlinear. In this case a linear scaling method can be used to effectively reduce the solution time to reach the onset of crack growth.
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Suppose that an applied “trial” load at increment $t = t _ { i }$ is just a fraction of the critical load at the onset time of crack growth, $t = t _ { c r i t }$ . The following algorithm is used in Abaqus/Standard to quickly converge to the critical load state:
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$$
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\Delta t _ {i + 1} = \left(\beta_ {i} \sqrt {\frac {G _ {e q u i v C}}{G _ {e q u i v}}} - 1\right) t _ {i},
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$$
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where initially $\beta _ { i }$ would be set between 0.7 and 0.9 depending on the degree of nonlinearity (the default value is 0.9). When $\Delta t _ { i + 1 }$ becomes smaller than 0.5% (indicating that the load is within 0.5% of its critical value), the next $\beta _ { i }$ is automatically set to 1.0 to cause the most critical crack-tip node to precisely reach the critical value at the next increment. After the first crack-tip node releases, the linear scaling calculations are no longer valid and the time increment is set to the default value. Cutback is then allowed.
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Input File Usage: $\scriptstyle * \mathrm { C O N T R O L S } , \mathrm { T Y P E = V C C T \ L I N E A R \ S C A L I N G }$
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<!-- source-page: 990 -->
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# Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: step name, VCCT Linear Scaling
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# Tips for using the VCCT or enhanced VCCT criterion in Abaqus/Standard
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Crack propagation problems using the VCCT or enhanced VCCT criterion are numerically challenging. The following tips will help you create a successful Abaqus/Standard model:
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• An analysis with the VCCT or enhanced VCCT criterion requires small time increments. Abaqus/Standard tracks the location of the active crack front node by node when the VCCT or enhanced VCCT criterion is used. Therefore, the crack front is allowed to advance only a single node forward in any single increment (although such an advance may take place across the entire crack front in three-dimensional problems). Because an analysis using the VCCT or enhanced VCCT criterion provides detailed results of the growth of the crack, you will need small time increments, especially if the mesh is highly refined.
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• Three different types of damping can be used to aid convergence for a model using the VCCT or enhanced VCCT criterion: contact stabilization, automatic or static stabilization, and viscous regularization. Contact and automatic stabilization are not specific to VCCT; they are built into Abaqus/Standard and are compatible with VCCT. Setting the value of the damping parameters is often an iterative procedure. If your VCCT model fails to converge due to unstable crack propagation, set the damping parameters to relatively high values and rerun the analysis. If the parameters are high enough, stable incrementation should return. However, the crack propagation behavior may have been modified by the damping forces and may not be physically correct. To monitor the energy absorbed by viscous damping, plot the damping energy and compare the results to the total strain energy in the model (ALLSE). When set properly, the value of the damping energy should be a small fraction of the total energy. Monitor the damping energy to ensure that the results of the VCCT simulation are reasonable in the presence of damping. When you use contact or automatic stabilization, Abaqus writes the damping energy to the variable ALLSD in the output database (.odb) file. When you use viscous regularization, Abaqus writes the damping energy to the variable ALLVD.
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• To maximize the accuracy of the debonding simulation, try to use matched meshes between the slave and master surfaces of the debonding contact pair.
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• If you do use a mismatched mesh, you can maximize the accuracy of the simulation by using the small-sliding, surface-to-surface formulation for the contact pair (see “Contact formulations in Abaqus/Standard,” Section 38.1.1).
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• Printing contact constraint information to the data (.dat) file allows you to review the status of the debonding contact pair at the beginning of the analysis. By printing detailed contact conditions to the message (.msg) file, you can track the incremental behavior of the advancing crack front during the analysis. For more information about these output requests, see “Output,” Section 4.1.1.
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• You can add a small clearance to the initially unbonded portion of the debonding contact pair (“Adjusting initial surface positions and specifying initial clearances in Abaqus/Standard contact pairs,” Section 36.3.5). The small clearance will help to eliminate unnecessary severe discontinuity iterations during incrementation as the crack begins to progress.
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