Difference between revisions of "Mooney Material"

From OSUPDOCS
Jump to navigation Jump to search
 
(51 intermediate revisions by the same user not shown)
Line 1: Line 1:
== Constitutive Law ==
== Constitutive Law ==


The Mooney-Rivlin material [[Material Models|Mooney-Rivlin]] is an isotropic, elastic material in large strains using the hyperelastic formulation.  
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.


Within the framework of hyperelasticity, the existence of a stored-energy ''W'' (per unit deformed or indeformed volume), function of a deformation gradient tensor (<i>'''F'''</i>, is postulated (Weichert et al., 2000; Truesdell and Noll 1965 and Ogden, 1984) and the constitutive law derived from a strain energy function that is a function of the deformation gradient tensor  '''F''', according to the inequality of Clausius-Duheim. Regarding to the objectivity conditions and using the representation theorem (Truesdell and Noll, 1965), the strain energy function is a function of the invariants of a strain tensor. With the left Cauchy-Green strain tensor, 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 &amp; 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:  


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<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 29: Line 29:
<math>U(J) ={\kappa\over 2 }(\ln J)^2 \qquad\qquad\qquad({\rm UJOption}=2)</math>
<math>U(J) ={\kappa\over 2 }(\ln J)^2 \qquad\qquad\qquad({\rm UJOption}=2)</math>


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. See below for an alternate compressibility terms. 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 (or true stress) stress tensor is determined by differentiating the strain energy function. It is calculated by the addition of press 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:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math> \mathbf{\sigma} ={\kappa\over 2}\left(J-{1\over J}\right)\mathbf{I} + {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>p = -{\kappa\over 2}\left(J-{1\over J}\right)</math>
 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<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
Line 45: Line 48:


Note that UJOption=0 is the default option because it is the only pressure that correctly becomes infinite for both ''J'' approaching 0 and ''J'' approaching infinity.
Note that UJOption=0 is the default option because it is the only pressure that correctly becomes infinite for both ''J'' approaching 0 and ''J'' approaching infinity.
=== Residual Stress ===
In the presence of temperature or concentration changes, this material accounts for residual stresses by replacing <i>'''F'''</i> with <i>'''F'''</i>/&lambda;<sub>res</sub>, where &lambda;<sub>res</sub> is the free residual expansion in one direction (and all three directions are the same for this isotropic material). The net effect is <i>'''B'''</i> and ''J'' are replaced with
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\mathbf{B}_{eff} = {\mathbf{B}\over \lambda_{res}^2} \qquad {\rm and} \qquad J_{eff} = {J\over J_{res}}</math>
where ''J<sub>res<sub>'' = &lambda;<sub>res</sub><sup>3</sup> is the ratio of the free expansion volume to the reference volume. Thus ''J<sub>eff</sub>'' is the ratio of the current volume to the free expansion volume while ''J'' is the ratio of current volume to the initial volume. The Cauchy stress becomes:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math> \mathbf{\sigma} = -p(J_{eff})\mathbf{I} + {G_{1} \over J_{eff}J^{2/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J_{eff}J^{4/3}} (I_{1}    \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I})</math>
where  ''p''(''J<sub>eff</sub>'') is one of the above pressure options evaluated using ''J<sub>eff</sub>'' instead of ''J''. Finally, the Kirchhoff stress used in the MPM time step multiplies by actual ''J'' to get:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<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 assumes thermal and moisture expansion coefficients are constant. Integrating volume change due to temperature change (for example), gives
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<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. A large deformation material will expand faster, shrink slower, and prohibit shrinkage beyond 100%.


