Nonlinear finite elements/Homework 6/Hints

Hints for Homework 6: Problem 1: Section 8 edit

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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






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




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




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