Neo-Hookean Material - Revision history

Nairnj: /* Constitutive Law */

2021-10-07T02:24:46Z

Constitutive Law

← Older revision		Revision as of 02:24, 7 October 2021
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	Note that UJOption=0 is the default option because it is the only one where pressure correctly becomes infinite for both ''J'' approaching 0 and ''J'' approaching infinity. When using this material in 2D plane stress calculations, the code has to solve for the z direction deformation required to get zero stress in that direction. For UJOption = 0 or 1, the z direction deformation can be found analytically; for UJOption = 2, however, the z direction deformation requires a numerical solution and therefore is less effficient. Note that the neo-Hookean option within the [[Mooney Material\|Mooney-Rivilin material]] needs numerical solutions in plane stress modeling for all UJOptions. Thus, the most efficient neo-Hookean material for plane stress modeling is this material with UJOption = 0 or 1.		Note that UJOption=0 is the default option because it is the only one where pressure correctly becomes infinite for both ''J'' approaching 0 and ''J'' approaching infinity. When using this material in 2D plane stress calculations, the code has to solve for the z direction deformation required to get zero stress in that direction. For UJOption = 0 or 1, the z direction deformation can be found analytically; for UJOption = 2, however, the z direction deformation requires a numerical solution and therefore is less effficient. Note that the neo-Hookean option within the [[Mooney Material\|Mooney-Rivilin material]] needs numerical solutions in plane stress modeling for all UJOptions. Thus, the most efficient neo-Hookean material for plane stress modeling is this material with UJOption = 0 or 1.

	The stresses can alternatively be divided into pressure and deviatoric stress, <math> {~~\bf~~\sigma} = -p \mathbf{I} + \bar{\bf\sigma} </math> resulting in:		The stresses can alternatively be divided into pressure and deviatoric stress, <math> \symbf{\sigma} = -p \mathbf{I} + \bar{\symbf{\sigma}} </math> resulting in:


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	<math> \bar{\bf\sigma} = {G \over J }{\rm dev}(\mathbf{B})</math>		<math> \bar{\symbf{\sigma}} = {G \over J }{\rm dev}(\mathbf{B})</math>

	The first term in the pressure changes for the other two UJOption settings. Note that the pressure depends on both terms in the strain energy function. This dependence contrasts with the [[Mooney Material\|Mooney-Rivilin material]] where the pressure depends only on the ''U(J)'' term.		The first term in the pressure changes for the other two UJOption settings. Note that the pressure depends on both terms in the strain energy function. This dependence contrasts with the [[Mooney Material\|Mooney-Rivilin material]] where the pressure depends only on the ''U(J)'' term.

Nairnj: /* Residual Stress */

2021-10-07T02:24:12Z

Residual Stress

← Older revision		Revision as of 02:24, 7 October 2021
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	<math> \bar{\symbf{\sigma}}~~\sigma~~ = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>		<math> \bar{\symbf{\sigma}} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>

	== Material Properties ==		== Material Properties ==

Nairnj: /* Residual Stress */

2021-10-07T02:23:41Z

Residual Stress

← Older revision		Revision as of 02:23, 7 October 2021
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	<math> \bar{\symbf{\sigma}} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>		<math> \bar{\symbf{\sigma}}\sigma = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>

	== Material Properties ==		== Material Properties ==

Nairnj: /* Residual Stress */

2021-10-07T02:23:22Z

Residual Stress

← Older revision		Revision as of 02:23, 7 October 2021
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	<math> \bar{\bf\sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>		<math> \bar{\symbf{\sigma}} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>

	== Material Properties ==		== Material Properties ==

Nairnj: /* Constitutive Law */

2021-10-07T02:13:59Z

Constitutive Law

← Older revision		Revision as of 02:13, 7 October 2021
Line 38:		Line 38:
	Note that UJOption=0 is the default option because it is the only one where pressure correctly becomes infinite for both ''J'' approaching 0 and ''J'' approaching infinity. When using this material in 2D plane stress calculations, the code has to solve for the z direction deformation required to get zero stress in that direction. For UJOption = 0 or 1, the z direction deformation can be found analytically; for UJOption = 2, however, the z direction deformation requires a numerical solution and therefore is less effficient. Note that the neo-Hookean option within the [[Mooney Material\|Mooney-Rivilin material]] needs numerical solutions in plane stress modeling for all UJOptions. Thus, the most efficient neo-Hookean material for plane stress modeling is this material with UJOption = 0 or 1.		Note that UJOption=0 is the default option because it is the only one where pressure correctly becomes infinite for both ''J'' approaching 0 and ''J'' approaching infinity. When using this material in 2D plane stress calculations, the code has to solve for the z direction deformation required to get zero stress in that direction. For UJOption = 0 or 1, the z direction deformation can be found analytically; for UJOption = 2, however, the z direction deformation requires a numerical solution and therefore is less effficient. Note that the neo-Hookean option within the [[Mooney Material\|Mooney-Rivilin material]] needs numerical solutions in plane stress modeling for all UJOptions. Thus, the most efficient neo-Hookean material for plane stress modeling is this material with UJOption = 0 or 1.

