Eigenstrain controlled deformation- and stress-states

European Journal of Mechanics A/Solids 23 (2004) 1–13

Eigenstrain controlled deformation- and stress-states ✩ Franz Ziegler Civil Engineering Department, Vienna University of Technology, Wiedner-Hstr. 8-10/E201, A-1040 Vienna, Austria Received 9 October 2003; accepted 30 October 2003

Abstract (Quasi-)Static shape control of large space structures and of smart structures are the main topics of current interest. Imposed strains (eigenstrains) are of thermal nature or mainly make use of the piezoelectric effect in composites containing conventional ferroelectric polycrystals, natural crystals or special polymers. In the present paper, the direct solution of the quasistatic shape control problem in the context of the multiple field approach (i.e., considering the given structure in the background) is discussed in detail and the proper extension to dynamic problems (be connected with vibration suppression) is sketched. The imposed eigenstrain load is not assumed regionally concentrated, but distributed throughout the structure, however, with its intensity unlimited. The inverse problem of shape control can be exactly solved under these conditions. The results derived so far for static and dynamic piezoelectric actuation of isotropic or anisotropic linear elastic beam-, plate- and shell-type structures in the context of static stress-free eigenstrains (so called impotent eigenstrains) provide deep insight into the characteristic features of deformation control. It is demonstrated, that such an eigenstrain analysis can be applied directly and successfully to ‘intelligent’, ‘smart’, or ‘adaptive’ structures, which utilize piezoelectricity for the sake of structural actuation, sensing and control in an integrated circuit. However, the inverse problem associated with shape control is ill-posed. So called nilpotent eigenstrains (also known in the field of thermoelasticity) produce stresses but no deformations and thus, when properly defined, can be used to redistribute the load stresses without influencing shape control. The stress control procedure, however, is subjected to severe constraints given by the local conditions of equilibrium. For discretized or discrete structures (e.g., trusses) the general solution is worked out in detail and the two orthogonal subspaces of the impotent and nilpotent eigenstrains in Hilbert space are mentioned. Further, vibration suppression is discussed briefly under the condition of separation in space and time of the forcing function. In those cases, knowledge of the quasistatic load deformation suffices to define the distributed actuators producing impotent eigenstrain.  2003 Elsevier SAS. All rights reserved. Keywords: Shape control; Thermal strain; Piezoelectric strain; Impotent eigenstrain; Nilpotent eigenstrain; Stress redistribution

1. Introduction Since (Haftka and Adelman, 1985) presented an analytical procedure for computing a temperature field in the supporting structure to minimize deviations of large space structures from their original shape, such combinations of Structural Mechanics and Control Theory are summarized under the headline “shape control of structures”. Their detailed analysis considered a 55 m radiometer antenna supported by tetrahedral truss modules, see Fig. 1. One of the first attempts to apply shape control in Aeronautics in the open literature by Austin et al. (1994) considered an adaptive wing of a fighter plane. An application to rotary wings to suppress their vibrations is given by Nitzsche and Breitbach (1994). With the technology of smart materials developing at a vast stage, aerospace applications changed from futuristic ✩ This paper has been presented as the EUROMECH-Solid-Mechanics-Prize-Lecture at the 5th EUROMECH Solid Mechanics Conference in Thessaloniki, Greece, August 2003. E-mail address: [email protected] (F. Ziegler).

0997-7538/$ – see front matter  2003 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2003.10.005

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F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

Fig. 1. Shape control of a radiometer antenna by means of a temperature field in the supporting truss (Haftka and Adelman, 1985).