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


This material provides an acceptable material model for rubbers or elastomers, but it will not get the thermodynamics correct unless you indicate the material is an ideal rubber (using the IdealRubber property). An ideal rubber means that during isothermal loading, there is no change in internal energy. In other words, all the work is turned into heat. In adiabatic loading, this heat causes the temperature to rise. For the thermal calculations to be correct, the material must specify a valid heat capacity.
This material provides an acceptable material model for rubbers or elastomers, but it will not get the thermodynamics correct unless you indicate the material is an ideal rubber (using the IdealRubber property). An ideal rubber means that during isothermal loading, there is no change in internal energy. In other words, all the work is turned into heat. In adiabatic loading, this heat causes the temperature to rise. For the thermal calculations to be correct, the material must specify a valid [[Common Material Properties#Basic Properties|heat capacity]].
 
The entropy for this material is the net effect of heat flow by external heating, by conduction, by internal mechanisms, or by work when it is set to be an ideal rubber. For example, when an ideal rubber is loaded isothermally in tension in the ''x'' direction, the force is related to entropy by
 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>F = A\sigma_{xx} = -T \left({\partial S\over\partial L}\right)_T</math>
 
where ''A'' is cross-sectional area and ''L'' in length. An ideal rubber in [[NairnMPM]] correctly follows this law.


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


The material properties are enters using:
The material properties are entered using:


{| class="wikitable"
{| class="wikitable"
Line 58: Line 92:
! Property !! Description !! Units !! Default
! Property !! Description !! Units !! Default
|-
|-
| K || Bulk modulus || MPa || none
| K || Bulk modulus || [[ConsistentUnits Command#Legacy and Consistent Units|pressure units]] || none
|-
|-
| G1 || The ''G''<sub>1</sub> shear modulus || MPa || none
| G1 || The ''G''<sub>1</sub> shear modulus || [[ConsistentUnits Command#Legacy and Consistent Units|pressure units]] || none
|-
|-
| G2 || The ''G''<sub>2</sub> shear modulus || MPa || 0
| G2 || The ''G''<sub>2</sub> shear modulus || [[ConsistentUnits Command#Legacy and Consistent Units|pressure units]] || 0
|-
|-
| UJOption || Set to 0, 1, or 2, to select the energy term from [[#Constitutive Law|above]]. || none || 0
| E || The ''E'' tensile modulus || [[ConsistentUnits Command#Legacy and Consistent Units|pressure units]] || 0
|-
|-
| alpha || Thermal expansion coefficient || ppm/M || 0
| nu || The Poisson's ratio || none || 0
|-
|-
| kCond || Thermal conductivity || W/(m-K) || 0
| UJOption || Set to 0, 1, or 2, to select the energy term from [[#Constitutive Law|above]]. || none || 0
|-
|-
| beta || Solvent expansion coefficient || 1/(wt fraction) || 0
| alpha || Thermal expansion coefficient || ppm/K || 0
|-
|-
| D || Solvent diffusion constant || mm<sup>2</sup>/sec || 0
| IdealRubber || Set to 0 to be an elastic material and to 1 to be an [[#Ideal Rubber|ideal rubber]]. || none || 0
|-
|-
| ([[Common Material Properties|other]]) || Properties common to all materials || varies || varies
| ([[Common Material Properties|other]]) || Properties common to all materials || varies || varies
|}
|}
You must enter K and G1 OR E and nu. When you enter E and nu, K and ''G'' = ''G''<sub>1</sub>+''G''<sub>2</sub> are calculated from the relations between [[Isotropic Material#Material Properties|these properties]]. The partition of ''G'' into ''G''<sub>1</sub> and ''G''<sub>2</sub> is determined by your entered value for G2 (''i.e.'', ''G''<sub>1</sub> = ''G'' - ''G''<sub>2</sub>).
See [[Isotropic Material#Material Properties|these relations]] to covert other properties (such as Lame modulus) to bulk and shear moduli.


