Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

7.2 Green’s identities

Back

’The principle of virtual displacements is nothing else than integration by

parts’ and so we start this section with repeating the rules for integration by

parts before we formulate Green’s identities which are based on integration

by parts. These identities encapsulate the basic principles of mechanics and

play a fundamental role in finite element analysis.

Integration by parts

Let u and ˆu be two functions with continuous first derivatives in the interval

(0, l) then

_ l

_ ˆudx = [u ˆu] l

−

_ l

u ˆu

dx (7.39)

and in higher dimensions with functions u and ˆu from C1(Ω),

u,xi ˆudΩ =

uni ˆuds −

u ˆu,xi dΩ , (7.40)

where ni is the i-th component of the normal vector n on the edge Γ of the

domain Ω.

For example let ˆu = 1 and u_ = ε the strain in a rod then

_ l

ε dx = u(l) − u(0) . (7.41)

In a plate where εxx = ux,x the same statement is

εxx dΩ =

ux nx ds . (7.42)

If for example Ω = a×b is a rectangle with nx = ±1 on the vertical edges ΓL

and ΓR and nx = 0 on the horizontal edges then the result resembles the 1-D

result

εxx dΩ =

ΓR

ux ds −

ΓL

ux ds . (7.43)

7.2 Green’s identities 509

of a bar element dN+pdx = 0

Bars

The equilibrium condition

H = 0 for a bar element dx (see Figure 7.1)

leads to the differential equation

− EAu

__ = p , (7.44)

where u(x) is the longitudinal displacement, p is the applied load, and EA

is the (constant) stiffness of the bar. If the left-hand side of the differential

equation is multiplied by a virtual displacement δu and we integrate by parts

_ l

−EAu

δu dx = [−EAu

δu] l

−

_ l

−EAu

δu

dx , (7.45)

the result is Green’s first identity:

G(u, δu) =

_ l

−EAu

δu dx

_ _ _

start

+[N δu] l

−

_ l

EAu

δu

_ _ _

transformed terms

_ l

−EAu

δu dx + [N δu] l

_ _ _

δWe

−

_ l

EAu

δu

_ _ _

δWi

= 0. (7.46)

The terms in brackets

[N δu]l

0 = N(l) δu(l) − N(0) δu(0) (7.47)

are the virtual external work done by the normal forces N = EAu_ at the

ends of the bar.

The expression B(u, ˆu) = G(u, ˆu) − G(ˆu, u) = 0 − 0 = 0 is Green’s second

identity,

B(u, ˆu) =

_ l

−EAu

__ ˆudx + [N ˆu] l

_ _ _

W1,2

−[u ˆN ] l

−

_ l

u (−EA ˆu

__) dx

_ _ _

W2,1

= 0

(7, .48)

and it formulates Betti’s theorem.

Fig. 7.1. Static equilibrium

510 7 Theoretical details

Fig. 7.2. Archimedes’ dilemma: all effort is consumed by the strain energy a) the

Earth will not move one iota b) the rubber band will stretch and stretch and ...

Green’s first identity basically is of the form

G(u, ˆu) = p(ˆu) − a(u, ˆu) = δWe(u, ˆu) − δWi(u, ˆu) = 0 (7.49)

or if it is solved for the strain energy product

a(u, ˆu) = p(ˆu) . (7.50)

Note that a(u, ˆu) is the first-order derivative of the quadratic form

F(u) =

2 a(u, u) , (7.51)

that is

_(u) :=

dε

2 a(u + εˆu, u + εˆu)

ε=0

= a(u, ˆu) . (7.52)

Beam

The differential equation of a beam with constant bending stiffness EI is

EI wIV (x) = p(x) . (7.53)

The bending moment is M(x) = −EI w__(x) and the shear force is V (x) =

−EI w___(x). Green’s first identity for the beam equation is

G(w, ˆ w) =

_ l

EI wIV ˆ wdx + [V ˆ w −M ˆ w

_] l

_ _ _

δWe

−

_ l

M ˆM

_ _ _

δWi

= 0, (7.54)

and B(w, ˆ w) = G(w, ˆ w) − G( ˆ w,w) = 0 is Betti’s theorem.

7.2 Green’s identities 511

An application—Archimedes’ dilemma

A place to stand does not suffice to move the Earth. Archimedes needs also a

lever with EI = ∞; see Fig. 7.2 a. Otherwise the lever will only bend. Because

of Green’s first identity G(w,w) = We − Wi = 0 the exterior work is at any

moment equal to the strain energy in the beam

We = Pr · wr − Pl · wl = a(w,w) =

_ l

dx = Wi (7.55)

Pr · wr = Pl · wl + a(w,w) . (7.56)

So that all of Archimedes’ effort, Pr · wr, will be consumed by the internal

energy a(w,w) and very little—effectively nothing—remains to lift the Earth.

