Parametric multiphysics finite-volume theory for

Original Article

Parametric multiphysics finite-volume theory for periodic composites with thermo-electro-elastic phases

Journal of Intelligent Material Systems and Structures 1–23 Ó The Author(s) 2017 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1045389X17711789 journals.sagepub.com/home/jim

Qiang Chen1, Wenqiong Tu2, Ruonan Liu1 and Xuefeng Chen1

Abstract The zeroth-order multiphysics finite-volume micromechanics has been proposed to model coupled thermo-electromechanical behaviors of unidirectional composites embedded with piezoelectric phases. Parametric mapping is implemented within the multiphysics finite-volume theory’s framework, facilitating modeling of multiphase piezoelectric materials with complex microstructures with relatively coarse unit cell discretization. The resulting theory admits piezoelectric materials with complete anisotropy and arbitrary poling direction and enables rapid generation of the entire set of coupled thermo-mechanical, piezoelectric properties, figures of merits, as well as the local fluctuations of fields within the composite microstructures with greater fidelity than its predecessor. The proposed method is verified extensively by comparison with the finite-element homogenization technique, which produces an excellent agreement in a wide range of volume fractions but offers much better stability and efficiency. The contrast with the rectangular theory is also presented and discussed, demonstrating the advantage and the need for the development of parametric formulation. This extension further increases the finite-volume direct averaging micromechanics theory’s range of applicability, providing an attractive standard for investigating multiphase and multiphysics problems with different microstructural architectures and scales against which other approaches may be compared. Keywords Piezoelectric composites, micromechanics, finite-volume theory, homogenization, thermo-electro-mechanical coupling

Introduction In the last decades, piezoelectric materials continue to attract considerable attention due to their distinct characteristics that provide unique coupling between mechanical and electric properties. Specifically, piezoelectric materials generate mechanical deformation when subjected to electric loads (direct piezoelectric effect) and electric field when subjected to mechanical loads (inverse piezoelectric effect). Monolithic piezoelectric materials, however, have several drawbacks, including brittleness, limited range of coupled properties, and pronounced directionality of sensing or actuation abilities (Kar-Gupta and Venkatesh, 2005, 2007). These drawbacks may be mitigated through the use of piezoelectric composites which offer better technological solutions in applications such as sensors, actuators, and ultrasonic transducers by providing enhanced properties when compared to their monolithic counterparts (Kar-Gupta et al., 2008; Kar-Gupta and Venkatesh, 2013; Kyung Ho and Yoon Young, 2010). The use of piezoelectric materials in the form of composites requires the development of appropriate

analysis techniques that can determine the response of this class of materials in thermal, electric, and mechanical environments. Numerical approaches like finiteelement method (FEM) are the prevailing ways to describe the behavior of piezoelectric composites because the complex local structural details and interaction, and arbitrary material properties can be easily addressed via commercial finite-element software (Sladek et al., 2017). However, large property contrast in composite materials produces high stress/electric field gradients within the composite microstructures, requiring detailed mesh discretization in the affected regions. Because finite-element analysis is sensitive to mesh discretization, computationally intensive three-dimensional

1

State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, P.R. China 2 Engineering Technology Associates, Inc, Troy, MI, USA Corresponding author: Xuefeng Chen, State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R. China. Email: [email protected]

2 (3D) mesh refinement is often required to ensure converged results. The finite-volume method has proved a potential alternative to the prevalent finite-element technique for the solution of boundary-value problems in solid mechanics (Berezovski et al., 2008; Leveque, 2002; Versteeg and Malalasekera, 2007), following its original development and application focused on applications to fluid mechanics problems. At present, three versions of this technique may be identified. The first two, namely, the cell centered and cell vertex finite-volume techniques (Fallah, 2004), have been developed for the structural mechanics problems of homogeneous materials, while the third version has evolved independently for use in the mechanics of multiphase and heterogeneous materials. The defining feature of the finitevolume method which distinguishes it from the FEM is the local satisfaction of governing field equations in a volumetric sense within subvolumes of discretized domain of interest. The reader is referred to the review article by Cavalcante et al. (2012) for additional references in this area. In particular, the finite-volume direct averaging micromechanics (FVDAM) homogenization theory (Bansal and Pindera, 2005, 2006) was developed explicitly to simulate the response of heterogeneous materials with periodic microstructures. The accuracy of homogenized and local responses has been shown to be comparable to the FEM but with greater efficiency. However, the standard FVDAM theory employs rectangular discretization of the repeating unit cell, yielding undesirable stress concentrations at the fiber/matrix interphase. Therefore, the parametric mapping capability has been incorporated into the structural version of the finite-volume theory by Cavalcante et al. (2007a, 2007b) originally developed by Bansal and Pindera (2003), which maps a reference square subvolume onto a quadrilateral subvolume in the actual microstructure. The mapping capability offers the same flexibility as the FEM for modeling inclusions with curved boundaries while retaining its quick convergence feature in the presence of highly heterogeneous microstructures (Cavalcante et al., 2007b). Subsequently, the homogenized version had been constructed by Gattu et al. (2008) and Khatam and Pindera (2009), who introduced parametric mapping into the original rectangular FVDAM theory of Bansal and Pindera (2006). As described in a sequence of papers by Pindera and his coworkers (Cavalcante et al., 2011, 2012; Pindera et al., 2009), the parametric FVDAM is particularly wellsuited for multiscale analysis of heterogeneous media with two-dimensional (2D) microstructures. An important feature and product of the theory that differentiate it from the FEM is the homogenized Hooke’s law valid under arbitrary 3D loading of 2D material architectures. To duplicate this capability with commercial finite-element codes, fully 3D unit cell models are required, with concomitant increase in execution times.

Journal of Intelligent Material Systems and Structures 00(0) Most recently, Tu and Pindera (2014, 2016) incorporated the cohesive-zone model into the parametric FVDAM framework to simulate progressive fiber/ matrix debonding phenomenon observed in SiC/Ti composites and damage evolution in cross-ply polymer composites. The standard FVDAM theory cannot easily accommodate such capability. Chen et al. (2016a) extended FVDAM to 3D domains, hence increasing the breadth of applicability to composites with triply periodic microstructures. The objective of this article is to present the equations and framework for a general coupled thermoelectro-elastic homogenization theory and provide a versatile tool for the analysis of underlying theoretical issues and underpinning mechanisms to develop materials and structures with superior properties. Towards this end, the zeroth-order parametric FVDAM theory is further extended to accommodate coupled thermo-electro-mechanical loading. The coupling between mechanical and electric fields is characterized by piezoelectric coefficients in the generalized material constitutive equations. The parametric mapping implementation is carried out within the finite-volume theory’s framework, which is based on a two-scale representation of the displacement field and electric potential in the reference square subvolume and then mapped onto a quadrilateral subvolume in the actual microstructures. The proposed micromechanical method not only predicts a complete set of effective moduli, but also describes the distribution of local fields with fidelity necessary to simulate damage, which is unattainable in most of the mean-field approaches such as Eshelby (Fakri et al., 2003) and Mori-Tanaka (MT) (Dunn, 1993; Li and Dunn, 1998) schemes. Given the use of piezoelectric composites as actuators, among other applications, the ability to predict high-fidelity stress fields is important in life-time assessment of these devices. The approach presented in this study also supersedes the so-called high-fidelity generalized method of cells of Aboudi (2001) for the analysis of multiphysics behavior of heterogeneous media based on the unnecessary two-level unit cell discretization and the concomitant satisfaction of higher-order moments of equilibrium equations in a volume-averaged sense.

Theoretical formulation In the standard version of the coupled thermo-electroelastic FVDAM theory, the repeating unit cell of a continuously reinforced multiphase material is discretized into Nb 3 Ng rectangular subvolumes of hb 3 lg dimensions, which span the unit cell along the y2- and y3-axes in the plane normal to the continuous reinforcement, Figure 1(a). Each subvolume is occupied by a specific homogeneous material to mimic the internal structure of a heterogeneous material.

Chen et al.

3 surface-averaging in the solution of unit cell problem based on the local/global stiffness matrix approach. The transformation is given by equation (2) y(k) i (h, j) =

4 X

Nj (h, j)y(i j, k) , i = 2, 3

ð2Þ

j=1

where N1 (h, j) = (1=4)(1  h)(1  j), N2 (h, j) = (1=4) (1 + h)(1  j), N3 (h, j) = (1=4)(1 + h)(1 + j), and N4 (h, j) = (1=4)(1  h)(1 + j).

Generalized local stiffness matrix construction

Figure 1. Unit cells with the same 48 3 48 subvolume discretization using (a) rectangular subvolumes (standard FVDAM) and (b) quadrilateral subvolumes (parametric FVDAM).

Following the zeroth-order homogenization theory (Bensoussan et al., 2011; Charalambakis, 2010), the dis(k) in the kth placement field u(k) i and electric potential a subvolume are represented by the two-scale expansion involving macroscopic and fluctuating components u(k) eij xj + u0(k) i = i  j x j + a0 a(k) =  E

ð3Þ

(k)

 j are the macroscopic strains and electric where eij and E fields, respectively. The fluctuating displacement components u0i (k)(i = 1, 2, 3) and electric potential components a0(k) are represented by the second-order Legendre polynomial expansion in the reference coordinates (h, j), respectively, which are given by u0 i

(k)

Figure 2. Mapping transformation in parametric FVDAM.

