4.19 Influence functions for mixed formulations

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To simplify the formulation, let us switch to a 1-D problem, an elastic bar.

The governing equations

u

_ − ε = ε+

EAε − N = N+ (4.151)

−N

_ = p

400 4 Plane problems

have the same structure as before, S = {u, ε,N},

G(S, ˆ S) = _A(S), ˆ S_ + [N ˆu]l

0

− a(S, Sˆ) = 0 (4.152)

and

a(S, Sˆ) =

_ l

0

(u

_ − ε) ˆ N dx +

_ l

0

EAε ˆε dx +

_ l

0

N (ˆu

_ − ˆε) dx . (4.153)

First we need Betti’s theorem

B(S, ˆ S) = _A(S), ˆ S_ + [N ˆu]l

0

− [u ˆN ]l

0

− _S,A( ˆ S)_ = 0. (4.154)

Next let S = {u, ε,N} the solution of A(S) = [ε+,N+, p ]T (see Fig. 4.63 a),

and let ˆ S = S0 = {G0, ε0,N0} the point solution (see Fig. 4.63 b) where

ε0 = d

dy

G0(y, x) N0 = EAε0 . (4.155)

Because the point solution lacks the necessary regularity, we formulate Green’s

second identity (4.154) on the punctured interval Iε := [0, x−ε]∪[x+ε, l] and

take the limit. Note that on Iε the right-hand side of the Green’s function S0

is zero, A(S0) = [0, 0, 0]T, so that

lim

ε→0

B(S, S0)Iε = _A(S), S0_ + [N G0]l

0

− [uN0]l

0

− u(x) = 0, (4.156)

where the single term u(x) is the limit of

lim

ε→0

{N0(x − ε) u(x − ε) − N0(x + ε) u(x + ε)} = 1· u(x) . (4.157)

If we take the boundary conditions into account, it follows

u(x) = _A(S), S0_ =

_ l

0

[N0 ε+ + ε0 N+ + G0 p ] dx , (4.158)

and in the same way we obtain the influence function for the normal force

N(x) = _A(S), S1_ =

_ l

0

[N1 ε+ + ε1 N+ + G1 p ] dx . (4.159)

It is not possible to derive an influence function for the strain ε, because

there are only two boundary operators in the “boundary integral” [N ˆu]—the

identity and the normal force operator, N = EAdu/dx; see Sect. 7.6, p. 529.

But of course N and ε are—up to the factor EA—the same. This problem and

its solution (ε _ N) has its origin in the nature of the system (4.151): first we

define the strains (one differential operator), then the stresses (no differential

operator only the “elasticity tensor” is applied!), and finally we define what

equilibrium means (one differential operator).

4.21 Nonlinear problems 401

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