Effects of interphase strength on the damage modes and mechanical behaviour of metal–matrix composites

Composites: Part A 30 (1999) 257–266

Effects of interphase strength on the damage modes and mechanical behaviour of metal–matrix composites X.F. Su a, H.R. Chen a,1, D. Kennedy b,*, F.W. Williams b a

Research Institute of Engineering Mechanics, Dalian University of Technology, Dalian 116024, People’s Republic of China b Cardiff School of Engineering, Cardiff University, Cardiff CF2 3TB, UK Received 15 March 1998; revised 1 July 1998; accepted 1 July 1998

Abstract A numerical version of the generalized self-consistent method previously developed by the authors is combined with the Gurson model to undertake a parametric investigation of the damage mechanisms and their relations with the macroscopic tensile properties of SiC reinforced aluminium, for three different interphase strengths. The results show that the interphase strength is a governing factor for damage propagation in the composite. Thus, transformation of the failure mechanism from reinforcement fracture to void nucleation and growth can be achieved by reducing the interphase bond strength, although the strengthening effects on the composite decrease unfavourably. q 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: B. Interphase strength; C. Damage mechanics; C. Finite element analysis (FEA); A. Metal–matrix composites (MMCs)

1. Introduction Ductile metals (e.g. aluminium or other alloys) reinforced by SiC particulate or whiskers can constitute composites with high specific stiffness and strength. However, the presence of whiskers or particulate decreases the composite ductility and fracture toughness dramatically, which limits the structural applications of such composites in, for example, the aerospace and automotive industries. In order to develop such composites with improved strengthening and fracture properties, it is of great importance to establish the relationship between the microscopic damage and failure mechanisms and the macroscopic mechanical behaviour of the composites. The strengthening and damage mechanisms of the composites depend to a great extent on the strength of the interphase, which in turn is governed by the chemical and physical nature of its constituents, specifically by the reactions which may occur during the fabrication process and/or heat treatment. In fibre-reinforced metal–matrix composites, it is generally essential to weaken the interphase so as to increase the toughness by exploiting the friction energy necessary for the pull-out of the fibres from the matrix [1]. * Corresponding author. 1 Currently Visiting Professor, Division of Stuctural Engineering, Cardiff School of Engineering, Cardiff University, Cardiff CF2 3TB, UK.

In contrast, for whiskers or particulate-reinforced ductile metals, a high interphase strength is desirable to derive the maximum strengthening properties [2, 3]. Unfortunately, a high strength bond can often result in the degradation of the composite toughness [4]. Therefore, the design of metal–matrix composites should be based upon the interaction between strengthening and toughening effects. Generally speaking, the damage and failure mechanisms are of three types, namely: reinforcement fracture, which causes brittle failure of the composites [5, 6]; matrix-reinforcement interfacial debonding [7, 8]; and ductile failure in the matrix caused either by the nucleation, growth and coalescence of voids from the cracking of the intermetallic inclusions and dispersoids in the matrix [9, 10], or by ductile tearing of the matrix between the reinforcements [11]. Transformation of the failure mode from brittle fracture to ductile fracture is usually determined by the properties of the interphase, which depend greatly on the artificial ageing process of the matrix [12, 13]. In spite of these findings, measures to improve the composite ductility or toughness are not sufficiently understood, because the mechanisms described above usually coexist and the overall behaviour is a result of their synergistic contribution. In addition, improving the ductility of the composite should not reduce the strengthening effect of the brittle reinforcements, which is the main benefit of metal–matrix composites. Thus, studies of strengthening,

1359-835X/99/$ – see front matter q 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S1359-835 X( 98)00 158-4

