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@mmcdara SIr! your guide did help m...

Aung 60
September 09 2024
0 0

@mmcdara SIr! your guide did help my problem.. i am sorry that i was a little hurry.
.....Here is another problem.. i change model equation  and it give again error....i dont' know what error means

my constraints for parameters are 
E[isnt] is between 13*10^9 and 16*10^9
P[i] and lambda[i] values are positive

i dont' know constraints for lambda[90] 

can you check and help? please.


 

> 

restart
> 

with(Statistics)with(plots)with(Optimization)with(LinearAlgebra)
> 

E[1] := 126*10^9E[2] := 11*10^9G[12] := 6.6*10^9G_0__12 := 10.1*10^9nu[12] := .28E_0__2 := 15.5*10^9
> 

true_strain := [0, .406915, .710106, .989362, 1.28457, 1.53989, 1.86702, 2.21011, 2.625, 2.99202]; true_stress := [0, 46.0227*10^6, 81.8182*10^6, 109.091*10^6, 138.068*10^6, 163.636*10^6, 194.318*10^6, 219.886*10^6, 248.864*10^6, 267.614*10^6]; epsilon_dot := 10^(-4)
> 

true_strain := true_strain[2..-1];
true_stress := true_stress[2..-1];


N := nops(true_strain);

sigma_t := strain -> E[1]/E[inst]*
(1-lambda[90]*
(1-(epsilon_dot/strain)*
(add((P[i]/lambda[i])*(1-exp(-lambda[i]*strain/epsilon_dot), i=1..N)))))
*strain;

[.406915, .710106, .989362, 1.28457, 1.53989, 1.86702, 2.21011, 2.625, 2.99202]

  

[46022700.00, 81818200.00, 109091000.0, 138068000.0, 163636000.0, 194318000.0, 219886000.0, 248864000.0, 267614000.0]

  

9

  

proc (strain) options operator, arrow; E[1]*(1-lambda[90]*(1-epsilon_dot*add(P[i]*(1-exp(-lambda[i]*strain/epsilon_dot), i = 1 .. N)/lambda[i])/strain))*strain/E[inst] end proc

 (1)
> 

Digits := 15:
obj := add(
         (
           sigma_t(true_strain[i])
           -
           true_stress[i]
         )^2
         , i=1..N
       );

opt := NLPSolve(obj, {seq(lambda[j] >= -1, j=1..N), seq(lambda[j] <= 0, j=1..N)} );