The same happens if you try to pull a heavy weight across the wet sand

on the beach (see Fig. 7.2 b)

We = Pr · _ _u_r

your effort

−Pl · ul = a(u, u) =

_ l

dx = Wi . (7.57)

The rubber band (EA) will stretch and stretch and stretch, ur → 1, 2, 3, . . .,

that is a(u, u) will increase but the weight will hardly move, ul _ 0.

Poisson equation

Green’s first identity for the differential equation −Δu = p is

G(u, ˆu) =

−Δu ˆudΩ +

∂u

∂n

ˆuds

_ _ _

δWe

−

∇u •∇ˆudΩ

_ _ _

δWi

= 0, (7.58)

where

a(u, ˆu) =

∇u •∇ˆudΩ =

(u,x ˆu,x +u,y ˆu,y ) dΩ (7.59)

is the strain energy product.

Kirchhoff plate

The differential equation of the Kirchhoff plate is the biharmonic equation

KΔΔw =p K= E h3

12 (1 − ν2) . (7.60)

512 7 Theoretical details

The curvature tensor K = [κij ] has elements κij = w,ij , and the bending

moment tensor is M = K {(1 − ν)K + ν(trK) I}, or

mxx = −K (w,xx +ν w,yy ), myy = −K (w,yy +ν w,xx ) , (7.61)

mxy = −(1 − ν)Kw,xy . (7.62)

The shear forces are

qx = −K (w,xxx +w,yyx ) qy = −K (w,xxy +w,yyy ) , (7.63)

and the resultant stresses on the boundary are, in indicial notation,

mn = mij ni nj mnt = mij ni tj (7.64)

qn = qi ni vn = d

mnt + qn . (7.65)

Green’s first identity is

G(w, ˆ w) =

KΔΔw ˆ wdΩ +

[vn ˆ w − mn ˆ w,n ] ds +

Fi ˆ w(xi)

_ _ _

δWe

− a(w, ˆ w)

_ _ _

δWi

= 0, (7.66)

where the strain energy product is the expression

a(w, ˆ w) =

[w,xx ( ˆ w,xx +ν ˆ w,yy ) + 2(1 − ν) w,xy ˆ w,xy

+w,yy ( ˆ w,yy +ν ˆ w,xx )] dΩ =

M• ˆK dΩ (7.67)

and the Fi are the corner forces resulting from the jumps in the twisting

moment mnt:

Fi := F(w)(xi) = mnt(x+

i ) − mnt(x

−

i ) . (7.68)

Reissner–Mindlin plate

The terms of a Reissner–Mindlin plate are the rotations ϕ = [ϕx, ϕy]T, the

deflection w, the shearing strains γ = [γx, γy]T , the curvature tensor K, the

bending moment tensor M, and the shear forces q = [qx, qy]T which are

governed by the equations

strains: K(ϕ) −K = 0 ϕ + ∇w − γ = 0 (7.69)

material law: C[K] −M = 0 a γ − q = 0 (7.70)

equilibrium: −divM + q = b∇p −div q = p (7.71)

7.2 Green’s identities 513

where

K(ϕ) =

(∇ϕ + ∇ϕT) =

2 ϕx,x ϕx,y +ϕy,x

ϕy,x +ϕx,y 2 ϕy,y

(7.72)

C[K] = K (1 − ν)E +ν K (trK) I (7.73)

and

K = E h3

12 (1 − ν2), a= K

1 − ν

￣λ

2, b= ν

1 − ν

￣λ

, λ￣2 =

h2 , (7.74)

where h is the plate thickness. These equations are equivalent to the system

− divC[K(ϕ)] + a(ϕ + ∇w) = b∇p (7.75)

−div (a (ϕ + ∇w)) = p (7.76)

or in indicial notation

− mαβ,β +a qα = b p,α, α= 1, 2 (7.77)

−a qβ,β = p (7.78)

where

mαβ = K(1 − ν)

(ϕα,β +ϕβ,α ) +ν K ϕγ,γ δαβ (7.79)

qα = ϕα + w,α . (7.80)