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damage and failure mechanisms, and the relationships between them, are of great concern to materials scientists and engineers. Damage and failure mechanisms are often investigated by experimental methods, starting from metallographic observations of strained composites. However, numerical study by mesomechanics of the effects of these mechanisms on the macroscopic deformation and ductility of metal–matrix composites is required, because they cannot be quantitatively investigated by experiments alone. Mesomechanical study of damage and failure mechanisms is usually based on the unit cell model and includes the investigation of matrixreinforcement interfacial debonding [14, 15], nucleation, growth and coalescence of the voids in the matrix [16], and reinforcement fracture [17]. The authors have previously [18–21] developed a finite element version of the Generalized Self-Consistent Method (GSCM), which proved powerful and effective in the study of the strengthening mechanisms and macroscopic elasto– plastic properties of metal–matrix composites. This numerical version can be used to derive the detailed stress–strain fields in each phase of composites with complicated phase geometry. In this paper, GSCM is combined with the Gurson model [22] to undertake a parametric investigation of the damage mechanisms mentioned above, and their relationship with the macroscopic tensile properties of SiC-reinforced Al5456. This forms a part of an overall approach to the derivation of damage mechanisms of metal–matrix composites.

2. Fundamental theories This section outlines fundamental theories for damage accumulation in the constituents of metal–matrix composites and for the macroscopic tensile properties of the composites. 2.1. Damage and failure of the matrix for ductile metal due to nucleation, growth and coalescence of voids The yield condition for porous plastic solids in terms of an internal variable f, the void volume fraction, proposed by Gurson [22] and developed by Tvergaard [23], is:

f…se ; sm ; s~ ; f † ˆ …se =s~ †2 1 2fql cosh…q2 sm =2s~ † 2 1 2 q3 f 2 ˆ 0

(1)

where se and sm are invariants (defined below) of the macroscopic stress state acting on the voided material, while s~ is the tensile yield strength in the matrix material. The constants q1, q2 and q3 introduced by Tvergaard [23] are: q1 ˆ 1:5; q2 ˆ 1; q3 ˆ

q21

ˆ 2:25

(2)

se and sm are defined in terms of the Cauchy stress tensor

s ij and metric tensor Gij, by:

s 2e ˆ

3 1 G G s 0ij s 0kl ; sm ˆ Gij s ij ; s 0ij ˆ s ij 2 Gij sm 2 ik jl 3 (3)

Here, s 0 ij is the deviator of the Cauchy stress tensor s ij, and G ij is the inverse of the metric tensor Gij. In general, the evolution of the void volume fraction rate arises from the growth of existing voids and the nucleation of new voids: f_ ˆ … f_†growth 1 … f_†nucleation

(4)

Since the matrix material is plastically incompressible, … f_†growth ˆ …l 2 f †Gij h_ pij

(5)

h_ pij

is the plastic strain rate tensor of the porous plastic where solid. Following Gurson [22], the rate of void nucleation controlled by plastic strain can be expressed by: … f_†nucleation ˆ As_

(6)

where s_ is the effective stress rate and A is a parameter. Following Chu and Needleman [24], A is chosen so that void nucleation follows a normal distribution. Thus, for a volume fraction of void nucleating inclusions fN, a mean strain for nucleation 1N, and a standard deviation sN: "     # 1 1 fN 1 1 p 2 1N 2 p exp 2 2 (7) Aˆ sN Et E sN 2p 2 Here, 1 p is the effective plastic strain, E is Young’s modulus and Et is the current tangent modulus. Considering the coalescence effects of two neighbouring voids when the void volume fraction f exceeds a critical value fc, Tvergaard and Needleman [25] proposed replacing f by the value: 9 f f # fc > = p (8) f 2 fc fc 1 u …f 2 fc † f . fc > ; fF 2 f c where fc ˆ 0.l5, fF ˆ 0.25 and fup ˆ 1=q1 is the ultimate value at which the macroscopic stress carrying capacity vanishes. 2.2. Constitutive relationship of the ductile metal–matrix with damage and failure The plastic strain rates of the porous plastic solid taken in a direction normal to the flow potential satisfy

h_ pij ˆ l

2f 2sij

(9)

By setting the plastic work rates equal to the matrix dissipation …1 2 f †s 1_ p ˆ s ij h_ pij

(10)

where s and 1_ p are, respectively, the effective stress and plastic strain rate of the matrix, and the scalar multiplier l is

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259

Fig. 1. Schematic generalized self-consistent model and corresponding finite element mesh: (a) quarter model; (b) enlarged finite element mesh.

