Comp Mech-elast Linear

8/2/2019 Comp Mech-elast Linear

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Elementi di Meccanica Computazionale

Corso di Laurea in Ingegneria CivilePavia, 2010

A quick review:

three-dimensional elasticity

Ferdinando Auricchio

January 25, 2011

F.Auricchio (DMS - ROSE - IMATI) Review: notation & 3D elasticity January 25, 2011 1 / 15
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3D elasticity: compact notation I

Three-dimensional body: R3

Deformable body: variable distance between points

Field equations:

equilibrium eq. div + b = 0 in

constitutive eq. = D in

compatibility eq. = su in

with:

: stress (second-order) tensoru : displacement vectordiv : divergence operatorD : elasticity (fourth-order) tensor

: strain (second-order) tensorb : body force vectors : symmetric gradient operator

Boundary conditions:imposed displ. u = u on u

imposed forces n = t on t

with u & t: assigned data; n: outward normal to t; u t =

Linear elasticity assumption. Easy to generalize = () !F.Auricchio (DMS - ROSE - IMATI) Review: notation & 3D elasticity January 25, 2011 2 / 15
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3D elasticity: indicial notation I

Recall: summation convention, i.e. repeated indices imply summation

Recall: comma indicates derivation

Field equations:

equilibrium eq. ij,j + bi = 0 in

constitutive eq. ij = Dijklkl in

compatibility eq. ij =1

2

(ui,j + uj,i) in

with:

ij : stress (second-order) tensorui : displacement vectorDijkl : elasticity (fourth-order) tensor

ij : strain (second-order) tensorbi : body force vector

Boundary conditions:imposed displ. ui = ui on u

imposed forces ijnj = ti on t

with ui & ti: assigned data; ni : outward normal to t; u t = .F.Auricchio (DMS - ROSE - IMATI) Review: notation & 3D elasticity January 25, 2011 3 / 15
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3D elasticity: engineering notation I

We take advantage of symmetry, i.e.:

= T , = T , D = DT , D = Dtij = ji , ij = ji , Dijkl = Dklij , Dijkl = Dijlk

, stress (second-order) tensor, in vector notation:

= {11, 22, 33, 12, 23, 31}T

b, body force, in vector notation:b = {b1, b2, b3}

T

, strain (second-order) tensor, in vector notation:

= {11, 22, 33, 12, 23, 31}T

D elasticity (fourth-order) tensor, in matrix notation:

D =

D1111 D1122 D1133 D1112 D1123 D1113

D1122 D2222 D2233 D2212 D2223 D2213

D1133 D2233 D3333 D3312 D3323 D3313

D1112 D2212 D3312 D1212 D1223 D1213

D1123 D2223 D3323 D1223 D2323 D2313

D1113 D2213 D3313 D1212 D2313 D1313

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Field equations:

equilibrium eq.

11,1 + 12,2 + 13,3 + b1 = 0

12,1 + 22,2 + 23,3 + b2 = 0

13,1 + 23,2 + 33,3 + b3 = 0

constitutive eq.

11 = D111111 +D112222 + . . .

22 = D221111 +D222222 + . . .

33 =D

331111 +D

332222 + . . .

compatibility eq.

11 = u1,1

22 = u2,2

33 = u3,3

,

12 =1

2(u1,2 + u2,1)

23 =1

2(u2,3 + u3,2)

13 =1

2(u1,3 + u3,1)

Boundary conditions:

. . .

Exercise. Write in explicit form the boundary conditions for the elastic problem in engineering notation

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Take advantage of vector Voigt notation for second-order tensors (and matrixnotation for fourth-order tensors)

Field equations:

equilibrium eq. [L]T{} + {b} = {0} in

constitutive eq. {} = [D][M]{} in

compatibility eq. {} = [L]{u} in

with:

{} : stress (second-order) tensor{u} : displacement vector[D] : elasticity (fourth-order) tensor

{} : strain (second-order) tensor{b} : body force vector

Boundary conditions:imposed displ. {u} = {u} on u

imposed forces {}??{n} = {t} on t

with {u} & {t}: assigned data; {n} : outward normal to t; u t = .