aspects into applicability, and reviewing papers summarizing the state-of-the-art appeared, (Crawley, 1994), with emphasis on piezoelectricity and its application in disturbance sensing and control of flexible structures (Rao and Sunar, 1994), including active vibration control of laminated piezoelectric beams, plates, and shells (Saravanos and Heyliger, 1999), with emphasis on control, sensors and actuators (Tzou, 1998; Lee, 1990). A most recent review of static and dynamic shape control by piezoelectric actuation is presented by Irschik (2002). Smart composite structures are considered by (Tauchert et al., 2000). The overall developments are documented in the Proceedings of various Symposia focused on this area, (Gabbert, 1998; BaheiEl-Din and Dvorak, 1998; Gabbert and Tzou, 2001; Watanabe and Ziegler, 2003). With respect to shape control of smart structures, this problem has been tackled by nonlinear optimization techniques, where a finite number of (shaped) actuator patches has been applied to the structure. The shaping problem is tackled by Pichler (2001). For an intelligent plate see Agrawal et al. (1994), Varadajan et al. (1996), for composite plates and shells (Koconis et al., 1994). Since a finite number of actuator patches is considered, or localized temperature fields are applied, such a collocation produces the desired deflection of a flexible distributed-parameter system only approximately. In the present paper, the direct solution of the quasistatic shape control problem in the context of the multiple field approach (i.e., considering the given structure in the background) is discussed in detail (Irschik and Ziegler, 2001), an outlook on stress control is given (Nyashin et al., 2000; Nyashin and Lokhov, 2003), and the extension to dynamic problems is sketched (Irschik et al., 2003). In the latter case, under conditions of time and space separable loads it is sufficient to control the much simpler quasistatic force deformation (Irschik and Pichler, 2001). Emphasis is led upon the definition of proper eigenstrain distributions, for precise definitions see Reissner (1938), however under the assumptions of unlimited intensity of the sources and their continuous distribution. In this sense benchmark solutions are derived and their practical application becomes necessarily approximate. “Eigenstrain” is a generic name originally given by Mura (1991) to inelastic strains resulting from thermal expansion, phase transformation, initial strains, plastic strains, and misfit strains. Other imposed strains produced, e.g., by electrical fields in piezoelectric materials, are of the same nature and in this context understood as eigenstrains. Shape control is ideally performed by impotent eigenstrains which produce deformation but no stress. The separated stress control procedure, however, is subjected to severe constraints given by the local conditions of equilibrium, and is achieved by nilpotent eigenstrains. The latter produce stress but no deformations. Classical examples are known in thermoelasticity, (Ziegler and Irschik, 1987), or have been encountered in the case of flexural vibrations of piezoelectric beams and attributed to an electric field without deflection (Irschik et al., 1999). When properly defined, they can be used to redistribute the load stresses without influencing shape control. All of these methods have an important purpose, design technological implementation and apply the proper structural actuation and achieve vibration suppression.

2. The multiple field concept of static shape control Subsequently, the control of the deformations of linear elastic structures produced by force loading, body forces and/or surface traction, by means of imposed eigenstrains is discussed in general terms of the associated boundary value problems. Assuming the Green’s stress dyadic known, the class of impotent eigenstrains is determined by inspection and identified

F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

3

as compatible strains, i.e., they are equal to the strains produced by the given load, or alternatively, in case of a desirable deformation, the latter must be related to a distribution of a fictitious force load. This class of eigenstrains when activated by distributed “actuators”, produce deformations but no stresses in the structure under consideration, say in the background. Interaction of the force strain field and the eigenstrains, properly produced by these actuators, interact in the background in the sense of the multiple field concept. 2.1. Force load We consider the general boundary value problem of a linear elastic, possibly anisotropic body loaded by body forces and/or surface traction. The local equilibrium requires div σ + b = 0.

(1)

On part of the boundary, kinematic boundary conditions apply, Γu : u = 0, . . .

(2a)

on the remaining part, the traction are prescribed, Γσ : σ · n = t (n) .

(2b)

Within the validity of linearized geometric relations, 1 εij (F ) = (ui,j + uj,i ) 2

(3)

and Hooke’s law, εij (F ) = Cij lm σlm

(4)

the solution of the force-displacements by means of the principle of virtual forces, see Ziegler (1998), – the Green’s stress dyadic σ˜ ij (k) produced by a unit single force applied in a direction k is assumed known, – is given in the form of the volume integral, a complementary Green’s formula,
(5) 1 · uk(F ) (x) = σ˜ ij (k) (ξ , x)εij (F ) (ξ ) dV (ξ ) V

which represents a virtual work relation. The representation of the solution (5) is chosen for the sake of simplicity for identifying impotent eigenstrains in Section 2.2. 2.2. Identification of impotent eigenstrains Since we are interested in eigenstrain distributions which do not produce stress σlm(ε) , the equilibrium condition (1), in absence of body forces has the trivial solution, σlm(ε) = 0.

