Mooney Material - Revision history

Nairnj: /* Residual Stress */

2023-01-30T01:05:48Z

Residual Stress

← Older revision		Revision as of 01:05, 30 January 2023
Line 71:		Line 71:
	<math>\lambda_{res} = \exp\left(\alpha\Delta T\right)</math>		<math>\lambda_{res} = \exp\left(\alpha\Delta T\right)</math>

	As a consequence, a temperature change corresponding to <math>\alpha\Delta T=100\%</math> would result in <math>\lambda_{res}=\exp(1)\gt2</math>. In other words, large-deformation material expansion is different than small-strain material expansion.		As a consequence, a temperature change corresponding to <math>\alpha\Delta T=100\%</math> would result in <math>\lambda_{res}=\exp(1)\gt2</math>. In other words, large-deformation material expansion is different than small-strain material expansion. A large deformation material will expand faster, shrink slower, and prohibit shrinkage beyond 100%.

	=== Ideal Rubber ===		=== Ideal Rubber ===

Nairnj: /* Residual Stress */

2023-01-30T01:04:12Z

Residual Stress

← Older revision		Revision as of 01:04, 30 January 2023
Line 66:		Line 66:
	<math> \mathbf{\tau} = -Jp(J_{eff})\mathbf{I} + {G_{1} J^{1/3}\over J_{eff} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J_{eff}J^{1/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I})</math>		<math> \mathbf{\tau} = -Jp(J_{eff})\mathbf{I} + {G_{1} J^{1/3}\over J_{eff} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J_{eff}J^{1/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I})</math>

	Note the the current material model ~~assume~~ thermal and moisture expansion coefficients are constant. Integrating volume change ~~give~~		Note the the current material model assumes thermal and moisture expansion coefficients are constant. Integrating volume change due to temperature change (for example), gives


			<math>\lambda_{res} = \exp\left(\alpha\Delta T\right)</math>

			As a consequence, a temperature change corresponding to <math>\alpha\Delta T=100\%</math> would result in <math>\lambda_{res}=\exp(1)\gt2</math>. In other words, large-deformation material expansion is different than small-strain material expansion.

	=== Ideal Rubber ===		=== Ideal Rubber ===

Nairnj: /* Residual Stress */

2023-01-30T00:59:25Z

Residual Stress

← Older revision		Revision as of 00:59, 30 January 2023
Line 65:		Line 65:

	<math> \mathbf{\tau} = -Jp(J_{eff})\mathbf{I} + {G_{1} J^{1/3}\over J_{eff} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J_{eff}J^{1/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I})</math>		<math> \mathbf{\tau} = -Jp(J_{eff})\mathbf{I} + {G_{1} J^{1/3}\over J_{eff} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J_{eff}J^{1/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I})</math>

			Note the the current material model assume thermal and moisture expansion coefficients are constant. Integrating volume change give

	=== Ideal Rubber ===		=== Ideal Rubber ===

Nairnj: /* Constitutive Law */

2021-10-07T02:15:42Z

Constitutive Law

← Older revision		Revision as of 02:15, 7 October 2021
Line 6:		Line 6:


	<math>\~~mathbf{~~\sigma} =2 {\delta W \over {\delta \mathbf{B}}} \mathbf{B}</math>		<math>{\bf\sigma} =2 {\delta W \over {\delta \mathbf{B}}} \mathbf{B}</math>

	In a Mooney-Rivlin material, stored stain energy is give by the expression:		In a Mooney-Rivlin material, stored stain energy is give by the expression:
Line 31:		Line 31:
	where <math>\kappa</math> is bulk modulus. At low strains, this material is equivalent to a linear elastic, isotropic material with shear modulus ''G'' = ''G''<sub>1</sub> + ''G''<sub>2</sub> and bulk modulus <math>\kappa</math>. If ''G''<sub>2</sub> = 0, the material is a neo-Hookean material (note that an alternate [[Neo-Hookean Material\|neo-Hookean]] material is also available). Some hyperelastic rubber models assume incompressible materials, which corresponds to <math>\kappa\to\infty</math>; such models do not work in dynamic code (because the dilational wave speed is infinite).		where <math>\kappa</math> is bulk modulus. At low strains, this material is equivalent to a linear elastic, isotropic material with shear modulus ''G'' = ''G''<sub>1</sub> + ''G''<sub>2</sub> and bulk modulus <math>\kappa</math>. If ''G''<sub>2</sub> = 0, the material is a neo-Hookean material (note that an alternate [[Neo-Hookean Material\|neo-Hookean]] material is also available). Some hyperelastic rubber models assume incompressible materials, which corresponds to <math>\kappa\to\infty</math>; such models do not work in dynamic code (because the dilational wave speed is infinite).