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3D elasticity: engineering notation II

Voigt representation requires the introduction of some special matrices

[L]T =

x0 0

y0

z

0

y0

x

z0

0 0

z0

y

x

[M] =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

0 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2



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3D elasticity I

For a linear isotropic material, constitutive equations specializes as follows:

= tr() 1 + 2

where

tr() = : 1 = 11 + 22 + 33

such that

D = [ (1 1) + 2I]

or in engineering notation

D =

+ 2 0 0 0 + 2 0 0 0

+ 2 0 0 00 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2



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3D elasticity II

For linear isotropic material, constitutive equations written also in different forms

Split stress and strain into volumetric and deviatoric components

= p1 + s

=

31 + e

with

p=

1

3

tr() =1

3

: 1 =1

3

(11 + 22 + 33)

= tr() = : 1 = 11 + 22 + 33 Accordingly

s = p1

e =

31

Constitutive equation simplifies as follows:p= K

s = 2e



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Principle of virtual work I

Given a system F (, b, t) and a system D (u, ), if for any system

D which satisfies compatibility the following equality holds:

Lext = Lint

then system F satisfies equilibrium.

Some definitions:Lint =

( : )dV

Lext =

(b u) dV +

t

(t u)dA

Principle of virtual work constitutes an integral (weak) formulation of equilibrium,i.e. it is proper for the development of FE discretization

Exercise. Prove the equivalence between principle of virtual work and equilibrium equation. Discuss alsothe assumptions under such a statement.



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Total potential energy I

Consider total potential energy functional:

(u) =

1

2

[D(u) : (u)] dV

[b u] dV

with

(u) = su =1

2

u + (u)T

where the term relative to external surface forces is neglected

Stationarity of gives

d(u, u) =

[D(u) : (u)] dV

[b u] dV = 0 (1)

Equation 1 implies equilibrium for system F (, b, t) with computed from

constitutive and compatibility equation, i.e. = Dsu

Stationarity of Total Potential energy constitutes an integral (weak) formulation ofequilibrium, i.e. it is proper for development of FE discretization

Exercise. Prove the equivalence between stationarity of and equilibrium equation under the positionthat is computed from the constitutive equation and the compatibility equation. Discuss also theassumptions introduced to prove such a statement.



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Hu-Washizu functional I

Observe that in potential energy approach we enforce in weak form only equilibrium

equation and not constitutive nor compatibilityIt is possible to consider more general weak formulations

Consider Hu-Washizu functional

(u,,) =1

2 [ : D] dV [ : ( s

u)] dV ext

Term ext represents the contribution due to external loads

Taking the variation of the functional wrt u, and it is possible to recoverequilibrium equation, constitutive equation and compatibility equation.

Exercise. Prove the equivalence between stationarity of and the full set of elasticity equations. Discussalso the assumptions introduced to prove such a statement.

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Hellinger-Reissner functional I

It is possible to consider other weak formulations

Consider Hellinger-Reissner functional

(u,) = 1

2 : D1 dV + [ :

su] dV ext

Term ext represents the contribution due to external loads

Taking the variation of the functional wrt u and it is possible to recoverequilibrium equation and a combination of constitutive and compatibility equation

Exercise. Prove the equivalence between stationarity of and a partial set of elasticity equations.Discuss also the assumptions introduced to prove such a statement.

Exercise. Derive Hellinger-Reissner functional from Hu-Washizu functional.



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Potential energy: reduction to 2D case I

For a 2D elastic problem adopting an engineering notation we may assume:

= {11, 22, 12}T

= {11, 22, 12}T

with D now a 3 3 matrix

In particular for a plane strain problem we have:

D = + 2 0 + 2 0

0 0 2

In particular for a plane stress problem we have:

D = + 2 0 + 2 0

0 0 2

with

=2

+ 2


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