Nonlinear constitutive law for ferroelectric/ ferroelastic material and its finite element realization

Transcription

1 Siene in China Series G: Physis, Mehanis & Astronomy 2007 Siene in China Press Springer-Verlag Nonlinear onstitutive law for ferroeletri/ ferroelasti material and its finite element realization LI YaoChen Department of Aerospae Engineering and Applied Mehanis, ongji University, Shanghai , China ( he hysteresis phenomena of ferroeletri/ferroelasti material in polarization proedure are investigated. Some assumptions are presented based on the published experimental data. he eletrial yielding riterion, mehanial yielding riterion and isotropi hardening model are established. he flow theory in inremental forms in polarization proedure is presented. he nonlinear onstitutive law for eletrial-mehanial oupling is proposed phenomenologially. Finally, the nonlinear onstitutive law expressed in a form of matries and vetors, whih is immediately assoiated with finite element analysis, is formulated. In the example problem of a retangular speimen subjeted to a uniaxial eletri field, the proedure from virgin state to fully polarized state is simulated. Afterward, a uniaxial ompressive loading is applied to depolarizing the speimen. Results are in agreement with the experimental data. ferroeletri/ferroelasti material, onstitutive law, nonlinear, finite element method. Piezoeletri materials, due to their oupling behavior between eletri and mehanial fields, are extensively used for eletromehanial and eletroni devies, suh as atuators, sensors and transduers in intelligent systems. If suh a material is polarized by an eletri field above the oerive field at a temperature below the Curie point, its response to a small signal may be haraterized as linear. Nowadays, appliation of piezoeletri materials involves rigorous loading and ompliated geometry of the omponents so that the assumption of small signals is no longer justified. Instead, the nonlinear behavior of the materials may beome dominant. Hene, it is signifiant to study the numerial method for simulating the nonlinear onstitutive behavior of the eletri devies made of piezoeletri materials. Curves haraterizing piezoeletriity and ferroeletriity may be found in many textbooks and artiles. hese hystereses demonstrate ferroelasti and ferroeletri nonlinearity and the underlying physial mehanisms. Attempts for analysis and desription of the ferroeletriity and Reeived Deember 3, 2005; aepted July 17, 2006 doi: /s Supported by the Researh Fund for the Dotoral Program of Higher Eduation of China (Grant No ) Si China-Phys Meh Astron Feb 2007 vol. 50 no

2 ferroelastiity were made by Chen and Hagood [1], Hwang, Lynh and MMeeking [2], Bassiouny, Ghaleb and Maugin [3], and Bassiouny and Maugin [4], based on the mehanisms of poling and domain-swithing. Kamlah and sakmakis [5], and Kamlah and Böhle [6] onstruted a phenomenologial nonlinear model for the ferroeletri and ferroelasti behavior of PZ eramis. However, further work is needed for finite element realization. Attempts for establishing a nonlinear onstitutive law for piezoeletri materials are made in this work, where some new assumptions and viewpoints are proposed. his onstitutive law is history dependent, involving reversible-irreversible nonlinear behavior of the material. Expressions for the onstitutive law in a matrix-and-vetor form and in the inremental form are formulated for finite element analysis. he variational priniple and the orresponding finite element implementation in an inrement form are established. he example problem of a retangular speimen subjeted to either a uniaxial eletri field or a uniaxial mehanial loading is onsidered to simulate the polarization and depolarization proedure. Results are ompared with the experimental data. he analysis for nonlinear onstitutive law may start with the linear onstitutive equations. here are four types of them as follows: E ype d: S = s + d E, D = d + ε E, (1) ij ijkl kl nij n m mij ij mn n ype e: ype g: E ij ijkl kl nij n = S e E, D ij ijkl kl nij n S = s + g D, S m mij ij εmn n D = e S + E, (2) n nij ij βmn m E = g + D, (3) ype h: D ij ijkl kl mij m S n nij ij βnm m = S h D, E = h S + D, (4) where the standard notation of IEEE is adopted. and S are the stress and strain tensors respetively, E and D are the vetors of eletri field and eletri displaement respetively, s and are the elasti oeffiient tensors, ε is the dieletri permittivity tensor, β is the inverse dieletri tensor, d, e, g and h are the piezoeletri onstant tensors, respetively, in different types of onstitutive laws. hese onstitutive equations are also true in inremental forms. 1 Hysteresis loops and onstitutive laws for piezoeletri materials Figure 1 shows the hysteresis loops of the eletri displaement and strain under uniaxial eletri or mehanial loading in the orientation of the x 3 -axis. he multi-linearly approximated hystereses are shown in Figure 2, where the key-points indiated by apital Latin letters orrespond to one another in the four figures. he eletri displaement and strain in the nonlinear ase may be written as D = D (r) + D (p), (5) D (p) = D (E) + D (σ ), (6) S = S (r) + S (p), (7) S (p) = S (E) + S (σ ), (8) where the supersripts (r) and (p) denote, respetively, the reversible and irreversible portions for the orresponding quantities. he irreversible eletri displaement, known as the remnant polarization, and the irreversible strain, known as the remnant strain, may be again lassified into the portions due, respetively, to the eletri field (180 domain swithing) and the stress (90 domain swithing). hey are denoted by the supersripts (E) and (σ ), respetively. LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

