Isotropic Softening Material - Revision history

Nairnj: /* References */

2026-03-29T17:35:35Z

References

← Older revision		Revision as of 17:35, 29 March 2026
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	<ref name="dmFM">J. A. Nairn, "Direct Comparison of Anisotropic Damage Mechanics to Fracture Mechanics of Explicit Cracks," Eng. Fract. Mech., 203, 197-207 (2018). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/ADAMvsFM.pdf See PDF])</ref>		<ref name="dmFM">J. A. Nairn, "Direct Comparison of Anisotropic Damage Mechanics to Fracture Mechanics of Explicit Cracks," Eng. Fract. Mech., 203, 197-207 (2018). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/ADAMvsFM.pdf See PDF])</ref>

	<ref name="dmGen">J. A. Nairn, "Generalization of Anisotropic Damage Mechanics Modeling in the Material Point Method," ''Int. J. for Numerical Methods in Engineering'', '''123''', 5072-5097 (2022). (https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/GenDamage.pdf See PDF])</ref>		<ref name="dmGen">J. A. Nairn, "Generalization of Anisotropic Damage Mechanics Modeling in the Material Point Method," ''Int. J. for Numerical Methods in Engineering'', '''123''', 5072-5097 (2022). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/GenDamage.pdf See PDF])</ref>

	</references>		</references>

Nairnj: /* References */

2026-03-29T17:31:35Z

References

← Older revision		Revision as of 17:31, 29 March 2026
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	<ref name="chaboche">J. Chaboche (1979). Le concept de contrainte effective appliqu ́e a` l’ ́elasticit ́e et a` la viscoplasticit ́e en pr ́esence d’un endommagement anisotrope. In Boehler, J.-P., editor, Mechanical Behavior of Anisotropic Solids / Comportment M ́echanique des Solides Anisotropes, pages 737–760. Springer Netherlands.</ref>		<ref name="chaboche">J. Chaboche (1979). Le concept de contrainte effective appliqu ́e a` l’ ́elasticit ́e et a` la viscoplasticit ́e en pr ́esence d’un endommagement anisotrope. In Boehler, J.-P., editor, Mechanical Behavior of Anisotropic Solids / Comportment M ́echanique des Solides Anisotropes, pages 737–760. Springer Netherlands.</ref>

	<ref name="dmref">J. A. Nairn, C. C. Hammerquist, and Y. E. Aimene, "Numerical Implementation of Anisotropic Damage Mechanics," Int. J. for Numerical Methods in Engineering, 112(12), 1846-1868 (2017). [http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MPMSoftening.pdf PDF]</ref>		<ref name="dmref">J. A. Nairn, C. C. Hammerquist, and Y. E. Aimene, "Numerical Implementation of Anisotropic Damage Mechanics," Int. J. for Numerical Methods in Engineering, 112(12), 1846-1868 (2017). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MPMSoftening.pdf See PDF])</ref>

	<ref name="dmFM">J. A. Nairn, "Direct Comparison of Anisotropic Damage Mechanics to Fracture Mechanics of Explicit Cracks," Eng. Fract. Mech., 203, 197-207 (2018). [http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/ADAMvsFM.pdf PDF]</ref>		<ref name="dmFM">J. A. Nairn, "Direct Comparison of Anisotropic Damage Mechanics to Fracture Mechanics of Explicit Cracks," Eng. Fract. Mech., 203, 197-207 (2018). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/ADAMvsFM.pdf See PDF])</ref>

			<ref name="dmGen">J. A. Nairn, "Generalization of Anisotropic Damage Mechanics Modeling in the Material Point Method," ''Int. J. for Numerical Methods in Engineering'', '''123''', 5072-5097 (2022). (https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/GenDamage.pdf See PDF])</ref>

	~~<ref name="dmGen">J. A. Nairn, "Generalization of Anisotropic Damage Mechanics Modeling in the Material Point Method," Int. J. for Numerical Methods in Engineering, in press (2022).</ref>~~
	</references>		</references>

Nairnj: /* Constitutive Law */

2023-11-18T17:48:17Z

Constitutive Law

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	== Constitutive Law ==		== Constitutive Law ==

	This [[Material Models\|MPM ~~Material~~]] is an isotropic, elastic material, but once it fails, it develops anisotropic damage. The constitutive law for this material is		This [[Material Models#Softening Materials\|MPM softening material]] is an isotropic, elastic material, but once it fails, it develops anisotropic damage. The constitutive law for this material is