(6)

The kinematic boundary condition (2a) applies and in absence of prescribed surface traction Eq. (2b) becomes Γσ : σ(ε) · n = 0.

(7)

Linearized geometric relations apply as well, 1 εij (ε) = (ui,j + uj,i )(ε) 2

(8)

and the generalized Hooke’s law εij (ε) − ε¯ ij = Cij lm σlm(ε)

(9)

reduces according to Eq. (6) to εij (ε) − ε¯ ij = 0 that means, the total strain is equal to the imposed impotent eigenstrain throughout the structure.

(10)

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F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

The deformations produced by any field of eigenstrains in a body subjected to the above mentioned boundary conditions are given by the volume integral, which is a virtual work expression with the force Green’s stress dyadic, σ˜ ij (k) , understood to be known,
(11) 1 · uk(ε) (x) = σ˜ ij (k) (ξ , x)¯εij (ξ ) dV (ξ ). V

Eq. (11) is recognized to be a generalization of Maysel’s formula of linear isotropic thermoelasticity. For its derivation by means of the principle of virtual displacements and its workability, e.g., with respect to the derivation of Green’s stress dyadic under relaxed kinematic boundary conditions, see Ziegler and Irschik (1987). Prescribing the left-hand side, Eq. (11) becomes the integral equation of static shape control. It is easily recognized that the inverse problem associated to shape control is ill-posed, since any field of eigenstrains which produce stresses but no deformation, so called nilpotent eigenstrains, can be superposed to the solution of the specified Eq. (11). However, if we require the force displacements of Eq. (5) to be equal to the eigenstrain produced displacements of Eq. (11), the eigenstrains in the latter equation can be identified equal to the load strains apparent in Eq. (5). Thus, the eigenstrains are a compatible field of strains, they are impotent and do not produce stresses, for a definition see Mura (1991). Hence, annihilation of the deformations by force loading is achieved by producing eigenstrains (which are the total strains, see again Eq. (10)) equal to the force produced strains with reversed sign. No additional stresses result in this case. For detailed derivations see Irschik and Ziegler (2001). For discrete or discretized structures it will be shown, that nilpotent portions of the imposed eigenstrain can be constructed to redistribute load stresses, however severely constrained by the equilibrium conditions of the superposed loading. 2.3. Identification of nilpotent eigenstrains Such a field of eigenstrains produces no deformation but a statically admissible stress field, σ (ε) , (in absence of body forces and surface traction, likewise to Section 2.2), div σ (ε) = 0.

(12)

Thus the boundary value problem in this case is given by Eq. (7) and Γu : u = 0.

(13)

The generalized Hooke’s law (9) in absence of total strain, εij (ε) = 0, reduces to −¯εij = Cij lm σlm(ε) .

(14)

The stresses σlm(ε) when combined with the load stresses σlm(F ) are subjected to the local equilibrium conditions of complemented Eq. (1). Thus, control of load stresses is severly constrained and merely becomes a redistribution strategy.

3. Shape control of frames and assemblies of beams In each of the beam members an uniaxial stress state is considered. For layered beams a generalized modulus of elasticity applies in the uniaxial generalized Hooke’s law of normal stress, σxx(ε) = E ∗ (εxx(ε) − ε¯ xx ).

(15)

Consequently, on the beam level, the imposed eigenstrain, ε¯ xx , acts in form of the mean eigenstrain, e, ¯ and the imposed curvature, κ, ¯ defined by cross-sectional integration,

1 1 E ∗ ε¯ xx dA, κ¯ = E ∗ ε¯ xx z dA, (16) e¯ = D B A

A

where effective stiffness are

D = E ∗ dA, B = E ∗ z2 dA. A

(17)

A

Performing the cross-sectional integration of Eq. (11) with Eqs. (15) to (17) taken into account, yields Lj

n
   (k) (ξ, x)e(ξ (k) (ξ, x)κ(ξ ¯ )+N ¯ ) dj ξ M 1 · uk (x) = j =1 0

(18)

