Softening Laws - Revision history

Nairnj: /* Softening Laws */

2021-07-20T20:16:38Z

Softening Laws

← Older revision		Revision as of 20:16, 20 July 2021
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	Click each one for more details and information on their properties. After picking the law, all its properties are set using property commands within the [[Material_Models#Softening_Materials\|softening material]] definition by prefacing the law property with appropriate codes. For example, to set mode I and mode II toughnesses in an [[Isotropic Softening Material\|isotropic softening material]], the properties are entered as I-Gc and II-Gc where I- or II- preface the Gc softening law property.		Click each one for more details and information on their properties. After picking the law, all its properties are set using property commands within the [[Material_Models#Softening_Materials\|softening material]] definition by prefacing the law property with appropriate codes. For example, to set mode I and mode II toughnesses in an [[Isotropic Softening Material\|isotropic softening material]], the properties are entered as I-Gc and II-Gc where I- or II- preface the Gc softening law property.

	If needed, it is relatively easy to [[Softening Law Implementation\|implement custom softening laws]]. In general, the specific choice of softening law is less important than implied in literature on damage mechanics. Perhaps some problems will arise where the law's details matter. It is possible, ~~however~~, that relative mode I and mode II laws will influence mixed-mode failure simulations.		If needed, it is relatively easy to [[Softening Law Implementation\|implement custom softening laws]]. In general, the specific choice of softening law is less important than implied in literature on damage mechanics. Perhaps some problems will arise where the law's details matter. It is possible, for example, that relative mode I and mode II laws will influence mixed-mode failure simulations.

	== References ==		== References ==

Nairnj: /* Softening Laws */

2021-07-20T20:11:52Z

Softening Laws

← Older revision		Revision as of 20:11, 20 July 2021
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	\| [[Cubic Step Function Softening\|CubicStep]] \|\| 3 \|\| Cubic function softening law \|\| effectively from 0.79 to 4/3		\| [[Cubic Step Function Softening\|CubicStep]] \|\| 3 \|\| Cubic function softening law \|\| effectively from 0.79 to 4/3
	\|-		\|-
	\| [[Double Exponential Softening\|DoubleExponential]] \|\| 4 \|\| Two exponential softening law \|\| effectively 1~~±0~~.5		\| [[Double Exponential Softening\|DoubleExponential]] \|\| 4 \|\| Two exponential softening law \|\| effectively 1 to 1.49
	\|}		\|}

Nairnj: /* Softening Laws */

2021-07-20T19:34:58Z

Softening Laws

← Older revision		Revision as of 19:34, 20 July 2021
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	\| [[Exponential Softening\|Exponential]] \|\| 2 \|\| Exponential softening law \|\| 1		\| [[Exponential Softening\|Exponential]] \|\| 2 \|\| Exponential softening law \|\| 1
	\|-		\|-
	\| [[Cubic Step Function Softening\|CubicStep]] \|\| 3 \|\| Cubic function softening law \|\| ~~≤~~4/3		\| [[Cubic Step Function Softening\|CubicStep]] \|\| 3 \|\| Cubic function softening law \|\| effectively from 0.79 to 4/3
	\|-		\|-
	\| [[Double Exponential Softening\|DoubleExponential]] \|\| 4 \|\| Two exponential softening law \|\| effectively 1±0.5		\| [[Double Exponential Softening\|DoubleExponential]] \|\| 4 \|\| Two exponential softening law \|\| effectively 1±0.5

Nairnj: /* Softening Laws */

2021-07-20T19:13:39Z

Softening Laws

← Older revision		Revision as of 19:13, 20 July 2021
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	\| [[Cubic Step Function Softening\|CubicStep]] \|\| 3 \|\| Cubic function softening law \|\| ≤4/3		\| [[Cubic Step Function Softening\|CubicStep]] \|\| 3 \|\| Cubic function softening law \|\| ≤4/3
	\|-		\|-
	\| [[Double Exponential Softening\|DoubleExponential]] \|\| 4 \|\| Two exponential softening law \|\| 1&pm;0.5		\| [[Double Exponential Softening\|DoubleExponential]] \|\| 4 \|\| Two exponential softening law \|\| effectively 1±0.5
	\|}		\|}

