Roger

tensiones p.xmcd

AHP

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Tensiones
Rotación de ejes:
Datos :

4   v1 :=  −1  5  
v2 :=
3

v2 := 1
1

v2 := 5
2

2 0 0  σ :=  0 −6 0   0 0 −6  

Cálculos :

−1 ⋅ v1 ⋅ v2 + v1 ⋅ v2 1 1 2 2 v1
3

(

)

v2 = 0.2
3

v3 := v1 × v2

 −25.2   v3 =  4.2   21   0.617   α 1 =  −0.154   0.772    0.196   α 2 =  0.98   0.039    −0.762   α 3 =  0.127   0.635  

α 1 :=

v1 v1

α 2 :=

v2v2

α 3 :=

v3 v3

A := augment α 1 , α 2 , α 3

(

)

 0.617 0.196 −0.762   A =  −0.154 0.98 0.127   0.772 0.039 0.635  

σ t := A⋅ σ⋅ A

T

 −2.952 −0.762 3.81   σ t =  −0.762 −5.81 −0.952   3.81 −0.952 −1.238   2   σ p =  −6   −6  

σ p := eigenvals ( σ )

tensionesp.xmcd

AHP

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np := eigenvecs ( σ )

1 0 0   np =  0 1 0  0 0 1    −6   σ tp =  2   −6    −0.787 −0.617 0.355   ntp = −0.121 0.154 0.93   0.605 −0.772 −0.098  

σ tp := eigenvals ( σ t)

ntp := eigenvecs σ t

( )

Invariantes :
J1 :=

∑σ

p

J1 = −10

J2 := σp ⋅ σ p + σ p ⋅ σ p + σ p ⋅ σ p 1 2 2 3 3 1 J3 := σ p ⋅ σ p ⋅ σ p 1 2 3

J2 = 12 J3 = 72

Tensor desviador :
σ m :=
1 J1 3 j := 1 .. 3 0 0   5.333 s= 0 −2.667 0   0 0 −2.667  

σ m = −3.333

i := 1 .. 3

si , j := σ i , j − σ m⋅ δ ( i , j)

sp := eigenvals ( s)

I2 := − sp ⋅ sp + sp ⋅ sp +sp ⋅ sp 1 2 2 3 3 1

(

)

 5.333   sp =  −2.667   −2.667  
I2 = 21.333

Tensión efectiva (von Mises):
σ ef :=
3I2

σ ef = 8