F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

5

and k denotes commonly the axial and the lateral direction, respectively. Analogously, the displacement due to the actual force loading follows by integration of (5), L  n
j  M(F ) (ξ ) N(F ) (ξ )   M(k) (ξ, x) 1 · uk(F ) (x) = + N(k) (ξ, x) dj ξ. B(ξ ) D(ξ )

(19)

j =1 0

Impotent curvature (rendering no cross-sectional moment, M(ε) = 0) and impotent mean strain (producing no normal force N(ε) = 0) thus follow from inspection of Eqs. (18) and (19), κ¯ =

M(F ) , B

e¯ =

N(F ) . D

(20)

Further details and specialization for piezoelectric layers are given in Irschik et al. (1994). Since Eqs. (18) and (19) are simply extended to the second in-plane dimension, plate-like structures are not further considered. 3.1. Desired deflection of a cantilevered beam, extension of shape control Agrawal et al. (1994) and Irschik et al. (1998) have considered with quite different strategies to produce by means of distributed imposed impotent curvatures the desired deflections, w1 and w2 , respectively, of a C–S beam of length L (with the support taken into account, actually having a single redundancy) without the simple support, and without creating resultant cross-sectional moments,      2  3  4  απx x x x −1 , w2 (x) = w0 3 . (21) −5 +2 w1 (x) = w0 cos L L L L The desired deflections are produced by proper fictitious lateral loads which, for constant flexural stiffness, are given by fj (x) = B

d4 wj dx 4

,

j = 1, 2,

(22)

and additional edge loads Mj (F ) (x = L) = −B

d2 wj dx 2

,

Qj (F ) (x = L) = −B

d3 wj dx 3

.

(23)

Along these lines the solution was established by Irschik et al. (1997). Thus, the linearized desired curvature is defined κj (F ) (x) = −

d2 wj

(24)

dx 2

and set equal to the imposed impotent curvature
1 κj (F ) (x) = κ¯ j = zE(z)¯εxx dA. B

(25)

A

Maysel’s formula may be used to verify the production of the desired deflection, for both cases,
L 1 · w(ε) =

 x)κ(ξ M(ξ, ¯ ) dξ,

(26)

0

where the moment influence function of the cantilevered beam is simply given by,  x) = ξ − x, M(ξ,  x) = 0, M(ξ,

0
ξ
x,

x
ξ
L.

(27)

Note, self-equilibrating eigenstresses distributed over the cross-sectional area at x may remain especially in the layered beam, σ(ε) = E(zκ¯ − ε¯ ).

(28)

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F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

Fig. 2. C–C beam under uniformly distributed nilpotent curvature, κ¯ = const, w(ε) ≡ 0.

Fig. 3. C–S beam under linearly distributed nilpotent curvature, w(ε) ≡ 0.

Fig. 4. Double span beam under nilpotent curvature, w(ε) ≡ 0.

3.2. Illustration of imposed nilpotent curvature Redundant beams, when subjected to an imposed curvature which is affin to the bending moment distribution remain undeflected (Ziegler and Irschik, 1987). This statement is confirmed through Eq. (26) or easily checked by considering the generalized Hooke’s law for the linearized curvature 

= − M(ε) + κ¯ → 0. w(ε) (29) B Hence, imposed nilpotent curvatures may be used for stiffening beams and to redistribute moments from lateral force loads, illustrations for beams with single redundancy in Figs. 2–4 are self-explanatory.