(k) (k) (k) = Wi(00) + hWi(10) + jWi(01) +

(k) + Wi(20)

In contrast, in the parametric finite-volume theory, the unit cell is divided into Nb 3 Ng quadrilateral subvolumes designated by the index k whose location is specified by the subvolume vertices (y2( p, k) , y3( p, k) ), Figure 1(b). Following the convention of Cavalcante et al. (2007a), the vertices are numbered in counterclockwise manner starting from the lower left corner k) (1, k) , y3 ). Accordingly, the face Fp is defined by (y(1, 2 the endpoints (y2( p, k) , y3( p, k) ) and (y2( p + 1, k) , y(3p + 1, k) ) for p = 1, 2, 3, and 4 such that p + 1 ! 1 when p = 4 (Figure 2). The components of the unit normal vector n( p, k) = ½ n2( p, k) n3( p, k)  for the face Fp are n(2p, k)

y( p + 1, k)  y(3p, k) ( p, k) y2( p + 1, k) + y(2p, k) = 3 n3 = lp lp

ð1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where lp = (y2( p + 1, k)  y2( p, k) ) + (y3( p + 1, k)  y3( p, k) ) . The kth quadrilateral subvolume is created by mapping the reference square subvolume in the h  j plane bounded by 1  h  1 and 1  j  1 to its actual location in the unit cell, which is used to perform the

a0

(k)

1 2 (k) (3j  1)Wi(02) 2

(k) (k) (k) = W4(00) + hW4(10) + jW4(01) +

(k) + W4(20)

1 (3h2  1) 2

1 (3h2  1) 2

ð4Þ

1 2 (k) (3j  1)W4(02) 2

(k) where Wi(: :) are the unknown micro-variables associated with each subvolume. The local strains and electric fields are then obtained in terms of the macroscopic and fluctuating strain and electric components, respectively

e(k) ij

  ∂u0 j (k) 1 ∂u0 i = eij + + 2 ∂yj ∂yi

Ei(k)

0 (k)  i  ∂a =E ∂yi

ð5Þ

Integrating interfacial displacements and electric potentials for the kth subvolume in the reference coordinates produces the surface-averaged displacements and electric potentials (the subscript k is omitted for convenience), which provides the fundamental unknowns for the parametric multiphysics FVDAM

4

Journal of Intelligent Material Systems and Structures 00(0)

^ui0(1, 3)

1 = 2

+ ð1

0

u i (h, 71)dh

1 = 2

^ui0(2, 4)

1

a ^

0 (1, 3)

1 = 2

+ ð1

u0 i (61, j)dj

1

+ ð1

0

a (h, 71)dh

a ^

0 (2, 4)

1 = 2

1

+ ð1

a0 (61, j)dj



eth 11

1

Using the definitions in equation (6), the first- and (k) (k) , Wi(01) , and second-order unknown coefficients, Wi(10) (k) (k) , Wi(02) in equation (4) are expressed in terms of Wi(20) the fluctuating surface-averaged displacements, electric (k) (see potentials, and the zeroth-order coefficients Wi(00) (k) are Appendix 1). The zeroth-order coefficients Wi(00) subsequently obtained in terms of the fluctuating surface-averaged displacements and electric potentials upon satisfying the equilibrium equations and the charge conservation equation within each subvolume in the large in the manner described in the sequel. ( p) The surface-averaged traction components ^ti and normal electric displacement ^f ( p) on each oblique face Fp of the kth subvolume with the unit normal vector n( p) are obtained in terms of interfacial stresses and electric displacements upon use of Cauchy’s relations ti( p) = sji( p)  nj( p) and f ( p) = Dj( p) nj( p) , respectively ð ð 1 ( p) 1 ^ti ( p) = ti dl = s(ji p)  n(j p) dl lp lp ð ð ð7Þ ^f ( p) = 1 ^f ( p) dl = 1 D( p)  n( p) dl j j lp lp

The coupling between the mechanical and electric fields is characterized by piezoelectric coefficients in the piezoelectric constitutive equations. Allowing for complete anisotropy of each subvolume material, the subvolume piezoelectric constitutive equation is given by 3(k)

s11 6 s22 7 7 6 6 s33 7 7 6 6 s23 7 7 6 6 s13 7 7 6 6 s12 7 7 6 6 D1 7 7 6 4 D2 5 D3

2

C11 6 C21 6 6 C31 6 6 C41 6 =6 6 C51 6 C61 6 6 e11 6 4 e21 e31

C12 C22 C32 C42 C52 C62 e12 e22 e32

C13 C23 C33 C43 C53 C63 e13 e23 e33

eth 22

½ a11 ð6Þ

2

thermal strain and thermal field components are related to a change in temperature from a given referenced temperature by

C14 C24 C34 C44 C54 C64 e14 e24 e34

C15 C25 C35 C45 C55 C65 e15 e25 e35

(k) (k) (k) where s(k) ij , Di , eij , and Ei are the stress components, electric displacement components, strain components, and electric field components for the kth subvolume, (k) respectively. Cij(k) , e(k) ij , and kij are the material stiffness components, piezoelectric components, and dielectric components for the kth subvolume, respectively. The

eth 33

a22

2eth 23 a33

2eth 13

a23

2eth 12

a13

a12

(k)

E1th

E2th

E3th

z1

z2

z3 (k) DT

=

where a(k) ij are subvolume coefficients of thermal expansion and z(k) are subvolume pyroelectric constants for i the kth subvolume. The surface-averaged strains on the faces of quadrilateral subvolumes in the actual microstructure needed in the expressions for the surface-averaged tractions are generated using the following relations between surface-averaged partial derivatives of the displacement field in the two coordinates 2 4

∂^u0 i ∂y2 ∂^u0 i ∂y3

3( p) 5

2 ^4 =J

∂^u0 i ∂h ∂^u0 i ∂j

3(^p) 5

ð9Þ

Similarly, the surface-averaged electric field components on the faces of quadrilateral subvolumes in the actual microstructure needed in the expressions for the surface-averaged electric displacements are generated using the relations between the partial derivatives of electric potential in the two coordinates " ∂^a0 #( p) " ∂^a0 #(^p) ∂y2 ^ ∂h =J ð10Þ ∂^ a0 ∂y3

∂^ a0 ∂j

^=J 1 = ½J^22 , J^23 ; J^32 , J^33  is the inverse of the where J volume-averaged Jacobian of the transformation Ð Ð  = 1 + 1 + 1 Jdhdj. The superscripts p and ^p denote J 1 4 1

C16 C26 C36 C46 C56 C66 e16 e26 e36

e11 e12 e13 e14 e15 e16 k11 k21 k31

e21 e22 e23 e24 e25 e26 k12 k22 k32

3(k)

e31 e32 7 7 e33 7 7 e34 7 7 e35 7 7 e36 7 7 k13 7 7 k23 5 k33

3(k) e11  eth 11 7 6 6 e22  eth 22 7 7 6 7 6 e33  eth 33 7 6 6 2e23  2eth 7 6 23 7 7 6 6 2e13  2eth 13 7 7 6 7 6 2e12  2eth 12 7 6 6 E  Eth 7 1 6 1 7 7 6 4 E2  E2th 5 E3  E3th 2

ð8Þ

the faces of quadrilateral and reference subvolumes, respectively, with the following correspondence ^p = 1, 3 ! j = 71 and ^p = 2, 4 ! h = 61. The surface-averaged strains and electric fields are (k) obtained in terms of the unknown coefficients Wi(mn) which are then explicitly expressed in terms of the

Chen et al.

5

surface-averaged fluctuating displacements and electric potentials upon use of definitions given by equation (6) and the application of the three static equilibrium equations and the charge conservation equation within each subvolume in the large (which provide a total of four equations for the determination of zeroth-order coeffi(k) ) cients Wi(00) ð

ð sij nj dS =

ti dS =

Sk

Sk

ð

ð Di ni dS =

Sk

ð

4 X p=1

p=1

Sk

^ti(2, k1) + ^ti(4, k) = 0 ^f (2, k1) + ^f (4, k) = 0

lp^ti( p) = 0



f ( p) dlp =

4 X



( p)

lp ^f

lp

ð11Þ

k1) k) ^u0(2, = ^u0(4, i i

(1)  (k) = ½ ^ Y p

 = ½ e11 X

^p(2)

e22

(3) ^ p

(4) ^ p 

(k)T

e33

2e23

^0 (2) u

^0 (3)

^u0(2 p)

^ u0(3 p)

a ^

N(k) = ½ n(1) n(2) n(3) 2 0 0 0 n3 6n 6 2 0 n3 0 =6 4 0 n3 n2 0 0 0 0 0 2 C21 C22 C23 6 6 C31 C32 C33 6 6 C41 C42 C43 6 6 Cðk Þ = 6 C51 C52 C53 6 6 C61 C62 C63 6 6 4 e21 e22 e23 e31 e32 e33

n(4) 

^ (k) = ½ ^u0 (1) U ^ u

0 ( p)

=½^ u0(1 p)

u

with ^ p

2e13

( p)