== History Variables ==
== History Variables ==


This material uses history #1 to store the volumetric strain (''i.e.'', the determinant of the deformation gradient). The total strain, which is elastic, is stored in the elastic strain variable, while the plastic strain stores the left Cauchy Green tensor.
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 ==
Line 85: Line 127:
These commands model polymer as an isotropic, hyperelastic material with G1=G2 =G/2 (using scripted or XML commands):
These commands model polymer as an isotropic, hyperelastic material with G1=G2 =G/2 (using scripted or XML commands):


  Material "polymer","polymer","Mooney"
  Material "polymer","Ideal Elastomer","Mooney"
    E 2500
  K 3000
    nu .4
  G1 10
    alpha 60
  G2 10
    rho 1.2
  alpha 60
  Done
  rho 1.2
 
  IdealRubber 1
  &nbsp;
  Cv 1200
<Material Type="8" Name="polymer">
  Done
    <rho>1.2</rho>
    <G1>35.714285714</G1>
    <G2>35.714285714</G2>
    <K>166.66666666</K>
    <alpha>60</alpha>
  </Material>


== References ==
== References ==
• Ogden R. W., (1984), Non-linear elastic deformations. Wiley et Sons, New York.
• Truesdell C. and W. Noll W (1965), The nonlienar field theories of mechanics, Edition Handbuch der Physik, Vol. III. Spinger, Berlin.
• Weichert D. et Y. Basar, (2000), Nonlinear continuum mechanics of solids, Springer, New York.


<references/>
<references/>

Latest revision as of 17:05, 29 January 2023

Constitutive Law

This 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 (F), is postulated[1][2][3] 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,[1] the strain energy function is a function of invariants of a strain tensor. Using the left Cauchy-Green strain tensor (B = FFT), the Cauchy stress is given by:

      [math]\displaystyle{ {\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:

      [math]\displaystyle{ W =U(J) + {G_{1} \over 2 } (\bar I_{1}-3) + {G_{2} \over 2 }(\bar I_{2}-3) }[/math]

where J (= det F) is relative volume change, G1 and G2 are shear material properties, and [math]\displaystyle{ \bar I_{1} }[/math] and [math]\displaystyle{ \bar I_{2} }[/math] are the strain invarients:

      [math]\displaystyle{ \bar I_{1} ={B_{xx}+B_{yy}+B_{zz} \over J^{2/3}} \qquad {\rm and} \qquad \bar I_{2} ={1 \over 2} (\bar I_{1}^2-{{B_{xx}^2+B_{yy}^2+B_{zz}^2+2B_{xy}^2+2B_{xz}^2+2B_{yz}^2} \over J^{2/3}}) }[/math]

This material allows three options for the volumetric energy term:

      [math]\displaystyle{ U(J) ={\kappa\over 2 }({1\over 2 }(J^2-1)-\ln J) \qquad({\rm UJOption}=0) }[/math]

      [math]\displaystyle{ U(J) ={\kappa\over 2 }(J-1)^2 \qquad\qquad\qquad({\rm UJOption}=1) }[/math]

      [math]\displaystyle{ U(J) ={\kappa\over 2 }(\ln J)^2 \qquad\qquad\qquad({\rm UJOption}=2) }[/math]

where [math]\displaystyle{ \kappa }[/math] is bulk modulus. At low strains, this material is equivalent to a linear elastic, isotropic material with shear modulus G = G1 + G2 and bulk modulus [math]\displaystyle{ \kappa }[/math]. If G2 = 0, the material is a neo-Hookean material (note that an alternate neo-Hookean material is also available). Some hyperelastic rubber models assume incompressible materials, which corresponds to [math]\displaystyle{ \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]\displaystyle{ {\bf\sigma} = -p \mathbf{I} + \bar {\bf\sigma} }[/math] resulting in:

      [math]\displaystyle{ p = -{\kappa\over 2}\left(J-{1\over J}\right) }[/math]

      [math]\displaystyle{ \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]\displaystyle{ I_{1} = J^{2/3} \bar I_{1} }[/math] and [math]\displaystyle{ 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

      [math]\displaystyle{ p = -\kappa(J-1) \qquad\qquad({\rm UJOption}=1) }[/math]

      [math]\displaystyle{ p = -\kappa {\ln J\over J} \qquad\qquad({\rm UJOption}=2) }[/math]

Note that UJOption=0 is the default option because it is the only pressure that correctly becomes infinite for both J approaching 0 and J approaching infinity.

