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 Part-1 Explicit Model Micromechanical Viscoplastic Model Explicit Model Viscoplastic Model Proposed by Gates and Sun t = e + p e = /E p = A()n The elastic portion of the strain is The plastic portion of the strain is represented by determined by Hook’s Law, where E is this non-linear equation, where A and n are Young’s Modulus material constants found from experimental data Gates, T.S., Sun, C.T., 1991. An elastic/viscoplastic constitutive model for fiber reinforced thermoplastic composites. AIAA Journal 29 (3), 457–463. Explicit Model The total strain rate is composed of elastic and plastic components dt/dt = de/dt + dp/dt de/dt = (d/dt)/E dvp /dt = dvp’/dt + dvp’’/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, Explicit Model The first component of the plastic strain rate is the plastic strain differentiated with respect to time dvp’/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. dvp’’/dt = (( - *)/K)1/m Tensile Tests Tensile Tests at Different Temperatures 400 350 T= 25 C T= 35 C 300 T= 45 C T= 50 C T= 55 C 250 Stress (MPa) T= 60 C T= 65 C 200 T= 75 C Poly. (T= 150 75 C) Poly. (T= 35 C) Poly. (T= 100 25 C) Poly. (T= 45 C) Poly. (T= 50 55 C) Poly. (T= 60 C) Poly. (T= 0 65 C) Poly. (T= 0 0.001 50 C) 0.002 0.003 0.004 0.005 0.006 Strain (mm/mm) Figure 1 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, = /E + A()n 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 model’s fit to the experimental data. Log(s) vs. Log(e - s/E) @ 45 0 -1 -2 2.3 2.35 2.4 2.45 2.5 y = 7.1791x - 21.826 2.55 A-n R2 = 0.925 Log(s) -3 Log( - /E) = -4 -5 -6 Log(e - s/E) nLog() + LogA Figure 2 Stress vs. Strain @ 45 Stress vs. Strain @ 45 350 140 300 120 250 100 Stress 200 Exper. Stress 80 y = 65922x 150 Modeled 60 R2 = 0.9989 100 40 50 20 0 0 0 0.001 0.002 0.003 0.004 0.005 0.006 0 0.0005 0.001 0.0015 0.002 0.0025 Strain Strain Figure 3 Figure 4 Table 1 Stress* T \ %Strain 30% 40% 50% 60% 70% 80% 25 124.054 160.19 200.82 242.14 285.19 322.12 35 101.93 140.73 177.8 229.92 276.68 316.46 45 85.878 135.66 173.91 201.93 224.98 277.78 50 81.821 117.35 154.97 181.73 196.57 245.75 55 69.657 110.49 132.5 173.12 189.18 235.28 60 68.61 103.32 138.24 145.5 187.69 215.91 65 63.11 94.13 102.65 135.77 150.44 178.86 75 58.604 76.479 101.95 125.25 136.44 167.97 Load Relaxation Tests Stress Relaxation @ 45 C 350 300 250 30% Strength Stress [ MPa] 200 40% Strength 50% Strength 150 60% Strength 70% Strength 100 80% Strength 50 0 0 1000 2000 3000 4000 5000 6000 7000 8000 tests The data from the load relaxationTime [s] 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. 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). Figure 6 Figure 7 0 = d/dt = (d/dt)/E + (( - *)/K)1/m => Log(-(d/dt)/E) = (1/m)(Log( - *) – (1/m)Log K 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 Temp (Deg) 25 35 45 50 55 60 65 75 A (Mpa) 10^-12.479 10^-6.8025 10^-12.156 10^-41.478 10^-19.4 10^-19.257 10^-28.563 10^-9.1163 n 3.6026 1.296 3.2313 15.305 6.3367 6.6161 10.165 2.5115 K 1.44E+07 9.21E+06 1.80E+13 3.50E+11 2.24E+07 4.39E+07 3.61E+06 4.00E+07 m 0.64654 0.74623 1.1965 1.0403 0.58915 0.71173 0.54771 0.8108 E(Mpa) 81081 72514 65922 65224 62014 60527 58331 46611 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 Newton’s method. The quasistaitc stress values are displayed in Table 1. 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 Part 2: Implicit Model Generalizing an Implicit Stress Function Using Neural Networks 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 neuron’s 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. 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. Pre-processing Before training, the data vectors were put into random order and were normalized by the equation Training NN Training a feed-forward backpropagating neural network consists of giving the network a vectorized training data set each epoch. Each individual vector’s inputs (temperature, strain level, time) are propagated through the network, and the output is incorporated with the vector’s experimental output in the error equation above. Training the network consists of minimizing this error function in weight’s space, adjusting the network’s weights using unconstrained local optimization methods. An example of a training session’s 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 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, 359–366. 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 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. 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 network’s 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 Figure 16 16. 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- Figure 17 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 18 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 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 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 22 Figure 23 Figure 24 displays the final performance of both models compared to the experimental data. The Quasi-Newton Comparing BFGS algorithm was used for the Implicit model, as it performed the best. The Implicit model ultimately Explicit outperformed the Explicit, and required only the load relaxation data to generate the solution. and Implicit Models Figure 24 Conclusion The Implicit model(NN+GA) ultimately outperformed the Explicit( Gates), and required only the load relaxation data to generate the solution.
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