2e12

(k)T ^0 (4) 

u

a ^0

= ½ ^t1( p) ^t2( p) ^t3( p) ^f ( p) T  1 E  2 E  3 T E

with

(2, k1)

=a ^0



(4, k)

a ^0

(3, k1)

=a ^0

(1, k)

ð16Þ

herein, i = 1, 2, 3. The superscripts (k  1), k, and (k  1), k are associated with adjacent subvolumes along rows and columns. Consequently, the resulting systems of equations can be symbolically represented by ^ = DCX  + DGDT Kglobal U

0 ( p) T

ð15Þ



k1) k) ^u0(3, = ^u0(1, i i

ð12Þ

where

ð14Þ

Similarly, the direct enforcement of displacement and electric potential continuity conditions takes the form

The final relations between the surface-averaged tractions, electric displacements, and the displacements, and electric potentials for the kth subvolume can be written in generic form as follows  (k) = N(k) C(k) X  + K(k) U ^ (k)  N(k) sth (k) Y



^f (3, k1) + ^f (1, k) = 0

=0

p=1

ð13Þ



^ti(3, k1) + ^ti(1, k) = 0

p=1

lp

4 ð X

f dS =

ti( p) dlp =

4 X

subvolumes, followed by direct enforcement of displacement and electric potential continuity conditions. Proceeding from the left to right and then upward, the enforcement of traction and normal electric displacement continuity conditions takes the form

ð17Þ

Finally, K(k) are the generalized local stiffness matrices which can be explicitly expressed in terms of the subvolume geometry and material properties (see Appendix 2).

^ contains all unknown interfacial and boundwhere U ary surface-averaged displacements and electric potentials. The global DC matrices are comprised of the differences in the local stiffness matrices of adjacent subvolumes. The vector DGDT contains thermal contribution. It should be noted that the generalized global stiffness matrix is singular because the applied periodicity conditions that connect boundary subvolumes on the opposite sides of the unit cell cannot explicitly eliminate the unit cell’s rigid body motion. The singularity is eliminated by constraining the four corner subvolume faces in the manner described by Bansal and Pindera (2006). The remaining interfacial surface-averaged displacements and electric potentials are then determined by solving the reduced global systems of equations iteratively, enabling calculation of the microvariables W1 , W2 , W3 , and W4 , hence the associated displacement, electric potential, strain, stress, and electric fields in each subvolume.

Global stiffness matrix assembly

Homogenization

The unknown interfacial surface-averaged displacements and electric potentials are determined by solving a global system of equations generated by first enforcing traction and normal electric displacement continuity conditions at each interface of the adjacent

Solution of the global system of equations for the ^ vector enables us to surface-averaged unknown U calculate the elements of the generalized Hill’s electroelastic concentration matrices A(k) for the kth subvolume in the localization relations (Hill, 1963)

(k)T

n2 0 0 0

with n( p) 3 0 0 ( p) 0 07 7 7 0 05

n2

n3

C24 C34 C44

C25 C35 C45

C26 C36 C46

e12 e13 e14

e22 e23 e24

C54 C64

C55 C65

C56 C66

e15 e16

e25 e26

e24 e34

e25 e35

e26 e36

k21 k31

k22 k32

e32 e33 e34

3(k)

7 7 7 7 7 7 e35 7 7 e36 7 7 7 k23 5 k33

6

Journal of Intelligent Material Systems and Structures 00(0)  (k) = A(k) X  + Ath(k) X

ð18Þ

where the matrices A(k) can be determined by successive application of one macroscopic strain or electric field at a time. The matrices Ath(k) represent thermal contributions in the kth subvolume. Subsequently, the macroscopic stresses and electric displacements for the unit cell of volume V can be expressed as weighted sum of the average stresses and electric displacements over all subvolumes = s

ð Nk ð Nk X 1 1X  (k) (x)dV (k) =  (k) v(k) s s s(x)dV = V V k=1 k=1 Vk

ð

Nk ð Nk X 1 1X   (k) (x)dV (k) =  (k) v(k) D D= D D(x)dV = V V k=1 k=1 Vk

ð19Þ

where v(k) is the volume fraction of the kth subvolume. Taking the localization relations in the expression for the average composite stresses and electric displacements, in conjunction with the volume-averaged constitutive relation in each subvolume, the global (or effective) constitutive equation for the multiphase thermo-electro-elastic composite is obtained 

    s e  eth  = Z th   E  D E

ð20Þ

where the matrix Z of coefficients of the electro-elastic composite is given in terms of the subvolume geometry, material properties, and the electro-elastic concentration matrices. The structure of the matrix Z is of the form Z =



C e

eT k



where C , e , and k are the effective elastic, piezoelectric, and dielectric coefficients, respectively Z =

Nk X

v(k) Z(k) A(k)

ð21Þ

k=1



 (k) !  Nk 1 1 X G eT (k) (k) th(k) Z v Z A  DT T =   p E V k=1 ð22Þ

(k) (k) where G(k) = Cijkl akl are the subvolume thermal stress coefficients. p(k) are the subvolume pyroelectric constants.

Finite-element homogenization for periodic piezoelectric composites The FEM is a ‘‘golden’’ standard for analysis of almost all kinds of physical problems in diverse area and provides the main vehicle for constructing micromechanics

Figure 3. Flow chart of fully coupled electromechanical solution of unit cell problem using commercial finite-element package ABAQUS.

theories for fully coupled electromechanical analyses of multiphase materials with arbitrary periodic microstructures under arbitrary loading. The common procedure of modeling piezoelectric composites in a commercial finite-element package involves four steps (see Figure 3) (Tu et al., 2015): (a) create a fully 3D repeating unit cell with specified volume fraction; (b) apply periodic boundary conditions so that deformation modes are the same on the opposite sides of repeating unit cells; (c) apply unit strain/electric field nine times and solve the corresponding problems to obtain local stress and electric fields; and (d) compute the generalized effective stiffness matrix by averaging local stress and electric fields. All the finite-element calculations were made with the commercial finite-element package ABAQUS. 3D multifield 20-node quadratic piezoelectric brick elements with displacement degree of freedom and an additional electric potential degree of freedom which are capable of fully coupled electromechanical analysis were employed herein for comparison with the multiphysics FVDAM approach. It should be noted that the modeling procedure in commercial finite-element software is quite involved. Repeating this process on an interactive session would be very time-consuming and prone to error if one wishes to perform a parametric study. Therefore, the ABAQUS Python script was used to automate the modeling process, reducing a lot of tedious manual work and saving time.

Numerical results In this section, we first demonstrate the convergence behavior of the parametric multiphysics FVDAM model by examining the homogenized properties for hypothetical composites with various combinations of

Chen et al.

7

Figure 4. Classification of continuously reinforced piezoelectric composites based on relative orientation of poling direction: (a) ‘‘longitudinal’’ composites and (b) ‘‘transverse’’ composites.

fiber/matrix properties and fiber volume contents. Subsequently, the accuracy of the proposed theory is tested by determining the homogenized thermo-electromechanical properties and figures of merit of piezoelectric composites comprised of constituents with different moduli contrast at a wide range of fiber volume fractions. The results are compared with finite-element and analytical predictions which provide the correct answer. Finally, the local stress and electric fields for composites with different architectural microstructures are provided for evaluation with finite-element predictions. We note that depending on the relative orientation of poling direction of constituent materials, two types of piezoelectric composites, namely, ‘‘longitudinal’’ and ‘‘transverse’’ composites, can be recognized in the study (Figure 4). Specifically, in ‘‘longitudinal’’ composites, fiber and matrix phases are both poled in the directions that are parallel with x1-direction (i.e. fiber direction). In ‘‘transverse’’ composites, the fiber is poled in the direction that is parallel with x1-direction while the matrix is poled in the direction orthogonal to the poling direction of the fiber phase. By examining these materials, the ability of the multiphysics FVDAM to correctly solve multiphase and multiphysics unit cell problems could be verified.

Convergence study We start with investigating the proposed theory’s convergence behavior by calculating homogenized moduli of two hypothetical composite materials with very large fiber/matrix modulus contrasts, namely, Zfiber =Zmatrix = 0:01, 100, as a function of mesh discretizations in the range of 8 3 8 to 160 3 160. The effective moduli were generated for three fiber volume fractions, namely, Vf = 0:05, 0:35, 0:60, to cover a wide range of fiber contents. The fiber spaced in square array that exhibits a circular cross-section shape shown in Figure 1(b) was used in the calculations. Figure 5 illustrates the convergence of homogenized properties, E22 , G12 , and k22 , for the two combinations of fiber/ matrix properties. The moduli have been normalized

by the corresponding moduli generated using the finest 160 3 160 mesh since it captures the circular fiber shape sufficiently well for both the dilute (Vf = 0:05) and non-dilute (Vf = 0:35, 0:60) fiber volume fraction cases. Hence, converged results correspond to moduli ratio of 1.0. It is observed that converged values are obtained with as few mesh as 30 3 30 for each combination and that further increasing the mesh refinement does not yield substantial improvement for the homogenized property calculations. Similar observations are found for other effective generalized stiffness matrix elements (not presented herein). This implies that the rate of convergence with mesh discretization is very rapid by this technique and the homogenized properties can be calculated accurately using relatively low mesh discretizations.

