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Creep Curve Univ.-Prof. Dr.-Ing. habil. J. BETTEN RWTH University Aachen Mathematical Models in Materials Science and Continuum Mechanics Augustinerbach 4-20 D-52056 A a c h e n , Germany betten@mmw.rwth-aachen.de Abstract Using MAPLE V, Release 10, a creep curve has been formulated, which represents the mechanical behavior of metals, polymers, and ceramics, for instance. Keywords: exponential description; primary, secondary, and tertiary creep stage; creep rate and acceleration

Creep tests are carried out on specimens loaded, e.g., in tension or compression, usually at constant load, inside a furnace which is maintained at a constant temperature. The extension of the specimen is measured as a function of time. A typical creep curve for metals, polymers, and ceramics exsists of three parts and is represented in this Maple worksheet. The primary or transient creep is characterized by a monotonic decrease in the rate of creep. Creep deformations of the secondary stage are large and of a similar character to "pure" plastic deformations. The tertiary creep phase is accompanied by the formation of microscopic cracks on the grain boundaries, so that damage- accumulation occurs. In the past three decades there has been considerable progress and significant advances made in the development of fundamental concepts of creep and damage mechanics an their application to solve practical engineering problems (BETTEN, J.: Creep Mechanics, 2nd Edition, Springer - Verlag, Berlin / Heidelberg / New York 2005).

restart: epsilon[creep](t):=A[11]*(1-exp(-A[12]*sqrt(t)))+ A[21]*t+A[31]*(exp(A[32]*t^n)-1);

NiM+LSYlKGVwc2lsb25HNiMlJmNyZWVwRzYjJSJ0RywoKiYmJSJBRzYjIiM2IiIiLCYiIiIiIiItJSRleHBHNiMsJComJiUiQUc2IyIjNyIiIiklInRHIyIiIiIiIyIiIiEiIiEiIiIiIiIiIiomJiUiQUc2IyIjQCIiIiUidEciIiIiIiIqJiYlIkFHNiMiI0oiIiIsJi0lJGV4cEc2IyomJiUiQUc2IyIjSyIiIiklInRHJSJuRyIiIiIiIiIiIiEiIiIiIiIiIg==

Digits:=5: epsilon[c](t):= subs({A[11]=0.4,A[12]=5,A[31]=0.02,A[32]=3,n=10},);

NiM+LSYlKGVwc2lsb25HNiMlImNHNiMlInRHLCokIiNRISIjIiIiKiYkIiIlISIiIiIiLSUkZXhwRzYjLCQqJiIiJiIiIiklInRHIyIiIiIiIyIiIiEiIiIiIiEiIiomJiUiQUc2IyIjQCIiIiUidEciIiIiIiIqJiQiIiMhIiMiIiItJSRleHBHNiMsJComIiIkIiIiKSUidEciIzUiIiIiIiIiIiIiIiI=

epsilon[c](0):=evalf(subs(t=0,));

NiM+LSYlKGVwc2lsb25HNiMlImNHNiMiIiEkIiIhIiIh

epsilon[c](1):=evalf(subs(t=1,));

NiM+LSYlKGVwc2lsb25HNiMlImNHNiMiIiIsJiQiJi16KCEiJiIiIiYlIkFHNiMiI0AiIiI=

A[21]:=solve(epsilon[c](1)=1,A[21]);

NiM+JiUiQUc2IyIjQCQiJik0QSEiJg==

alias(H=Heaviside,th=thickness): plot1:=plot(epsilon[c](t),t=0..1,color=black,th=2): plot2:=plot({epsilon[c](1),epsilon[c](1)*H(t-1)}, t=0..1.001,color=black, title="Creep Curve # exponential description"): plots[display]({plot1,plot2});

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

time_derivative(t):=diff(epsilon[c](t),t);

NiM+LSUwdGltZV9kZXJpdmF0aXZlRzYjJSJ0RywoKigkIiYrKyIhIiUiIiIpJSJ0RyMiIiIiIiMhIiItJSRleHBHNiMsJComIiImIiIiKSUidEcjIiIiIiIjIiIiISIiIiIiIiIiJiUiQUc2IyIjQCIiIiooJCIjZyEiIyIiIiklInRHIiIqIiIiLSUkZXhwRzYjLCQqJiIiJCIiIiklInRHIiM1IiIiIiIiIiIiIiIi

time_derivative(0):=infinity;

NiM+LSUwdGltZV9kZXJpdmF0aXZlRzYjIiIhJSlpbmZpbml0eUc=

time_derivative(1):=evalf(subs({A[21]=0.22098,t=1},%%));

NiM+LSUwdGltZV9kZXJpdmF0aXZlRzYjIiIiJCImIUc3ISIk

plot3:=plot(time_derivative(t),t=0..1,0..2,color=black,th=2, title="Time Derivative of the Creep Curve"): plot4:=plot({2,2*H(t-1)},t=0..1.001,color=black): plots[display]({plot3,plot4});

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

plots[display]({plot1,plot2,plot3,plot4});