defined in terms of s^ ij , the Jaumann derivative of the Cauchy stress tensor, as:

lˆ

1 p s^ ij h ij

Cijkl ˆ Lijkl e 2 (11)

where hˆ "

h 2f ij 2f 2f 2f ij 2f G 1 1 …A †s 2 …1 2 f † ij …1 2 f †s 2f 2f 2s 2s 2s ij

#

(12) and pij ˆ

2f 2s ij

(13)

The quantity h in Eq. (12) is the equivalent tensile hardening rate of the matrix material, given by: 1 1 1 h ˆ Et 2 E

(14)

The relationship between the Jaumann derivative of the Cauchy stress tensor s^ ij and the Lagrangian strain tensor increment h_ kl given in the form:

s^ ij ˆ Cijkl h_ kl

where

(15)

rskl Lijmn e pmn prs Le h 1 prs Lrsmn pmn e

(16)

with Lijkl e representing the elastic tensor. In the numerical computation of the finite element scheme, Eq. (15) is generally transformed into the incremental constitutive relation, in terms of the convected rate of Kirchhoff stress and Lagrangian strain rate [26]. This transformation has been used for the computing process of this paper. 2.3. Fracture of brittle solids: damage and failure of interphase and reinforcements For metal–matrix composites, the presence of precipitates on the reinforcement surface during the ageing process, and the reaction between the reinforcement and matrix, usually embrittle the interphase. As Li and Wisnom [27] proposed, the crack criterion for a brittle interphase with definite thickness could take the form of a maximum principal stress criterion. This criterion is also appropriate for the brittle reinforcement. If s1, s2 and s3 are used to represent the three principal stresses, cracking occurs when: Max…s1 ; s2 ; s3 † $ s i0 or sf0

s i0

sf0

(17)

and are the tensile strengths of the interphase where and reinforcement, respectively.

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as:  ep  f 1 …Vi 2 Vdi † 1=Ecep ˆ 1=E ep m 1 …Vf 2 Vdf †…1=Ef 2 1=E m †a  i 1 …Vdf 1 Vdi †…1=Ed 2 1=E ep d  …1=Ei 2 1=E ep m †a m †a (18) is the composite incremental tensile modulus, E ep Here, m represents the incremental tensile modulus of the porous matrix; Ef and Ei are Young’s moduli of the reinforcements and interphase, respectively; for computational stability, the cracked parts of the reinforcements and interphase are regarded as a new phase with a rather weak modulus Ed ˆ 0.0001Ef; Vf and Vi are, respectively, the initial volume fractions of reinforcements and interphase; Vdf and Vdi represent the volume fractions of their cracked parts that evolve with the external loading; a¯f and a¯i are, respectively, the average stress concentration factors for residual reinforcements and interphase; and a¯d is the average stress concentration factor for the cracked parts. A derivation of Eq. (18) is given in Appendix A. The incremental tensile modulus of the matrix E ep m can be written as: Ecep

Fig. 2. Composite stress—strain curves for three different interphase strengths, showing the cases of no damage in the composite and an experimental result [28] as comparators.

1

2.4. Tensile properties of metal–matrix composites based on constituent damage initiation and propagation

E ep m ˆ

The authors have previously [21] developed a finite element version of GSCM to predict elasto–plastic tensile properties of short fibre-reinforced metal–matrix composites (SiC/Al). The model assumed that a composite inclusion, i.e. reinforcement (e.g. short fibre, whisker or particulate), which is surrounded by a multilayered shell consisting of the matrix and interphase layers, is embedded in an infinite uniform composite medium, called the effective composite medium, as shown in Fig. 1(a). The size of the outer composite shell adopted in this paper was 10 times that of the composite inclusion. Numerical tests demonstrated that this approach effectively simulated an infinite medium, avoiding the influence of the boundary of the outer composite shell on the stress–strain fields in the composite inclusion. As in the well-known Periodic Array Model (PAM) [14–17, 28], GSCM defines a representative unit cell, the composite inclusion mentioned above. However, in this model, the effective composite medium is employed to replace the unit cell’s side wall constraint conditions which are commonly used in PAM. As the fibre array forms in GSCM are rather implicit, the results of GSCM lie between the predictions of two PAMs, namely the aligned fibre ends model and the staggered fibre ends model [28]. The iterative process of GSCM has previously been described in detail [18, 19]. Considering the cracked parts of the reinforcements and interphase with the external loading, the relationship giving the composite tensile modulus with constituent damage, in terms of the material properties of the constituents’ damaged parts and their volume fractions, can be deduced