Validation on homogenized electromechanical constants This section presents a comparison of the effective electromechanical constants as a function of fiber volume fraction, which was predicted by the parametric multiphysics FVDAM theory, the unit-cell-based finiteelement homogenization method introduced in section ‘‘Finite-element homogenization for periodic piezoelectric composites’’, as well as literature data from KarGupta and Venkatesh (2007) for two ‘‘longitudinal’’ piezoelectric composites. The materials examined include piezoelectric composites with piezoceramic (PZT-7A) fibers embedded in a piezoceramic matrix (BaTiO3) and a soft-non-piezoelectric epoxy matrix (PVDF), respectively. The electromechanical properties that are given for the x1 poling direction are listed in Table 1. All the constituent materials are transversely isotropic. It should be noted that the maximum volume fraction used in FVDAM prediction is about 0.7 since a square array arrangement of circular fibers can only give a maximum fiber volume fraction of p=4 which occurs when the fiber spacing reaches fiber diameter. In such case, we used dashed line to extrapolate to the maximum volume fraction of 1.0. Figure 6 reveals that

8

Journal of Intelligent Material Systems and Structures 00(0)

Figure 5. Convergence of the selected homogenized moduli as a function of mesh discretization for composites with volume fractions of 0.05, 0.35, and 0.60: (a) Zf /Zm = 0.01 and (b) Zf /Zm = 100.

Table 1. Electromechanical properties for the matrices and fiber. Properties

PZT-7A

BaTiO3

PVDF

r (kg/m) C11 (GPa) C12 (GPa) C13 (GPa) C22 (GPa) C23 (GPa) C33 (GPa) C44 (GPa) C55 (GPa) C66 (GPa) e11 (C/m2) e12 (C/m2) e26 (C/m2) k11 (n C/Vm) k22 (n/Vm)

7700 131 74.2 74.2 148 76.2 148 35.9 25.3 25.3 10.99 22.324 9.31 2.081 3.984

5700 145.5 65.94 65.94 150.4 65.63 150.4 42.37 43.86 43.86 17.36 24.322 11.4 15.1 12.8

1770 4.63 2.22 2.22 4.84 2.72 4.84 1.06 0.0526 0.0526 20.1099 0.004344 20.001999 0.07083 0.06641

Material properties are given for the x1 poling direction.

the electromechanical constants generated by the multiphysics FVDAM theory with the 48 3 48 representative mesh discretization shown in Figure 1(b) correlate well with finite-element results in a wide range of fiber volume fractions for the two material systems. The effective electro-mechanical constants vary linearly with PZT-7A fiber volume fraction for BaTiO3-based system, but nonlinearly for PVDF-based system due to

the large fiber/matrix modulus mismatch in PVDFbased system. The electromechanical constants converge to matrix properties in the limit as the fiber volume fraction goes to zero. Results for the homogenized moduli of ‘‘longitudinal’’ PZT-7A/BaTiO3 and PZT-7A/PVDF composites generated by the MT (Dunn, 1993), asymptotic homogenization (AHM) (Guinovart-Dıa¨ az et al., 2001) and FEMs are also provided herein (Tables 2 and 3) for quantitative comparison with the multiphysics FVDAM predictions. In this case, we limit calculation of effective properties for composites containing 0.1 and 0.5 fiber contents. As observed, there is a good agreement between the predictions of four approaches for both the dilute and non-dilute cases. To get more evidence on the accuracy of the proposed multiphysics FVDAM theory, the predictions of the FVDAM model for the ‘‘transverse’’ BaTiO3and PVDF-based systems (assuming the matrix phase is poled in the x2-direction) are compared with the corresponding finite-element results. As observed in Figure 7, the correlation between the FVDAM and FEM predictions is remarkable. In addition, we observe the elastic constants Cij are virtually the same as in the case of ‘‘longitudinal’’ material systems, indicating that the elastic constants are barely influenced by the poling directions of the matrix for the two material systems.

Chen et al.

9

Figure 6. Comparison of the electromechanical constants for two ‘‘longitudinal’’ piezoelectric composites as a function of the PZT-7A fiber volume fraction predicted by the parametric FVDAM with the finite-element method.

Validation on homogenized thermal expansion coefficients and pyroelectric constants The sensing capability of piezoelectric materials is influenced by two factors in thermal fields, that is, thermal expansion coefficients and pyroelectric constants (Kumar and Chakraborty, 2009). The performance of

the present parametric FVDAM in predicting thermal constants is validated by comparison with the corresponding finite-element results obtained by Jayendiran and Arockiarajan (2014). The material considered herein is a ‘‘longitudinal’’ PZT-5A/epoxy piezoelectric composite and the thermo-electro-elastic properties of constituent materials, which are given for the x1 poling

10

Journal of Intelligent Material Systems and Structures 00(0)

Table 2. Quantitative comparison of the homogenized properties for composites containing 0.1 fiber contents with analytical and numerical predictions. PZT-7A/BaTiO3

C11 (GPa) C22 (GPa) C23 (GPa) C44 (GPa) e11 (C/m2) e12 (C/m2) k11 (nC/Vm)

PZT-7A/PVDF

AHM

MT

FEM

FVDAM

AHM

MT

FEM

FVDAM

144.01 150.06 66.77 41.65 16.71 24.13 213.80

144.01 150.09 66.74 41.66 16.71 24.13 213.80

144.01 150.09 66.74 41.65 16.71 24.13 13.80

144.01 150.09 66.73 41.65 16.71 24.13 13.80

12.71 5.54 3.04 1.23 1.15 20.01 0.27

12.71 5.54 3.05 1.24 1.15 20.01 0.28

12.70 5.55 3.03 1.23 1.15 20.01 0.28

12.70 5.55 3.03 1.23 1.15 20.01 0.28

AHM: asymptotic homogenization; MT: Mori-Tanaka; FEM: finite-element method; FVDAM: finite-volume direct averaging micromechanics.

Table 3. Quantitative comparison of the homogenized properties for composites containing 0.5 fiber contents with analytical and numerical predictions. PZT-7A/BaTiO3

C11 (GPa) C22 (GPa) C23 (GPa) C44 (GPa) e11 (C/m2) e12 (C/m2) k11 (nC/Vm)

PZT-7A/PVDF

AHM

MT

FEM

FVDAM

AHM

MT

FEM

FVDAM

138.14 148.90 71.16 38.91 14.15 23.34 8.58

138.14 149.01 71.06 38.97 14.15 23.34 8.60

138.14 149.07 70.99 38.90 14.15 23.34 8.60

138.14 149.07 70.99 38.90 14.15 23.34 8.60

45.86 11.81 4.64 2.24 6.15 20.09 1.05

45.86 10.86 5.58 2.64 6.15 20.09 1.10

45.90 12.05 4.56 2.24 6.15 20.09 1.10

45.84 12.03 4.56 2.24 6.15 20.09 1.10

AHM: asymptotic homogenization; MT: Mori-Tanaka; FEM: finite-element method; FVDAM: finite-volume direct averaging micromechanics.

direction, are listed in Table 4. Note in the parametric FVDAM theory, the effective thermal stress and pyroelectric constants are obtained using the results of Levin (1967) 

   X Nk (k) T G (k) G (k) v A = p p k=1 

a =C

1



G

ð23Þ

Determination of figures of merit Figures of merit are used to assess the utility of piezoelectric composites for use in specific applications. Various parameters have been used to evaluate the performance of the piezoelectric composites, which mainly include the following: Table 4. Electromechanical property of the matrix and fiber.

ð24Þ

Figure 8 presents the variation of effective thermal expansion coefficients and pyroelectric constants as a function of PZT-5A volume fraction, respectively. It is evident that the effective transverse thermal expansion coefficient a22 of the considered composites leads to the fiber coefficient in a nonlinear manner with the increasing fiber volume fraction. The effective longitudinal thermal expansion coefficient a11 exhibits the same trend after reaching a maximum value just below the volume fraction of 0.1. The pyroelectric constant z increases linearly with the fiber volume fraction. In general, the prediction of pyroelectric constant z shows better agreement than those of thermal expansion coefficients a11 and a22 with the finite-element results (Jayendiran and Arockiarajan, 2014), which may be worthy of further study.

Properties

PZT-5A

Epoxy

C11 (GPa) C12 (GPa) C13 (GPa) C22 (GPa) C23 (GPa) C33 (GPa) C44 (GPa) C55 (GPa) C66 (GPa) e11 (C/m2) e12 (C/m2) e26 (C/m2) k11 (n C/Vm) k33 (nC/Vm) a11 (1026/oC) a33 (1026/oC) z (1026C/m2/oC)

116.8 87.1 87.1 129.3 91.6 129.3 18.85 9.7 9.7 19.2 22.7 5.5 16.4 17.3 2 8.53 600

3.9 1.68 1.68 3.9 1.68 3.9 1.1 1.1 1.1 0 0 0 0.04 0.04 60 60 0

Material properties are given for the x1 poling direction.

Chen et al.

11

Figure 7. Comparison of the electromechanical constants for two ‘‘transverse’’ piezoelectric composites as a function of the PZT7A fiber volume fraction predicted by the parametric FVDAM with the finite-element method.

1. 2.