3 Figure 1 Hysteresis phenomena observed in piezoeramis. (a) Hysteresis of eletri field E 3 vs. eletri displaement D 3 ; (b) butterfly hysteresis of eletri field E 3 vs. strain S 33 ; () hysteresis of stress 33 vs. strain S 33 ; (d) hysteresis of stress 33 vs. eletri displaement D 3. Figure 2 Multi-linear approximation for hystereses of piezoeramis. (a) Hysteresis of eletri field E 3 vs. eletri displaement D 3 ; (b) butterfly hysteresis of eletri field E 3 vs. strain S 33 ; () hysteresis of stress 33 vs. strain S 33 ; (d) hysteresis of stress 33 vs. eletri displaement D LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

4 1.1 Relation between the eletri displaement and the eletri field he simplified dieletri hysteresis is shown in Figure 2(a). When the eletri field is added to the unpoled piezoeletri material, the loading path will be line OB'. he eletri field reahes its oerive value at point B', whih indiates the onset of eletri domain swithing. Afterward the loading path will turn to line B'B. If the eletri field is removed gradually at point G on the loading path B'B, the unloading path will be GG'G", parallel to the line OB' or DB. Portion OG' on the ordinate is the irreversible eletri displaement, or the remnant polarization D (E) of point G. On line GG'G", the eletri displaement hange, denoted by D (r), is linear and reversible, and the remnant polarization remains unhanged. he remnant polarization reahes its maximum (or saturation) value at point B. If eletri field inreases ontinuously, the loading path will be BC, whih indiates the saturated region. If the eletri field is removed gradually from point C, the unloading path will return along CB to point D. Point D on the ordinate orresponds to the saturated polarization D (s). If eletri field is added in the opposite diretion, the unloading path will reah point D'. he eletri displaement hange on portion CBDD' is also reversible. We define B' as the first yielding point (domain swithing point), B'B as the eletri yielding range, B as the seond yielding point (saturation point), BC as the eletri saturation range. In a general loading ase, the onditions of reahing the first and seond yielding points may be respetively written as * E = E or f 1 = E E * = 0, (9) and D (p) = D (s) or f 2 = D (p) D (s) = 0. (10) On the path B'B, the eletri yielding riterion may be expressed as f E (E, κ) = E ( E * + κ ) = 0, (11) where E = EE i i ; κ may be known as the parameter of "eletri hardening" and assumed to be the funtion of the aumulation of the remnant polarization indued by the eletri field, i.e. κ = F D ( dd (E) ), (12) and E * is the oerive eletri field. Aording to the data from Li and Fang [7], the ompressive stress in the orientation of poling will ause the hysteresis loop flat. Referring to the formula of the oerive stress vs. the eletri field by Shäufele and Härdtl [8], a linear dependene of the oerive eletri field on a superposed ompressive stress field may be appropriately assumed as * p E = E + m, (13) σ where the pair of angle parenthesis means that <x> = 0 for the negative ontent x, otherwise x = x ; E is the oerive eletri field in absene of the ompressive stress field; σ is the oerive stress without the superposed eletri field; m is a onstant parameter; p is the omponent of the stress deviator in the diretion of poling, whih an be expressed as p = g (p) g (p) = ij g (p) i g (p) j, (14) where g (p) is the unit vetor in the diretion of poling. Eq. (13) implies that the stati hydrauli stress has no influene on the oerive eletri field. Aording to eq. (14), the riteria for loading and unloading an be written as LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

5 fe f E = 0 and dei > 0, loading; E i fe f E = 0 and dei = 0, neutral loading; E i fe f E = 0 and dei < 0, unloading (reversible). E i hese riteria are also true for the yielding onditions in the following setions. Atually, eq. (11) gives a model of isotropi eletri hardening. In this model, the yielding surfae is a spherial surfae entered at the origin of the oordinate system formed by the eletri field omponents E 1, E 2 and E 3 with the radius of E * + κ. However, the model of kineti eletri hardening is more pratial, as indiated by the eletri fields at points G and G" in Figure 2(a). his model an be represented by the following equation: f E (E, κ ) = E κ E * = 0. (15) Eq. (15) represents a spherial surfae with the enter at κ i (i = 1, 2, 3) and the radius of E *. However, this model is not onvenient for obtaining the nonlinear onstitutive laws and the finite element expressions in an inremental form. o make the model of isotropi hardening available, weak hardening behavior of the material is assumed. he flow theory in the eletri yielding range is ( E) ( E) g ddi = dλd, Ei where g may be known as the poling potential. For the aligned flow theory, g is set to be the yielding funtion f E, as shown in eq. (11). hus, we have ( E) ( E) fe ( E) Ei ddi = dλd = d λd. (16) Ei E his equation means that the inrement of poling has the same diretion as the eletri field. he geometrial explanation of eq. (16) is that the vetor of the inrement of poling is perpendiular to ( E) ( E) the yielding surfae. aking the square for both sides of eq. (16), we obtain dλ = dd, and eq. (16) beomes ( E) ( E) Ei dd i = d D. (17) E he differentiation form of eq. (11) is fe m fe dfe = dei dp dκ = 0. (18) Ei σ κ It is obtained due to eq. (12) in the ase of isotropi hardening that fe ( E) ( E) d κ = FD ( d ) d. κ D D (19) Let ε' be the slope of line B'B. Usually, ε' is muh larger than the slope of line OB', the dieletri permittivity ε 33. Considering that eq. (12) is also true under the uniaxial eletri field E 3, we have F D ( d D ) =. ε (20) ( E) 1 D 74 LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

