Viscoplastic Models for Polymeric Composite

1
Viscoplastic Models for Polymeric Composite

• Mentee
• Chris Rogan
• Department of Physics
• Princeton University
• Princeton, NJ 08544
• Mentors
• Marwan Al-Haik M.Y Hussaini
• School of Computational Science
• Florida State University
• Tallahassee, FL 32306

2
Part-1 Explicit Model

• Micromechanical Viscoplastic Model

3
Explicit Model

• Viscoplastic Model Proposed by Gates and Sun

?t ?e ?p
?p A(?)n
?e ?/E
The plastic portion of the strain is represented
by this non-linear equation, where A and n are
material constants found from experimental data
The elastic portion of the strain is determined
by Hooks Law, where E is Youngs Modulus
Gates, T.S., Sun, C.T., 1991. An
elastic/viscoplastic constitutive model for fiber
reinforced thermoplastic composites. AIAA Journal
29 (3), 457463.
4
Explicit Model
The total strain rate is composed of elastic and
plastic components

• d?t/dt d?e/dt d?p/dt

d?e/dt (d?/dt)/E
d?vp /dt d?vp/dt d?vp/dt
The elastic portion of the strain rate is the
elastic component of the strain differentiated
with respect to time. The component of the
strain rate is further divided into two
viscoplastic terms,
5
Explicit Model
The first component of the plastic strain rate is
the plastic strain differentiated with respect to
time
d?vp/dt A(n)(?)n-1(d?/dt)
The second component utilizes the concept of
overstress, or ?- ?, where ? is the
quasistatic stress and and ? is the dynamic
stress. K and m are material constants found from
experimental data.
d?vp/dt ((? - ?)/K)1/m
6
Tensile Tests
Figure 1
7
Methodology

• Firstly, the tensile test data (above) was used
  to determine the material constants A,
• n and E for each temperature. E was calculated
  first,
• fitting the linear portion of the tensile test
  curve to reflect the elastic component of
• the equation as shown in Figure 2. Next, the
  constants A and n were calculated by
• plotting Log(?) vs. Log(? - ?/E) and extracting n
  and A from a linear fit as the
• slope and y-intercept respectively. Figure 3
  displays the resulting models fit to the
• experimental data.

? ?/E A(?)n
8
A-n

• Log(? - ?/E) nLog(?) LogA

Figure 2
Figure 3
Figure 4
Figure 4
9
Table 1
10
Load Relaxation Tests
The data from the load relaxation tests was used
to determine the temperature-dependent material
constants K and m. For each temperature, the load
relaxation test was conducted at 6 different
stress levels, as shown in Figure 4.
11
Curve Fitting of Load Relaxation
Figure 5
Firstly, the data from each different strain
level at each temperature was isolated. The noise
in the data was eliminated to ensure that the
stress is monotonically decreasing, as dictated
by the physical model (Figure 5). The data was
then fit into two different trends exponential
and polynomial of order 9 functions (Figures 6
and 7).
12
Figure 7

• Figure 6

0 d?/dt (d?/dt)/E ((? - ?)/K)1/m
gt  Log(-(d?/dt)/E) (1/m)(Log(? - ?)
(1/m)Log K
13
From the exponential fits the constants K and m
were calculated by plotting Log(-(d?/dt)/E) vs.
Log(? - ?), and calculating the linear fit, as
shown in Figures 8 and 9. The tabulated material
constants for each temperature are pictured below.
Table 2
14
Figure 9
Figure 8
? ?/E A(?)n
For each temperature and strain level, the
quasistatic stress was found by solving the above
non-linear equation using Newtons method. The
quasistaitc stress values are displayed in Table
1.
Table 1
15
Simulation of Explicit Model
-(d?/dt)/E ((? - ?)/K)1/m
The total strain rate is zero during the load
relaxation test, leading to the differential
equation above. The explicit model solution was
generated by solving this differential equation
using the fourth order Runge-Kutta method.
Different step-sizes were experimented with, and
an example solution is shown in Figure 10.
Figure 10
16
Part 2 Implicit Model

• Generalizing an Implicit Stress Function Using
  Neural Networks

17
Neural Networks (NN)
The Implicit Model consists of creating an
implicit, generalized stress function, dependent
on vectors of temperature, strain level and time
data. A generalized neural network and one
specific to this model are shown in Figure 11. A
neural network consists of nodes connected by
links. Each node is a processing element which
takes weighted inputs from other nodes, sums
them, and then maps this sum with an activation
function, the result of which becomes the
neurons output. This output is then propagated
along all the links exiting the neuron to
subsequent neurons. Each link has a weight value
to which traveling outputs are multiplied.
18
Procedures for NN

• Based on the three phases of neural networks
  functionality (training, validation and
  testing),the data sets from the load relaxation
  tests were split into three parts. The data sets
  for three temperatures were set aside for
  testing. The other five temperatures were used
  for training, excluding five specific
  combinations of temperature and strain levels
  used for validation.