The piezoelectric coupling constant kt = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  =C  11 = C  + e2 =k .  11 , where C 1  C11 11ffiffiffiffiffiffiffiffiffiffi 11 11 p  11 r, where r is The acoustic impedance Z = C the density of the composite given by r = rf Vf + rm (1  Vf ); and rf , rm , and Vf are

3.

the fiber density, matrix density, and fiber volume fraction, respectively. The piezoelectric charge coefficient dh = d11 + d12 + d13 , where d1i = e1j Sji ; and Sji are the effective compliance components.

12

Journal of Intelligent Material Systems and Structures 00(0)

Figure 8. Comparison of the effective coefficients as a function of fiber volume fraction predicted by the parametric FVDAM with the FEM results.

Figure 9. Variations of figures of merit kt , dh , and Z as a function of fiber volume fraction for longitudinal PZT-7A/BaTiO3 and PZT7A/PVDF systems. Comparison with finite-element and Mori-Tanaka (MT) predictions.

To improve the performance of piezoelectric composites in practical applications, high piezoelectric coupling constant kt and low acoustic impedance Z are desired. The parametric multiphysics FVDAM admits arbitrarily fully anisotropic subvolume materials and provides all 45 independent electromechanical constants of piezoelectric composites. The knowledge of the effective electromechanical constants makes it possible for quick generation of figures of merit of piezoelectric composites. Figure 9 illustrates the variations of three figures of merit, that is, the piezoelectric

coupling constant kt , the acoustic impedance Z, and the piezoelectric charge coefficient dh , for the ‘‘longitudinal’’ PZT-7A/BaTiO3 and PZT-7A/PVDF piezoelectric composites as a function of PZT-7A fiber volume fraction, respectively. Once again, the MT predictions are included in the figures for comparison. It is evident that the correlation between FVDAM, FEM and MT predictions is extremely good, further verifying the validity of the proposed multiphysics theory. Also noted is that the PVDF-matrix system shows improved performance characteristics with higher piezoelectric

Chen et al.

13

Figure 10. Selected electric fields within hexagonal unit cell domain generated by unit electric loading E2 = 1 MV=m. Comparison of the parametric FVDAM predictions (right column) with the FEM solutions (left column): (a) E2 (MV/m) and (b) E3 (MV/m).

coupling constant kt and lower acoustic impedance Z as compared with BaTiO3-matrix system. The PVDF, therefore, is recommended to improve the performance of piezoelectric composites.

Local stress/electric field recovery within the composite microstructures We proceed to investigate the multiphysics FVDAM theory’s predictive capability to accurately capture coupled stress and electric fields within composite microstructures. Moreover, to demonstrate the FVDAM model’s ability to model complex microstructures, a multi-inclusion unit cell with ‘‘longitudinal’’ circular PZT-7A fibers spaced in hexagonal arrays embedded in BaTiO3 matrix containing total volume fraction of 0.3 was first employed in the calculation. Figure 10 presents comparison between the full E2 and E3 electric field distributions generated by the multiphysics finite-element (left columns) and FVDAM models (right columns) by applied electric loading  2 = 1 MV=m, simulating the operation of a piezoelecE tric device in the actuator mode. The multiphysics FVDAM model was generated with 9216 2D quadrilateral subvolumes while the FEM employed 9115 3 6 3D quadratic brick elements. Cursory examination of the two electric field components in Figure 10 reveals excellent agreement between the FVDAM and FEM

predictions. It should be pointed out that the above results could not be reproduced easily using the standard multiphysics FVDAM theory which requires a substantially finer mesh for modeling curved inclusion boundaries than the parametric version. The parametric multiphysics FVDAM does not suffer from this limitation, hence providing an evidence of the parametric model’s capability to analyze complex geometric arrangements of fibers. The full-field effective stress, s12 and s13 , distribu 2 = 1 MV=m tions at the macroscopic electric field of E predicted by the parametric multiphysics FVDAM and FEM models are presented in Figure 11. The corresponding effective stress distributions in the horizontal and vertical mid-plane of the hexagonal unit cell are shown in Figure 12. At the center of the hexagonal unit cell, the stress distributions generated by the parametric FVDAM matches well with finite-element results. But relatively large differences are observed occur near the boundary of the unit cell. They are due to the differences in the applied boundary conditions. Because explicit expressions for the generalized stiffness matrix are available in the multiphysics FVDAM formulation, full periodic conditions can be applied in a direct manner by setting the surface-averaged displacements and electric potentials on the external RUC surface to common unknown quantities. Therefore, the local field distributions generated by the multiphysics FVDAM are

14

Journal of Intelligent Material Systems and Structures 00(0)

2 = 1 MV=m. Comparison of Figure 11. Selected stress fields within hexagonal unit cell domain generated by unit electric loading E the parametric FVDAM predictions (right column) with the FEM solutions (left column): (a) effective stresses (MPa), (b) s12 (MPa), and (c) s13 (MPa).

strictly periodic as observed in the figures. However, the commercial FEM packages do not include full periodic conditions. Imposition of periodic conditions is partially achieved through constraining equations over the node potential and displacement variables on the external RUC surface, resulting in the differences in the coupled local stresses along the boundaries. Next, we examine a hypothetical ‘‘longitudinal’’ PZT-7A/BaTiO3 composite material whose unit cell is comprised of two sets of diagonally placed fibers with radically different radii. The total volume fraction for the large fibers is 0.5, while the total volume fraction for the radically smaller fibers is 0.05. This is a more demanding case because of the more pronounced stress/electric concentrations around the fiber/matrix interface. Similarly,

 2 = 1 MV=m is applied in the x2an electric field of E direction. We focus on the E2 and s13 distributions within the square unit cell domain, which are presented in Figure 13. The parametric multiphysics FVDAM employed 25,600 subvolumes, while the finite-element model was generated with 26,602 3 6 elements. Comparison with finite-element predictions shows excellent agreement both around the large and smaller fibers, validating the multiphysics FVDAM model’s accuracy and reliability to capture local fields in composites with radically different microstructures. Particularly worthy of notice is the fact that the solution times taken to solve the preceding problems in an uncompiled MATLAB environment are remarkably faster for the parametric multiphysics FVDAM analysis

Chen et al.

15

Figure 12. Effective stresses in the horizontal (x2) and vertical (x3) mid-plane of unit cell. Comparison of FVDAM predictions with finite-element solutions.

2 = 1 MV=m. Figure 13. Selected stress/electric fields within square unit cell domain generated by unit electric loading E Comparison of the parametric FVDAM predictions (right column) with the FEM solutions (left column): (a) E2 (MV/m) and (b) s13 (MPa).

relative to the finite-element solution using commercial package ABAQUS, as is shown in Table 5. The computational times are averaged based on five runs. It is important to point out that the solutions for the square unit cell with diagonally placed radically different fibers were performed on different PCs due to the exceedingly large memory required by the FEM for the employed arrays.

There are several factors contributing to the differences in execution times. First, because the multiphysics FVDAM theory uses generalized plane strain/electric field constraint, the homogenized generalized Hooke’s law is valid under arbitrary 3D loading of 2D material architectures. To duplicate this capability using finiteelement approach, computationally intensive 3D unit cell models are required, resulting in a dramatic increase

16

Journal of Intelligent Material Systems and Structures 00(0)

Table 5. Comparison of execution times (averaged based on five runs) required by the parametric multiphysics FVDAM and FEM.

Hexagonal unit cell Square unit cell

Methods

Subvolumes/elements

Execution time (s)

FVDAMa FEMa FVDAMa FEMb

9216 9115 3 6 25,600 26,602 3 6

33 1716 89 6296

FEM: finite-element method; FVDAM: finite-volume direct averaging micromechanics. a The results are generated on a 12 Gbyte PC with 3.3 GHz Intel (R) Core (TM) i3-3220 CPU. b The results are generated on a 128 Gbyte PC with 2.6 GHz Intel (R) Xeon (R) E5-2650 v2 CPU.

in execution times. Furthermore, as will be shown in the sequel, in the multiphysics FVDAM approach most of the solution time is consumed in assembling the generalized global stiffness matrices. The FVDAM theory facilitates explicit derivation of the generalized local stiffness matrix elements, while these elements are evaluated numerically in finite-element technique, considerably slowing down the global stiffness matrix assembly in FEM. This provides an additional explanation for the observed difference in execution times.