4. Shape control of thin shells of revolution Deformations of shells of revolution under axial symmetric loading can be controlled by axi-symmetrically imposed impotent strains without producing additional stresses or, relaxed for thin shells, without rendering additional cross-sectional resultants (forces per unit of length). Since Eq. (11) presents virtual work terms, it is easily adjusted to keep the axial symmetry by changing the unit force load in the Green’s function problem to a unit ring load, see Fig. 5, without intermediate ring supports, 

2πr u(ε) (φ) sin ψ + w(ε) (φ) cos ψ = 2π


φ2   ˜ φ (φ ∗ , φ)κ¯ φ (φ ∗ ) − m ˜ θ (φ ∗ , φ)κ¯ θ (φ ∗ ) r ∗ R ∗ dφ ∗ . n˜ φ (φ ∗ , φ)¯εφ (φ ∗ ) + n˜ θ (φ ∗ , φ)¯εθ (φ ∗ ) − m

(30)

φ1

For thin cylindrical shells, Eq. (30) reduces to 1 · w(ε) (x) =


L  0

 ¯ ) − m(ξ, ˜ x)κ(ξ ¯ ) dξ, n˜ θ (ξ, x)e(ξ

(31)

F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

7

Fig. 5. Geometry of a shell of revolution, intermediate ring supports, number j indicated, removed. Cross-sectional resultants and, for auxiliary problem, unit ring load under angle ψ , shown.

where the Green’s function, the deflection of an infinitely long shell under a unit ring load is simply given by    2 κξ κξ R3 κξ 4 = 3 1 − ν 2  R exp − , κ cos . w (ξ, x = 0) = + sin R R R h 8Kκ 3

(32)

Thickness of the shell is denoted h, its radius is R and K denotes the flexural stiffness. For ring stiffened shells, Fig. 5, see Irschik and Ziegler (1996). In this complicating case, the virtual work of the coupling forces, X(n) and X(t ) , sketched in Fig. 5, has to be added to Eq. (30).

5. Shape control of discretized or discrete structures The Finite Element Method is the common tool for discretizing continuous structures. A priori discrete structures are, e.g., idealized trusses. In case of linear elasticity and linearized geometric relations a constant stiffness matrix, K, relates the external loads, F , with the nodal displacements, u. Thus, Hooke’s law (4) of the load case becomes, F = K uF .

(33)

The solution of Eq. (33) is done by the efficient algorithms developed for inverting the stiffness matrix. Another matrix multiplication of the nodal displacements renders the strains, εij (F ) , which become constant within the finite element for the important choice of linear shape functions for the distribution of the displacements within the finite element. Nodal displacements are the generalized coordinates in such a Ritz approximation of deformation. According to the definition of impotent eigenstrains, Section 2.2, these load strains are to be selected for shape control. Since hundreds or thousands of degrees-of-freedom are necessary to reflect the properties of the continuous structure, it becomes necessary in the course of shape control to form hyper elements of proper size and to approximate the strain distribution even further. Subsequently, the procedure is illustrated for idealized trusses, considering both, impotent strains for shape control without additional stress and nilpotent strains for control (or redistribution) of load stress. 5.1. Truss field with single redundancy In Fig. 6, the X-braced square shaped idealized truss is shown loaded by four self-equilibrating forces. To include an anisotropic effect, the extensional stiffness of one diagonal, member rod number 6, is chosen different from the common stiffness of the remaining five rods, C6 = αC,

C = (EA)−1 .

(34)

Since the analysis of the load case is quite simple, say by application of Menabrea’s theorem, see, e.g., Irschik and Ziegler (2001), we list the result on the load strains numbered according to the member rods in Fig. 6 as follows, √ √ √ √ √   (35) ε(F ) = CF (−β/ 2), (β − 2), (−β/ 2), (1 − β/ 2), (1 − β/ 2), αβ , where

 β = 1+

−1 α √ 1+ 2

(36)

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F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

denotes the anisotropy factor. It is easily checked that the eigenstrain εn = εn(F ) ,

n = 1, 2, . . . , 6,

(37)

do not produce any stresses in the member rods, Sn(ε) = 0, n = 1, 2, . . . , 6. Hence, they are impotent strains and with their sign reverted annihilate the load deformations. 5.2. Shape control of a truss field with single redundancy and the nilpotent strains The general solution of the impotent eigenstrain is derived by means of inverting the stiffness matrix. The latter for a plane truss is composed from the stiffness of a single member rod of length, 1, in local coordinates,  EA k 0 (38) , k0 = k= 0 0 1 considering the rotational matrix, A, and the nodal shift matrix, B, in Hooke’s law, F = ku,  x2 x y −xj2i −xj i yj i  ji ji ji  yj2i −xj i yj i −yj2i  Eij Aij  k ii k ij  .  k ij = =  , 2 k j i k jj . xj i xj i yj i  13  . ij

.