If this system is interpreted as the application of an operator −L to the

vector-valued function u = [ϕx, ϕy, w]T we have the identity

G(u,ˆu) =

−Lu•ˆu dΩ +

[Mn•ˆϕ + q •n ˆ w ] ds − a(u,ˆu) = 0

(7.81)

where

a(u,ˆu) =

[M• ˆK + q •ˆγ ] dΩ . (7.82)

Linear elasticity

The governing equation is

Lu := −

μΔ+ μ

1 − 2 ν

∇div

u = p , (7.83)

or in tensor notation

514 7 Theoretical details

separately on each element

− μui,jj − μ

1 − 2 ν

uj ,ji = pi i = 1, 2 (7.84)

which is equivalent to

− σij ,j = pi i = 1, 2 . (7.85)

Green’s first identity is

G(u,ˆu) =

−Lu•ˆu dΩ +

τ (u) •ˆu dΩ − a(u,ˆu) = 0, (7.86)

where τ (u) = Sn is the traction vector on the boundary and

a(u,ˆu) =

[σxx ˆεxx + 2σxy ˆεxy + σyy ˆεyy] dΩ =

S • ˆE dΩ (7.87)

is the strain energy product. For more identities see [115].

Regularity

Because the identities are based on integration by parts, the functions u and

ˆu must be sufficiently regular. If that is not the case, the interval (0, l) or the

domain Ω can be subdivided into as many intervals or partitions as necessary:

G(u, ˆu)(0,l) = G(u, ˆu)(0,l1) + G(u, ˆu)(l1,l2) + . . . + G(u, ˆu)(ln,l) = 0.

(7.88)

Typically the partitions are the individual elements; see Fig. 7.3.

Green’s first identity and stiffness matrices

Substituting two nodal unit displacements (not necessarily the actual displacements

but “any” displacements ϕi) into Green’s first identity for a beam

yields

G(ϕi, ϕj) =

_ l

EIϕIV

i ϕj dx +

Vi ϕj −Mi ϕ

_ _ _

p ij

−

_ l

EIϕ

i ϕ

j dx

_ _ _

k ij

= 0

(7.89)

Fig. 7.3. Problems of regularity

can be overcome by

formulating Green’s identity

7.2 Green’s identities 515

which means that

δWe(pi, ϕj) = p ij = k ij = δWi(ϕi, ϕj) (7.90)

or that the strain energy product (virtual internal energy) k ij between two

such nodal unit displacements ϕi and ϕj is equal to the virtual external work

p ij done by the unit load case pi via the virtual displacements ϕj .

The load case pi simply consists of all forces that produce the shape ϕi, i.e.,

the distributed load EI ϕIV

i , the shear forces Vi(0), Vi(l), and the moments

Mi(0),Mi(l) at the ends of the beam. The double subscripted term p ij is the

virtual external work δWe(pi, ϕj) corresponding to the load case pi and the

virtual displacement ϕj .

With uh =

j uj ϕj and the n-fold identity

G(uh, ϕi) = 0 i = 1, 2, . . . n (7.91)

this is equivalent to

P u−Ku = 0 or fh

−Ku = 0 (7.92)

where fh := P u.

Strain energy = nodal forces × nodal displacements

It should be obvious by now that the strain energy in a single element

a(uh,uh)Ωe =

Ωe

σij · εij dΩ =

Ωe

•uh dΩ +

Γe

th •uh ds = fTe

(7.93)

is the same as the scalar product between the equivalent nodal forces of that

element

i =

Ωe

•ϕei

dΩ +

Γe

th •ϕei

ds (7.94)

—the th are the tractions on the edge of the element—and the vector ue of

nodal displacements. Summing the contributions from all elements we obtain

the well known formula for the strain energy stored in a structure

a(uh,uh) = uT Ku = fT u. (7.95)

Green’s first identity and projections

The FE solution uh is the projection of the exact solution u onto Vh

uh ∈ Vh : a(u − uh, ϕi) = 0 ϕi ∈ Vh (7.96)

516 7 Theoretical details

uh ∈ Vh : a(uh, ϕi) = a(u, ϕi) = δWe(u, ϕi) = fi ϕi ∈ Vh (7.97)

where δWe(u, ϕi) is short for

G(u, ϕi) =

_ l

−EAu

ϕi dx + [N ϕi]l

_ _ _

δWe(u,ϕi)

−a(u, ϕi) = 0 (7.98)

so that Green’s first identity allows to replace the term a(u, ϕi) by an expression

of external virtual work. This is the vector f.

Пресс-релиз популярных книг

7.2 Green’s identities

Популярные книги

Популярные статьи

Навигация