Residual Stress

In the presence of temperature or concentration changes, this material accounts for residual stresses by replacing F with Fres, where λres is the free residual expansion in one direction (and all three directions are the same for this isotropic material). The net effect is B and J are replaced with

      [math]\displaystyle{ \mathbf{B}_{eff} = {\mathbf{B}\over \lambda_{res}^2} \qquad {\rm and} \qquad J_{eff} = {J\over J_{res}} }[/math]

where Jres = λres3 is the ratio of the free expansion volume to the reference volume. Thus Jeff is the ratio of the current volume to the free expansion volume while J is the ratio of current volume to the initial volume. The Cauchy stress becomes:

      [math]\displaystyle{ \mathbf{\sigma} = -p(J_{eff})\mathbf{I} + {G_{1} \over J_{eff}J^{2/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J_{eff}J^{4/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I}) }[/math]

where p(Jeff) is one of the above pressure options evaluated using Jeff instead of J. Finally, the Kirchhoff stress used in the MPM time step multiplies by actual J to get:

      [math]\displaystyle{ \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 assumes thermal and moisture expansion coefficients are constant. Integrating volume change due to temperature change (for example), gives

      [math]\displaystyle{ \lambda_{res} = \exp\left(\alpha\Delta T\right) }[/math]

As a consequence, a temperature change corresponding to [math]\displaystyle{ \alpha\Delta T=100\% }[/math] would result in [math]\displaystyle{ \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

This material provides an acceptable material model for rubbers or elastomers, but it will not get the thermodynamics correct unless you indicate the material is an ideal rubber (using the IdealRubber property). An ideal rubber means that during isothermal loading, there is no change in internal energy. In other words, all the work is turned into heat. In adiabatic loading, this heat causes the temperature to rise. For the thermal calculations to be correct, the material must specify a valid heat capacity.

The entropy for this material is the net effect of heat flow by external heating, by conduction, by internal mechanisms, or by work when it is set to be an ideal rubber. For example, when an ideal rubber is loaded isothermally in tension in the x direction, the force is related to entropy by

      [math]\displaystyle{ F = A\sigma_{xx} = -T \left({\partial S\over\partial L}\right)_T }[/math]

where A is cross-sectional area and L in length. An ideal rubber in NairnMPM correctly follows this law.

Material Properties

The material properties are entered using:

Property Description Units Default
K Bulk modulus pressure units none
G1 The G1 shear modulus pressure units none
G2 The G2 shear modulus pressure units 0
E The E tensile modulus pressure units 0
nu The Poisson's ratio none 0
UJOption Set to 0, 1, or 2, to select the energy term from above. none 0
alpha Thermal expansion coefficient ppm/K 0
IdealRubber Set to 0 to be an elastic material and to 1 to be an ideal rubber. none 0
(other) Properties common to all materials varies varies

You must enter K and G1 OR E and nu. When you enter E and nu, K and G = G1+G2 are calculated from the relations between these properties. The partition of G into G1 and G2 is determined by your entered value for G2 (i.e., G1 = G - G2). See these relations to covert other properties (such as Lame modulus) to bulk and shear moduli.

History Variables

This material uses two history variables:

  1. J or the determinant of the deformation and equal to V/V0.
  2. Jres 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

These commands model polymer as an isotropic, hyperelastic material with G1=G2 =G/2 (using scripted or XML commands):

Material "polymer","Ideal Elastomer","Mooney"
  K 3000
  G1 10
  G2 10
  alpha 60
  rho 1.2
  IdealRubber 1
  Cv 1200
Done

References

  1. 1.0 1.1 C. Truesdell and W. Noll, The nonlienar field theories of mechanics, Edition Handbuch der Physik, Vol. III. Spinger, Berlin (1965).
  2. R. W. Ogden, Non-linear elastic deformations. Wiley & Sons, New York (1984).
  3. D. Weichert and Y. Basar, Nonlinear continuum mechanics of solids, Springer, New York (2000).