Material Models LS-DYNA Theory Manual

19.40

and the prime denotes the deviatoric component. The yield stress is defined as

( )0p

y pEσ σ ε ε= + (19.19.2)

where pε is the effective plastic strain and hE is given in terms of Young’s modulus and the

tangent modulus by

tp

t

E EE

E E=

− . (19.19.3)

Both Young's modulus and the tangent modulus may optionally be made functions of strain rate by specifying a load curve ID giving their values as a function of strain rate. If these load curve ID's are input as 0, then the constant values specified in the input are used.

Note that all load curves used to define quantities as a function of strain rate must have the same number of points at the same strain rate values. This requirement is used to allow vectorized interpolation to enhance the execution speed of this constitutive model. This model also contains a simple mechanism for modeling material failure. This option is activated by specifying a load curve ID defining the effective stress at failure as a function of strain rate. For solid elements, once the effective stress exceeds the failure stress the element is deemed to have failed and is removed from the solution. For shell elements the entire shell element is deemed to have failed if all integration points through the thickness have an effective stress that exceeds the failure stress. After failure the shell element is removed from the solution. In addition to the above failure criterion, this material model also supports a shell element deletion criterion based on the maximum stable time step size for the element, maxtΔ . Generally,

maxtΔ goes down as the element becomes more distorted. To assure stability of time integration,

the global LS-DYNA time step is the minimum of the maxtΔ values calculated for all elements in

the model. Using this option allows the selective deletion of elements whose time step maxtΔ has

fallen below the specified minimum time step, crittΔ . Elements which are severely distorted

often indicate that material has failed and supports little load, but these same elements may have very small time steps and therefore control the cost of the analysis. This option allows these highly distorted elements to be deleted from the calculation, and, therefore, the analysis can proceed at a larger time step, and, thus, at a reduced cost. Deleted elements do not carry any load, and are deleted from all applicable slide surface definitions. Clearly, this option must be judiciously used to obtain accurate results at a minimum cost.

Material Type 20: Rigid The rigid material type 20 provides a convenient way of turning one or more parts

comprised of beams, shells, or solid elements into a rigid body. Approximating a deformable body as rigid is a preferred modeling technique in many real world applications. For example, in sheet metal forming problems the tooling can properly and accurately be treated as rigid. In the design of restraint systems the occupant can, for the purposes of early design studies, also be treated as rigid. Elements which are rigid are bypassed in the element processing and no storage is allocated for storing history variables; consequently, the rigid material type is very cost efficient.

LS-DYNA Theory Manual Material Models

19.41

Two unique rigid part IDs may not share common nodes unless they are merged together using the rigid body merge option. A rigid body may be made up of disjoint finite element meshes, however. LS-DYNA assumes this is the case since this is a common practice in setting up tooling meshes in forming problems. All elements which reference a given part ID corresponding to the rigid material should be contiguous, but this is not a requirement. If two disjoint groups of elements on opposite sides of a model are modeled as rigid, separate part ID's should be created for each of the contiguous element groups if each group is to move independently. This requirement arises from the fact that LS-DYNA internally computes the six rigid body degrees-of-freedom for each rigid body (rigid material or set of merged materials), and if disjoint groups of rigid elements use the same part ID, the disjoint groups will move together as one rigid body.

Inertial properties for rigid materials may be defined in either of two ways. By default, the inertial properties are calculated from the geometry of the constituent elements of the rigid material and the density specified for the part ID. Alternatively, the inertial properties and initial velocities for a rigid body may be directly defined, and this overrides data calculated from the material property definition and nodal initial velocity definitions. Young's modulus, E, and Poisson's ratio, are used for determining sliding interface parameters if the rigid body interacts in a contact definition. Realistic values for these constants should be defined since unrealistic values may contribute to numerical problem in contact.

Material Model 21: Thermal Orthotropic Elastic In the implementation for three-dimensional continua a total Lagrangian formulation is used. In this approach the material law that relates second Piola-Kirchhoff stress S to the Green-St. Venant strain E is

tlS C E T C T E= ⋅ = ⋅ (19.21.1)

where T is the transformation matrix [Cook 1974].

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

2 2 21 1 1 1 1 1 1 1 12 2 22 2 2 2 2 2 2 2 22 2 23 3 3 3 3 3 3 3 3

1 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 2 1

2 3 2 3 2 3 2 3 3 2 2 3 3 2 2 3 3 2

3 1 3 1 3 1 3 1 1 3 3 1 1 3 3 1 1 3

2 2 2

2 2 2

2 2 2

l m n l m m n n l

l m n l m m n n l

l m n l m m n n lT

l l m m n n l m l m m n m n n l n l

l l m m n n l m l m m n m n n l n l

l l m m n n l m l m m n m n n l n l

=+ + ++ + ++ + +

(19.21.2)

il , im , in are the direction cosines

'1 2 3 1, 2,3i i i ix l x m x n x for i= + + = (19.21.3)