TMM4100 Materialteknikk

Introduksjon Parameterstudie og kurvetilpasning for atombinding Dimensjonering av en trykktank Kurvetilpasning av en strekktest Materialvariabilitet i tre RSS for en tilfeldig lastretning Testing av bruddenergi Bruddseighet for glassfiber Kurvetilpasning for en relaksasjonstest

Kurvetilpasning av en strekktest¶

Eksperimentelle data for spenning (sig_exp) versus tøyning (eps_exp) er gitt under:

eps_exp = [0.0000,0.0005,0.0010,0.0015,0.0020,0.0025,0.0030,0.0035,0.0040,0.0045,0.0050,0.0055,0.0060,0.0065,
           0.0070,0.0075,0.0080,0.0085,0.0090,0.0095,0.0100,0.0105,0.0110,0.0115,0.0120,0.0125,0.0130,0.0135,
           0.0140,0.0145,0.0150,0.0155,0.0160,0.0165,0.0170,0.0175,0.0180,0.0185,0.0190,0.0195,0.0200]

sig_exp = [0.000,39.001,77.995,117.023,155.891,194.676,217.633,227.000,233.000,237.098,240.000,242.599,245.000,
           247.128,248.988,250.604,252.000,253.201,254.243,255.164,256.000,256.783,257.521,258.216,258.868,259.481,
           260.055,260.592,261.094,261.563,262.000,262.407,262.786,263.138,263.465,263.769,264.052,264.314,264.559,
           264.787,265.000]

Grafisk representasjon av datapunkt:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

plt.figure(figsize=(4,4))
plt.plot(eps_exp, sig_exp,'.-',c='red')
plt.xlim(0,0.02)
plt.ylim(0,)
plt.grid()
plt.show()

Tilpasser parameter i Ramberg-Osgood sammenhengen som er introdusert i Plastisitet:

\begin{equation} \varepsilon = \frac{\sigma}{E} + a \frac{\sigma}{E} \bigg( \frac{\sigma}{\sigma_0} \bigg)^{n-1} \tag{1} \end{equation}

hvor

\begin{equation} a=\varepsilon_{a} \frac{E}{\sigma_0} \tag{2} \end{equation}

Finner et sett av $E$, $\varepsilon_a$, $\sigma_0$ og $n$ som visuelt ser ut til å passe best:

sig=np.linspace(0,275,100)

E=78000
epsa=0.001
sig0=233
n=21

a=epsa*E/sig0
eps0 = sig0/E + epsa

eps=sig/E + a*(sig/E)*(sig/sig0)**(n-1)

plt.figure(figsize=(10,8))
plt.plot(eps_exp, sig_exp,'.',c='red', label='Eksperimentelt')
plt.plot(eps, sig, '-', c='black', linewidth=0.5, label='Ramberg-Osgood')
plt.plot((epsa, epsa+sig0/E),(0,sig0), '--', c='black', linewidth=1, label='{}% offset'.format(epsa*100))
plt.legend()
plt.xlim(0,0.02)
plt.ylim(0,)
plt.grid()
plt.show()

For løsningen over, er $\varepsilon_a = 0.001$ som betyr at $\sigma_0$ tilsvarer flytestyrken ved 0.1% offset av tøyning fra den lineære delen av kurven.

Flytestyrken ved 0.2% offset blir høyere som illustrert under:

plt.figure(figsize=(10,8))
plt.plot(eps_exp, sig_exp,'.',c='red', label='Eksperimentelt')
plt.plot(eps, sig, '-', c='black', linewidth=0.5, label='Ramberg-Osgood')
plt.plot((epsa, epsa+sig0/E),(0,sig0), '--', c='black', linewidth=1, label='{}% offset'.format(epsa*100))
plt.plot((0.002, 0.002+250/E),(0,250), '--', c='blue', linewidth=1, label = '0.2% offset')
plt.legend()
plt.xlim(0,0.02)
plt.ylim(0,)
plt.grid()
plt.show()

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