(51271290000.0000*(1-lambda[90]*(1-0.245751569738152e-3*P[1]-0.245751569738152e-3/lambda[1]+(-0.245751569738152e-3+0.245751569738152e-3*exp(-4069.150000*lambda[1]), -0.245751569738152e-3*(1 = 1 .. 9))))/E[inst]-46022700.00)^2+(89473356000.0000*(1-lambda[90]*(1-0.140824045987500e-3*P[2]-0.140824045987500e-3/lambda[2]+(-0.140824045987500e-3+0.140824045987500e-3*exp(-7101.060000*lambda[2]), -0.140824045987500e-3*(2 = 1 .. 9))))/E[inst]-81818200.00)^2+(124659612000.000*(1-lambda[90]*(1-0.101075238385950e-3*P[3]-0.101075238385950e-3/lambda[3]+(-0.101075238385950e-3+0.101075238385950e-3*exp(-9893.620000*lambda[3]), -0.101075238385950e-3*(3 = 1 .. 9))))/E[inst]-109091000.0)^2+(161855820000.000*(1-lambda[90]*(1-0.778470616626575e-4*P[4]-0.778470616626575e-4/lambda[4]+(-0.778470616626575e-4+0.778470616626575e-4*exp(-12845.70000*lambda[4]), -0.778470616626575e-4*(4 = 1 .. 9))))/E[inst]-138068000.0)^2+(194026140000.000*(1-lambda[90]*(1-0.649397034853139e-4*P[5]-0.649397034853139e-4/lambda[5]+(-0.649397034853139e-4+0.649397034853139e-4*exp(-15398.90000*lambda[5]), -0.649397034853139e-4*(5 = 1 .. 9))))/E[inst]-163636000.0)^2+(235244520000.000*(1-lambda[90]*(1-0.535612901843580e-4*P[6]-0.535612901843580e-4/lambda[6]+(-0.535612901843580e-4+0.535612901843580e-4*exp(-18670.20000*lambda[6]), -0.535612901843580e-4*(6 = 1 .. 9))))/E[inst]-194318000.0)^2+(278473860000.000*(1-lambda[90]*(1-0.452466166842374e-4*P[7]-0.452466166842374e-4/lambda[7]+(-0.452466166842374e-4+0.452466166842374e-4*exp(-22101.10000*lambda[7]), -0.452466166842374e-4*(7 = 1 .. 9))))/E[inst]-219886000.0)^2+(330750000000.000*(1-lambda[90]*(1-0.380952380952381e-4*P[8]-0.380952380952381e-4/lambda[8]+(-0.380952380952381e-4+0.380952380952381e-4*exp(-26250.000*lambda[8]), -0.380952380952381e-4*(8 = 1 .. 9))))/E[inst]-248864000.0)^2+(376994520000.000*(1-lambda[90]*(1-0.334222364823765e-4*P[9]-0.334222364823765e-4/lambda[9]+(-0.334222364823765e-4+0.334222364823765e-4*exp(-29920.20000*lambda[9]), -0.334222364823765e-4*(9 = 1 .. 9))))/E[inst]-267614000.0)^2

  

Error, (in simpl/reloprod) invalid terms in product: 1 .. 9

  
> 

indets(obj, name)

Error, (in simpl/reloprod) invalid terms in product: 1 .. 9

  
> 

display(
  ScatterPlot(true_strain, true_stress, symbol=circle, color=blue),
  pointplot([[0.001$2]], symbol=circle, color=blue),
  plot(eval(sigma_t(Sigma), opt[2]), Sigma=0.001..max(true_strain), color=red)
)

Error, invalid input: eval received opt[2], which is not valid for its 2nd argument, eqns

  
> 

display(
  ScatterPlot(true_strain, true_stress, symbol=circle, color=blue),
  pointplot([[0.001$2]], symbol=circle, color=blue),
  plot(eval(sigma_t(Sigma), opt[2]), Sigma=0.001..max(true_strain), color=red),
  axis[2]=[mode=log]
)

Error, invalid input: eval received opt[2], which is not valid for its 2nd argument, eqns

  
> 

opt := NLPSolve(
         obj,
         {
           seq(P[j] >= 10, j=1..N),
           seq(lambda[j] >= -1, j=1..N),
           seq(lambda[j] <= 0, j=1..N)
         }
       );

display(
  ScatterPlot(true_strain, true_stress, symbol=circle, color=blue),
  pointplot([[0.001$2]], symbol=circle, color=blue),
  plot(eval(sigma_t(Sigma), opt[2]), Sigma=0.001..max(true_strain), color=red)
)

Error, (in simpl/reloprod) invalid terms in product: 1 .. 9

  

Error, invalid input: eval received opt[2], which is not valid for its 2nd argument, eqns

  
> 

 


 

Download 90As4_mmcdara.mw

@mmcdara Sir....it not give symboli...

