MMO A03

import numpy as np import matplotlib.pyplot as plt !pip install uncertainties from uncertainties import ufloat

epsilonNom= np.linspace(0,10,1001) sigmaTrue = 30*epsilonNom**0.8 sigmaNom = sigmaTrue/(1+epsilonNom) sigmaNomNeck = max(sigmaNom) #sigmaList = sigmaNom.tolist() #epsilonNeck = sigmaList.index(sigmaNeck) epsilonNomNeckIndex = sigmaNom.tolist().index(sigmaNomNeck) epsilonNomNeck = epsilonNom[epsilonNomNeckIndex] sigmaTrueNeck = sigmaTrue[epsilonNomNeckIndex] plt.plot(epsilonNom,sigmaNom,epsilonNomNeck, sigmaNomNeck, "o") #plt.hlines(sigmaNeck,0,10, 'C2') print("The nominal strain at necking is", epsilonNomNeck) print("The true stress at necking is", np.round(sigmaTrueNeck,2), "Mpa")

epsilonNomNeg = np.linspace(-0.99,10,1100) epsilonTrue = np.log(1 + epsilonNomNeg) # (i) plt.plot(epsilonNomNeg, epsilonTrue) plt.xlabel('\u03B5_nom') #epsilon plt.ylabel('\u03B5_t') #epsilon

# (ii) epsilonSmallDeviation = np.zeros(100) counter = 0 deviation = abs((epsilonTrue-epsilonNomNeg)/epsilonNomNeg)*100 for i in range(len(deviation)): if(deviation[i] <= 5.0): #print(deviation[i]) epsilonSmallDeviation[counter] = epsilonNomNeg[i] counter += 1 print("\u03B5_t deviates with more than 5% from \u03B5_nom when:") print("\u03B5_nom <", np.round(epsilonSmallDeviation[0],4), "and \u03B5_nom >", np.round(epsilonSmallDeviation[counter-1],4), "\n\n\n")

epsilonTrueApprox = epsilonNomNeg - epsilonNomNeg**2/2 print(epsilonTrueApprox) plt.plot(epsilonNomNeg, epsilonTrueApprox) plt.xlabel('\u03B5_nom') #epsilon plt.ylabel('\u03B5_t_approx') #epsilon

# (iii) epsilonSmallDeviation2 = np.zeros(100) counter2 = 0 deviation2 = abs((epsilonTrueApprox-epsilonTrue)/epsilonTrue)*100 for i in range(len(deviation)): if(deviation2[i] <= 5.0): #print(deviation[i]) epsilonSmallDeviation2[counter2] = epsilonNomNeg[i] counter2 += 1 print("\u03B5_t_approx deviates with more than 5% from \u03B5_t when:") print("\u03B5_nom <", np.round(epsilonSmallDeviation2[0],4), "and \u03B5_nom >", np.round(epsilonSmallDeviation2[counter2-1],4), "\n\n\n")

# b) (i) E_c = 1000 #Gpa E_s = 1 #Gpa f = 0.5 E = E_c*f + E_s*(1-f) print("b) (i)\n E(f = 0.5) =", E, "Gpa\n") # b) (ii) f = 0.9 E = E_c*f + E_s*(1-f) print("b) (ii)\n E(f = 0.9) =", E, "Gpa\n") # c) (i) E = 73 #Gpa f = (E - E_s)/(E_c - E_s) print("c) (i)\n f(E =", E, "Gpa) =", np.round(f,2), "\n") # c) (ii) E = 235 #Gpa f = (E - E_s)/(E_c - E_s) print("c) (ii)\n f(E =", E, "Gpa) =", np.round(f,2), "\n")