Nairnj: /* Softening Laws */

2021-07-20T19:12:49Z

Softening Laws

← Older revision		Revision as of 19:12, 20 July 2021
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	\| [[Cubic Step Function Softening\|CubicStep]] \|\| 3 \|\| Cubic function softening law \|\| ≤4/3		\| [[Cubic Step Function Softening\|CubicStep]] \|\| 3 \|\| Cubic function softening law \|\| ≤4/3
	\|-		\|-
	\| [[Double Exponential Softening\|DoubleExponential]] \|\| 4 \|\| ~~Exponential~~ softening law \|\| ≤1		\| [[Double Exponential Softening\|DoubleExponential]] \|\| 4 \|\| Two exponential softening law \|\| 1&pm;0.5
	\|}		\|}

Nairnj: /* Softening Laws */

2021-07-20T17:28:23Z

Softening Laws

← Older revision		Revision as of 17:28, 20 July 2021
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	Click each one for more details and information on their properties. After picking the law, all its properties are set using property commands within the [[Material_Models#Softening_Materials\|softening material]] definition by prefacing the law property with appropriate codes. For example, to set mode I and mode II toughnesses in an [[Isotropic Softening Material\|isotropic softening material]], the properties are entered as I-Gc and II-Gc where I- or II- preface the Gc softening law property.		Click each one for more details and information on their properties. After picking the law, all its properties are set using property commands within the [[Material_Models#Softening_Materials\|softening material]] definition by prefacing the law property with appropriate codes. For example, to set mode I and mode II toughnesses in an [[Isotropic Softening Material\|isotropic softening material]], the properties are entered as I-Gc and II-Gc where I- or II- preface the Gc softening law property.

	If needed, it is relatively easy to [[Softening Law Implementation\|implement custom softening laws]]. In general, the specific choice of softening law is less important than implied in literature on damage mechanics. Perhaps some problems will arise where the law's details matter.		If needed, it is relatively easy to [[Softening Law Implementation\|implement custom softening laws]]. In general, the specific choice of softening law is less important than implied in literature on damage mechanics. Perhaps some problems will arise where the law's details matter. It is possible, however, that relative mode I and mode II laws will influence mixed-mode failure simulations.

	== References ==		== References ==

Nairnj: /* Normalized Softening Law */

2021-07-20T17:26:04Z

Normalized Softening Law

← Older revision		Revision as of 17:26, 20 July 2021
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	<math>s{G}_c = \Omega(\delta_{max},s) = \int_0^{\delta_{max}}f(\delta,s)d\delta</math>		<math>s{G}_c = \Omega(\delta_{max},s) = \int_0^{\delta_{max}}f(\delta,s)d\delta</math>

	for δ<sub>max</sub>. The reason for this approach as that δ<sub>max</sub> is not a material property, but rather depends on mesh size. In ~~contract~~ ''G<sub>c</sub>'' is treated as a material property for damage (although it may have ~~have~~ mesh dependencies as well).		for δ<sub>max</sub>. The reason for this approach as that δ<sub>max</sub> is not a material property, but rather depends on mesh size. In contrast, ''G<sub>c</sub>'' is treated as a material property for damage (although it may have some mesh dependencies as well).

	=== Law Stability ===		=== Law Stability ===
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	<math>\Delta x \le \eta \frac{G_c}{\sigma_0\varepsilon_0} \qquad{\rm where} \qquad		<math>\Delta x \le \eta \frac{G_c}{\sigma_0\varepsilon_0} = \eta\left(\frac{K_c}{\sigma_0}\right)^2 \qquad{\rm where} \qquad
	\eta = \frac{1}{sG_c\max\bigl(-f'(\delta,s)\bigr)}</math>		\eta = \frac{1}{sG_c\max\bigl(-f'(\delta,s)\bigr)}</math>

	is a softening-law-dependent stability factor. It is always possible to get stable results by high enough resolution (or small enough particle size). The need for high resolution is most common for brittle materials (low <math>G_c</math> with high <math>\sigma_0\varepsilon_0</math>). The most stable softening law is [[Linear Softening]] for which <math>\eta=2</math>.		is a softening-law-dependent stability factor, <math>\sigma_0</math> and <math>\varepsilon_0</matH> are [[Damage Initiation Laws\|damage initiation properties]], and <math>K_c</math> is the corresponding stress-intensity factor toughness. It is always possible to get stable results by high enough resolution (or small enough particle size). The need for high resolution is most common for brittle materials (low <math>G_c</math> with high <math>\sigma_0\varepsilon_0</math>). The most stable softening law is [[Linear Softening]] for which <math>\eta=2</math>.

