Nonlinear finite elements/Homework 10/Solutions



Problem 1: Kinematics and Stress RatesEdit

Given:

Figure 1 shows a linear three-noded triangular element in the reference configuration.

 
Figure 1. Three-noded triangular element.

The motion of the nodes is given by:

 

The configuration ( ) of the element at time   is given by

 

SolutionsEdit

Part 1Edit

Write down expressions for  ,  , and   in terms of the initial configuration ( ) ?

In the initial configuration  . Therefore,

 

Therefore, the initial configuration is given by

 

Substituting the values of   and  , we get

 

Hence

 

Also, the shape functions must satisfy the partition of unity condition

 

Therefore,

 

The required expressions are

 

Part 2Edit

Derive expressions for the deformation gradient and the Jacobian determinant for the element as functions of time.

The deformation gradient is given by

 

Before computing the derivatives, let us express   in terms of  . Recall

 

Therefore,

 

Substituting in the expressions for   and  , we get

 

In the above expression, the parent coordinates   are no longer useful. Therefore, we write

 

where  . Taking derivatives, we get

 

Therefore,

 

The Jacobian determinant is

 

or

 

Part 3Edit

What are the values of   and   for which the motion is isochoric?

For isochoric motion,  . Therefore,

 

One possibility is

 

This is a pure rotation.

Another possibility is that

 

This is a combination of shear and rotation where the volume remains constant.

Part 4Edit

For which values of   and   do we get invalid motions?

We get invalid motions when  . Let us consider the case where  . Then

 

This is possible when

 

If   then

 

That is,

 

Therefore the values at which we get invalid motions are

 

Part 5Edit

Derive the expression for the Green (Lagrangian) strain tensor

for the element as a function of time.

The Green strain tensor is given by

 

Recall

 

Let us make the following substitutions

 

Then

 

Therefore,

 

Hence,

 

The Green strain is

 

Part 6Edit

Derive an expression for the velocity gradient as a function of

time.

The velocity is the material time derivative of the motion.

Recall that the motion is given by

 

Therefore,

 

We could compute the velocity gradient using

 

after expressing   in terms of  . However, that makes the expression quite complicated. Instead, we will use the relation

 

The time derivative of the deformation gradient is

 

The inverse of the deformation gradient is

 

Using the substitutions

 

we get

 

and

 

The product is

 

Note that the first matrix is symmetric while the second is skew-symmetric.

Therefore, the velocity gradient is

 

Part 7Edit

Compute the rate of deformation tensor and the spin tensor.

The rate of deformation is the symmetric part of the velocity gradient:

 

The rate of deformation is the skew-symmetric part of the velocity gradient:

 

Part 8Edit

Assume that   and  . Sketch the undeformed configuration and the deformed configuration at   and  .  Draw both the deformed and undeformed configurations on the same plot

and label.

Recall that in the initial configuration

 

Also, the motion is

 

Plugging in the values of   and  , we get

 

At  ,

 

At  ,

 

The deformed and undeformed configurations are shown below.

 
Deformed and undeformed configurations.

Part 9Edit

Compute the polar decomposition of the deformation gradient with the above values of   and  ,

The deformation gradient is

 

The right Cauchy-Green deformation tensor is

 

The eigenvalue problem is

 

This problem has a solution if

 

i.e.,

 

The eigenvalues are (as expected)

 

The principal stretches are

 

The principal directions are (by inspection)

 

Now, the right stretch tensor is given by

 

Therefore,

 

Hence the right stretch is

 

At  , we have

 

Now

 

and

 

Therefore, the rotation is

 

At  , we have

 

Part 10Edit

Assume an isotropic, hypoelastic constitutive equation for the material of the element. Compute the material time derivative of the Cauchy stress at   using (a) the Jaumann rate and (b) the Truesdell rate.

