Appendix A: Basic Relations of Elasticity

Isotropic Material

Hooke Law 3D (Principal Stress and Strain)

{ σ }=[ D ]{ ε }     σ 1 = D 11 ε 1 + D 12 ε 2 + D 13 ε 3     σ 1 =( λ+2μ ) ε 1 +λ( ε 2 + ε 3 )     σ 1 =λ( ε 1 + ε 2 + ε 3 )+2μ ε 1     σ 1 =K ε kk +2μ e 1       with  ε kk =( ε 1 + ε 2 + ε 3 ) and e 1 = ε 1 −1/3( ε 1 + ε 2 + ε 3 ) { σ }=[ D ]{ ε } ; [ D ]= E ( 1+ν )( 1−2ν ) [ 1−ν ν ν 0 0 0 1−ν ν 0 0 0 1−ν 0 0 0 1−2ν 2 0 0 Symm. 1−2ν 2 0 1−2ν 2 ] { ε }=[ C ]{ σ } ; [ C ]=[ 1 E −ν E −ν E 0 0 0 1 E −ν E 0 0 0 1 E 0 0 0 2( 1+ν ) E 0 0 Symm. 2( 1+ν ) E 0 2( 1+ν ) E ]

Hooke Law for 2D Plan Stress

{ σ } = [ H ] { ε }

σ 1 = H 11 ε 1 + H 12 ε 2

{ σ } = [ H ] { ε } ; [ H ] = E 1 − ν 2 [ 1 ν 0 0 0 ν 1 0 0 0 0 0 1 − ν 2 0 0 0 0 0 1 − ν 2 0 0 0 0 0 1 − ν 2 ]

{ ε } = [ C ] { σ } ; [ C ] = 1 E [ 1 − ν 0 0 0 − ν 1 0 0 0 0 0 2 ( 1 + ν ) 0 0 0 0 0 2 ( 1 + ν ) 0 0 0 0 0 2 ( 1 + ν ) ]

Hooke Law for 2D Plane Strain

{ σ } = [ H ] { ε } ; [ H ] = E ( 1 + ν ) ( 1 − 2 ν ) [ 1 − ν ν 0 ν 1 − ν 0 0 0 1 − 2 ν 2 ]

{ ε } = [ C ] { σ } ; [ C ] = 1 + ν E [ 1 − ν − ν 0 − ν 1 − ν 0 0 0 2 ]

Table 1. Material Constants Relations
  E, ν E,G E,B G, ν G, B B, ν λ , μ
E E E E 2(1+v)G 9 B G 3 B + G 3(1-2v)B ( 3 λ + 2 μ ) μ λ + μ
G = μ E 2 ( 1 + v ) G 3 E B 9 B − E G G 3 ( 1 − 2 v ) B 2 ( 1 + v ) μ
B=K E 3 ( 1 − 2 v ) E G 9 G − 3 E B 2 ( 1 + v ) G 3 ( 1 − 2 v ) B B 3 λ + 2 μ 3
ν ν E − 2 G 2 G 3 B − E 6 B ν 3 B − 2 G 6 B + 2 G ν λ 2 ( λ + μ )
λ E v ( 1 + v ) ( 1 − 2 v ) ( E − 2 G ) G 3 G − E ( 3 B − E ) 3 B 9 B − E 2 G v 1 − 2 v 3 B − 2 G 3 3 B v ( 1 + v ) λ

Orthotropic Material

General 3D Orthotropic Case

The strain-stress relations are defined using 9 material constants:
  • Three Young's modulus in orthotropic directions 1, 2 and 3: E 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIXaaabeaaaaa@37A7@ , E 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIXaaabeaaaaa@37A7@ , E 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIXaaabeaaaaa@37A7@
  • Three shear modulus in planes 12, 13 and 23: G 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIXaaabeaaaaa@37A7@ , G 13 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIXaaabeaaaaa@37A7@ , G 23 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIXaaabeaaaaa@37A7@
  • Three Poisson ratio's satisfying the relations:

    ν 12 E 1 = ν 21 E 2   ;   ν 13 E 1 = ν 31 E 3   ;   ν 23 E 2 = ν 32 E 3

    1 − ν 12 ν 21 > 0 ; 1 − ν 13 ν 31 > 0 ; 1 − ν 23 ν 32 > 0

    1 − ν 12 ν 21 − ν 13 ν 31 − ν 23 ν 32 − ν 12 ν 23 ν 31 − ν 21 ν 13 ν 32 > 0

    { ε } = [ C ] { σ } ; [ C ] = [ 1 E 1 − ν 21 E 2 − ν 31 E 3 0 0 0 − ν 12 E 1 1 E 2 − ν 32 E 0 0 0 − ν 13 E 1 − ν 23 E 2 1 E 3 0 0 0 0 0 0 1 G 12 0 0 0 0 0 0 1 G 13 0 0 0 0 0 0 1 G 23 ]

2D In-plane Orthotropic Material

  • Orthotropic plane 1-2, isotropic plane 2-3
    • Orthotropy coefficients in the plane 1-2: E 1 ,   E 2 ,   ν 12 ,   G 12
    • Isotropy coefficients in plane 2-3: E 2 ,   ν
  • Five independent coefficients

    { ε } = [ C ] { σ } ; [ C ] = [ 1 E 1 − ν 12 E 1 − ν 12 E 1 0 0 0 − ν 12 E 1 1 E 2 − ν E 2 0 0 0 − ν 12 E 1 − ν E 2 1 E 2 0 0 0 0 0 0 1 G 12 0 0 0 0 0 0 1 G 12 0 0 0 0 0 0 2 ( 1 + ν ) E 2 ]

Stiffness Matrix of Beam Element

Terms of the stiffness matrix:

[ k ]=[ EA L 0 0 0 0 0 − K 11 0 0 0 0 0 12E I 3 L 3 ( 1+ ϕ 2 ) 0 0 0 L 2 K 22 0 − K 22 0 0 0 K 26 12E I 2 L 3 ( 1+ ϕ 2 ) 0 − L 2 K 33 0 0 0 − K 33 0 K 35 0 GJ L 0 0 0 0 0 − K 44 0 0 ( 4+ ϕ 3 )E I 2 L( 1+ ϕ 3 ) 0 0 0 − K 35 0 2− ϕ 3 4+ ϕ 3 K 55 0 ( 4+ ϕ 2 )E I 3 L( 1+ ϕ 2 ) 0 − K 26 0 0 0 2− ϕ 2 4+ ϕ 2 K 66 K 11 0 0 0 0 0 K 22 0 0 0 − K 26 Symm. K 33 0 − K 35 0 K 44 0 0 K 55 0 K 66 ]

For a rectangle cross-section:

ϕ 2 = 144( 1+ν ) I 3 5A L 2

ϕ 3 = 144( 1+ν ) I 2 5A L 2

I = b h 3 12