Solid continuum element

Introduction

To be implemented ..

Formulation

Integrating the governing equation by parts using the Green-Gauss theorem gives the so-called weak form of the equilibrium equation as

\[ \int_\Omega B^T.\sigma d\Omega - \int_{\Gamma} N^T.t d\Gamma-\int_{\Omega} N^T.b d\Omega = 0 \]

where:

• \( B^T \) is the element strain-displacement matrix
• \( N \) is a matrix of shape functions

Plugin options

The mechanical continuum element supports the plugin options detailed in Mechanical Fem Plugin options and additional plugin options only for continuum elements are detailed below:

Plugin object types

The Mechanical Fem Physic plugin publishes four objects type depending on the nature of the element formulation (Plane strain, Plane strain, Axissymetric, and 3D) and so references to the plugin name while creating a physics object requires the type name.

Objects type for continuum elements

• PlaneStress
• PlaneStrain
• Axissymetric
• 3D

Example:

PhysicalMethod {
  id       = 'block', 
  typeName = 'MechanicalFemPhysics.3D', 
  type     = 'fem',
  mesh = 'mesh',
  materials = {'elastic'}, 
  ruleSet = 1, 
} 

State variables

The Continuum element plugin requires the model to contain the displacement state variable 'u', which is a vector with size equal to the node coordinate dimension. Further details see: Mechanical state variables.

Continuum material types

The mechanical continuum elements support several materials as detailed below:

Material type	Material name
Elastic	elastic
Von Mises	vonMises
Drucker Prager	druckerPrager
Modified Mohr Coulomb	modifiedMohrCoulomb
Modified Mohr Coulomb with cap	capMohrCoulomb
Fractured Rock model	fracturedRock
Isotropic Damage model	isotropicDamage
Solidifying Kelvin Chain	solidifyingKelvinChain

Other materials are under development such as: Lade kim, Hoek-Brown, ect.

The mechanical continuum materials are fully described in Mechanical Continuum Material

Physics attributes

When defining plugin object of Mechanical fem Physic, the following fields are available for usage during the definition:

Attribute	Description	Type	Required	Def. Value
id	The physics name.	String	Yes	-
	Example: id = 'myPhysicsName'
typeName	Physics defines the plugin name according to the type name such as 'planeStress','planeStrain', etc.	String	Yes	-
	Example: typeName = 'MechanicalFemPhysics.3D'
type	Type of physics object. Should be equal to 'fem'.	String	Yes	-
	Required value: type= 'fem'
ruleSet	Integration rule set selected for simulation elements.	String	No	Empty
	Example: ruleSet = 1

Like other Fem physics, stateVar, property and attribute renaming fields are supported. Likewise, restricting the physics application domain to a set of cell groups, setting the desired integration rule and unit system remapping are also available. Those common optional fields are detailed at the GeMA Common Fem Porocess.

Example

PhysicalMethod {
  id       = 'Plate',
  typeName = 'MechanicalFemPhysics.PlaneStrain',
  type     = 'fem',
  mesh = 'mesh',
  ruleSet = 1,
  boundaryConditions = {'bc2'},
  materials = {'elastic', 'von Mises'},
}

Further details see Mechanical Physic Attributes.

Boundary conditions

The Continuum element plugin supports the Boundary condititons detailed in Mechanical Boundary Conditions.

Supported elements

The Mechanical Fem physics supports 2D and 3D continuum elements. Supported elements are:

• quad4
• quad8
• quad9
• tri3
• tri6
• tet4
• hex8
• hex20

Unit system

The Mechanical Fem physics uses internally the international system of units for its calculations, as can be seen above by the default units for properties and attributes. The whole unit system can be replaced through the unitSystem physics attribute, as described at the gemaFemProcessCommonFemPhysicsOptions page.

When making the substitution, the units on the following table should be replaced by the desired units, forming a coherent unit system. The unit system of the material should be also replaced.

local newUnitSystem = {
  time     = 's',
  coord    = 'm',
  u        = 'm',
  F        = 'kN',
  E        = 'kPa',
  pressure = 'kPa',
  Sws      = 'kN/m3',
  gr       = 'm/2',
  pl       = 'kPa',
}

References

1.Potts, D. M. and Zdravkovic, L. (1999). Finite elements analysis in geotechnical engineering - Theory. 68-70. Thomas Telford.

2.Zienkiewicz OC. (1980). Finite element methods. Editiorial Reverte: Barcelona.

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