Aung 60
September 08 2024
0 0

@mmcdara 
Sir....it not give symbolically answer....it should need to define n terms...
is there any way can i get general answer...i don't want to define n  at this step..here is my mfile..
it is ok no matter what ...i just want the symbolic final answer...may be my calculation process is not right...so it give back with integration term


 

> 

restart:
> 

with(IntegrationTools):
> 

``
> 


eq1 := int(sum(A[i]*lambda[i]*exp(-lambda[i]*(t - tau)), i = 1 .. n)*tau, tau = 0 .. t);

int((sum(A[i]*lambda[i]*exp(-lambda[i]*(t-tau)), i = 1 .. n))*tau, tau = 0 .. t)

 (1)
> 

eq2 := IntegrationTools:-Expand(eq1);

int((sum(A[i]*lambda[i]*exp(lambda[i]*tau)/exp(lambda[i]*t), i = 1 .. n))*tau, tau = 0 .. t)

 (2)
> 

eq2 := subsindets(eq2, specfunc({specfunc(sum),name=range},int),
                   u->Sum(Int(op([1,1],u),op(2,u)),op([1,2],u)));

int((sum(A[i]*lambda[i]*exp(lambda[i]*tau)/exp(lambda[i]*t), i = 1 .. n))*tau, tau = 0 .. t)

 (3)
> 

simplify(eq2);

int((sum(A[i]*lambda[i]*exp(-lambda[i]*(t-tau)), i = 1 .. n))*tau, tau = 0 .. t)

 (4)
> 

value(%);

int((sum(A[i]*lambda[i]*exp(-lambda[i]*(t-tau)), i = 1 .. n))*tau, tau = 0 .. t)

 (5)
> 

simplify(combine(%));

int((sum(A[i]*lambda[i]*exp(-lambda[i]*(t-tau)), i = 1 .. n))*tau, tau = 0 .. t)

 (6)
> 

 

> 

NULL
> 

NULL
> 

NULL


 

Download solution_of_integral.mw

@mmcdara  sir...i still make a mist...

Aung 60
September 08 2024
0 0

@mmcdara 
sir...i still make a mistake....i think..model...
here is my integration....
i try normally it gives zero...but i try watching from maple Help 
it can't solve...please check this and then i can recaculate the model 

 

> 

restart:
> 

with(IntegrationTools):
> 

``
> 


eq1 := int(sum(A[i]*lambda[i]*exp(-lambda[i]*(t - tau)), i = 1 .. n)*tau, tau = 0 .. t);

int((sum(A[i]*lambda[i]*exp(-lambda[i]*(t-tau)), i = 1 .. n))*tau, tau = 0 .. t)

 (1)
> 

eq2 := IntegrationTools:-Expand(eq1);

int((sum(A[i]*lambda[i]*exp(lambda[i]*tau)/exp(lambda[i]*t), i = 1 .. n))*tau, tau = 0 .. t)

 (2)
> 

eq2 := subsindets(eq2, specfunc({specfunc(sum),name=range},int),
                   u->Sum(Int(op([1,1],u),op(2,u)),op([1,2],u)));

int((sum(A[i]*lambda[i]*exp(lambda[i]*tau)/exp(lambda[i]*t), i = 1 .. n))*tau, tau = 0 .. t)

 (3)
> 

simplify(eq2);

int((sum(A[i]*lambda[i]*exp(-lambda[i]*(t-tau)), i = 1 .. n))*tau, tau = 0 .. t)

 (4)
> 

value(%);

int((sum(A[i]*lambda[i]*exp(-lambda[i]*(t-tau)), i = 1 .. n))*tau, tau = 0 .. t)

 (5)
> 

simplify(combine(%));

int((sum(A[i]*lambda[i]*exp(-lambda[i]*(t-tau)), i = 1 .. n))*tau, tau = 0 .. t)

 (6)
> 

 

> 

NULL
> 

NULL
> 

NULL


 

Download solution_of_integral.mw
it is only integration part....

@mmcdara Yes sir ..it is an indeed ...

Aung 60
June 17 2024
0 0

@mmcdara Yes sir ..it is an indeed viscoelastic problem. thank you i have experimental data and i'm gonna try like the guidelines you described. actually..i already fit the data using NonLinearFit command but i'm a little bit not faimliar with numberical calculation or optimization problem behind the sense..so i try to get this..thank u Sir.

@acer Sir, i have the same solution...