6 Substituting it into eq. (19), we have fe m 1 ( E) dei = dp + d D. (21) E σ ε 1.2 Relation between eletri field and strain i he simplified butterfly hysteresis of strain versus eletri field is shown in Figure 2(b). he points B', B, C, D, G, et. have the same meanings as those in Figure 2(a). Due to experimental data, the slope of loading path OB' is set to be zero. he first yielding point B', as well as point E', is also the turning point between strain inreasing and dereasing. he strain is zero and the eletri field reahes the oerive one at the two points. he strain hange on the unloading path GG'G" is reversible, and OG' is the orresponding remnant strain denoted by S (E). It is assumed that GG'G" is a straight line, and its extension passes the intersetion between line BD and the absissa. If the negative eletri field inreases ontinuously at point G", the unloading path will be G"E'. he seond yielding point B has the largest remnant strain S (s), orresponding to the ordinate of point D. he loading path at point B is line BC. he unloading path is CBDD', where the strain hange is reversible. Sine the eletri field auses poling and domain swithing, whih immediately indue the remnant strain, it may be assumed that the inrement of maximum prinipal remnant strain ds is parallel to the eletri field. Furthermore, it is assumed that the remnant strain is volume preserving. Beause of this assumption, it is not neessary to analyze the relation between E 3 and S 11. We an immediately write the other two inrements of the prinipal remnant strains indued by the eletri field as ( E) ( E) 1 ( E) ds2 = ds3 = d S1, (22) 2 where the subsripts 1, 2 and 3 represent the order of the inrement of the prinipal strains respetively from the largest to the smallest. Hene, the inrement of the remnant strain omponents an be expressed in terms of the prinipal remnant strains as ( ) ( ) (1) (1) ( ) (2) (2) ( ) (3) (3) ds E = ds E g g + ds E g g + d S E g g, (23) ij 1 i j 2 i j 3 i j (k) where g i represents the diretion osines of the k-th remnant prinipal strain inrement on the x i -axis. Substitution of (22) into (23) reads ( E) ( E) 3 ( E) ( E) 1 dsij = d S gi g j δij, (24) 2 3 (E) where g i is the diretion osines of the eletri field on the x i -axis; ds is the inrement of the equivalent remnant strain, defined by where ( E ) 1/2 ( E) 2 ( E) ( E) ds = dsij d Sij, 3 (E) ds is equal to ds 1 if eq. (22) is satisfied. he parameter of the eletri hardening κ an also be expressed as Hene, aording to eq. (11) we have ( E) ( E ) ( E ) 1 (25) κ = F ( d S ). (26) ε LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

7 f E ( E) ( E) d κ = F ε ( d S )ds κ. (27) It should be noted that the piezoeletri onstant d 333 is atually the slope of the oblique line DB shown in Figure 2(b). Let d' be the slope of the oblique line B'B. Usually d' is muh larger than d 333, hene, we an approximately write ( E) dfε 1+α F ε ( d S ) = =, (28) ( E) ds d where α = d 333 E * /S (s). (29) With the use of eqs. (27) and (28), eq. (18) beomes fe m 1 α ( E) dei = dp + + d S. (30) E σ d Eqs. (30) and (21) must be the same, thus we have 1 ( E) 1+ α ( E) dd = d S. ε d 1.3 Relation between stress and strain i When the stress level exeeds the oerive stress σ, the strain due to the stress an be divided into elasti (reversible) strain and remnant strain indued by 90 domain swithing, as shown by path GG'G" in Figure 2(). he material an be onsidered to be isotropi for elasti strain. he inrement of the remnant strain omponent an be onsidered to be proportional to the orresponding deviate stress omponent. his relation is the same as that of flow theory in plastiity. hus, the Von-Mises riterion an be employed as the domain-swithing riterion, and an be written as * fσ (, κ) = ( σ + χ) = 0, (32) ij where is the equivalent stress and χ is the parameter of strain hardening defined respetively by 3 = ij ij 2 where ij is the deviate stress omponent and 1/2 ( ) S σ, (31) (33) χ = H ( d ), (34) ε ( ) ds σ is the equivalent remnant strain inrement ( ) indued by stress. he definition of ds σ is similar to eq. (25) with the supersript (E) replaed by (σ). Shäufele and Härdtl [8] proposed the following formula for the oerive stress σ * : * E p σ = σ + n, (35) E where n is a onstant parameter, E p is the omponent of the eletri field in the poling diretion, whih an be expressed as E p = E g (p) = E i g (p) i. (36) he well-known Prandtl-Reuss flow theory an be adopted for the onstitutive law, whih is ( σ ) dsij = dλ σ. (37) ij 76 LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

