the last one only being valid for isotropic constitutive relations.6.4 The Principle of Stationary Potential Energy As we remarked above, the Principle of Virtual Work can be used for an
Trang 1wherewVLM = w LM and wVLM are the components of the incremental second Piola-Kirchho stress tensor; it is important to recognize that the three tensors
in Eq (6.33) are referred to the spatial configuration at timew
Also,
w+w
becausew%LM = 0
Replacing with Eqs (6.33) and (6.34) in Eq (6.32), we get
Z
w Y
âw
LM+ wVLMđ
(w%LM) wdY = w+wZh{w> (6.35)
we can write an incremental constitutive equation referred to thewconfiguration,
and get,
Z
w Y
âw
LM+ wFLMNO w%NOđ
(w%LM) wdY = w+wZh{w= (6.37)
In a fixed Cartesian system we can show that (Bathe 1996),
w%= 1
2(wx> + wx> + wx> wx>) (6.38) where wx> = Cx
C w } =
We can decompose the strain increment into a linear and a nonlinear increment in the unknown incremental displacement; that is to say,
w% = wh + w
wh =1
2(wx> + wx>) (6.39)
w =1
2(wx> wx>) = Hence we can write Eq (6.37) as,
Z
w Y
êw
+ wF â
wh + wđô
â
wh + wđ w
dY = w+wZh{w=
(6.40) The above is the momentum balance equation at time w + w; which is
a nonlinear equation in the incremental displacement vector Proceeding in the same way as in the total Lagrangian formulation we obtain the linearized momentum balance equation (Bathe 1996):
Z
w Y w
F wh wh wdY +
Z
w Y
w w wdY (6.41)
= w+wZh{w
Z
w Y
w wh wdY =
Trang 2194 Nonlinear continua
It is easy to show that
rVLM =
r
w wV
lm ¡w
r[1¢L l
¡w
r[1¢M
r%LM= w%lm wr[lL wr[mM (6.43) and therefore if the same material is considered in both formulations the incremental constitutive tensors should be related,
FLMNO=
r
w wF
pqst ¡w
r[1¢L p
¡w
r[1¢M q
¡w
r[1¢N s
¡w
r[1¢O
t = (6.44) Any problem can be alternatively solved using either the total or the up-dated Lagrangian formulations and the results should be identical (Bathe 1996)
For solving finite-strain elastoplastic problems, in Sect 5.2.6 we introduced
an adhoc incremental formulation, the total Lagrangian-Hencky formulation
6.3 The Principle of Virtual Power
There are formulations where the primary unknowns are the material veloci-ties rather than the material displacements (e.g fluid problems, metal-forming Eulerian (Dvorkin, Cavaliere & Goldschmit 1995, Dvorkin & Petöcz 1993) or ALE formulations (Belytschko, Liu & Moran 2000), etc.) For these cases the momentum conservation leads to,
Z
w Y
wb· wvw wdY +
Z
w V
wt· wvwdV =
Z
w Y
w : wdwdY = (6.45)
In the above equation wv is the material velocity at a point andwdis the strain-rate tensor
Of course, we can use, for formulating the Principle of Virtual Power, other energy conjugated stress/strain rate measures, for example:
Z
Y
wb· wv r dY +
Z
V
wt· wvwMV dV =
Z
Y
w : wddY > (6.46a) Z
Y
wb· wvr dY +
Z w
V
wt· wvwMV dV =
Z
Y
w
rS:wr%· dY > (6.46b) Z
Y
wb· wv r dY +
Z
V
wt· wvwMV dV =
Z
Y
w
rPW :wrX· dY >
(6.46c) Z
Y
wb· wvr dY +
Z
V
wt· wvwMV dV =
Z
Y
w : wrH· dY > (6.46d)
Trang 3the last one only being valid for isotropic constitutive relations.