Aung 60
June 01 2024
0 0

@acer Sir, i have the same solution to this..this is right...but can you use a little simpler command..i mean i don't understand behind the idea of this line
temp := subsindets(temp, specfunc({specfunc(sum),name=range},int),
                   u->Sum(Int(op([1,1],u),op(2,u)),op([1,2],u)));

@Preben Alsholm Sir ..what you mean...

Aung 60
April 11 2024
0 0

@Preben Alsholm Sir ..what you mean is that ..it can be used directly in the model and without changing to standard function we can get the values of k from curve fitting?..right?

@Preben Alsholm thank u Sir i just ...

Aung 60
April 10 2024
0 0

@Preben Alsholm thank u Sir
i just want k as symbolic parameters..is there any other way k is as symbolic k?

@dharr thank u Sir.  i don...

Aung 60
April 10 2024
0 0

@dharr thank u Sir.

 i don't know k , k is a material constant and i actually think k can be get from curve fitting .(may be i'm wrong)...and can we get standard functinos ..with k symbolic...

@Scot Gould Sir.... actually...the ...

Aung 60
April 09 2024
0 0

@Scot Gould Sir.... actually...the equation is like this......this is the governing equation for damage accumulation model of composite under fatigue loading...i consider with sinusodial loading case
  i have some derivation...dD/dt = some function and....finally we get   

D(t) = 1-(1-C(1+k) integral 0 to N (sigma_a * sin(omega*t))^k dt........

if i can get closed form solution..i can plot damage accumlation vs number of load cylce ...
and and loading parttern is a choice .. ....i can get closed form with different type of loading history..for eaxmple ( constant trapezodial pattern)..but sinusoidal loading are actually easy and cheap to set up  experiment and ...the problem is i don't get closed form solution for integral term.

@Preben Alsholm Sir.... actually......

Aung 60
April 06 2024
0 0

@Preben Alsholm Sir.... actually...the equation is like this......this is the governing equation for damage accumulation model of composite under fatigue loading...i consider with sinusodial loading case
  i have some derivation...dD/dt = some function and....finally we get   

D(t) = 1-(1-C(1+k) integral 0 to N (sigma_a * sin(omega*t))^k dt........

if i can get closed form solution..i can plot damage accumlation vs number of load cylce ...
and and loading parttern is a choice .. ....i can get closed form with different type of loading history..for eaxmple ( constant trapezodial pattern)..but sinusoidal loading are actually easy and cheap to set up  experiment and ...the problem is i don't get closed form solution for integral term.

@aroche it's still show integra...

Aung 60
April 06 2024
0 0

@aroche it's still show integral..not solution...here is the problemintegral.mwintegral.mw

@Preben Alsholm yes.Sir..but the in...

Aung 60
April 04 2024
0 0

@Preben Alsholm yes.Sir..but please check this equation.....
d(n)=int( (sin(omega*t))^k,t=0..N
i have eqautino like this...and maple didn't solve that equation by default...is that maple crack..or this eqaution doesn't converge....is there any other way..to solve this eqaution

@acer thank you sir...but in my pro...

Aung 60
April 04 2024
0 0

@acer yes.Sir..but please check this equation.....
d(n)=int( (sin(omega*t))^k,t=0..N
i have eqautino like this...and maple didn't solve that equation by default...is that maple crack..or this eqaution doesn't converge....is there any other way..to solve this eqaution

@Preben Alsholm thank you sir...but...

Aung 60
April 04 2024
0 0

@Preben Alsholm thank you sir...but in my problem k is parameter have to decide..from experimental data with curve fitting..is there any way..to solve symbolically with k prescence...N is loading cycle ...and n is cycle to failure ..this is cyclic loading problem....
the problem is k is material constant.....can you help ..without getting rid of k....MaplePrimes24-04-04_int_bug.mw

i go to the axes properties......

Aung 60
December 27 2023
0 0

i go to the axes properties. and used tick marks..it doesn't change..can you please check my plot here is my m.file

45.mw

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