.

.

(39)

yj2i

where, A=

1 1ij



yj i xj i

xj i −yj i



,

−1 0 , 0 −1

B=

k ii = AT k 0 A,

k ij = B k ii ,

k jj = B k ij = k ii

(40)

have been applied consecutively with the global nodal coordinates taken into account in xj i = xj − xi ,

yj i = yj − yi .

(41)

Application of the general stiffness matrix of a single member (39) to the (quadratic) field of Fig. 7 yields the symmetric stiffness matrix, vector of nodal displacements is indicated and the kinematic boundary conditions are built in,

(1) u(1) = vb

(1)

uc

(1)

vc

(1)

ud

(1) T

vd

,

Fig. 6. Single redundant truss field under self-equilibrating external load F .

Fig. 7. Single redundant truss field, variable stiffness of all six rods, support and global coordinates illustrated.

F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

 (1) (1) kbc + kba   .   (1) K = .    .

(1)

(1)

(1)

kac + kbc .

0

0

(1)

0

0

0

−kdc

(1) (1) kbd + kad

(1) kad (1) (1) kad + kcd

−kbc (1)

(1)

kbc + kdc .

.

.

(1)

−kbc

kbc

(1)

.

9

     ,   

(42)

. . √ (1) (1) (1) (1)∗ (1)∗ (1) (1)∗ (1)∗ where kij = Eij Aij /1ij , 1ad = 21ad = 12 2 = 1bc , lengths of the outer member rods remain unchanged, 1ij = 1. Inversion of Eq. (42) yields the nodal displacements for any load case F . Choosing consecutively all possible unit nodal force loads yields the matrix of associated nodal displacements, identical to the columns of the flexibility matrix, u(F ) = K −1 F = K −1 .

(43)

Using so called code-vectors nij , the 4 × 5 matrix of the nodal displacements in global coordinates of all these unit load cases is formed, the indicator of load case (F ) is suppressed, uT ij = (ui , v i , uj , v j )

(44)

and subsequently transformed to local coordinates by matrix multiplication, u0ij = Aˆ ij uij ,

1 Aˆ ij = (xj i 1ij

yj i ).

(45)

Consequently, the strain vector of a single member rod for all unit load cases is given by εij (F ) =

1 (u − u0i ) 1ij 0j

(46)

and the strain matrix of all member rods and all unit load cases εT (F ) = (ε ab

ε ad

. . .)

(47)

defines the general solution of impotent eigenstrains which are eligible for shape control without additional stress. Assuming for sake of simplicity a constant extensional stiffness of the rods in Fig. 7 and considering imposed eigenstrain in the absence of external force load, the redundant force is selected to be Sbc(ε) = X1(ε) = Sad(ε) , the nodal equilibrium conditions render the axial forces in the member rods to be proportional, note the statically determined support conditions,  1 1 1 1 1 −√ −√ 1 −√ . (48) S (ε) = X1(ε) − √ 2 2 2 2 Hence, there is no size effect, and the nilpotent eigenstrains, causing no deformations (total strains vanish), are distributed according to  1 1 1 1 (49) ε = d1 √ √ −1 √ −1 √ . 2 2 2 2 Nyashin and Lokhov (2003) defined the inner product of strain tensors in the Hilbert space,
(B · D ) =

B · C −1 · D dV =

N  j =1

V

Bj Dj Ej Aj Lj = EA

N 

Bj Dj L j

(50)

j =1

which is shown reduced to a sum for discrete structures and further simplified for constant extensional stiffness. Thus, considering the norm of Eq. (49), the unit base element of the subspace Hσ in Hilbert space of nilpotent strains is defined, its order equals the grade of redundancy,  1 1 1 1 1 1 1 (1) ε= √  −1 √ −1 √ . (51) φ = √ √ √ ε EA1 2 + 2 2 2 2 2 2 Any given distribution of eigenstrain, when projected on the base unit N  

(1) εj φj 1j a = ε · φ (1) = EA j =1

(52)

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F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

yields elements in Hσ . Hence, taking the difference renders the impotent part of the eigenstrain distribution, the decomposition into impotent and nilpotent components ε impot = ε − aφ (1) .