6.4 The Principle of Stationary Potential Energy
As we remarked above, the Principle of Virtual Work can be used for any material constitutive relation, for any type of loading and for any nonlinearity
in the case to be analyzed
In the present section we will specialize the Principle of Virtual Work for:
• Hyperelastic materials
• Conservative external loads
For a hyperelastic material we have seen in Chap 5 (Eq (5.3d) that,
w
rVLM = r C
wU(wr%)
Cw
The external conservative loads are the external loads that can be derived from a potential Hence, a load field is said to be conservative in a region if the net work done around any closed path in that region is zero (Crandall 1956)
A typical conservative load system can be represented as,
Following the definitions introduced above, the load system in Eq (6.48)
is a body attached load system with constant direction
For conservative loads per unit mass, we write
wb=C
wJ (wu)
and for conservative surface loads
wt=C
wj (wu)
Note that the above-defined surface loads are defined as loads per unit reference surface; therefore, its resultant at timew isR
V
wtdV
We now define a functional of the function wucalled the potential energy functional:
w
r =
Z
Y
¡w
U + wJ¢
dY + Z
V
Trang 4196 Nonlinear continua
Therefore,
w =
Z
Y
"
C
wU
Cw
%:
w
% + C
wJ
Cu · wu
#
dY + Z
V
Cwj
Cu ·wudV = (6.52)
In the above,wuare admissible variationsâ
wu= 0 onVx see Fig 6.1đ andw% is derived from the displacement variations Therefore,
w =
Z
Y
h
w
S:w% wb· wui
dY
Z
V
wt· wudV = (6.53)
Hence, for a hyperelastic material under a conservative load system, the principle of virtual work, in Eq (6.18), can be written as
The above equation states that when thewconfiguration is in equilibrium the potential energy functional reaches a stationary value; i.e it fulfills the necessary requirements for being an extreme (Fung 1965)
In what follows we show that in the case of infinitesimal strains the po-tential energy not only is stationary at the equilibrium configuration but it actually attains there a minimum
Using the nomenclature introduced in Eq (6.1) we write the potential energy functional for an admissible configuration close to the equilibrium one as
w
r0 =
Z
Y
êw U(w% + w%) wb·âw
u+wuđô
Z
V
wt·âw
u+wuđ
dV = Using a Taylor expansion,
wU(w% + w%) = wU(w%) + C
wU
Cw
%
É
É
Éw
%
: w% + 1
2w% : C
2wU
Cw
% Cw
%
É
É
Éw
%
: w% + · · · =
(6.56) Hence,
w
r0 w
r = w +
Z
Y
1 2
w% : C
2wU
Cw
% Cw
%
É
É
Éw
%
: w%dY + · · · = (6.57)
Since at equilibrium w = 0, the sign of the l.h.s is the sign of the integrand on the r.h.s
Trang 5In the case of infinitesimal strains case we can assume thatw
% 0 and we havewU(0) = 0 (convention) and wS¯
0= C w U
C w
%
¯
¯
0= 0; hence, from Eq (6=56)
wU(w%) = 1
2 w% : C
2wU
Cw
% Cw
%
¯
¯
¯
0
: w% + · · · = (6.58)
Since, in a stable material the value of the elastic strain energy is positive for any strain tensor (the elastic strain energy is a positive-definite function)
we conclude that,
w
and the potential energy is a local minimum at the equilibrium configura-tion In the infinitesimal strains case we call it the minimun potential energy principle (Washizu 1982)
Conservative and nonconservative loading
(a) Conservative loading
Let us consider a linear elastic, cantilever beam under the conservative end-load shown in the figure,
Conservative load
The elastic energy stored in the beam is,
wU =
Z O 0
H L 2
µ
d2 wx2
dw} 2
¶2
dw}1
whereH is Young’s modulus and L is the beam cross section moment of inertia The Principle of Virtual Work states,
Trang 6198 Nonlinear continua
wU = wS x2
¡w UwS wx2¢
= 0 wherew = wU wS wx2 is the potential energy of the system
(b) Nonconservative loading
We now consider the same linear elastic cantilever beam but under a follower load, as shown in the figure
Body-attached follower load The principle of virtual work states,
wU =wS sin¡w
¢
x1 + wS cos¡w
¢
x2= For small displacement derivatives we can approximate
sin¡w
¢
w
µdwx
2
dw}1
¶
O
cos¡w
¢
1 hence,
wU = wS
µ
dwx2
dw}1
¶
O
wx1+wx2
¸
= Since
wS
µ
dwx2
dw}1
¶
O
wx1+wx2
¸ 6= C
wG
Cu · wu the load is nonconservative and the principle of stationary potential energy
Trang 7Example 6.4 JJJJJ Stability of the equilibrium configuration (buckling) (Ho 1956)
Let us consider the system shown in the following figure, in equilibrium at timew, in the straight configuration:
wS : axial conservative load ; O : length of the rigid bar ; n : sti ness of the linear spring;nW : sti ness of the torsional spring
Assume that the equilibrium configuration is perturbed with a rotation
?? 1 The axial load displacement is obtained from the following scheme:
Trang 8200 Nonlinear continua
For ?? 1 , S O 22 and O sin() O
The potential of the external load is,
wJ = wS S =
wS O 2
The only deformable bodies are the springs; hence
wU =1
2 n (O )2 + 1
2 nW 2 = Therefore the potential energy functional of the system is
w
r = 1
2 n O2 2+1
2 nW 2+
wS O 2
2 and the equilibrium configuration is defined by
w = 0 which leads to, £
n O2 + nW + wS O ¤
= 0 = Since is arbitrary the bracket has to be zero Two solutions are possible: (i) = 0 ; that is to say, the straight (undeformed) configuration
(ii)wS = ¡
n O + nW
O
¢ For the second solution is undefined We call this load value the critical value,Sfu, because at this load there are two branching solutions ( = 0 and
6= 0 )
Trang 9Sfudefines the bifurcation or buckling load.