	The stresses can alternatively be divided into pressure and deviatoric stress, <math> {\bf\sigma} = -p \mathbf{I} + \bar \~~mathbf{~~\sigma} </math> resulting in:		The stresses can alternatively be divided into pressure and deviatoric stress, <math> {\bf\sigma} = -p \mathbf{I} + \bar{\bf\sigma} </math> resulting in:

Nairnj: /* Constitutive Law */

2021-10-07T02:13:23Z

Constitutive Law

← Older revision		Revision as of 02:13, 7 October 2021
Line 38:		Line 38:
	Note that UJOption=0 is the default option because it is the only one where pressure correctly becomes infinite for both ''J'' approaching 0 and ''J'' approaching infinity. When using this material in 2D plane stress calculations, the code has to solve for the z direction deformation required to get zero stress in that direction. For UJOption = 0 or 1, the z direction deformation can be found analytically; for UJOption = 2, however, the z direction deformation requires a numerical solution and therefore is less effficient. Note that the neo-Hookean option within the [[Mooney Material\|Mooney-Rivilin material]] needs numerical solutions in plane stress modeling for all UJOptions. Thus, the most efficient neo-Hookean material for plane stress modeling is this material with UJOption = 0 or 1.		Note that UJOption=0 is the default option because it is the only one where pressure correctly becomes infinite for both ''J'' approaching 0 and ''J'' approaching infinity. When using this material in 2D plane stress calculations, the code has to solve for the z direction deformation required to get zero stress in that direction. For UJOption = 0 or 1, the z direction deformation can be found analytically; for UJOption = 2, however, the z direction deformation requires a numerical solution and therefore is less effficient. Note that the neo-Hookean option within the [[Mooney Material\|Mooney-Rivilin material]] needs numerical solutions in plane stress modeling for all UJOptions. Thus, the most efficient neo-Hookean material for plane stress modeling is this material with UJOption = 0 or 1.

	The stresses can alternatively be divided into pressure and deviatoric stress, <math> \~~mathbf{~~\sigma} = -p \mathbf{I} + \bar \mathbf{\sigma} </math> resulting in:		The stresses can alternatively be divided into pressure and deviatoric stress, <math> {\bf\sigma} = -p \mathbf{I} + \bar \mathbf{\sigma} </math> resulting in:


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	<math> \bar\~~mathbf{~~\sigma} = {G \over J }{\rm dev}(\mathbf{B})</math>		<math> \bar{\bf\sigma} = {G \over J }{\rm dev}(\mathbf{B})</math>

	The first term in the pressure changes for the other two UJOption settings. Note that the pressure depends on both terms in the strain energy function. This dependence contrasts with the [[Mooney Material\|Mooney-Rivilin material]] where the pressure depends only on the ''U(J)'' term.		The first term in the pressure changes for the other two UJOption settings. Note that the pressure depends on both terms in the strain energy function. This dependence contrasts with the [[Mooney Material\|Mooney-Rivilin material]] where the pressure depends only on the ''U(J)'' term.

Nairnj: /* Residual Stress */

2021-10-07T02:12:41Z

Residual Stress

← Older revision		Revision as of 02:12, 7 October 2021
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	~~<math> \bar\mathbf{\sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>~~		<math> \bar{\bf\sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>

	~~     ~~
	<math> \bar{\bf ~~\sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>~~
	~~     ~~
	~~<math> \bar{~~\sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>

	== Material Properties ==		== Material Properties ==

Nairnj: /* Residual Stress */

2021-10-07T02:12:02Z

Residual Stress

← Older revision		Revision as of 02:12, 7 October 2021
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	<math> \bar{\bf \sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>		<math> \bar{\bf \sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>

			<math> \bar{\sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>

	== Material Properties ==		== Material Properties ==

Nairnj: /* Residual Stress */

2021-10-07T02:11:42Z

Residual Stress

← Older revision		Revision as of 02:11, 7 October 2021
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	<math> \bar{bf \sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>		<math> \bar{\bf \sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>

	== Material Properties ==		== Material Properties ==

Nairnj: /* Residual Stress */

2021-10-07T02:11:31Z

Residual Stress

← Older revision		Revision as of 02:11, 7 October 2021
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	<math> \bar\sigma = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>		<math> \bar{bf \sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B})</math>

	== Material Properties ==		== Material Properties ==