8 aking the square for both sides of eq. (37), we obtain ds ( σ ) ( ) ij = ds σ ij With the use of eqs. (33) (35), differentiation of eq. (32) reads where ( ) S σ ( ) dλ = ds σ. hen eq. (37) beomes σ. (38) n ( σ) ( σ) dij = d Ep + H ε ( d S )d S, E (39) ij H ε ( d ) is the slope 1/s' of the oblique line BB' in the mehanial yielding range as shown in Figure 2(). Hene, eq. (39) beomes 1.4 Influene of stress on polarization n 1 ( σ ) dij = dep + d S. (40) E s ij When the tensile stress 33 exeeds the oerive stress, the eletri domains in all the orientations perpendiular to the x 3 -diretion will swith or tend to swith toward either positive or negative diretion of the x 3 -axis with an idential probability (Figure 3). Hene, in the marosopi point of view, the poling effets indued by the domain swithing are eliminated. In other words, the tensile stress 33 does not generate marosopi polarization. But it indues remnant strain due to the 90 domain swithing. If there is no poling in the x 1 - or x 2 -diretion, this 90 domain swithing does not generate polarization. In ontrary, if material is polarized in the x 1 - or x 2 -diretion, this 90 domain swithing will redue the number of the domains in this poling diretion (Figure 4). Hene, it redues the remnant polarization in this diretion. However, at most, this redution may ause the remnant polarization to vanish. It annot yield inverse polarization. he eletri domains an swith toward all the orientations perpendiular to the x 3 -axis with an equal probability if ompressive stress 33 is applied. Hene, no additional marosopi remnant polarization an be observed in the diretions of x 1 - and x 2 -axes. If there is no remnant polarization in the x 3 -axis, it may be onsidered that the numbers of eletri domains in both positive and negative diretions of the x 3 -axis are equal. Also, it may be onsidered that the numbers of domains in both diretions to swith toward the orientations parallel to the x 1 - and x 2 -axes are also equal. Hene, the 90 domain-swithing indued by the ompressive stress 33 does not generate polarization in the x 3 -diretion. However, it generates remnant strain in this diretion, as mentioned Figure 3 Domain swithing in the unpoled material indued by tensile stress. he domains swith toward the positive and negative diretions of the x 3 -axis with equal probability. LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

9 Figure 4 Domain swithing in the polarized material indued by ompressive stress. he domains swith toward all the diretions perpendiular to the x 3 -axis with equal probability. previously. In ontrary, if the material is polarized in the x 3 -diretion, the ompressive stress 33 will redue the polarization. Nevertheless, the redution of polarization at most auses the remnant polarization in the x 3 -diretion to vanish, and it does not yield any inverse poling. Aording to the analysis above, the relation between 33 and D 3 an be expressed as in Figure 2(d). he initial polarization is generated by eletri field. he eletri displaement hange on the line DD' is reversible and an be determined by eq. (1). On the line D'E', the eletri displaement inrement an be divided into reversible part and irreversible part. he former an be omputed by using eq. (1), and the latter an be determined due to the summary of the previous analysis as follows: Atually, only the deviate stress indues the 90 domain swithing. he stati hydrauli stress does not ause the 90 domain swithing. Stress is always reduing the remnant polarization. However, the polarization redution due to stress auses at most the remnant polarization to vanish; If there is no polarization in a diretion, the stress does not indue polarization in this diretion either. Positive deviate stress (tension) does not indue polarization, and negative deviate stress (ompression) redues the magnitude of remnant polarization. hen, we an assume that the polarization due to the stress is parallel to the orientation of the minimum (ompressive) prinipal deviate stress, namely ( σ) ( σ) (3) dd = d D g, (41) i where g (3) is the unit vetor in the orientation of the minimum prinipal stress, whih satisfies the ondition D (p) g (3) < 0. (42) It should be noted that when D (p) i + dd (σ) i has an opposite sign to D (p) i, we have to put D (p) (σ) i + dd i = 0. In Figure 2(d), point D' is the first yielding point. Aording to the data provided by Kamlah and Böhle [6], the ordinate of point D' represents the oerive stress σ. Point E is the seond yielding point (saturation point), orresponding to the saturation stress σ s. If removing the ompressive stress at point G", the unloading path is G"G', whose extension is assumed to pass the intersetion of line D'D and the ordinate axis. On the line G"G' the hange of eletri displaement is reversible. On the line D'E', the stress 33 indues new remnant polarization. Let us onsider the yielding i 78 LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