Nairnj: /* History Variables */

2023-09-21T21:17:12Z

History Variables

← Older revision		Revision as of 21:17, 21 September 2023
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	# For 2D it is cos(θ), but for 3D it is Euler angle α.		# For 2D it is cos(θ), but for 3D it is Euler angle α.
	# For 2D it is sin(θ), but for 3D it is Euler angle β.		# For 2D it is sin(θ), but for 3D it is Euler angle β.
	# For ~~2D it is not used, but for~~ 3D it is Euler angle γ.		# For 3D it is Euler angle γ. In 2D has total dissipated energy by damage mechanics (in [[ConsistentUnits Command#Legacy and Consistent Units\|energy units]])
	# ''A<sub>c</sub>''/''V<sub>p</sub>'' where ''A<sub>c</sub>'' is crack area within the particle and ''V<sub>p</sub>'' is particle volume.		# ''A<sub>c</sub>''/''V<sub>p</sub>'' where ''A<sub>c</sub>'' is crack area within the particle and ''V<sub>p</sub>'' is particle volume.
	# Relative strength derived at the start by <tt>coefVariation</tt> and <tt>coefVariationMode</tt> properties.		# Relative strength derived at the start by <tt>coefVariation</tt> and <tt>coefVariationMode</tt> properties.

Nairnj: /* White Noise Statistical Variations */

2023-04-10T23:15:48Z

White Noise Statistical Variations

Nairnj: /* White Noise Statistical Variations */

2023-04-10T23:12:43Z

White Noise Statistical Variations

← Older revision		Revision as of 23:12, 10 April 2023
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	<math>p = \frac{1}{2}\left[1 + {\rm erf}\left( \frac{r-1}{C_v\sqrt{2}}\right)\right]</math>		<math>p = \frac{1}{2}\left[1 + {\rm erf}\left( \frac{r-1}{C_v\sqrt{2}}\right)\right]</math>

	where <math>C_v</math> is coefficient of variation of the distribution (entered with <tt>coefVariation</tt> parameter). When using this distribution, a simulation starts by randomly picking <math>p</math> (over the interval 0.001 to 0.999 that corresponds to &pm;3.09 standard deviations) and then numerical solving for <math>r</math>. This calculation is done for each particle. Note that if entered <tt>coefVariation</tt> is greater than 32.4%, the range of relative strengths will include negative numbers. To avoid negative strengths, the minimum relative strength allowed is 0.05. To avoid this truncation giving poorly-scaled results, problems that need very high coefficient of variation should use the [[#Weibull Distribution\|Weibull distribution]] described in the next section.		where <math>C_v</math> is coefficient of variation of the distribution (entered with <tt>coefVariation</tt> parameter). When using this distribution, a simulation starts by randomly picking <math>p</math> (over the interval 0.001 to 0.999 that corresponds to &pm;3.09 standard deviations) and then numerical solving for <math>r</math>. This calculation is done for each particle. Note that if entered <tt>coefVariation</tt> is greater than 32.4%, the range of relative strengths will include negative numbers. To avoid zero or negative strengths, the minimum relative strength allowed is 0.05. To avoid this truncation giving poorly-scaled results, problems that need very high coefficient of variation should use the [[#Weibull Distribution\|Weibull distribution]] described in the next section.

	=== Weibull Distribution ===		=== Weibull Distribution ===
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	<math>r = \frac{1}{\Gamma\left(1+\frac{1}{\alpha}\right)}\left[-\frac{V_0}{V_p}\ln(1-p)\right]^{1/\alpha} </math>		<math>r = \frac{1}{\Gamma\left(1+\frac{1}{\alpha}\right)}\left[-\frac{V_0}{V_p}\ln(1-p)\right]^{1/\alpha} </math>

	This calculation is done for each particle.		This calculation is done for each particle. To avoid zero strengths, the minimum relative strength allowed is 0.05.

	Unlike a [[#Normal Distribution\|normal distribution]], a Weibull distribution is only defined for positive <math>r</math>, which means it can be used for high-coefficient-of-variation distributions without truncating the curve. Because a [[#Normal Distribution\|normal distribution]] starts truncating when CV is greater than 32.4%, switching to a Weibull distribution might work better. Some high coefficients of variation can be modeled with the following Weibull shape parameters:		Unlike a [[#Normal Distribution\|normal distribution]], a Weibull distribution is only defined for positive <math>r</math>, which means it can be used for high-coefficient-of-variation distributions without truncating the curve. Because a [[#Normal Distribution\|normal distribution]] starts truncating when CV is greater than 32.4%, switching to a Weibull distribution might work better. Some high coefficients of variation can be modeled with the following Weibull shape parameters:

Nairnj: /* Weibull Distribution */

2023-04-10T23:11:18Z

Weibull Distribution

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	\|}		\|}

	Although higher shape parameters are allowed, that option does not add anything to stochastic modeling of failure. If one plots the [[#Normal Distribution\|normal distribution]] and the [[#Weibull Distribution\|Weibull distribution]] cumulative distribution functions (<math>p</math> defined above), the curves are essentially identical for coefficient of variation less than 30%. The only time a Weibull distribution might benefit a simulations is when modeling materials with a higher coefficient of variation and that could be done by using a low shape parameter.		Although higher shape parameters are allowed, that option does not add anything to stochastic modeling of failure. If one plots the [[#Normal Distribution\|normal distribution]] and the [[#Weibull Distribution\|Weibull distribution]] cumulative distribution functions (<math>p</math> defined above), the curves are essentially identical for coefficient of variation less than 30%. The only time a Weibull distribution might benefit simulations is when modeling materials with a very higher coefficient of variation and that could be done by using a Weibull distribution with a low shape parameter.