Parametric versus standard multiphysics FVDAM models The parametric multiphysics FVDAM model provides superior modeling capability relative to the standard theory, capturing local structural details more efficiently via quadrilateral subvolume discretization. Moreover, the standard multiphysics FVDAM is obtained as a special case of the parametric theory when a reference square subvolume is mapped onto a rectangular subvolume. In this section, we provide a justification for the development of the parametric multiphysics FVDAM model by comparing homogenized properties and local fields generated by the parametric and rectangular multiphysics FVDAM models, respectively. Specifically, the effective properties generated for a BaTiO3 matrix weakened by holes which are simulated by very compliant fibers that yield the stiffness ratio Zfiber =Zmatrix = 1 3 106 are compared, while the local fields are compared for a ‘‘transverse’’ PZT-7A/BaTiO3 material system containing 0.4 fiber volume fraction for the unit cell with increasingly finer discretizations. Figure 14 presents the comparison of the homogenized generalized stiffness elements of a porous BaTiO3 system for the porosity content in the range of 0.0520.7 in 0.05 increments, which were generated by the parametric FVDAM (PFVDAM in the figure), rectangular FVDAM (RFVDAM in the figure) and the finiteelement approaches, respectively. Moreover, to clearly demonstrate the difference in the three methods, enlarged view of the stiffness elements C11 , C22 , C66 , C12 , and C23 at higher volume fractions are presented in Figure 15. In the calculations, the unit cells were discretized into

98 3 98 subvolumes because modeling of inclusions with curved boundaries requires a much finer mesh using the rectangular subvolume discretization. We note that the effective moduli predicted by the parametric FVDAM model show excellent agreement with those predicted by FEM. The rectangular FVDAM model, however, consistently slightly underestimates the effective values of C22 , C12 , and C13 relative to the parametric FVDAM predictions (Figure 15). The small differences are due to the factor that the staircase fiber/matrix boundary approximation in the rectangular FVDAM produces artificial stress concentrations (which will be shown in the sequel). These disturbances may potentially lead to the lower effective modulus predictions relative to the parametric version results through smaller average subvolume stresses. This issue remains to be addressed in a future study. Figure 16 presents the transverse normal stress/electric field distributions generated by the rectangular and parametric multiphysics FVDAM models for the ‘‘transverse’’ PZT-7A/BaTiO3 system containing 0.4 fiber volume fraction produced by macroscopic electric  1 = 1 MV=m in the longitudinal direction, loading E demonstrating the advantage and the need for the parametric formulation. The constituent material properties are the same as those given in Table 1. Due to the stepwise discretization of the circular interface in the rectangular FVDAM, high stress/electric field concentrations are expected at the fiber/matrix boundary. In contrast, the parametric FVDAM produces smoothly varying local fields along the fiber/matrix boundary with rather coarse 24 3 24 unit cell discretizations. No discontinuity and artificial stress/electric field concentrations are observed as is the case with the rectangular FVDAM results seen in the figure. While the rectangular FVDAM theory generates very similar local stress fields as those of parametric FVDAM with highly refined unit cell discretization (96 3 96) within most of the unit cell domain, the interfacial stress disturbances are not completely eliminated. Hence, the coupled electromicromechanical theory based on quadrilateral subvolume discretization developed herein yields more accurate prediction of local fields in piezoelectric composites. This in turn enables identification of

Chen et al.

17

Figure 14. Effective electric-elastic constants as a function of porosity volume fraction. Comparison of rectangular, parametric FVDAM, and FEM predictions.

Figure 15. Enlarged view for the stiffness C11 , C22 , C66 , C12 , and C23 . Comparison of rectangular, parametric FVDAM, and FEM predictions.

underpinning mechanisms of damage initiation and propagation in this class of composites along the interface (Tu and Pindera, 2014), as well as within fiber and matrix phases that will be addressed in future studies. The execution times for different levels of mesh discretizations required by the parametric multiphysics FVDAM to form the generalized global stiffness equation (equation (17)) and to obtain the entire sets of effective thermo-mechanical and piezoelectric constants (equation (21)) for unit cell with square array of circular fibers are illustrated in Figure 17, wherein most of

the solution time is consumed in assembling the global stiffness matrices, demonstrating the FVDAM model’s computational efficiency. The results indicate that the model can be used as an efficient tool in advanced composite material development through rapid identification and selection of constituent materials, as well as incorporating with larger structural codes for further development of multiphysics FVDAM theory to achieve better understanding of underpinning deformation mechanisms of large composite structures in a multiscale setting (Chen et al., 2016b).

18

Journal of Intelligent Material Systems and Structures 00(0)

1 = 1 MPa=m within the ‘‘transverse’’ Figure 16. Selected stress/electric field distributions generated by the actuator type loading E PZT-7A/BaTiO3 systems with fiber volume fraction of 0.4. Comparison of the parametric FVDAM predictions (left column) with the rectangular FVDAM solutions (right column): (a) s22 (MPa), (b) s33 (MPa), (c) E2 (MV/m), and (d) E3 (MV/m).

Discussion The standard computational theory for the fully coupled thermo-electro-mechanical analysis of periodic materials is the FEM. The development of the parametric multiphysics FVDAM theory herein provides an attractive alternative to the prevailing finite-element approach, as well as an efficient standard for solving the multiphase and multiphysics problems against which other methods can be compared. The multiphysics FVDAM shares a lot of similarities with the FEM, while also having distinct differences as described in the

sequel. We end this article by drawing a clear distinction between the multiphysics FVDAM and variational principle-based FEM since at a first glance one might assume they are essentially the same. Both approaches are capable of analysis of multiphase unit cells with arbitrary anisotropic piezoelectric properties with arbitrarily distributed phases and arbitrary shapes. The manner of volume discretization and the formation of generalized local and global stiffness matrices are common to both approaches. However, the multiphysics FVDAM is based on direct satisfaction of equilibrium equations in the large and the continuity of

Chen et al.

Figure 17. Comparison of execution times (averaged based on five runs) as a function of subvolume discretization required by the parametric multiphysics FVDAM theory for square fiber arrays. The results are generated on a 12 Gbyte PC with 3.3GHz Intel (R) Core (TM) i3-3220 CPU.

displacements/electric potentials and forces/electric displacements imposed in an integral sense at the subvolume interfaces. This is contrast with the FEM wherein minimization of global potential energy employed in the discretized unit cells and the continuity of displacements/electric potentials and forces/electric displacements imposed at point-wise sense. In summary, a beneficial result in our parametric multiphysics FVDAM theory is the rapid convergence of the homogenized properties and local fields with relatively coarse unit cell discretization.

19 concentration matrices, and hence the homogenized composite properties. This work provides a versatile tool to researchers and engineers in their future analysis of multiphysics behavior of coupled thermo-electric-mechanical materials. It can provide not only a complete set of global properties, but also local fields with very high resolution at the targeted locations for composites with very complicated microstructures. Numerical results obtained in the study indicate that the FVDAM predictions match well with the FEM but with much greater efficiency. Finally, the contrast with the rectangular FVDAM theory is illustrated, which justifies the development of parametric formulation and provides motivation for its further development, including incorporation of plastic effects and damage evolution. Acknowledgements The lead author would like to thank China Scholarship Council for providing partial support for the research reported herein.

Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding This work was supported by National Natural Science Foundation of China (Nos 51505364, 51405369, and 51605365).

Summary and conclusion

References

To further enhance its range of applicability, the zeroth-order FVDAM theory has been extended by incorporating coupled thermo-electro-mechanical capability into the theory’s analytical framework, enabling efficient and accurate analysis of both local and global responses of heterogeneous materials embedded with piezoelectric phases with different microstructural architectures and scales. The present approach is based on parametric mapping, which is used to map a referenced square subvolume onto a quadrilateral subvolume resident in the actual microstructure. The parametric mapping also facilitates the construction of generalized local stiffness matrices in the actual microstructure that relate surface-averaged tractions and electric displacements to corresponding displacements and electric potentials. The traction and electric displacement continuity conditions at each interface of the adjacent subvolumes, as well as displacement and electric potentials, are enforced in a surface-averaged sense, which are then assembled efficiently into a global system of equations whose solution enables the determination of thermo-electro-mechanical

Aboudi J (2001) Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multiphase composites. Smart Materials and Structures 10: 867–877. Bansal Y and Pindera MJ (2003) Efficient reformulation of the thermoelastic higher-order theory for functionally graded materials. Journal of Thermal Stresses 26: 1055–1092. Bansal Y and Pindera MJ (2005) A second look at the higher-order theory for periodic multiphase materials. Journal of Applied Mechanics: Transactions of the ASME 72: 177–195. Bansal Y and Pindera MJ (2006) Finite-volume direct averaging micromechanics of heterogeneous materials with elastic-plastic phases. International Journal of Plasticity 22: 775–825. Bensoussan A, Lions JL and Papanicolaou G (2011) Asymptotic Analysis for Periodic Structures. Providence, RI: American Mathematical Society. Berezovski A, Engelbrecht J and Maugin GA (2008) Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. Hackensack, NJ: World Scientific. Cavalcante MAA, Khatam H and Pindera MJ (2011) Homogenization of elastic–plastic periodic materials by FVDAM