	The Cauchy stress tensor is determined by differentiating the strain energy function. It is calculated by the addition of pressure and deviatoric stress, <math> \~~mathbf{~~\sigma} = -p \mathbf{I} + \bar \~~mathbf{~~\sigma} </math> resulting in:		The Cauchy stress tensor is determined by differentiating the strain energy function. It is calculated by the addition of pressure and deviatoric stress, <math> {\bf\sigma} = -p \mathbf{I} + \bar {\bf\sigma} </math> resulting in:


Line 37:		Line 37:


	<math> \bar \~~mathbf{~~\sigma} = {G_{1} \over J^{5/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J^{7/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I})</math>		<math> \bar {\bf\sigma} = {G_{1} \over J^{5/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J^{7/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I})</math>

	where <math>I_{1} = J^{2/3} \bar I_{1}</math> and <math> I_{2} = J^{4/3} \bar I_{2}</math>. The above stress assumes the default UJOption=0. For the other options, the pressure term changes to		where <math>I_{1} = J^{2/3} \bar I_{1}</math> and <math> I_{2} = J^{4/3} \bar I_{2}</math>. The above stress assumes the default UJOption=0. For the other options, the pressure term changes to

Nairnj: /* Examples */

2021-04-23T17:59:50Z

Examples

← Older revision		Revision as of 17:59, 23 April 2021
Line 123:		Line 123:
	K 3000		K 3000
	G1 10		G1 10
			G2 10
	alpha 60		alpha 60
	rho 1.2		rho 1.2

Nairnj: /* Examples */

2021-04-23T17:58:33Z

Examples

← Older revision		Revision as of 17:58, 23 April 2021
Line 123:		Line 123:
	K 3000		K 3000
	G1 10		G1 10
	~~nu .48~~
	alpha 60		alpha 60
	rho 1.2		rho 1.2

Nairnj: /* Constitutive Law */

2017-11-16T22:42:34Z

Constitutive Law

← Older revision		Revision as of 22:42, 16 November 2017
Line 1:		Line 1:
	== Constitutive Law ==		== Constitutive Law ==

	This [[Material Models\|MPM Material]] is an isotropic, elastic material in large strains using a hyperelastic formulation. This model can be described as a generalized neo-Hookean model because it has three material properties instead of two. See [[Comparison of Neo-Hookean Materials]] for ~~some more~~ details.		This [[Material Models\|MPM Material]] is an isotropic, elastic material in large strains using a hyperelastic formulation. This model can be described as a generalized neo-Hookean model because it has three material properties instead of two. See [[Comparison of Neo-Hookean Materials]] for details on available neo-Hookean materials.

	In hyperelasticity, the existence of a stored-energy ''W'' as a function of a deformation gradient tensor (<i>'''F'''</i>), is postulated<ref name="TN">C. Truesdell and W. Noll, The nonlienar field theories of mechanics, Edition Handbuch der Physik, Vol. III. Spinger, Berlin (1965).</ref><ref>R. W. Ogden, Non-linear elastic deformations. Wiley & Sons, New York (1984).</ref><ref>D. Weichert and Y. Basar, Nonlinear continuum mechanics of solids, Springer, New York (2000).</ref> and the constitutive law is derived from that strain energy function according to the inequality of Clausius-Duheim. Regarding to the objectivity conditions and using the representation theorem,<ref name="TN"/> the strain energy function is a function of invariants of a strain tensor. Using the left Cauchy-Green strain tensor ('''B''' = '''FF'''<sup>T</sup>), the Cauchy stress is given by:		In hyperelasticity, the existence of a stored-energy ''W'' as a function of a deformation gradient tensor (<i>'''F'''</i>), is postulated<ref name="TN">C. Truesdell and W. Noll, The nonlienar field theories of mechanics, Edition Handbuch der Physik, Vol. III. Spinger, Berlin (1965).</ref><ref>R. W. Ogden, Non-linear elastic deformations. Wiley & Sons, New York (1984).</ref><ref>D. Weichert and Y. Basar, Nonlinear continuum mechanics of solids, Springer, New York (2000).</ref> and the constitutive law is derived from that strain energy function according to the inequality of Clausius-Duheim. Regarding to the objectivity conditions and using the representation theorem,<ref name="TN"/> the strain energy function is a function of invariants of a strain tensor. Using the left Cauchy-Green strain tensor ('''B''' = '''FF'''<sup>T</sup>), the Cauchy stress is given by:

Nairnj: /* Constitutive Law */

2017-11-16T22:26:35Z

Constitutive Law

← Older revision		Revision as of 22:26, 16 November 2017
Line 1:		Line 1:
	== Constitutive Law ==		== Constitutive Law ==

	This [[Material Models\|MPM Material]] is an isotropic, elastic material in large strains using a hyperelastic formulation. ~~Two other neo-Hookean materials are~~ a ~~[[Neo-Hookean Material\|~~neo-Hookean material~~]] and a~~ [[~~Clamped~~ Neo-Hookean ~~Material\|clamped neo-Hookean material~~]] ~~with <tt>CritTens</tt> less than zero. This material gives a third neo-Hookean formation with slightly different stresses~~.		This [[Material Models\|MPM Material]] is an isotropic, elastic material in large strains using a hyperelastic formulation. This model can be described as a generalized neo-Hookean model because it has three material properties instead of two. See [[Comparison of Neo-Hookean Materials]] for some more details.