where Em is the Young’s modulus of the matrix and d1 pm =d s m is the derivative of the average von Mises effective plastic strain with respect to the average von Mises effective stress in the matrix, which can be obtained by an incremental procedure from GSCM, united with the finite element analysis and the constitutive relationship of the ductile metal–matrix with damage and failure provided in Eqs. (9)–(16). A detailed description of this method has been given previously [21] and will not be repeated here.

1 d1 pm 1 d s m Em

(19)

3. Numerical results Fig. 1(b) shows an enlarged finite element mesh for a part of the representative model of GSCM, which consists only of reinforcement, interphase and matrix. The tensile properties and mesoscopic distributions of constituent damage for SiC-Al5456 at various loading stages are studied for three different interphase strengths, namely 300, 500 and 1300 MPa. The aspect ratio of the length to diameter and volume fraction of the cylindrical SiC reinforcements are 4 and 20%, respectively. The interphase thickness adopted here is 1% of the reinforcement radius. The nucleating parameters of the voids in the matrix are fN ˆ 0.05, sN ˆ 0.01 and 1N ˆ 0.05. The constituent properties are as follows. For the aluminium matrix (5456): E ˆ 73 GPa, n ˆ 0.33 (Poisson’s ratio), spl ˆ 24l MPa (proportional limit), s0.2 ˆ 259 MPa (0.2% offset yield) and n ˆ 0.01 (workhardening exponent). For the SiCw (whisker): E ˆ 485 GPa, strength ˆ l500 MPa and n ˆ 0.2. For the

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261

Fig. 3. Contours of interphase debonding and void distribution in matrix for weak interphase strength 300 MPa, for the three applied stresses shown. The zones encircled by black lines represent interphase debonding.

interphase: E ˆ 300 GPa and n ˆ 0.25. The practical strength of a SiC whisker is 2 GPa [29] or more [17]. However, since the effects of whisker clustering or residual stresses introduced during material processing on the whisker crack are neglected in the present continuum analysis, in order to guarantee the occurrence of the whisker crack in

the proper loading stage, the SiCw strength is reduced to 1.5 GPa. 3.1. Stress–strain curves of composites for three different interphase strengths The composite stress–strain curves in longitudinal

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Fig. 4. Contours of interphase debonding and void distribution in matrix for medium interphase strength 500 MPa, for the three applied stresses shown. The zones encircled by black lines represent interphase debonding.

tension for the three different interphase strengths of 300, 500 and 1300 MPa are shown in Fig. 2, with the perfect case of no damage in the composite and an experimental result [28] of SiCw-Al5456 used as comparators. It can be seen that the moduli of all the curves are the same, because in the elastic deformation stages there is no damage in the compo-

site. In the elasto–plastic deformation stages, the composite work-hardening rate for interphase strength 300 MPa begins to decrease sharply, making this the lowest curve on Fig. 2. The curve for interphase strength 500 MPa is not very much higher. However, the composite stress–strain curve for interphase strength 1300 MPa is roughly the same as for