20 and FEM approaches—an assessment. Composites Part B: Engineering 42: 1713–1730. Cavalcante MAA, Marques SPC and Pindera MJ (2007a) Parametric formulation of the finite-volume theory for functionally graded materials—Part I: analysis. Journal of Applied Mechanics: Transactions of the ASME 74: 935–945. Cavalcante MAA, Marques SPC and Pindera MJ (2007b) Parametric formulation of the finite-volume theory for functionally graded materials—Part II: numerical results. Journal of Applied Mechanics: Transactions of the ASME 74: 946–957. Cavalcante MAA, Pindera MJ and Khatam H (2012) Finitevolume micromechanics of periodic materials: past, present and future. Composites Part B: Engineering 43: 2521–2543. Charalambakis N (2010) Homogenization techniques and micromechanics: a survey and perspectives. Applied Mechanics Reviews 63: 030803. Chen Q, Chen X, Zhai Z, et al. (2016a) A new and general formulation of three-dimensional finite-volume micromechanics for particulate reinforced composites with viscoplastic phases. Composites Part B: Engineering 85: 216–232. Chen Q, Chen X, Zhai Z, et al. (2016b) Micromechanical modeling of viscoplastic behavior of laminated polymer composites with thermal residual stress effect. Journal of Engineering Materials and Technology: Transactions of the ASME 138: 031005. Dunn ML (1993) Micromechanics of coupled electroelastic composites: effective thermal expansion and pyroelectric coefficients. Journal of Applied Physics 73: 5131–5140. Fakri N, Azrar L and El Bakkali L (2003) Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements. International Journal of Solids and Structures 40: 361–384. Fallah N (2004) A cell vertex and cell centred finite volume method for plate bending analysis. Computer Methods in Applied Mechanics and Engineering 193: 3457–3470. Gattu M, Khatam H, Drago AS, et al. (2008) Parametric Finite-volume micromechanics of uniaxial continuouslyreinforced periodic materials with elastic phases. Journal of Engineering Materials and Technology: Transactions of the ASME 130: 031015. Guinovart-Dıa¨ az R, Bravo-Castillero J, Rodrıa¨ guez-Ramos R, et al. (2001) Overall properties of piezocomposite materials 1–3. Materials Letters 48: 93–98. Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 11: 357–372. Jayendiran R and Arockiarajan A (2014) Viscoelastic modeling and experimental characterization of thermoelectromechanical response of 1–3 piezocomposites. Journal of Applied Physics 116: 214103. Kar-Gupta R and Venkatesh TA (2005) Electromechanical response of 1–3 piezoelectric composites: effect of poling characteristics. Journal of Applied Physics 98: 054102. Kar-Gupta R and Venkatesh TA (2007) Electromechanical response of 1–3 piezoelectric composites: an analytical model. Acta Materialia 55: 1093–1108. Kar-Gupta R and Venkatesh TA (2013) Electromechanical response of (2–2) layered piezoelectric composites. Smart Materials and Structures 22: 025035.

Journal of Intelligent Material Systems and Structures 00(0) Kar-Gupta R, Marcheselli C and Venkatesh TA (2008) Electromechanical response of 1–3 piezoelectric composites: effect of fiber shape. Journal of Applied Physics 104: 024105. Khatam H and Pindera MJ (2009) Parametric finite-volume micromechanics of periodic materials with elastoplastic phases. International Journal of Plasticity 25: 1386–1411. Kumar A and Chakraborty D (2009) Effective properties of thermo-electro-mechanically coupled piezoelectric fiber reinforced composites. Materials & Design 30: 1216–1222. Kyung Ho S and Yoon Young K (2010) Layout design optimization for magneto-electro-elastic laminate composites for maximized energy conversion under mechanical loading. Smart Materials and Structures 19: 055008. Leveque RJ (2002) Finite Volume Methods for Hyperbolic Problems. Cambridge: Cambridge University Press. Levin VM (1967) On the coefficients of thermal expansion of heterogeneous materials. Mechanics of Solids 2: 58–61. Li JY and Dunn ML (1998) Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior. Journal of Intelligent Material Systems and Structures 9: 404–416. Pindera MJ, Khatam H, Drago AS, et al. (2009) Micromechanics of spatially uniform heterogeneous media: a critical review and emerging approaches. Composites Part B: Engineering 40: 349–378. Sladek J, Sladek V and Pan E (2017) Effective properties of coated fiber composites with piezoelectric and piezomagnetic phases. Journal of Intelligent Material Systems and Structures 28: 97–107. Tu W and Pindera MJ (2014) Cohesive zone-based damage evolution in periodic materials via finite-volume homogenization. Journal of Applied Mechanics: Transactions of the ASME 81: 101005. Tu W and Pindera MJ (2016) Damage evolution in cross-ply laminates revisited via cohesive zone model and finitevolume homogenization. Composites Part B: Engineering 86: 40–60. Tu W, Yang Y and Pindera MJ (2015) Evaluation of homogenized moduli of composite materials with finite-volume micromechanics and Abaqus. SIMULIA Regional User Meetings, Minneapolis, MN, USA. Versteeg HK and Malalasekera W (2007) An Introduction to Computational Fluid Dynamics: The Finite Volume Method. New York: Pearson Education.

Appendix 1 In this section, we present relations between the first-, second-order microvariables, and the surface-averaged interfacial displacements, electric potentials, and the zeroth-order microvariables. Using the generalized subvolume constitutive equations, equation (8), and noting that none of the stresses and electric displacements depends on the out-of-plane coordinates y1 , the four-field equations (equilibrium and their electric counterparts) in equation (11) that need to be satisfied in the kth subvolume become

Chen et al.

21

h

i h 2 i 2 2 2 C66 J^22 + C55 J^32 + ðC65 + C56 ÞJ^22 J^32 W1ð20Þ + C66 J^23 + C55 J^33 + ðC65 + C56 ÞJ^23 J^33 W1ð02Þ h 2 i h 2 i 2 2 + C62 J^22 + C54 J^32 + ðC64 + C52 ÞJ^22 J^32 W2ð20Þ + C62 J^23 + C54 J^33 + ðC64 + C52 ÞJ^23 J^33 W2ð02Þ h 2 i h 2 i 2 2 + C64 J^22 + C53 J^32 + ðC63 + C54 ÞJ^22 J^32 W3ð20Þ + C64 J^23 + C53 J^33 + ðC63 + C54 ÞJ^23 J^33 W3ð02Þ h 2 i h 2 i 2 2 + e26 J^22 + e35 J^32 + ðe25 + e36 ÞJ^22 J^32 W4ð20Þ + e26 J^23 + e35 J^33 + ðe25 + e36 ÞJ^23 J^33 W4ð02Þ = 0 h 2 i h 2 i 2 2 C26 J^22 + C45 J^32 + ðC25 + C46 ÞJ^22 J^32 W1ð20Þ + C26 J^23 + C45 J^33 + ðC25 + C46 ÞJ^23 J^33 W1ð02Þ h 2 i h 2 i 2 2 + C22 J^22 + C44 J^32 + ðC24 + C42 ÞJ^22 J^32 W2ð20Þ + C22 J^23 + C44 J^33 + ðC24 + C42 ÞJ^23 J^33 W2ð02Þ h 2 i h 2 i 2 2 + C24 J^22 + C43 J^32 + ðC23 + C44 ÞJ^22 J^32 W3ð20Þ + C24 J^23 + C43 J^33 + ðC23 + C44 ÞJ^23 J^33 W3ð02Þ h 2 i h 2 i 2 2 + e22 J^22 + e34 J^32 + ðe24 + e32 ÞJ^22 J^32 W4ð20Þ + e22 J^23 + e34 J^33 + ðe24 + e32 ÞJ^23 J^33 W4ð02Þ = 0 h 2 i h 2 i 2 2 C46 J^22 + C35 J^32 + ðC45 + C36 ÞJ^22 J^32 W1ð20Þ + C46 J^23 + C35 J^33 + ðC45 + C36 ÞJ^23 J^33 W1ð02Þ h 2 i h 2 i 2 2 + C42 J^22 + C34 J^32 + ðC44 + C32 ÞJ^22 J^32 W2ð20Þ + C42 J^23 + C34 J^33 + ðC44 + C32 ÞJ^23 J^33 W2ð02Þ h 2 i h 2 i 2 2 + C44 J^22 + C33 J^32 + ðC43 + C34 ÞJ^22 J^32 W3ð20Þ + C44 J^23 + C33 J^33 + ðC43 + C34 ÞJ^23 J^33 W3ð02Þ h 2 i h 2 i 2 2 + e24 J^22 + e35 J^32 + ðe23 + e34 ÞJ^22 J^32 W4ð20Þ + e24 J^23 + e35 J^33 + ðe23 + e34 ÞJ^23 J^33 W4ð02Þ = 0 h 2 i h 2 i 2 2 e26 J^22 + e35 J^32 + ðe25 + e36 ÞJ^22 J^32 W1ð20Þ + e26 J^23 + e35 J^33 + ðe25 + e36 ÞJ^23 J^33 W1ð02Þ h 2 i h 2 i 2 2 + e22 J^22 + e34 J^32 + ðe24 + e32 ÞJ^22 J^32 W2ð20Þ + e22 J^23 + e34 J^33 + ðe24 + e32 ÞJ^23 J^33 W2ð02Þ h 2 i h 2 i 2 2 + e24 J^22 + e33 J^32 + ðe23 + e34 ÞJ^22 J^32 W3ð20Þ + e24 J^23 + e33 J^33 + ðe23 + e34 ÞJ^23 J^33 W3ð02Þ h 2 i h 2 i 2 2  k22 J^22 + k33 J^32 + ðk23 + k32 ÞJ^22 J^32 W4ð20Þ  k22 J^23 + k33 J^33 + ðk23 + k32 ÞJ^23 J^33 W4ð02Þ = 0

Using the relations between the zeroth- and second-order microvariables, Wið00Þ , and Wi(20) , and the surfaceaveraged displacements ^u0i ( p) and electric potentials a ^ 0( p) , the zeroth-order microvariables are obtained directly in terms of surface-averaged displacements and electric potentials 2