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The creep curve exsists of three parts: restart: parameters_of_the_primary_term:=A[11], A[12];

NiM+JT9wYXJhbWV0ZXJzX29mX3RoZV9wcmltYXJ5X3Rlcm1HNiQmJSJBRzYjIiM2JiUiQUc2IyIjNw==

parameters_of_the_scondary_term:=A[21]=K*sigma^m; # NORTON-BAILEY

NiM+JUBwYXJhbWV0ZXJzX29mX3RoZV9zY29uZGFyeV90ZXJtRy8mJSJBRzYjIiNAKiYlIktHIiIiKSUmc2lnbWFHJSJtRyIiIg==

parameters_of_the_tertiary_term:=A[31],A[32],n;

NiM+JUBwYXJhbWV0ZXJzX29mX3RoZV90ZXJ0aWFyeV90ZXJtRzYlJiUiQUc2IyIjSiYlIkFHNiMiI0slIm5H

For the primary creep the sqrt(t)-law has been assumed [BETTEN, J.: Creep Mechanics, 2nd Edition, Springer-Verlag, Berlin / Heidelberg / New York, 2005]. The exponent n in the tertiary term regulates the tangent of the creep curve at the creep rupture time. The creep parameters can be determined by suitable Experiments, some of which are discussed in the above mentioned book.

restart: epsilon[creep](t):=A[11]*(1-exp(-A[12]*sqrt(t)))+ A[21]*t+A[31]*(exp(A[32]*t^n)-1);

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creep_rate(t):=diff(epsilon[creep](t),t);

NiM+LSUrY3JlZXBfcmF0ZUc2IyUidEcsKComIyIiIiIiIyIiIioqJiUiQUc2IyIjNiIiIiYlIkFHNiMiIzciIiIpJSJ0RyMiIiIiIiMhIiItJSRleHBHNiMsJComJiUiQUc2IyIjNyIiIiklInRHIyIiIiIiIyIiIiEiIiIiIiIiIiIiIiYlIkFHNiMiI0AiIiIqLiYlIkFHNiMiI0oiIiImJSJBRzYjIiNLIiIiKSUidEclIm5HIiIiJSJuRyIiIiUidEchIiItJSRleHBHNiMqJiYlIkFHNiMiI0siIiIpJSJ0RyUibkciIiIiIiIiIiI=

creep_rate(0):=infinity;

NiM+LSUrY3JlZXBfcmF0ZUc2IyIiISUpaW5maW5pdHlH

creep_rate(1):=subs(t=1,);

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Digits:=5: creep_rate(1):=evalf(subs({A[11]=0.4,A[12]=5, A[21]=0.22098,A[31]=0.02,A[32]=3,n=10},));

NiM+LSUrY3JlZXBfcmF0ZUc2IyIiIiQiJiFHNyEiJA==

Creep_rate(t):=evalf(subs({A[11]=0.4,A[12]=5, A[21]=0.22098,A[31]=0.02,A[32]=3,n=10},creep_rate(t)));

NiM+LSUrQ3JlZXBfcmF0ZUc2IyUidEcsKCooJCImKysiISIlIiIiKSUidEcjIiIiIiIjISIiLSUkZXhwRzYjLCQqJiQiIiYiIiEiIiIpJSJ0RyMiIiIiIiMiIiIhIiIiIiIiIiIkIiYpNEEhIiYiIiIqKCQiI2chIiMiIiIpJSJ0RyIiKiIiIi0lJGV4cEc2IywkKiYkIiIkIiIhIiIiKSUidEciIzUiIiIiIiIiIiIiIiI=

acceleration(t):=diff(epsilon[creep](t),t$2);

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Acceleration(t):=evalf(subs({A[11]=0.4,A[12]=5, A[31]=0.02,A[32]=3,n=10},));

NiM+LSUtQWNjZWxlcmF0aW9uRzYjJSJ0RywqKigkIiYrKyYhIiYiIiIpJSJ0RyMiIiQiIiMhIiItJSRleHBHNiMsJComJCIiJiIiISIiIiklInRHIyIiIiIiIyIiIiEiIiIiIiEiIiooJCImK10jISIlIiIiJSJ0RyEiIi0lJGV4cEc2IywkKiYkIiImIiIhIiIiKSUidEcjIiIiIiIjIiIiISIiIiIiISIiKigkIiRTJiEiIyIiIiklInRHIiIpIiIiLSUkZXhwRzYjLCQqJiQiIiQiIiEiIiIpJSJ0RyIjNSIiIiIiIiIiIiIiIiooJCIlKz0hIiMiIiIpJSJ0RyIjPSIiIi0lJGV4cEc2IywkKiYkIiIkIiIhIiIiKSUidEciIzUiIiIiIiIiIiIiIiI=

alias(H=Heaviside, th=thickness): plot1:=plot({Creep_rate(t),Acceleration(t)}, t=0..1,-10..10,color=black,th=2): plot2:=plot({10,-10,10*H(t-1),-10*H(t-1)}, t=0..1.001,color=black, title="Creep Rate and Acceleration"): plots[display]({plot1,plot2});

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