@Axel Vogt 

How did you get c? How did you exclude sign function in it? And how did you get this equation 1+1/2*(-1+1/(Pi^(1/2)))*sin(b)? And one more quation why this representation better for maple?

@Axel Vogt 

Any way thank you for help. I think need to solve concrete task with you recommendation.

S1,S2,S3 have concrete value that we know before calculation and do calculation after that.

@Axel Vogt 

Apologies for the long absence. 

1) Yes. I have tried to do some numerical calculation in a different cases and hypothesis plastic incompressibility are applied here, so k11+k22+k33=0. That was included to this sheet.

2) In the first post for simplification task I have used S2 not equal 0, S1=S3=0.

In the end of attachment file there are 3 integral k11, k22, k23 respectively. This file is initial from that I have copied previous sheet.

http://yadi.sk/d/Eb3cyZelHgHju

k11=Int(Int(Int(-(0.13808e-3*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w)))*cos(a)*cos(b)*(S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)+S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)+S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)+S2*sin(a)*cos(b)^2*sin(w)+S3*cos(b)*sin(w)*sin(b)-100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11-100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22+100.*sin(b)*cos(b)*sin(w)*(k11+k22)-200.*sin(a)*cos(b)^2*sin(w)*k23+58.5*signum(-1.*S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)-1.*S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)-1.*S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)-1.*S2*sin(a)*cos(b)^2*sin(w)-1.*S3*cos(b)*sin(w)*sin(b)+100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11+100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22-100.*sin(b)*cos(b)*sin(w)*(k11+k22)+200.*sin(a)*cos(b)^2*sin(w)*k23))*Heaviside(S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)+S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)+S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)+S2*sin(a)*cos(b)^2*sin(w)+S3*cos(b)*sin(w)*sin(b)-100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11-100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22+100.*sin(b)*cos(b)*sin(w)*(k11+k22)-200.*sin(a)*cos(b)^2*sin(w)*k23-58.5)*signum(-1.*S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)-1.*S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)-1.*S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)-1.*S2*sin(a)*cos(b)^2*sin(w)-1.*S3*cos(b)*sin(w)*sin(b)+100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11+100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22-100.*sin(b)*cos(b)*sin(w)*(k11+k22)+200.*sin(a)*cos(b)^2*sin(w)*k23+58.5)/(1.-.2178*sin(b)), a = 0 .. 2*Pi), b = 0 .. (1/2)*Pi), w = 0 .. 2*Pi),

k22=Int(Int(Int(-(0.13808e-3*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w)))*sin(a)*cos(b)*(S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)+S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)+S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)+S2*sin(a)*cos(b)^2*sin(w)+S3*cos(b)*sin(w)*sin(b)-100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11-100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22+100.*sin(b)*cos(b)*sin(w)*(k11+k22)-200.*sin(a)*cos(b)^2*sin(w)*k23+58.5*signum(-1.*S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)-1.*S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)-1.*S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)-1.*S2*sin(a)*cos(b)^2*sin(w)-1.*S3*cos(b)*sin(w)*sin(b)+100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11+100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22-100.*sin(b)*cos(b)*sin(w)*(k11+k22)+200.*sin(a)*cos(b)^2*sin(w)*k23))*Heaviside(S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)+S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)+S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)+S2*sin(a)*cos(b)^2*sin(w)+S3*cos(b)*sin(w)*sin(b)-100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11-100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22+100.*sin(b)*cos(b)*sin(w)*(k11+k22)-200.*sin(a)*cos(b)^2*sin(w)*k23-58.5)*signum(-1.*S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)-1.*S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)-1.*S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)-1.*S2*sin(a)*cos(b)^2*sin(w)-1.*S3*cos(b)*sin(w)*sin(b)+100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11+100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22-100.*sin(b)*cos(b)*sin(w)*(k11+k22)+200.*sin(a)*cos(b)^2*sin(w)*k23+58.5)/(1.-.2178*sin(b)), a = 0 .. 2*Pi), b = 0 .. (1/2)*Pi), w = 0 .. 2*Pi),

k23=Int(Int(Int(-(0.69039e-4*((-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)+sin(a)*cos(b)^2*sin(w)))*(S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)+S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)+S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)+S2*sin(a)*cos(b)^2*sin(w)+S3*cos(b)*sin(w)*sin(b)-100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11-100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22+100.*sin(b)*cos(b)*sin(w)*(k11+k22)-200.*sin(a)*cos(b)^2*sin(w)*k23+58.5*signum(-1.*S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)-1.*S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)-1.*S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)-1.*S2*sin(a)*cos(b)^2*sin(w)-1.*S3*cos(b)*sin(w)*sin(b)+100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11+100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22-100.*sin(b)*cos(b)*sin(w)*(k11+k22)+200.*sin(a)*cos(b)^2*sin(w)*k23))*Heaviside(S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)+S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)+S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)+S2*sin(a)*cos(b)^2*sin(w)+S3*cos(b)*sin(w)*sin(b)-100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11-100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22+100.*sin(b)*cos(b)*sin(w)*(k11+k22)-200.*sin(a)*cos(b)^2*sin(w)*k23-58.5)*signum(-1.*S1*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*cos(a)*cos(b)-1.*S1*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(a)*cos(b)-1.*S2*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*sin(b)-1.*S2*sin(a)*cos(b)^2*sin(w)-1.*S3*cos(b)*sin(w)*sin(b)+100.*cos(a)*cos(b)*(-1.*cos(a)*sin(w)*sin(b)-1.*sin(a)*cos(w))*k11+100.*sin(a)*cos(b)*(-1.*sin(a)*sin(b)*sin(w)+cos(a)*cos(w))*k22-100.*sin(b)*cos(b)*sin(w)*(k11+k22)+200.*sin(a)*cos(b)^2*sin(w)*k23+58.5)/(1.-.2178*sin(b)), a = 0 .. 2*Pi), b = 0 .. (1/2)*Pi), w = 0 .. 2*Pi)

@Axel Vogt 

Task 

This is the most common task that i could do in my experimental work.

