******* Lame MU ******* :: Lame MU = {model_name} {float_list} [M/ :math:`Lt^2`] ----------------------- **Description / Usage** ----------------------- This required card is used to specify the model for the Lame coefficient μ for the solid constitutive equation (see Sackinger, et. al. 1995, and *Solid Constitutive Equation* card); this coefficient is equivalent to the shear modulus *G* in most cases, as described below. Definitions of the input parameters are as follows: +-------------+---------------------------------------------------------------------------------------+ |{model_name} |Name of the Lame Mu coefficient model. This parameter can have one of the following | | |values: **CONSTANT, POWER_LAW, CONTACT_LINE, SHEAR_HARDEN, EXPONENTIAL, | | |DENSE_POWER_LAW, or USER.** | +-------------+---------------------------------------------------------------------------------------+ |{float_list} |One or more floating point numbers ( through ) whose values are | | |determined by the selection for {model_name}. These are identified in the discussion | | |of each model. | +-------------+---------------------------------------------------------------------------------------+ The details of each model option are given below: +-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------+ |**CONSTANT** |For the **CONSTANT** model, {float_list} is a single value: - Standard value of the | | |coefficient :math:`\mu`. (See Technical Discussion.) | +-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------+ |**POWER_LAW** |The **POWER_LAW** model is only to be used for deformable porous media where the shear modulus is | | |allowed to vary as a power of the porosity, :math:`\phi` (see Scherer, 1992): | | | | | |The {float_list} contains three values for this model, where: | | |.. figure:: /figures/360_goma_physics.png | | |:align: center | | |:width: 90% | | | | | | * - :math:`G_0` is the base shear modulus at the initial porosity (or μ0) | | | * - :math:`\phi_0` is the porosity in the stress free state | | | * - *m* is the powerlaw exponent | +-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------+ |**CONTACT_LINE** |The **CONTACT_LINE** model is a convenient way to control mesh deformation near a fixed point and | | |is normally used ONLY for *ARBITRARY Mesh Motion* types. This model enables the user to make the | | |shear modulus much larger near the contact line (fixed point) than far away from the contact line, | | |so that elements near the contact line are forced to retain their shape. The shear modulus in this | | |model varies inversely with distance from the contact line: | | | | | |.. figure:: /figures/361_goma_physics.png | | | :align: center | | | :width: 90% | | | | | |*r* is the distance from the fixed point, :math:`r_0` is a decay length, :math:`G_0`is the modulus | | |far from the contact line, and :math:`G_0 + G_1` is the modulus at the contact line. | | | | | |The {float_list} contains four values for this model, where: | | | | | | * - Node set number of the fixed point (converted to an integer by *Goma*) | | | * - :math:`G_0` (or :math:`\mu_0`) | | | * - :math:`G_1` | | | * - :math:`r_0` | +-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------+ |**MULTI_CONTACT_LINE** ... |The **MULTI_CONTACT_LINE** model is a based on the CONTACT_LINE model and allows multiple nodesets | | | | | |.. figure:: /figures/361_goma_physics.png | | | :align: center | | | :width: 90% | | | | | |*r* is the distance from the fixed point, :math:`r_0` is a decay length, :math:`G_0`is the modulus | | |far from the contact line, and :math:`G_0 + G_1` is the modulus at the contact line. | | | | | |The {float_list} contains four values for this model, where: | | | | | | * - :math:`G_0` (or :math:`\mu_0`) | | | * - :math:`G_1` | | | * - :math:`r_0` | | | * - Node set number of points to calculate distance | | | * - Optional second node set number calculate distance | | | * - Optional N'th node set number calculate distance | +-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------+ |**SHEAR_HARDEN** |The **SHEAR_HARDEN** model is: | | | | | |.. figure:: /figures/362_goma_physics.png | | | :align: center | | | :width: 90% | | | | | |where :math:`\chi` is the coefficient of variation, :math:`II_E` is the second invariant of the | | |strain tensor (see *Solid Constitutive Equation* card), :math:`G_0` is the modulus at zero shear. | | | | | |The {float_list} contains two values for this model, where: | | | | | | * - :math:`G_0` (or :math:`\mu_0`) | | | * - :math:`\chi` | +-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------+ |**EXPONENTIAL** |The **EXPONENTIAL** model is used exclusively for poroelastic problems, and allows for an | | |exponential dependence of the shear modulus :math:`\mu` (or G) on porosity: | | | | | |.. figure:: /figures/363_goma_physics.png | | | :align: center | | | :width: 90% | | | | | |where :math:`\lambda` is the rate of decay, :math:`\phi_0` is the porosity in the stress-free | | |state, :math:`G_0` is the modulus at zero shear. | | | | | | * - :math:`G_0` | | | * - :math:`\lambda` | | | * - :math:`\phi_0` | +-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------+ |**DENSE_POWER_LAW** |The **DENSE_POWER_LAW** model is used mostly for drying/consolidation problems for which it is | | |desired to have a plateau max-pack modulus behavior. This option requires input from the *Stress | | |Free Solvent Vol Frac* card (:math:`y_0` in equation below), and is used for solvent drying from a | | |condensed, gelled phase. The functional form for the shear modulus is | | | | | |.. figure:: /figures/364_goma_physics.png | | | :align: center | | | :width: 90% | | | | | |where *m* is the power law exponent, *F* is deformation gradient tensor (see *Solid Constitutive | | |Equation* card), and :math:`G_0` is the modulus at zero shear. This function is truncated or | | |clipped at the low end value at G=:math:`10^-12`. | | | | | | * - :math:`G_0` | | | * - :math:`\lambda` | | | * - :math:`\phi_0` | +-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------+ |**TABLE** {LINEAR | BILINEAR} [integer2] |Please see discussion at the beginning of the material properties chapter 5 for input description | | |and options. | +-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------+ |**USER** ,..., |For the **USER** model, {float_list} is of arbitrary length, and the values are used through the | | |param[] array in usr_lame_mu function to parameterize a user-defined model. See examples in | | |user_mp.c. | +-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------+ All modulus values in these equations have the same units as Lame Mu, i.e., M/Lt2. ------------ **Examples** ------------ The following is a sample card: :: Lame MU = CONSTANT 1. ------------------------- **Technical Discussion** ------------------------- Note that :math:`\mu` and :math:`\lambda`, (see the *Lame LAMBDA* card) are related to the more often used Young’s Modulus and Poisson’s Ratio by the following standard expressions: .. figure:: /figures/365_goma_physics.png :align: center :width: 90% where E is the Young’s modulus and υ is Poisson’s ratio. A significant limiting case is approached as :math:`\nu` approaches 0.5, in which case the solid becomes incompressible. The **POWER_LAW** option could easily be adapted to a concentration measure, viz. made dependent on the concentration of some species (see EQ = *species_bulk* card). This can be done through the user option, and in fact in usr_lame_mu function of file user_mp.c in the *Goma* distribution has an example that is appropriate. Also note that all of these models are available for the elastoviscoplastic option on the *Plasticity* card, and for the real-solid in *TOTAL_ALE* mesh motion. -------- **FAQs** -------- Important note that when one desires an incompressible solid through the use of INCOMP_PSTRAIN type models, by using an incompressible continuity equation in a LAGRANGIAN mesh region (see *EQ = continuity*), then the bulk modulus, or Lame Lambda expansion term is also added on. So to get a truly incompressible response, one must set the Lame LAMBDA coefficient to zero. -------------- **References** -------------- Sackinger, P. A., Schunk, P. R. and Rao, R. R. 1995. "A Newton-Raphson Pseudo-Solid Domain Mapping Technique for Free and Moving Boundary Problems: A Finite Element Implementation", J. Comp. Phys., 125 (1996) 83-103. Scherer, G.W., 1992, “Recent Progress in Drying of Gels”, J. of Non-Crystalline Solids, 147&148, 363-374 GT-001.4: GOMA and SEAMS tutorial for new users, February 18, 2002, P. R. Schunk and D. A. Labreche GT-019.1: Elastoviscoplastic (EVP) Constitutive Model in GOMA: Theory, Testing, and Tutorial, P. R. Schunk, A. Sun, S. Y. Tam (Imation Corp.) and K. S. Chen, January 11, 2001 GTM-027: Probing Plastic Deformation in Gelatin Films during Drying, M. Lu, S. Y. Tam, A. Sun, P. R. Schunk and C. J. Brinker, 2000 SAND96-2149: Drying in Deformable Partially-Saturated Porous Media: Sol-Gel Coatings, Cairncross, R. A., P. R. Schunk, K. S. Chen, S. S. Prakash, J. Samuel, A. J. Hurd and C. Brinker (September 1996) .. TODO - Lines 39, 54, 70, 85, 101 and 139 are photos that need to be replaced with the correct equations.