(53)

The impotent eigenstrains form the multidimensional subspace Hu in Hilbert space which is orthogonal to Hσ . Limitations of load stress redistribution by nilpotent eigenstrains is clearly indicated by Eq. (49) where size and sign of the parameter d1 can be selected with the distribution of strain and thus stress, Eq. (48) given. 5.3. Cantilevered redundant plane truss Adding similar fields to the single truss of Fig. 7, renders a cantilevered truss. Its stiffness matrix is easily determined by considering an intermediate field, say of number 2
m
n, shown in Fig. 8. In a hyper matrix form, the stiffness matrix is then given by  (m) (m)  K ab,ab K ab,cd (m) K = (54) (m) (m) K cd,ab K cd,cd to form an 8 × 8 matrix according to the arrangements of nodal deformations

(m) (m) (m) (m) (m) (m) (m) (m) T va ub vb uc vc ud vd . u(m) = ua

(55)

Hence, for n = 4 fields, arranged geometrically periodic in series, the superposition of individual fields becomes obvious, and the stiffness hyper matrix takes on the form,   (1) (1) K ab,ab K ab,cd 0 0 0   (1) (1) (2) (2) K K ab,cd 0 0    cd,ab K cd,cd + K ab,ab   (2) (2) (3) (3) (4)  (56) K = 0 K cd,ab K cd,cd + K ab,ab K ab,cd 0  .   (3) (3) (4) (4)   0 0 K cd,ab K cd,cd + K ab,ab K ab,cd   0

0

0

(4)

K cd,ab

(4)

K cd,cd

Inversion of Eq. (56), yields the symmetric flexibility matrix, for n fields its order is (4n + 1)(4n + 1), which contains all nodal displacements due to unit nodal forces. Thus, following the lines of Section 5.2, the nodal displacements are identified by codevectors and subsequently the matrix of compatible strains, its order is (5n + 1)(4n + 1) for n fields, is identified. When imposed by actuators, it defines the impotent eigenstrains. Since the redundancy of the periodically set-forth cantilevered truss is equal to the number n of fields, the subspace Hσ of nilpotent eigenstrains is of the same order. Applying nodal equilibrium and selecting the normal force in one of the two diagonals in each field as the redundant force yields at once the required solution. Since n parameters are encountered, an ortho-normalization of the nilpotent eigenstrain vectors should be applied subsequently. For a given force load, trusses are optimized with respect to stiffness and minimum weight. For such a procedure with subsequently applied shape control along these lines see Rosko and Ziegler (2003).

Fig. 8. Intermediate field adjacent to the truss field of Fig. 7.

F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

11

6. An outlook on dynamic problems of shape control The main goal of dynamic shape control is vibration suppression by means of controlled input of electrical voltage into shaped piezoelectric layers. The latter function as both, sensors and actuators. These piezoelectric strains are the (impotent) eigenstrains imposed in a time specific manner to annihilate the total transient strains, i.e., within the state-of-the-art without production of additional stress. For control of plate vibrations, see Irschik et al. (1997). Irschik et al. (2003) considered the boundary value problem where part of the surface is loaded by prescribed traction, a time dependent generalized load as of Eq. (2b) and considered a prescribed rigid body motion of the remaining part of the boundary, thus generalizing Eq. (2a) to Γu : u = 0,

plus rigid body motion.

(57)

The static Eq. (1) takes on the form of the Euler–Cauchy equation of motion with absolute acceleration a entering the inertia term on the right-hand side div σ + b = ρa.