10 riterion, eq. (32), and define the parameter of strain hardening as aking the differentiation of eq. (32), we obtain ( σ ) χ = H D ( d D ). (43) n ( σ) ( σ) d = d E + H ( d ) d. D D (44) ij ij p D E In Figure 2(d), the slope of line DD' is the inverse of the piezoeletri onstant d 333 shown in eq. (1), and the slope of line D'E' is 1/d'. Usually, d' is muh larger than d 333, hene it may be approximated that ( σ ) dh D 1 β HD ( d D ) = =, (45) ( σ ) dd d where hen, eq. (40) beomes β = d 333 σ * /D (s). (46) n 1 β ( σ ) dij = dep + d D. (47) E d ij Sine we use the same yielding riterion (32), eqs. (40) and (44) must be the same. Hene, we have 1 ( σ) 1 β ( σ) ds = d D. (48) s d 2 Finite element formulas From now on, the tensor notation is abandoned and the engineering notation is adopted for stress, strain and piezoeletri onstants. he onjugates of the subsripts between the tensor notations and engineering notations are 11-1, 22-2, 33-3, 23-4, 31-5, and Due to eq. (4), the linear onstitutive equations in an inremental form an be expressed in terms of matries and vetors as {d} = [ D ]{ds (r) } [h]{dd (r) }, (49) {de} = [h] {ds (r) } + [β S ]{dd (r) }. (50) he above equations are written in the global oordinate system. However, in many textbooks and artiles, the linear onstitutive equations are written in suh a manner that the x 3 -axis is in the poling diretion. In finite element modeling, poling diretions in the elements are usually different from the global x 3 -axis. Hene, the matries in eqs. (49) and (50) have to be transformed orrespondingly. Using eqs. (5) and (7), we an rewrite eqs. (49) and (50) as {d} = [ D ]({ds} {ds (p) }) [h]({dd} {dd (p) }), (51) {de} = [h] ({ds} {ds (p) }) + [β S ]({dd} {dd (p) }), (52) where the matries are defined as the transformed ones. Eqs. (6) and (8) an be rewritten as {ds (p) }= {ds (E) }+ {ds (σ) }, (53) {dd (p) }= {dd (E) }+ {dd (σ) }. (54) With the use of eqs. (17), (24), (38) and (41), the above equations beome {ds (p) ( ) }= ds E {N (E) ( ) }+ ds σ {A}, (55) {dd (p) }= dd (E) {g (E) }+ dd (σ) {g (3) }, (56) LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

11 where Let us put that ds ( E) ( p) ( E) 3 ( E) ( E) ( E) ( E) ( E) ( E) { N } = g1 g1 1/3, g2 g2 1/3, g3 g3 1/3, 2 ( E) ( E) ( E) ( E) ( E) ( E) 2 g2 g3, 2 g3 g1, 2 g1 g 2, { } {A}= =,,,,, ( σ ) ( p) = ξds, d S = (1 ξ )ds, From eqs. (31) and (32), we find η dd = 2 With the use of eq. (59), eqs. (45) and (46) beome d ( E) ( p) = η dd, (57). (58) ( σ ) ( p) d D = (1 η) dd. (59) ε s (1 + α)(1 β) ξ. (60) (1 ξ ) + ε s (1 + α)(1 β) ξ ( p) {ds (p) }= ds {Φ}, (61) {dd (p) }= dd (p) {Ψ}, (62) where {Φ} = ξ{n (E) } + (1 ξ){a}, {Ψ} = η{g (E) } + (1 η){g (3) }. fσ Eqs. (51) and (52), multiplied by { } and f E respetively, with the use of eqs. (21), { E} (30), (40) and (47), lead to {A} [ D ]({ds} {ds (p) }) {A} [h]({dd} {dd (p) }) n 1 ( ) n 1 ( ) = dep ds σ σ + = d Ep + d D (63) E s E d {g (E) } [h] ({ds} {ds (p) }) + {g (E) } [β S ]({dd} {dd (p) }) m 1 ( E) m 1+ α ( E) = d p + d D = dp + d S. σ ε σ d Substituting eqs. (61) and (62) into the above equations and employing eq. (59), we obtain ( p) ( p) 21d 22 d 2 (64) ( p) ( p) a11ds + a12 dd = b 1. (65) a S + a D = b Due to eqs. (63) and (64), there are two expressions for oeffiients a 11, a 12, a 21, a 22, i.e. D 1 a11 = { A} [ ]{ Φ} (1 ξ ), a12 = { A} [ h]{ Ψ}, s (66) ( E) ( E) S 1 a21 = { g } [ h] { Φ}, a22 = { g } [ β ]{ Ψ} η, ε and D 1 β a11 = { A} [ ] { Φ}, a12 = { A} [ h]{ Ψ} (1 η), d (67) ( E) 1+ α ( E) S a21 = { g } [ h] { Φ} ξ, a22 = { g } [ β ]{ Ψ}. d 80 LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