	== Using Gaussian Random Fields ==		== Using Gaussian Random Fields ==

Nairnj: /* Weibull Distribution */

2023-04-10T23:09:13Z

Weibull Distribution

← Older revision		Revision as of 23:09, 10 April 2023
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	\|-		\|-
	\| 4 \|\| 28.0544		\| 4 \|\| 28.0544
	\|-
	~~\| 5 \|\| 22.9053~~
	\|}		\|}

Nairnj: /* Weibull Distribution */

2023-04-10T23:08:53Z

Weibull Distribution

← Older revision		Revision as of 23:08, 10 April 2023
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	\|}		\|}

	~~Higher~~ shape parameters ~~can also be used~~, ~~but it it likely~~ that ~~differences between a Weibull~~ distribution and a [[#~~Normal~~ Distribution\|~~normal~~ distribution]] ~~with~~ the ~~corresponding~~ coefficient of variation ~~(if~~ less than 20%~~) would not~~ be ~~significant~~.		Although higher shape parameters are allowed, that option does not add anything to stochastic modeling of failure. If one plots the [[#Normal Distribution\|normal distribution]] and the [[#Weibull Distribution\|Weibull distribution]] cumulative distribution functions (<math>p</math> defined above), the curves are essentially identical for coefficient of variation less than 30%. The only time a Weibull distribution might benefit a simulations is when modeling materials with a higher coefficient of variation and that could be done by using a low shape parameter.

	== Using Gaussian Random Fields ==		== Using Gaussian Random Fields ==

Nairnj: /* Normal Distribution */

2023-04-10T21:21:57Z

Normal Distribution

← Older revision		Revision as of 21:21, 10 April 2023
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	<math>p = \frac{1}{2}\left[1 + {\rm erf}\left( \frac{r-1}{C_v\sqrt{2}}\right)\right]</math>		<math>p = \frac{1}{2}\left[1 + {\rm erf}\left( \frac{r-1}{C_v\sqrt{2}}\right)\right]</math>

	where <math>C_v</math> is coefficient of variation of the distribution (entered with <tt>coefVariation</tt> parameter). When using this distribution, a simulation starts by randomly picking <math>p</math> (over the interval 0.001 to 0.999 that ~~correspond~~ to &pm;3.09 standard deviations) and then numerical solving for <math>r</math>. This calculation is done for each particle. Note that if entered <tt>coefVariation</tt> is greater than 32.4%, the range of relative strengths will include negative numbers. To avoid negative strengths, the minimum relative strength allowed is 0.05. To avoid this truncation giving poorly-scaled results, problems that need very high coefficient of variation should use the [[#Weibull Distribution\|Weibull distribution]] described in the next section.		where <math>C_v</math> is coefficient of variation of the distribution (entered with <tt>coefVariation</tt> parameter). When using this distribution, a simulation starts by randomly picking <math>p</math> (over the interval 0.001 to 0.999 that corresponds to &pm;3.09 standard deviations) and then numerical solving for <math>r</math>. This calculation is done for each particle. Note that if entered <tt>coefVariation</tt> is greater than 32.4%, the range of relative strengths will include negative numbers. To avoid negative strengths, the minimum relative strength allowed is 0.05. To avoid this truncation giving poorly-scaled results, problems that need very high coefficient of variation should use the [[#Weibull Distribution\|Weibull distribution]] described in the next section.

	=== Weibull Distribution ===		=== Weibull Distribution ===

← Older revision		Revision as of 23:15, 10 April 2023
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	<math>p = \frac{1}{2}\left[1 + {\rm erf}\left( \frac{r-1}{C_v\sqrt{2}}\right)\right]</math>		<math>p = \frac{1}{2}\left[1 + {\rm erf}\left( \frac{r-1}{C_v\sqrt{2}}\right)\right]</math>

