Naloga 4

Podatki

In[54]:=

σij = {{σ11, 0, 0}, {0, (σ11 + σ33)/2, 0}, {0, 0, σ33}} ; σij // MatrixForm // Print["[σij]=", #] & ; en = {enx, eny, enz} ;

[σij]= ( ... 0 0 0 σ33

Rešitev

Napetosti vektor v ravnini z normalo e_n,   normalna in strižna napetost v tej ravnini .

In[57]:=

σn = σij . en ; σnn = σn . en // FullSimplify τn = (σn . σn - σnn^2 )^(1/2)// FullSimplify

Out[58]=

enx^2 σ11 + enz^2 σ33 + 1/2 eny^2 (σ11 + σ33)

Out[59]=

(enx^2 σ11^2 + enz^2 σ33^2 + 1/4 eny^2 (σ11 + σ33)^2 - (enx^2 σ11 + enz^2 σ33 + 1/2 eny^2 (σ11 + σ33))^2)^(1/2)

In[60]:=

s = Solve[{σnn == (σ11 + σ33)/2, en . en1, τn == (σ11 - σ33)/4}, {enx, eny, enz}] // Simplify

Out[60]=

{{enx -1/(2 2^(1/2)), enz -1/(2 2^(1/2)), eny -3^(1/2)/2}, {enx ... y -3^(1/2)/2}, {enx1/(2 2^(1/2)), enz1/(2 2^(1/2)), eny3^(1/2)/2}}

Kako izgledajo vse te ravnine?

In[61]:=

g1 = Polygon[{{1/enx, 0, 0}, {0, 1/eny, 0}, {0, 0, 1/enz}}] /. s ; Graphics3D[g1] // Show

[Graphics:../HTMLFiles/vaja04_97.gif]

Out[62]=

⁃Graphics3D⁃

In[63]:=

Clear[σij, en, σn, σnn, τn, enx, eny, enz, s, σ11, σ33, g1] ;

Created by Mathematica  (November 7, 2003)