	In hyperelasticity, the existence of a stored-energy ''W'' as a function of a deformation gradient tensor (<i>'''F'''</i>), is postulated<ref name="TN">C. Truesdell and W. Noll, The nonlienar field theories of mechanics, Edition Handbuch der Physik, Vol. III. Spinger, Berlin (1965).</ref><ref>R. W. Ogden, Non-linear elastic deformations. Wiley & Sons, New York (1984).</ref><ref>D. Weichert and Y. Basar, Nonlinear continuum mechanics of solids, Springer, New York (2000).</ref> and the constitutive law is derived from that strain energy function according to the inequality of Clausius-Duheim. Regarding to the objectivity conditions and using the representation theorem,<ref name="TN"/> the strain energy function is a function of invariants of a strain tensor. Using the left Cauchy-Green strain tensor ('''B''' = '''FF'''<sup>T</sup>), the Cauchy stress is given by:		In hyperelasticity, the existence of a stored-energy ''W'' as a function of a deformation gradient tensor (<i>'''F'''</i>), is postulated<ref name="TN">C. Truesdell and W. Noll, The nonlienar field theories of mechanics, Edition Handbuch der Physik, Vol. III. Spinger, Berlin (1965).</ref><ref>R. W. Ogden, Non-linear elastic deformations. Wiley & Sons, New York (1984).</ref><ref>D. Weichert and Y. Basar, Nonlinear continuum mechanics of solids, Springer, New York (2000).</ref> and the constitutive law is derived from that strain energy function according to the inequality of Clausius-Duheim. Regarding to the objectivity conditions and using the representation theorem,<ref name="TN"/> the strain energy function is a function of invariants of a strain tensor. Using the left Cauchy-Green strain tensor ('''B''' = '''FF'''<sup>T</sup>), the Cauchy stress is given by:

Nairnj: /* Constitutive Law */

2017-10-06T03:42:16Z

Constitutive Law

← Older revision		Revision as of 03:42, 6 October 2017
Line 1:		Line 1:
	== Constitutive Law ==		== Constitutive Law ==

	This [[Material Models\|MPM Material]] is an isotropic, elastic material in large strains using a hyperelastic formulation. In hyperelasticity, the existence of a stored-energy ''W'' as a function of a deformation gradient tensor (<i>'''F'''</i>), is postulated<ref name="TN">C. Truesdell and W. Noll, The nonlienar field theories of mechanics, Edition Handbuch der Physik, Vol. III. Spinger, Berlin (1965).</ref><ref>R. W. Ogden, Non-linear elastic deformations. Wiley & Sons, New York (1984).</ref><ref>D. Weichert and Y. Basar, Nonlinear continuum mechanics of solids, Springer, New York (2000).</ref> and the constitutive law is derived from that strain energy function according to the inequality of Clausius-Duheim. Regarding to the objectivity conditions and using the representation theorem,<ref name="TN"/> the strain energy function is a function of invariants of a strain tensor. Using the left Cauchy-Green strain tensor ('''B''' = '''FF'''<sup>T</sup>), the Cauchy stress is given by:		This [[Material Models\|MPM Material]] is an isotropic, elastic material in large strains using a hyperelastic formulation. Two other neo-Hookean materials are a [[Neo-Hookean Material\|neo-Hookean material]] and a [[Clamped Neo-Hookean Material\|clamped neo-Hookean material]] with <tt>CritTens</tt> less than zero. This material gives a third neo-Hookean formation with slightly different stresses.

			In hyperelasticity, the existence of a stored-energy ''W'' as a function of a deformation gradient tensor (<i>'''F'''</i>), is postulated<ref name="TN">C. Truesdell and W. Noll, The nonlienar field theories of mechanics, Edition Handbuch der Physik, Vol. III. Spinger, Berlin (1965).</ref><ref>R. W. Ogden, Non-linear elastic deformations. Wiley & Sons, New York (1984).</ref><ref>D. Weichert and Y. Basar, Nonlinear continuum mechanics of solids, Springer, New York (2000).</ref> and the constitutive law is derived from that strain energy function according to the inequality of Clausius-Duheim. Regarding to the objectivity conditions and using the representation theorem,<ref name="TN"/> the strain energy function is a function of invariants of a strain tensor. Using the left Cauchy-Green strain tensor ('''B''' = '''FF'''<sup>T</sup>), the Cauchy stress is given by:

Nairnj: /* History Variables */

2016-01-24T16:42:26Z

History Variables

← Older revision		Revision as of 16:42, 24 January 2016
Line 107:		Line 107:
	== History Variables ==		== History Variables ==

	This material uses history #~~1 to store the volumetric strain (''i.e.'',~~ the determinant of the deformation ~~gradient~~ or ~~''J'')~~. The total strain, which is elastic, is stored in the elastic strain variable, while the plastic strain stores the left Cauchy Green strain.		This material uses two history variables:

			# J or the determinant of the deformation and equal to V/V<sub>0</sub>.
			# J<sub>res</sub> or ratio of free thermal expansion volume to the initial volume.

			The total strain, which is elastic, is stored in the elastic strain variable, while the plastic strain stores the left Cauchy Green strain.

	== Examples ==		== Examples ==