A hypoelastic material behaves according to the relation

 

For an isotropic material

 

Therefore,

 

Recall that

 

Using the values of   and   from the previous part, at  ,

 

Therefore, the trace of the rate of defromation is

 

Therefore,

 

For the Jaumann rate

 

where the spin is

 

Therefore,

 

or,

 

For the Truesdell rate

 

Therefore,

 

Recall,

 

For  ,  ,  , we have

 

Therefore,

 

and

 

Hence,

 


Problem 2: Hyperelastic Pinched Cylinder ProblemEdit

Read the following paper on shells:

Buchter, N., Ramm, E., and Roehl, D., 1994, "Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept," Int. J. Numer. Meth. Engng., 37, pp. 2551-2568.

Answer the following questions:


SolutionEdit

Part 1Edit

What do the authors mean by "enhanced assumed strain"?

See Wikipedia article on [w:Enhanced assumed strain|enhanced assumed strain]] .

Part 2Edit

Example 8.2 (and Figures 3 and 4 and Table III) in the paper discusses the simulation of a hyperelastic cylinder. Perform the same simulation using ANSYS for a shell thickness of 0.2 cm. Use shell elements and the Neo-Hookean hyperelastic material model that ANSYS provides.

The following material properties are used:

  kN/cm ,  ,   kN/cm , and   cm /kN. Symmetry is used and only half of the model is meshed. At the support, the model is constraint in all directions. The load of 36 kN is applied on the top of the cylinder (Fig~2. ANSYS input listing is shown Fig~6 of Appendix.

 
Figure 2. Meshed model

Part 3Edit

Compare the total load needed to achieve an edge displacement of 16 cm with the results given in Table III. Comment on your results.

The plot of force vs. edge displacement (vertical) and the deformed model are shown in Fig 3. From the plot, one sees that a load of 35.1 kN is required to deform the edge by 16 cm. This is less than 1% difference compared to the result given in the paper using 7-parameter shell theory.

 
Figure 3. Left: Force vs. edge displacement. Right: Deformed model.

Problem 3: Elastic-Plastic Punch IndentationEdit

Read the following paper on elastic-viscoplastic FEA:

Rouainia, M. and Peric, D., 1998, "A computational model for elasto-viscoplastic solids at finite strain with reference to thin shell applications," Int. J. Numer. Meth. Engng., 42, pp. 289-311.

Answer the following questions:

SolutionEdit

Part 1Edit

Example 5.4 of the paper shows a simulation of the deformation of a thin sheet by a square punch. Perform the same simulation for 6061-T6 aluminum. Assume linear isotropic hardening and no rate dependence.

The following data are used for the thin plate:  GPa,  ,   Mpa,   Mpa, see Fig 4 for the stress-strain data. Symmetry is used and only half of the model is meshed (Fig 4). At the support, the model is constrained in all directions. A plastic finite strained shell (SHELL43) is chosen for this simulation.

The following material properties are used for the punch and die:  GPa,  . The value of Young's modulus is arbitrary chosen so long as it is high enough to remain rigid during the simulation.

The meshed model is illustrated in Fig 4. A load of 10 kN is applied on the top of the punch. Load steps are split into two steps. First load step the punch is moved close to the plate to activate the contact elements (CONTAC49). The second load step, 30 kN is applied on top of the punch. ANSYS input listing is shown Fig 7 of the Appendix.

 
Figure 4. Left: Bi-linear stress-strain data. Right: Meshed model.

Part 2Edit

Draw a plot of the punch force vs. punch travel and compare your result with the results shown in Figure 13 of the paper (qualitative comparison only).

The punch force vs. punch travel and the plot of the deformation at the final load step is shown in Fig 5. The curve displays similar pattern as those shown in the paper.

 
Figure 5. Left: Punch force vs. punch travel. Right: Sketch of deformation at final load step.

AppendixEdit

 
Figure 6. ANSYS input listing for Problem 3.
 
Figure 7. ANSYS input listing for Problem 3.