	== Softening Laws ==		== Softening Laws ==

Nairnj: /* Normalized Softening Law */

2021-07-20T17:21:20Z

Normalized Softening Law

← Older revision		Revision as of 17:21, 20 July 2021
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	<math>{G}_c = s\Omega(\delta_{max},s) = s\int_0^{\delta_{max}}f(\delta,s)d\delta</math>		<math>s{G}_c = \Omega(\delta_{max},s) = \int_0^{\delta_{max}}f(\delta,s)d\delta</math>

	for δ<sub>max</sub>. The reason for this approach as that δ<sub>max</sub> is not a material property, but rather depends on mesh size. In contract ''G<sub>c</sub>'' is treated as a material property for damage (although it may have have mesh dependencies as well).		for δ<sub>max</sub>. The reason for this approach as that δ<sub>max</sub> is not a material property, but rather depends on mesh size. In contract ''G<sub>c</sub>'' is treated as a material property for damage (although it may have have mesh dependencies as well).

Nairnj: /* Normalized Softening Law */

2021-07-20T17:20:37Z

Normalized Softening Law

← Older revision		Revision as of 17:20, 20 July 2021
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	<math>\Omega(\delta,s) = \int_0^{\delta}f(\delta,s)d\delta - {1\over2}\delta f(\delta,s)</math>		<math>\Omega(\delta,s) = \int_0^{\delta}f(\delta,s)d\delta - {1\over2}\delta f(\delta,s)</math>

	Each softening law requires ''G<sub>c</sub>'' and possibly other parameters to define the law's functional form. The laws do not need ~~to be provided~~ δ<sub>max</sub> because it can be calculated from the law shape, ''s'', ''G<sub>c</sub>'', and possible other parameters be ~~equating ''~~G<sub>c</sub>~~'' to total area under the law~~. ~~Furthermore,~~ δ<sub>max</sub> depends on mesh size ~~while~~ ''G<sub>c</sub>'' is a material property for damage.		Each softening law requires ''G<sub>c</sub>'' and possibly other parameters to define the law's functional form. The laws do not need provide δ<sub>max</sub> because it can be calculated from the law shape, ''s'', ''G<sub>c</sub>'', and possible other parameters be solving


			<math>{G}_c = s\Omega(\delta_{max},s) = s\int_0^{\delta_{max}}f(\delta,s)d\delta</math>

			for δ<sub>max</sub>. The reason for this approach as that δ<sub>max</sub> is not a material property, but rather depends on mesh size. In contract ''G<sub>c</sub>'' is treated as a material property for damage (although it may have have mesh dependencies as well).

	=== Law Stability ===		=== Law Stability ===

Nairnj: /* Normalized Softening Law */

2021-07-20T17:17:00Z

Normalized Softening Law

← Older revision		Revision as of 17:17, 20 July 2021
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	<math>\Omega(\delta,s) = \int_0^{\delta}f(\delta,s)d\delta - {1\over2}\delta f(\delta,s)</math>		<math>\Omega(\delta,s) = \int_0^{\delta}f(\delta,s)d\delta - {1\over2}\delta f(\delta,s)</math>

	Each softening law requires ''G<sub>c</sub>'' and possibly other parameters to define the law's functional form. The laws do not need to be provided δ<sub>max</sub> because it can be calculated from the law shape, ''s'', and ''G<sub>c</sub>'' ~~from~~ total area under the law. Furthermore, δ<sub>max</sub> depends on mesh size while ''G<sub>c</sub>'' is a material property for damage.		Each softening law requires ''G<sub>c</sub>'' and possibly other parameters to define the law's functional form. The laws do not need to be provided δ<sub>max</sub> because it can be calculated from the law shape, ''s'', ''G<sub>c</sub>'', and possible other parameters be equating ''G<sub>c</sub>'' to total area under the law. Furthermore, δ<sub>max</sub> depends on mesh size while ''G<sub>c</sub>'' is a material property for damage.

	=== Law Stability ===		=== Law Stability ===