^u0(2) u0(4) 1 +^ 1

3

7 6 0(1) 7 6 ^u1 + ^u0(3) 1 7 6 7 6 0(2) 7 6 ^u2 + ^u0(4) 2 3 2 7 6 W1(00) 6 0(1) 0(3) 7 7 6 6 W2(00) 7 6 7 = F1 Q6 ^u2 + ^u2 7 7 6 0(2) 4 W3(00) 5 0(4) 6 ^u3 + ^u3 7 7 6 W4(00) 7 6 0(1) 7 6 ^u3 + ^u0(3) 3 7 6 7 6 (2) (4) 4a ^ +a ^ 5 a ^ (1) + a ^ (3)

The elements of the matrices F and Q in equation (25) are given explicitly below h 2 2 i h 2 2 i

F11 = C55 J^32 + J^33 + C66 J^22 + J^23 + ðC56 + C65 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F12 = C54 J^32 + J^33 + C62 J^22 + J^23 + ðC52 + C64 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F13 = C53 J^32 + J^33 + C64 J^22 + J^23 + ðC54 + C63 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F14 = e35 J^32 + J^33 + e26 J^22 + J^23 + ðe25 + e36 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F21 = C45 J^32 + J^33 + C26 J^22 + J^23 + ðC25 + C46 Þ J^22 J^32 + J^23 J^33

ð25Þ

22

Journal of Intelligent Material Systems and Structures 00(0) h 2 2 i h 2 2 i

F22 = C44 J^32 + J^33 + C22 J^22 + J^23 + ðC24 + C42 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F23 = C43 J^32 + J^33 + C24 J^22 + J^23 + ðC23 + C44 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F24 = e34 J^32 + J^33 + e22 J^22 + J^23 + ðe24 + e32 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F31 = C35 J^32 + J^33 + C46 J^22 + J^23 + ðC45 + C36 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F32 = C34 J^32 + J^33 + C42 J^22 + J^23 + ðC44 + C32 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F33 = C33 J^32 + J^33 + C44 J^22 + J^23 + ðC43 + C34 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F34 = e35 J^32 + J^33 + e24 J^22 + J^23 + ðe23 + e34 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F41 = e35 J^32 + J^33 + e26 J^22 + J^23 + ðe25 + e36 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F42 = e34 J^32 + J^33 + e22 J^22 + J^23 + ðe24 + e32 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F43 = e33 J^32 + J^33 + e24 J^22 + J^23 + ðe23 + e34 Þ J^22 J^32 + J^23 J^33 h 2 2 i h 2 2 i

F44 =  k33 J^32 + J^33  k22 J^22 + J^23  ðk23 + k32 Þ J^22 J^32 + J^23 J^33



C55 J^32 + C66 J^22 C55 J^33 + C66 J^23 J^22 J^32 ðC56 + C65 Þ J^23 J^33 ðC56 + C65 Þ ; Y12 = + + Y11 = 2 2 2 2



C54 J^32 + C62 J^22 C54 J^33 + C62 J^23 J^22 J^32 ðC52 + C64 Þ J^23 J^33 ðC52 + C64 Þ ; Y14 = Y13 = + + 2 2 2 2



C53 J^32 + C64 J^22 C53 J^33 + C64 J^23 J^22 J^32 ðC54 + C63 Þ J^23 J^33 ðC54 + C63 Þ ; Y16 = Y15 = + + 2 2 2 2



e35 J^32 + e26 J^22 e35 J^33 + e26 J^23 J^22 J^32 ðe25 + e36 Þ J^23 J^33 ðe25 + e36 Þ ; Y18 = Y17 = + + 2 2 2 2



^ ^ ^ ^ ^ ^ ^ ^ C45 J 32 + C26 J 22 C45 J 33 + C26 J 23 J 22 J 32 ðC46 + C25 Þ J 23 J 33 ðC46 + C25 Þ ; Y22 = Y21 = + + 2 2 2 2



^ ^ ^ ^ ^ ^ J J C44 J^32 + C22 J^22 C + C J 22 J 32 ðC42 + C24 Þ J 23 J 33 ðC42 + C24 Þ 44 33 22 23 ; Y24 = Y23 = + + 2 2 2 2



C43 J^32 + C24 J^22 C43 J^33 + C24 J^23 J^22 J^32 ðC44 + C23 Þ J^23 J^33 ðC44 + C23 Þ ; Y26 = Y25 = + + 2 2 2 2



e34 J^32 + e22 J^22 e34 J^33 + e22 J^23 J^22 J^32 ðe24 + e32 Þ J^23 J^33 ðe24 + e32 Þ ; Y28 = Y27 = + + 2 2 2 2



C35 J^32 + C46 J^22 C35 J^33 + C46 J^23 J^22 J^32 ðC36 + C45 Þ J^23 J^33 ðC36 + C45 Þ ; Y32 = Y31 = + + 2 2 2 2



^ ^ ^ ^ ^ ^ ^ ^ C34 J 32 + C42 J 22 C34 J 33 + C42 J 23 J 22 J 32 ðC32 + C44 Þ J 23 J 33 ðC32 + C44 Þ ; Y34 = Y33 = + + 2 2 2 2



^ ^ ^ ^ ^ ^ J J C33 J^32 + C44 J^22 C + C J 22 J 32 ðC34 + C43 Þ J 23 J 33 ðC34 + C43 Þ 33 33 44 23 ; Y36 = Y35 = + + 2 2 2 2



e35 J^32 + e24 J^22 e35 J^33 + e24 J^23 J^22 J^32 ðe23 + e34 Þ J^23 J^33 ðe23 + e34 Þ ; Y38 = Y37 = + + 2 2 2 2



e35 J^32 + e26 J^22 e35 J^33 + e26 J^23 J^22 J^32 ðe36 + e25 Þ J^23 J^33 ðe36 + e25 Þ ; Y42 = Y41 = + + 2 2 2 2



e34 J^32 + e22 J^22 e34 J^33 + e22 J^23 J^22 J^32 ðe32 + e24 Þ J^23 J^33 ðe32 + e24 Þ ; Y44 = Y43 = + + 2 2 2 2



^ ^ ^ ^ ^ ^ ^ ^ e33 J 32 + e24 J 22 e33 J 33 + e24 J 23 J 22 J 32 ðe34 + e23 Þ J 23 J 33 ðe34 + e23 Þ ; Y46 = Y45 = + + 2 2 2 2



^ ^ ^ ^ ^ ^ J J k33 J^32 + k22 J^22 k + k J 22 J 32 ðk23 + k32 Þ J 23 J 33 ðk23 + k32 Þ 33 33 22 23 ; Y48 =  Y47 =    2 2 2 2

Chen et al.

23

Using the definitions in equation (6) and equation (25), the first- and second-order microvariables are written as follows (1)  ^ W = B½ u0

(2) ^ u0

(3) ^ u0

(4) ^ u0 

T

where

P = diag½ v1

v1

v1

v1 with v1 =

ð26Þ

where W = ½ W1 W2 W3 W4 T with Wi = ½ Wi(10) Wi(01) Wi(20) Wi(02) T  in equation (26) takes the form The matrix B  = P  NF1 QM B

2

ð27Þ

1

0

16 60 6 241

0 1

1 0

0

0

1

1

N = diag½ v2 v2 v2 v2 with v2 = ½ 0 0 1 1  M = diag½ v3 v3 v3 v3 v3 v3 v3 v3  with v3 = ½ 1

0

1 7 7 7 0 5 1 T

1

Appendix 2 The local stiffness matrix in equation (12) takes the form B  K=A

ð28Þ

 is the product of five matrices A  = DCEBA:  is given explicitly in equation (27). The matrix A where the matrix B ^ J ^ J ^ J ^ J ^ J ^ J ^ J ^ J ^ J ^ J ^ J ^ J ^ J ^ B = diag½ J (1) (2) (3) (4) D = diag½ n n n n  2 A4 0 0 0 A1 0 0 0 A3 0 0 6 0 A 0 0 0 A1 0 0 0 A3 0 4 6 A=6 4 0 0 A4 0 0 0 A1 0 0 0 A3 0 0 0 A4 0 0 0 A1 0 0 0  T  T 1 0 3 0 1 0 3 0 A2 = A1 = 0 1 0 0 0 1 0 0  T   1 0 0 0 1 0 0 0 T A3 = A3 = 0 1 0 3 0 1 0 3 2 C22 C23 C24 C25 6 C32 C33 C34 C35 6 6 C42 C43 C44 C45 6      C = diag½ C C C C  with C = 6 6 C52 C53 C54 C55 6 C62 C63 C64 C65 6 4 e22 e23 e24 e25 e32 e33 e34 e35 2 0 0 1 60 0 0 6 60 0 0 6      E = diag½ E E E E  with E = 6 60 1 0 61 0 0 6 40 0 0 0 0 0

^ J

^ J

0 0

A2 0

0 A2

0 0

0 A3

0 0

0 0

A2 0

C26 C36 C46 C56 C66 e26 e36 0 0 1 0 0 0 0

e22 e23 e24 e25 e26 k22 k32 0 0 1 0 0 0 0

0 1 0 0 0 0 0

0 0 0 0 0 1 0

3 e32 e33 7 7 e34 7 7 e35 7 7 e36 7 7 k23 5 k33 3 0 07 7 07 7 07 7 07 7 05 1

3 0 T 0 7 7 7 with 0 5 A2

3