(58)

Referring the force deformation quite naturally to the guiding system of the rigid body motion, with a body fixed reference point ˙ all three vectors are prescribed time functions, A with acceleration a A , and an angular velocity ω and angular acceleration ω, e.g., when considering the vibrations of an adaptive wing relative to the rigid body motion of the airplane, cruising under controlled flight conditions, the total acceleration can be expressed by superposition of the following three vectors, (Ziegler, 1998, p. 498), ¨ a = a g + a c + u,

a g = a A + ω˙ × p + ω × (ω × p),

a c = 2(ω × u), ˙

(59)

where p is the position vector from origin A in the guiding system. For small deformations, its length may be assumed timeinvariant, hence, approximately, |p| = const in Eq. (59). Considering a generalized body force, Eq. (58) is rewritten in the somewhat simpler form, div σ (F ) + b∗ = 2ρω × u˙ + ρ u, ¨

b∗ = b − ρa g .

(60)

Irschik and Pichler (2001) and, for the boundary value problem (57), Irschik et al. (2003), showed that for dynamic control of the deformation u(x, t) = u(F ) produced by prescribed body forces and surface traction, both separable in space and time, it suffices to control the much simpler quasistatic solution of the force problem, div σ (s) + b∗ = 0, (F )

(61)

where Hooke’s law (4) relates quasistatic force strain to quasistatic force stress, (F )

(F )

ε(s) = C σ (s)

(62)

and the linearized geometric conditions (3) apply as well (F )

ε(s) =

1

T . grad u(s) + grad u(s) 2

(63)

Fotiu et al. (1993) generalized Maysel’s formula to the dynamic case of eigenstrain loads, see also Irschik and Ziegler (1988) where the static Green’s force function is used. Case A, refers to eigenstrains which are here denoted ε∗A , and the dummy force problem is denoted B, which reads, in the Laplace transform domain (convolution understood) for quiet initial conditions,

A dV = ∗A bB u σB (64) i i kl ε kl dV . V

V

Thus, special care must be taken in case of nonhomogeneous initial conditions since additional terms appear in (64), when applying the control (F )

ε(s) + ε = 0. Illustrative examples, e.g., on the beam level, are included in Irschik et al. (2003) and thus, are not repeated here.

(65)

12

F. Ziegler / European Journal of Mechanics A/Solids 23 (2004) 1–13

7. Conclusions An important purpose of the methods discussed in this paper is applications to proper structural actuation and vibration suppression. To achieve an exact solution of static and dynamic shape control, the shaped actuators must produce the load strain in form of the impotent eigenstrain with reversed sign. In such an ideal setting, no additional stresses are produced. For the dynamic problems of vibration suppression the solution of the quasistatic problem suffices for time and space separable driving forces. Thus, the definition of the parametrically time-dependent impotent eigenstrain field is greatly simplified. In that case, the distributions of static shape control become applicable with a proper time variation assigned to suppress forced vibrations. As a byproduct, an extension of the definition of compatibility of eigenstrain from classical results, the latter based upon integration of the second order compatibility conditions, to (fictitiously) load produced strain is derived. In the case of layered structures, local self-equilibrating stresses may remain. Additionally, the control of load stresses is possible by superposition of so called nilpotent eigenstrains, the latter produce stress but no strain, and thus do not disturb shape control. Stress control, however, is under the severe restrictions of the local equilibrium conditions, and merely renders a redistribution of load stresses. A general solution of shape control with impotent eigenstrains for dicretized or discrete structures is presented based on the flexibility matrix. The nilpotent eigenstrains for possible stress control form a subspace of a Hilbert space of the order of redundancy of the discrete structure. Redundant trusses serve for illustration of this powerful technique. Linear elasticity and linearized geometric conditions are assumed to hold within the scope of this paper, however, visions of extension of the shape control to nonlinear structures based again on an incremental application of the generalized Maysel’s formula exist, for the latter see Irschik and Ziegler (1995). Open or closed loop control are not discussed here. Production of eigenstrain is by temperature fields in graded materials or, more promising, by shaped piezoelectric layers. In composite beams, plates and shells only cross-sectional resultants remain unchanged under action of impotent strains.

Acknowledgement Support through the INTAS-ESA Project no.99-0185 on “Optimal design of space structures: stress and strain control”, where F. Ziegler functions as the coordinator, and a travel grant from the European Society of Mechanics, EUROMECH, are gratefully acknowledged. Author thanks Hans Irschik and Yuriy Nyashin for long lasting and successful joint research efforts on shape control and Peter Rosko for advice on matrix structural mechanics.

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