The equilibrium path is
Since in the above derivation the terms higher than 2 were neglected, we cannot assess anything about the branching equilibrium path JJJJJ
Postbuckling behavior
We repeat the previous example derivation keeping terms higher than2 By doing this, we get
S = O (1 cos ) O
µ
2
2
4
4!
¶
wJ = wS S wS O
µ2
2
4
4!
¶
wUn= 1
2 n (O sin )2 1
2 n O2
µ
3
3!
¶2
wUW = 1
2 nW 2 = Hence
w
r = 1
2 n O2
µ
2
4
3 +
6 36
¶ +1
2 nW 2+ wS O
2
2 wS O
4
24 = For the equilibrium configurationw = 0 and therefore,
n O2
µ
2
3
5 12
¶ +nW + wS O wS O
3
6
¸
= 0 =
Trang 10202 Nonlinear continua
Since is arbitrary, we get, neglecting terms higher than 3,
n O2
µ
12
2
3
¶ +nW + wS O
µ
1
2
6
¶¸
= 0 which has again two possible solutions:
(i) = 0 the straight solution
(ii)wS = n O
³ 1 2 2 3
´ +nWO 1 2 6
In the second solution, for = 0> wS = Sfu=¡
n O +nW
O
¢ The bifurcation point is the same as the one calculated in the previous exam-ple; however, now is defined
We see that for A 0 Eq (ii) provideswS = wS ()
If we examine the case withnW = 0, we get
wS =
n O ³
12 2
3
´
1 2
6
and we can represent
If we examine the case withn = 0, we get
wS = nW
O³
1 2
6
´ and we can represent
Trang 11It is clear that the above cases represent two very di erent behaviors from a structural point of view
For the case nW = 0, the buckling is catastrophic because for A 0 the load-carrying capacity of the structure keeps dimishing: unstable postbuck-ling behavior
For the casen = 0, the load-carrying capacity of the structure increases after
Natural boundary conditions
Let us study the following linear elastic cantilever beam under conservative loads:
H : Young’s modulus; L : moment of inertia of the beam cross section Assumewx2=wx2(w}1) is the beam transversal displacement and using linear beam theory (Ho 1956)
Trang 12204 Nonlinear continua
w =
Z O 0
H L 2
µ
d2wx2
dw}12
¶2
dw}1
Z O 0
wtwx2 dw}1
wSO ¡w
x2¢
O wPO
µ
dwx2
dw}1
¶
O
= For the equilibrium configurationw = 0; hence
Z O
0
H L
µd2 wx2
dw}12
¶
µd2 wx2
dw}1 2
¶
dw}1
Z O 0
wt ¡w
x2
¢
dw}1 ((A))
wSO ¡w
x2¢
O wPO
µ
dwx2
dw}1
¶
O
¸
= 0 =
In the first integral we use (Fung 1965) ³
d2 wx 2
d w } 1 2
´
= dwd}21 2
¡
wx2
¢ and inte-grating by parts twice, we get
Z O
0
H L
µ
d2 wx2
dw}1 2
¶
d2
dw}12
¡
wx2¢
dw}1=
H L d
2 wx2
dw}1 2
d
dw}1
¡
wx2¢¸O
0
H L d
3 wx2
dw}1 3
wx2
¸O 0
+
Z O 0
H L d
4 wx2
dw}1 4
wx2 dw}1 =
At w}1 = 0 we have as boundary conditions wx2 = dwx2
d w } 1 = 0, hence,
¡
wx2¢
w } 1 =0=h
³
dwx 2
d w } 1
´i
w } 1 =0= 0
Replacing in (A), we get
Z O
0
H L d
4 wx2
dw}1 4 wt
¸
wx2 dw}1
µ
H L d
3 wx2
dw}13
¶
O
+ wSO
¸
¡w
x2¢
O
+
µ
H L d
2 wx2
dw}1 2
¶
O
wPO
¸
µ
dwx2
dw}1
¶
O
= 0=
Since wx2 is arbitrary at every point 0 6 w}1 6 O we must fulfill the following di erential equation
wt = H L d
4 wx2
dw}1 4
which is the well-known equation of beam theory
Atw} =O we get
Trang 13Essential (rigid) boundary conditions or Natural boundary condition Either (wx2)O is fixed and (wx2)O= 0 or wSO = H L ³
d 3 w x 2
d w } 1 3
´
O
Either³
dwx 2
d w } 1
´
O is fixed and³
dwx 2
d w } 1
´
O= 0 or wPO= H L ³
d2wx 2
d w } 1 2
´
O
The Rayleigh-Ritz method
In the previous example, from the principle of stationary potential energy we derived the di erential equation that governs the deformation of a cantilever beam
Usually, we need to work in the opposite direction: we know the di erential equations that govern the deformation of a continuum but we cannot integrate them and we resort to the principle of stationary potential energy to derive