19
Pre-processing
Before training, the data vectors were put into
random order and were normalized by the equation
20
Training NN

• Training a feed-forward backpropagating neural
  network consists of giving the network a
  vectorized training data set each epoch. Each
  individual vectors inputs (temperature, strain
  level, time) are propagated through the network,
  and the output is incorporated with the vectors
  experimental output in the error equation above.
  Training the network consists of minimizing this
  error function in weights space, adjusting the
  networks weights using unconstrained local
  optimization methods. An example of a training
  sessions graph is shown in Figure 12, in this
  case using a gradient descent method with
  variable learning rate and momentum terms to
  minimize the error function.

Figure 12
21
2 Hidden Layers NN
The architecture of the neural network is
difficult to decide. Research by Hornik et al.
(1989) suggests that a network with two hidden
layers can approximate any function, although
there is no indication as to how many neurons to
put in each of the hidden layers. Too many
neurons causes overfitting the network
essentially memorizes the training data and
becomes a look-up-table, causing it to perform
poorly with the validation and training data that
it has not seen before. Too few neurons leads to
poor performance for all of the data.
Hornik, K., Stinchocombe, M., White, H., 1989.
Multilayer feedforward networks are universal
approximators. Neural Networks, 359366.
22
Error Surface

• Figure 13 shows the resulting mean squared
  error performance values for neural networks with
  different numbers of neurons in each hidden layer
  after 1000 epochs of training.

Figure 13
23
Figure 14
Figure 15

• Figures 14 and 15 display similar data, except
  that only random data points are used in the
  neuron space and a cubic interpolation is
  employed in order to distinguish trends in the
  neuron space. As figure 15 shows, there appears
  to be a minimum in the area of about 10 neurons
  in the first hidden layer and 30 in the second. A
  minimum did in fact occur with a 10 31 1
  network.

24
Genetic Algorithm (GA) Pruning

• A genetic algorithm was used to try to determine
  an optimal network architecture. Based on the
  results of earlier exhaustive methods, a domain
  from 1 to 15 and 1 to 35 was used for the number
  of neurons in the first and second hidden layers
  respectively.
• A population of random networks in this domain
  was generated, each network encoded as a binary
  chromosome. The probability of a particular
  networks survival is a linear function of its
  rank in the population.
• Stochastic remainder selection without
  replacement was used in population selection. For
  crossovers, a two-point crossover of chromosomes
  reduced surrogates was used as shown in Figure
  16.

Figure 16
25
GA-Pruning

• This method allows pruning of not only neurons
  but links, as each layers of neurons is not
  necessarily completely connected to the next, and
  connections between non-adjacent layers is
  permitted. The genetic algorithm was run with
  varying parameter values and two different
  objective functions one seeking to minimize only
  the training performance error of the networks
  and another minimizing both the performance error
  and the number of neurons and links. Figure 17
  displays an optimal network when only the
  performance error is considered. Figure 18 shows
  and optimal network when the number of neurons
  and links was taken into account.

Figure 17
Figure 18
26
GA-Performance

• Figure 19 shows the results of an exhaustive
  architecture search in a smaller domain than
  earlier, the first arrow pointing to a minimum
  that coincides with the network architecture
  displayed in Figure 17.

Figure 19
27
Results of NN Implicit Model

• A network architecture of 10 31 1 was used
  for the training and testing of the neural
  networks. Several different minimization
  algorithms were tested and compared for the
  training of the network and are listed in Figures
  20 and 21. These two figures display the training
  performance error and gradient over 1000 epochs.

Figure 21
Figure 20
28
Training- Validation Testing of Final NN
Structure

• Figure 22 shows the testing, validation and
  training performance for the Gradient Descent
  algorithm while Figure 23 shows the plot of a
  linear least squares regression between the
  experimental data and network outputs for the
  Polack Ribiere Conjugate Gradient method.

Figure 23
Figure 23
Figure 22
29

• Figure 24 displays the final performance of both
  models compared to the experimental data. The
  Quasi-Newton BFGS algorithm was used for the
  Implicit model, as it performed the best. The
  Implicit model ultimately outperformed the
  Explicit, and required only the load relaxation
  data to generate the solution.

Comparing Explicit and Implicit Models
Figure 24
30
Conclusion

• The Implicit model(NNGA) ultimately
  outperformed the Explicit( Gates), and required
  only the load relaxation data to generate the
  solution.