	where <math>C_v</math> is coefficient of variation of the distribution (entered with <tt>coefVariation</tt> parameter). When using this distribution, a simulation starts by randomly picking <math>p</math> (over the interval 0.001 to 0.999 that corresponds to &pm;3.09 standard deviations) and then numerical solving for <math>r</math>. This calculation is done for each particle. Note that if entered <tt>coefVariation</tt> is greater than 32.4%, the range of relative strengths will include negative numbers. To avoid zero or negative ~~strengths~~, the minimum relative ~~strength~~ allowed is 0.05. To avoid this truncation giving poorly-scaled results, problems that need very high coefficient of variation should use the [[#Weibull Distribution\|Weibull distribution]] described in the next section.		where <math>C_v</math> is coefficient of variation of the distribution (entered with <tt>coefVariation</tt> parameter). When using this distribution, a simulation starts by randomly picking <math>p</math> (over the interval 0.001 to 0.999 that corresponds to &pm;3.09 standard deviations) and then numerical solving for <math>r</math>. This calculation is done for each particle. Note that if entered <tt>coefVariation</tt> is greater than 32.4%, the range of relative strengths will include negative numbers. To avoid zero or negative values, the minimum relative value allowed is 0.05. To avoid this truncation giving poorly-scaled results, problems that need very high coefficient of variation should use the [[#Weibull Distribution\|Weibull distribution]] described in the next section.

	=== Weibull Distribution ===		=== Weibull Distribution ===
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	<math>p = 1 - \exp\left(-\frac{V_p}{V_0}\left(r\Gamma\left(1+\frac{1}{\alpha}\right)\right)^\alpha\right)</math>		<math>p = 1 - \exp\left(-\frac{V_p}{V_0}\left(r\Gamma\left(1+\frac{1}{\alpha}\right)\right)^\alpha\right)</math>

	where <math>V_0</math> (entered using <tt>wV0</tt>) and <math>\alpha</math> (entered <tt>wShape</tt>) are Weibull reference volume and shape parameters. For simulations where particles volumes, <math>V_p</math>, may differ, this distribution introduces some size effects into the statistical calculations. When using this distribution, <math>r</math> is interpreted as ~~strength~~ relative to ~~strength~~ at the reference volume <math>V_0</math>.		where <math>V_0</math> (entered using <tt>wV0</tt>) and <math>\alpha</math> (entered <tt>wShape</tt>) are Weibull reference volume and shape parameters. For simulations where particles volumes, <math>V_p</math>, may differ, this distribution introduces some size effects into the statistical calculations. When using this distribution, <math>r</math> is interpreted as value relative to the value at the reference volume <math>V_0</math>.

	The mean and coefficient of variation for the relative value distribution are:		The mean and coefficient of variation for the relative value distribution are:
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	<math>r = \frac{1}{\Gamma\left(1+\frac{1}{\alpha}\right)}\left[-\frac{V_0}{V_p}\ln(1-p)\right]^{1/\alpha} </math>		<math>r = \frac{1}{\Gamma\left(1+\frac{1}{\alpha}\right)}\left[-\frac{V_0}{V_p}\ln(1-p)\right]^{1/\alpha} </math>

	This calculation is done for each particle. To avoid zero ~~strengths~~, the minimum relative ~~strength~~ allowed is 0.05.		This calculation is done for each particle. To avoid zero relative values, the minimum relative value allowed is 0.05.

	Unlike a [[#Normal Distribution\|normal distribution]], a Weibull distribution is only defined for positive <math>r</math>, which means it can be used for high-coefficient-of-variation distributions without truncating the curve. Because a [[#Normal Distribution\|normal distribution]] starts truncating when CV is greater than 32.4%, switching to a Weibull distribution might work better. Some high coefficients of variation can be modeled with the following Weibull shape parameters:		Unlike a [[#Normal Distribution\|normal distribution]], a Weibull distribution is only defined for positive <math>r</math>, which means it can be used for high-coefficient-of-variation distributions without truncating the curve. Because a [[#Normal Distribution\|normal distribution]] starts truncating when CV is greater than 32.4%, switching to a Weibull distribution might work better. Some high coefficients of variation can be modeled with the following Weibull shape parameters:
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	\|}		\|}

	Although higher shape parameters are allowed, that option does not add anything to stochastic modeling of failure. If one plots the [[#Normal Distribution\|normal distribution]] and the [[#Weibull Distribution\|Weibull distribution]] cumulative distribution functions (<math>p</math> defined above), the curves are essentially identical for coefficient of variation less than 30%. The only time a Weibull distribution might benefit simulations is when modeling materials with a very ~~higher~~ coefficient of variation and that could be done by using a Weibull distribution with a low shape parameter.		Although higher shape parameters are allowed, that option does not add anything to stochastic modeling of failure. If one plots the [[#Normal Distribution\|normal distribution]] and the [[#Weibull Distribution\|Weibull distribution]] cumulative distribution functions (<math>p</math> defined above), the curves are essentially identical for coefficient of variation less than 30%. The only time a Weibull distribution might benefit simulations is when modeling materials with a very high coefficient of variation and that could be done by using a Weibull distribution with a low shape parameter.

	== Using Gaussian Random Fields ==		== Using Gaussian Random Fields ==