# Solving the system of equations to find TS_inf and A (2eq) Mn_1 = 73800 # g/mol TS_1 = 16.7 # Mpa Mn_2 = 51300 # g/mol TS_2 = 14.4 # Mpa A = np.array([[1,-1/Mn_1],[1, -1/Mn_2]]) b = np.array([TS_1, TS_2]) X = np.linalg.solve(A,b) TS_inf = X[0] A = X[1] print("TS_inf =",np.round(TS_inf,2),"\nA=", np.round(A,2), "\n") # a) TS = 18.1 #Mpa Mn = A / (TS_inf - TS) print("a)\n Mn(TS =", TS, "Mpa) =", np.round(Mn,2), "g/mol\n") # b) TS = TS_inf * 0.95 Mn = A / (TS_inf - TS) print("b)\n TS = 95% TS_inf =", np.round(TS,2), "Mpa\n Mn(TS =", np.round(TS,2), "Mpa) =", np.round(Mn,2), "g/mol\n")

POM = np.loadtxt('polymer_sample.txt') n = POM[:,0] DP = POM[:,1] # a) total weight n_mono_tot = sum(n*DP) print("a)\n m_tot =", n_mono_tot, "g\n") # Vekt for ett mol monomerer [g/mol] M_c = 12 #g/mol M_h = 1 #g/mol M_o = 16 #g/mol M_pom_mol = M_c + 2*M_h + M_o mol = 6.022*10**23 # Vekt for ett monomer [g] M_pom = M_pom_mol/mol # Total masse mass_tot = M_pom * n_mono_tot

# b) # Weight fraction w_i = n*(M_pom*DP)/mass_tot # Numbers fraction x_i = n/sum(n) # Molecular weight for range M_i = DP*M_pom plt.figure(1) plt.plot(M_i, x_i, 'bo') plt.xlabel('M_i [g]') plt.ylabel('w_i') plt.figure(2) plt.plot(M_i, w_i, 'ro') plt.xlabel('M_i [g]') plt.ylabel('x_i') print("b)\n Se plot\n")

# c) M_w = sum(w_i*M_i) M_n = sum(x_i*M_i) DPI = M_w/M_n # Typical range 1.02 to 50+ print("c)\n M_w =", "%.4g" %M_w,"g\n M_n =", "%.4g" %M_n, "g\n DPI =", np.round(DPI,2)) #print(M_w * mol) #print(M_w/mol)

# b) p_a = 840 # density amorphous PP [kg/m^3] p_c = 950 # density crystalline PP [kg/m^3] p_s = 880 # density of PP specimen [kg/m^3] # The degree of crystallinity (DC) is given by: DC = (p_c*(p_s - p_a))/(p_s*(p_c - p_a))*100 # (i) Assuming no voids we have: print("b) (i) Assuming no voids we have:\n DC =", np.round(DC,2), "[%]\n") # (ii) Estimating uncertainty p_a = ufloat(840, 5) # p_a = 840+/-5 [kg/m^3] p_c = ufloat(950, 5) # p_a = 950+/-5 [kg/m^3] p_s = ufloat(880, 5) # p_s = 880+/-5 [kg/m^3] DC = (p_c*(p_s - p_a))/(p_s*(p_c - p_a))*100 print("b) (ii) With uncertainty we get:\n DC =",DC, "[%]\n") std_DC = DC.std_dev COV_DC = DC.std_dev/DC.nominal_value print(" Standard deviation =", np.round(std_DC,2), "[%]\n COV =",np.round(COV_DC,2), "%")

volumeTeaspoon = 4.93 # mL COV_teaspoon = 0.05 # 5% stdTeaspoon = COV_teaspoon*volumeTeaspoon # using uncertainty +/-5% areaAcre = 4046.86 # m COV_acre = 0.20 # 10% stdAcre = COV_acre*areaAcre # using uncertainty +/-10% volumeTeaspoon = ufloat(volumeTeaspoon, stdTeaspoon) # mL areaAcre = ufloat(areaAcre, stdAcre) # m area = areaAcre/2 # m volume = volumeTeaspoon*10**(-6) # L thickness = volume/area print("Thickness =", thickness, "m\n") COV_thickness = thickness.std_dev/thickness.nominal_value print("Standard deviation thickness =","%.2g" %thickness.std_dev, "m") print("COV thickness =",np.round(COV_thickness,4)*100, "%")