12 he inhomogeneous terms on the right hand sides of eq. (65) are n Ep b 1 = {A} ([ D ]{ds} [h]{dd}) + d, E (68) m p b 2 = {g (E) } ([h] {ds} [β S ]{dd}) + d. (69) σ Both groups of the oeffiients in eqs. (66) and (67) lead to the same solutions for ds and dd (p). For the onstant oerive eletri field and onstant oerive stress, the last terms on the right hand sides of eqs. (68) and (69) an be negleted. Hene, the solutions for ds and dd (p) in eq. (65) are ( p) d S = { CSS} {d S} + { CSD} {d D} ( p), (70) d D = { CDS} {d S} + { CDD} {d D} where 1 ( E) D { CSS} = ( a12{ g }[] h + a22{ A}[ ]) Δ 1 ( E) S { CSD} = ( a12{ g }[ β ] + a22{ A}[]) h Δ, (71) 1 ( E) D { CDS} = ( a11{ g } [ h] + a21{ A} [ ]) Δ 1 ( E) S { CDD} = ( a11{ g }[ β ] + a21{ A}[]) h Δ Δ = a 11 a 22 a 12 a 21. (72) ( p) Substituting ds and dd (p) shown in eq. (70) into eqs. (61) and (62), then into piezoeletri eqs. (51) and (52), we have D {d } = [ ] rp{d S} [ h] rp{d D} S, (73) {d E} = [ h] rp{d S} + [ β ] rp{d D} where D D D [ ] rp = [ ] [ ]{ Φ}{ CSS} + [ h]{ Ψ}{ C } DS D [ h] rp = [ h] [ h]{ Ψ}{ CDD} + [ ]{ Φ}{ C } SD S. (74) [ h] rp = [ h] [ h] { Φ}{ CSS} + [ β ]{ Ψ}{ CDS} S S S [ β ] rp = [ β ] [ β ]{ Ψ}{ CDD} + [ h] { Φ}{ CSD} hey are the reversible-irreversible matries for the onstitutive relations in an inremental form. Generally speaking, [ ] [ ] D D S S h rp h rp, [ ] rp [ ] rp and [ β ] rp [ β ] rp, therefore, relation (73) is non-symmetri. he form of eq. (2) is more onvenient for the finite element analysis. Hene, we transform eq. (73) into the following form: E {d } = [ ] rp{d S} [ e] rp{d E} S, (75) {d D} = [ e] rp{d S} + [ ε ] rp{d E} ( p) ( p) LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

13 where the reversible-irreversible matries are E D S [ ] rp = [ ] rp [ h] rp[ β ] rp[ h] rp S 1 [] e rp = [] h rp[ β ] rp S 1. (76) [] e rp = [ β ] rp[] h rp S S 1 [ ε ] rp = [ β ] rp For variable oerive eletri field and oerive stress, the last terms on the right hand sides of eqs. (68) and (69) should be reserved. he dedution for these reversible-irreversible matries is negleted here. 3 Disussion ij 3.1 he parameters in the reversible onstitutive relation Unpoled piezoeletri materials are isotropi. After polarization they beome transversely isotropi. However, it is noted that the elastiity oeffiients D 11 and D 33 only have a negligible differene after full polarization, and so do the eletri permittivity oeffiients ε S 11 and ε S 33. On the other hand, the reversible portion of eletri displaement inrement due to the eletri field is muh smaller than the irreversible portion, and the reversible portion of the strain inrement due to the stress is muh smaller than the irreversible portion. Hene, the elasti and dieletri behavior of the poled material an still be onsidered to be isotropi and independent of the remnant polarization. However, the piezoeletri oeffiients perform transversely isotropi. As shown in Figures 2(b) and (d), they are proportional to the remnant polarization, and an be written as ( p) D ( s) dij = d ( s) ij, (77) D ( s) where d is the piezoeletri oeffiient for fully poled material. For the PZ ferroeletri eramis, only the oeffiients with subsripts ij = 31, 32, 33, 24, 15 are different from zero. 3.2 Dependene of the hardening parameters on the stress and eletri field Experimental investigation [7] shows that when the ompressive stress parallel to the poling diretion is applied, the hardening parameters ε' and d' will derease. However, the onditions ε' ε 33 and d' d 333 must be satisfied. Hene, we have the following expressions: ε = ε 33 + (ε 0 ε 33 ) e α ε p, (78) d' = d 33 + (d' 0 d 333) e α d p, (79) where ε' 0 and d' 0 are respetively the values of ε' and d' in absene of the ompressive stress, α ε and α d are onstants. In fat, we only need either ε' or d'. he other one an be determined using eq. (31). 3.3 Material behavior after the seond yielding point When the seond yielding point is reahed, the onstitutive relations in inremental forms again beome reversible and linear as shown in eqs. (1)-(4). It may be noted that the material onstants in these equations are different from those before the seond yielding point. On the other hand, 82 LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

