Nonlinear finite elements/Homework 6/Hints

Hints for Homework 6: Problem 1: Section 8Edit

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The problem becomes easier to solve if we consider numerical values of the parameters. Let the local nodes numbers of element   be   for node  , and   for node  .

Let us assume that the beam is divided into six equal sectors. Then,

 

Let   and  . Since the blue point is midway between the two,  .

Also, let the components of the stiffness matrix of the composite be

 

Let the velocities for nodes   and   of the element be

 

The   and   coordinates of the master and slave nodes are

 
 
 

Since there are two master nodes in an element, the parent element is a four-noded quad with shape functions

 

The isoparametric map is

 

Therefore, the derivatives with respect to   are

 

Since the blue point is at the center of the element, the values of   and   at that point are zero. Therefore,

 

The local laminar basis vector   is given by

 

The laminar basis vector   is given by

 

To compute the velocity gradient, we have to find the velocities at the slave nodes using the relation

 

For master node 1 of the element (global node 5), the velocities of the slave nodes are

 

For master node 2 of the element (global node 6), the velocities of the slave nodes are

 

The interpolated velocity within the element is given by

 

The velocity gradient is given by

 

The velocities are given in terms of the parent element coordinates ( ). We have to convert them to the ( ) system in order to compute the velocity gradient. To do that we recall that

 

In matrix form

 

and

 

Therefore,

 

The rate of deformation is given by

 

The global base vectors are

 

Therefore, the rotation matrix is

 

Therefore, the components of the rate of deformation tensor with respect to the laminar coordinate system are

 

The rate constitutive relation of the material is given by

 

Since the problem is a 2-D one, the reduced constitutive equation is

 

The laminar  -direction maps to the composite  -direction and the laminar  -directions maps to the composite  -direction. Hence the constitutive equation can be written as

 

Rearranging,

 

The plane stress condition requires that   in the laminar coordinate system. We assume that the rate of   is also zero. In that case, we get

 

or,

 

To get the global stress rate and rate of deformation, we have to rotate the components to the global basis using