In it we have k11 not eqvial k22 but from phycal reason it steal k11+k22=-k33 that was included.

Besides it has now right phycal constant. 

S1, S2, S3 - strain, they have specific values. Uper we have S1=S3=0, S2=200.

Thank you for your interest and help.

@Axel Vogt 

Yes, its model of the plastic deformation. In it we have two coordination system. We know about relation between b31 and T31 in first coord.(T31 I find with first procedure)  In the second coord I need to find mean of b31, using integration. In integration I always have problems.

With_par.mw

Download With_par.mw

@Axel Vogt 

The full task is to find the value of integral in a different combination. In investment you could find program that I use. 

As you can see the parameters aren’t forming trivial expression: 1000*k1+1000*k2+1000*k3 (that we have in case k3=k2-0,5 k1). I have tried to make simpler expression, but hurry with that. Forgot about k3=k2-0,5 k1, it going from physical reasons, but I didn’t see where I could find it in integral.

P/S I take min and max of function because sometimes Maple didn’t want to calculate integral. 

http://yadi.sk/d/QJKHv8jtGmSDA

@Axel Vogt 

Now i understend it more completely.

@Carl Love 

Which method do u use to solve it?

@Axel Vogt 

But why i couldnt understand. How did u find relations between the unknown without nimerical calculation? 

@Markiyan Hirnyk 

Yes, its work. It s easy and very good way to solve this task.

In the first time it had no solution, but it had wrong physical constant (my mistake. one from one material other from another), so when I put it right it was solved.

Thanks!!!

@Carl Love 

Ranges for ki (i=1,2,3) is (-3;3). Usually it is not more than +-1. 

P/S One hour it s ok for me. Yesterday i have tried fsolve, but after 4 hour there was no solution, in the morning i have thought that it needs to be put ranges for it in fsolve procedure. In what way have you solved it? 

@acer 

Its need to find, from system.

@Carl Love

I have more complex problem. Under integral i have 3 parameters. So after integration i will have system of 3 linear equations with 3 unknown.

I don’t sure that Maple could do numerical integration with even one parameter so I do more simple task.

Here the full task:

k1=Int(Int(Int(max(0., (0.9483573506e-3*(-1.*sin(a)*cos(w)-1.*cos(a)*sin(w)*sin(b)))*cos(a)*cos(b)^2*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3+58.5*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3))*Heaviside(-(1.*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3))*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3)-58.5)*Heaviside(1.-.98*cos(b)^2)/sqrt(1.-.98*cos(b)^2)), a = 0. .. 6.283185308), b = 0. .. 1.570796327), w = 0. .. 6.283185308)+Int(Int(Int(min(0., (0.9483573506e-3*(-1.*sin(a)*cos(w)-1.*cos(a)*sin(w)*sin(b)))*cos(a)*cos(b)^2*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3+58.5*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3))*Heaviside(-(1.*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3))*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3)-58.5)*Heaviside(1.-.98*cos(b)^2)/sqrt(1.-.98*cos(b)^2)), a = 0. .. 6.283185308), b = 0. .. 1.570796327), w = 0. .. 6.283185308)

k2=Int(Int(Int(max(0., (0.9483573506e-3*(cos(a)*cos(w)-1.*sin(a)*sin(b)*sin(w)))*sin(a)*cos(b)^2*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3+58.5*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3))*Heaviside(-(1.*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3))*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3)-58.5)*Heaviside(1.-.98*cos(b)^2)/sqrt(1.-.98*cos(b)^2)), a = 0. .. 6.283185308), b = 0. .. 1.570796327), w = 0. .. 6.283185308)+Int(Int(Int(min(0., (0.9483573506e-3*(cos(a)*cos(w)-1.*sin(a)*sin(b)*sin(w)))*sin(a)*cos(b)^2*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3+58.5*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3))*Heaviside(-(1.*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3))*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3)-58.5)*Heaviside(1.-.98*cos(b)^2)/sqrt(1.-.98*cos(b)^2)), a = 0. .. 6.283185308), b = 0. .. 1.570796327), w = 0. .. 6.283185308)

k3=Int(Int(Int(max(0., 0.9483573506e-3*cos(b)^2*sin(w)*sin(b)*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3+58.5*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3))*Heaviside(-(1.*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3))*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3)-58.5)*Heaviside(1.-.98*cos(b)^2)/sqrt(1.-.98*cos(b)^2)), a = 0. .. 6.283185308), b = 0. .. 1.570796327), w = 0. .. 6.283185308)+Int(Int(Int(min(0., 0.9483573506e-3*cos(b)^2*sin(w)*sin(b)*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3+58.5*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3))*Heaviside(-(1.*(200.*cos(b)*sin(w)*sin(b)-1000.*k1-1000.*k2-1000.*k3))*signum(-200.*cos(b)*sin(w)*sin(b)+1000.*k1+1000.*k2+1000.*k3)-58.5)*Heaviside(1.-.98*cos(b)^2)/sqrt(1.-.98*cos(b)^2)), a = 0. .. 6.283185308), b = 0. .. 1.570796327), w = 0. .. 6.283185308) 

P\S I use epsilon=10^(-2).