14 sine the elements in the medium reah the seond yielding point suessively, the inremental method is still needed in the finite element modeling proedure. 4 Variational priniple and finite element method he variatioal priniple for eletro-mehanially oupled ferroeletri/ferroelasti materials is ( ijδ S ij D iδ E i) d V = F iδ u idγ + f iδ u id V q δφd Γ, (80) V Γ σ where ij is stress, S ij is strain, u i is displaement, F i is the tration exerted on the boundary, f i is the body fore per unit volume, D i is eletri displaement, E i is eletri field, φ is eletri portential, q is the eletri harge per unit area on the boundary, Γ σ represents the boundary with the known tration, Γ D represents the boundary where the eletri displaement is given. In eq. (80), the strain and eletri field are defined as S ij = (u i,j + u j,i )/2, E i = φ,i. (81) In addition, the displaement and eletri potential have to be equalized to the given values on the boundaries Γ u and Γ φ respetively. he inremental form for eq. (80) is also true, i.e. (d ijδ S ij d D iδ E i)d V = d F iδ u idγ + d f iδ u id V d q δφd Γ. (82) V Γ σ Inrements of displaement, eletri potential, strain and eletri field in an element an be expressed, respetively, as {du} = [N u ]{d û }, {dφ} = [N φ ]{d ˆ φ }, {ds} = [B]{d û }, {de} = [P]{d ˆ φ }, (83) where {d û } and {d ˆ φ } are, respetively, the inrements of the nodal displaement and the nodal eletri potential; [N u ] and [N φ ] are, respetively, the orresponding shape funtions defined in the element. Substituting eq. (83) into eq. (82) and using eq. (75), the finite element algebrai equations are obtained: K L duˆ dfˆ L G =, (84) dφ ˆ dqˆ where {du ˆ } and {dφ ˆ } are, respetively, the vetors of the inrements of the global nodal displaements and eletri potentials. he other matries and vetors are defined as E rp, [L] = [ ][] rp[ ]d e V e e Ve [K] = [ B] [ ] [ B]dV V V B e P V, [L ] = [ P] [ e ] rp[ B]dV Γ D Γ D, e Ve S [G] = [ P] [ ε ] rp[ P]dV, (85) e Ve u {d ˆQ } = [ N ] d d. Γ φ q Γ e Γ e e e {d ˆF } = [ N ] {d F}d Γ, (86) where {d ˆF } and {d ˆQ } are, respetively, the vetors of the inrements of the global nodal equivalent fores and eletri harges; [K] is the global stiffness matrix; [L] and [L ] are piezoeletri matries; [G] is the dieletri matrix; symbol denotes the assemblage of elemental matries. e LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

15 5 Examples Consider a retangular speimen made of PZ-4, 10 mm wide along the x 1 -axis and 16 mm high in the x 3 -diretion. he external eletri field E 3 is applied in the x 3 -diretion. We inrease the eletri field until the speimen enters the eletri yielding range, then the speimen is fully polarized and over-polarized. hereafter, we keep the eletri field unhanged, and exert a ompressive load in the x 3 -diretion to depolarize the speimen. We shall simulate this proedure. Material onstants are given as follows: In the virgin state, the material is isotropi and paraeletri. Young s modulus Y = 80 GPa, Poisson s ratio is ν = 0.35; piezoeletri oeffiients vanish, dieletri oeffiients ε = F/m (from refs. [5,6]). In the fully polarized state, material is transversely isotropi. Elasti onstants C 11 =139 GPa, C 12 =77.8 GPa, C 13 =74.3 GPa, C 33 =113 GPa, C 44 =25.6 GPa; piezoeletri onstants d 33 = C/N, d 31 = C/N, d 15 = C/N; dieletri oeffiients ε 11 = F/m, ε 33 = F/m (from ref. [9]). In the poling state, the above-mentioned material onstants an be obtained by interpolation between the virgin state and fully polarized state. Some other onstants are: dieletri oeffiient ε = F/m, piezoeletri oeffiient d = C/N, strain-hardening oeffiient s = Pa 1, oerive field E = 1.0 MV/m, oerive stress σ = 50 MPa, saturation polarization P sat = 0.2 C/m 2, saturation strain S sat = (measured in the diagrams provided in ref. [7]). Sine the speimen an deform freely, the external eletri field does not indue stress. Hene, the deformation of the speimen only has a negligible ontribution to the eletri displaement. hus, only the equation [ G]{d Φ ˆ} = {d Qˆ }, shown in eq. (84), needs to be onsidered. With the obtained solution of the inrements of nodal potential, the eletri fields and eletri displaements in the elements an be determined. Furthermore, the strains in the elements an be determined. It is found that the speimen is fully polarized when the field reahes 1.1 MV/m. Afterwards, the relations between the eletri field and the eletri displaement, and between the eletri field and strain are reversible and again, linear. In this stage, the material onstants are C 11 = 139 GPa, C 12 = 77.8 GPa, C 13 = 74.3 GPa, C 33 = 113 GPa, C 44 = 25.6 GPa, d 31 = C/N, d 33 = C/N, d 15 = C/N, ε 11 = F/m, ε 33 = F/m (from refs. [7,9]). he speimen is meshed into 40 4-noded quadrilateral isoparametri elements with 5 equal-spaings along the x 1 -axis and 8 equal-spaings along the x 3 -axis. he number of nodes is 54. Keep on inreasing the external eletri field until E z reahes 3 MV/m. Curves of E 3 vs. D 3 and E 3 vs. S 3 in this range are shown in Figures 5 and 6 respetively. It is seen that the urves are in good agreement with those shown in the first quadrant of Figures 2(a) and 2(b). Afterward, a uniaxial ompressive load along the x 3 -axis is exerted to the speimen. Sine the model introdued in this theory employs an isotropi hardening model instead of the kineti hardening one, the strain, eletri displaement and polarization at above-mentioned 3 MV/m eletri field must be treated as their initial values, and our omputation has to start again. he material behavior is linear before the stress in the speimen reahes the oerive stress σ. When it reahes the oerive stress, the strains are found to be S 1 = , S 3 = , and the eletri displaement is D 3 = C/m 2. Using the material onstants in this stage, we obtain 84 LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