an approximate solution One method for deriving approximate solutions is the Rayleigh-Ritz method (Ho 1956)
Let us consider again the linear case analyzed in the previous example and let us assume that we want to derive an approximate solution for the case
wt = wSO= 0 For this case
w =
Z O 0
H L 2
µ
d2 wx2
dw}1 2
¶2
dw}1 wPO
µ
dwx2
dw}1
¶
O
=
To derive an approximate solution we consider trial functions that fulfill the geometrical or essential boundary conditions,
wx2(0) =
µ
dwx2
dw}1
¶
0
= 0= For example
wx2ª(w}1) =d
µ
1 cos
w}1
2O
¶
where the parameterd will be determined by imposing the minimization con-dition on
wª= wª(d) = Using the adopted trial function, we get
wª= H L 4 d2
64O3
wPO d
Trang 14206 Nonlinear continua
The minimum value that can attain the above functional is, within the sidered set of trial functions, our best approximation to the equilibrium con-figuration Imposing
Cwª
Cd = 0>
we get
d = 16
wPO O2
H L 3 = Therefore our approximate solution is
wx2ª(w}1) = 16
wPO O2
H L 3
µ
1 cos
w}1
2O
¶
wª=4
wP2
O O
H L 2 = For the case we are analyzing the exact solution is
wx h{dfw 2
¡w }1¢
=
wPO (w}1)2
2H L
wh{dfw=
wP2
O O
2H L =
It is obvious thatwª A wh{dfw
If we want to improve our approximate solution we enrich the trial function set using, for example
wx2 ¡w
}1¢
=d
µ
1 cos
w}1
2O
¶ +e
µ
1 cos
w}1
O
¶
=
It is important to note that the above defined trial function:
Fulfills the essential (rigid) boundary conditions
Contains the previous one,wx2ª(w}1), as a particular case (e = 0)
Since we will determine the values of both constants by imposing onw the necessary conditions for attaining a minimum, it is obvious that
w 6 wª = That is to say, we will either find the same solution as before (e = 0 and
w = wª) or a better one (e 6= 0 and w ? wª) We cannot deteriorate the solution by adding more terms in the trial function
Usingwx2 (w}1)> we get
w = H L 3
2O3
µd e
32d2+
2e2
¶
wPO d
Imposing CwCd = CwCe = 0, we get
Trang 15d = 0=6294
wPO O2
H L
e = 0=0668
wPO O2
H L
w =0=4940
wP2
O O
H L =
It is clear from the derived values that, as expected:wh{dfw ? w ? wª, and thereforewx2 (w}1) is a “better” approximation thanwx2ª(w}1).JJJJJ
In the previous example we have introduced three relevant topics that we want to highlight:
1 When obtaining approximate solutions using the Rayleigh-Ritz method, based on the minimum potential energy principle (infinitesimal strains),
we can only rank the merit of di erent solution using their potential energy value; that is to say, if wD ? wE then the D-solution is a better approximation than the E-solution
2 The trial functions have to exactly satisfy the rigid boundary conditions (admissible functions) but not the natural boundary conditions
3 Approximate solutions do not need to fulfill exactly either the equilib-rium equation inside the dominium or the natural boundary conditions (equilibrium equations on the boundary)
6.5 Kinematic constraints
In the example shown in Fig 6.4, wherewS is a conservative load, the potential energy is,
w =1
For the inextensible string there is a kinematic constraint given by,
Hence, we have to minimize the functional of the potential energy given
in Eq.(6.60) under the constraint expressed in Eq (6.61)= Using the Lagrange multipliers technique (Fung 1965, Fung & Tong 2001), we define a new func-tional (w) and we perform on it an unconstrained minimization:
w=1
2 nwx2 wS wy + w¡
2wy wx¢
(6.62) wherew is the Lagrange multiplier