16 Figure 5 he urve of E 3 vs. D 3. Figure 6 he urve of E 3 vs. S 3. the Poisson s ratio ν = With the use of the equation, 2 2 ε = S S + S S + S S 21+ ν ( ) ( ) ( ) ( ) It is omputed that the equivalent strain ε = , when the stress reahes σ. he element is supposed to yield if the equivalent strain in the element reahes this value. From the first part of eq. (84), i.e. [ K]{d Uˆ} = {d Fˆ}, we may solve for the inrements of nodal displaement, and then obtain the inrements of the strain and eletri displaement. It is found that when the stress 3 reahes MPa, the polarization disappears, and the depolarization proedure finishes. Figure 3 shows the strain indued by stress. It does not inlude the strain indued by the above-mentioned 3 MV/m eletri field. It is seen that the urve in Figure 7 is in agreement with that shown in the fourth quadrant of Figure 2(). Otherwise, the strain at the end of depolarization proedure is he urve in Figure 8 also agrees with that in Figure 2(d).. Figure 7 Curve of 3 vs. S 3. Figure 8 Curve of 3 vs. D 3. 6 Conluding remarks Based on the analysis of the hysteresis loops of piezoeletri materials, some assumptions are presented for establishing the nonlinear onstitutive relations. he first and seond yielding points are defined. he yielding riteria are provided. In the yielding range, the inrements of eletri displaement and strain onsist of reversible and irreversible portions. he reversible portion an be determined by using the linear onstitutive relations. o determine the irreversible portions of LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no

17 the eletri displaement and strain, we presented the following assumptions: (1) he inrement of the remnant polarization indued by eletri field is in the same diretion as that of the eletri field. (2) he inrement of the maximum prinipal remnant strain indued by eletri field is parallel, in orientation, to the eletri field. (3) he inrement of the stress-indued remnant strain omponent is proportional to the orresponding stress deviator omponents. (4) he inrement of the stress-indued polarization is parallel, in orientation, to the maximum ompressive stress deviator, and always redues the magnitude of the total poling. But it does not indue inverse polarization. (5) Inrement of remnant strain is volume preserving. (6) he oerive stress depends on the internal material property as well as the eletri field omponent in the poling diretion. In turn, the oerive eletri field depends on the internal material property as well as the stress deviator omponent in the poling diretion. Stress and eletri field also influene the hardening parameters. Based on these assumptions, the reversible-irreversible matries for finite element appliation are formulated. It is found that the nonlinear onstitutive relation in a matrix and vetor form is non-symmetri. On the other hand, sine muh more parameters and influening fators than those in the linear ase are introdued to the nonlinear onstitutive relation, the finite element analysis for this problem beomes somewhat diffiult. he variational priniple and the finite element method based on the theory of nonlinear onstitutive law are proposed. A retangular speimen subjeted to a uniaxial eletri field and, suessively, a ompressive loading is onsidered as an example. he proedure that the speimen is from the virgin state to fully polarized, and further to over-polarized state, then depolarized under the uniaxial ompressive loading, is simulated. Results of strain and eletri displaement obtained are in agreement with experimental data. In a summary, this work provides the theory and method for analyzing historial-dependent nonlinear behaviors of eletromehanial and eletroni devies. 1 Chen K H, Hagood N. Modeling of nonlinear piezoeramis for strutural atuation. SPIE Conferene. Smart Mater Strut, 1994, 2190: Hwang S C, Lynh C S, MMeeking R M. Ferroeletri/ferroelasti interations and a polarization swithing modal. Ata Metall Mater, 1995, 43: [DOI] 3 Bassiouny E, Ghaleb A F, Maugin G A. hermodynamial formulation for oupled eletromehanial hysteresis effets (I): Basi equations, (II): Poling of eramis. Int J Engng Si, 1988, 26: [DOI] 4 Bassiouny E, Maugin G A. hermodynamial formulation for oupled eletromehanial hysteresis effets (III): Parameter identifiation, (IV): Combined eletromehanial loading. Int J Engng Si, 1989, 27: [DOI] 5 Kamlah M, sakmakis C. Phenomenologial modeling of the non-linear eletro-mehanial oupling in ferroeletris. Int J Solids Strut, 1999, 36: [DOI] 6 Kamlah M, Böhle U. Finite element analysis of piezoerami omponents taking into aount ferroeletri hysteresis behavior. Int J Solids Strut, 2001, 38: [DOI] 7 Li C Q, Fang D N. Experimental investigation for onstitutive relations of ferroeletri eramis PZ. Ata Meh Sinia, 2001, 32(1): Shäufele A, Härdtl K H. Ferroelasti properties of lead zironate titanate eramis. J Am Ceram So, 1996, 79: Park S, Sun C. Frature riterion for piezoeletri eramis. J Am Ceram So, 1995, 78(6): [DOI] 86 LI YaoChen Si China Ser G-Phys Meh Astron February 2007 vol. 50 no