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the perfect case, because there is no serious damage in the composite during most of the deformation stage. When the applied stress exceeds 630 MPa, cracking in the reinforcements results in a sudden decrease of the composite workhardening rate of this curve. 3.2. Effects of interphase strength on the propagation of mesoscopic damage to the composite Contours showing interphase debonding and void distribution in the matrix for the case of weak interphase strength 300 MPa are shown in Fig. 3, for the applied longitudinal tensile stresses of 350, 500 and 650 MPa. The zones encircled by black lines represent interphase debonding. From these contours, because of the weak interphase strength and stress concentration, it is seen that interphase debonding and voids in the matrix initiate during the early loading stage, particularly around the corner of the reinforcement. In the medium loading stage, the interphase debonds completely, while the nucleation and growth of voids develop progressively, spreading fast from the corner of the reinforcement to the whole matrix. In the latter loading stage, the distribution of void volume fractions in the matrix is roughly homogeneous, except for the regions around the corner of the reinforcement and ahead of the reinforcement end. No reinforcement cracking occurs. Fig. 4 repeats Fig. 3, except that it is for medium interphase strength 500 MPa. It can be seen that the propagation of interphase debonding and the nucleation and growth of voids is a little slower than for the cases of Fig. 3. In the latter loading stage (as for Fig. 3) the distribution of void volume fractions in the matrix is roughly homogeneous, except for the regions around the corner of the reinforcement and ahead of the reinforcement end. Again, no reinforcement cracking occurs. Fig. 5 repeats Figs 3 and 4, except that it is for high interphase strength 1300 MPa and includes an additional applied tensile stress of 750 MPa. The regions denoted X–X in Fig. 5(c) and (d) indicate the occurrence of reinforcement cracking. For this case, the propagation of interphase debonding and the nucleation and growth of voids are much slower than for the weaker interphase cases of Figs 3 and 4. The nucleation and growth of voids ahead of the reinforcement end propagate more rapidly than in the region between the side walls of the reinforcements. When the applied stress reaches 650 MPa, reinforcement cracking occurs, causing the void volume fraction around the fracture surface of the reinforcement to rise sharply. Also, the regions of interphase debonding at the end and at the cracked part of the reinforcement expand towards each other. The volume fractions of voids in the matrix are rather inhomogeneous. 4. Discussion Interphase strength plays a dominant role in the propagation of mesoscopic damage and, therefore, governs the

263

ductility of metal–matrix composites. Interphase debonding in the region of the reinforcement end decreases void nucleation and growth in the matrix ahead of the reinforcement end, because of stress relaxation. Interphase debonding in the region of the reinforcement side wall promotes void nucleation and growth in the matrix between the side walls of the reinforcements, because the shear lag transfer of load will be hindered by interphase debonding, causing more load to be carried by the matrix in this region. This is probably why the numerical predictions of composites with interphase strength 300 and 500 MPa deviate from experimental results [28] more sharply than that of the composite with interphase strength 1300 MPa, in which relatively little mesoscopic damage occurs during most of the loading stages. The rate of interphase debonding is nearly identical to that of the propagation of void nucleation. Further investigations of this relationship are in progress. The void propagation and distribution in the case of weak interphase strength tend to be homogeneous, becoming stable well before final failure. This is in agreement with Whitchouse and Clyne [10]. The damage is carried by the whole metal matrix. Thus, excellent composite ductility can be obtained. However, strengthening the composite results in an unfavourable degradation. To strengthen the ductile metal requires high interphase strength, which results in a rather inhomogeneous propagation of voids in the matrix. In this situation, cracking of the reinforcements occurs, forming a new source for void nucleation and causing brittle failure of the composite. Additionally, the presence of high hydrostatic tensile stress in the matrix promotes void nucleation and growth, leading to greatly reduced ductility of the composites, as has been observed by other authors [16, 30, 31]. Therefore, the choice of interphase strength remains a problem of material optimization for future investigations. As the literature reports, this choice is affected by coating the reinforcements [32] and by different matrix heat-treating conditions [33]. Several authors have argued that characterization of the stress–strain response for metal–matrix composites in terms of 0.2% offset yield s0.2 is incomplete [21, 28, 34]. The results of this paper show that, for composites with high interphase strength or perfect interphase, to characterize the composite stress–strain curves in terms of s0.2 fails to describe the initial work-hardening rate, while for cases with medium and weak interphase strength there is a risk of losing insight into both the initial and succeeding workhardening rates. A complete characterization of composite stress–strain response should contain several parameters, including the proportional limit, initial work-hardening rate and succeeding work-hardening rates, which require further examination. The generalized self-consistent method must define a representative element or a cell, as do most of the mesomechanical methods. Damage initiation and propagation in the representative element implies the same behaviour

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Fig. 5. Contours of interphase debonding and void distribution in matrix for high interphase strength 1300 MPa, for the four applied stresses shown. The zones encircled by black lines represent interphase debonding, while the region X–X in (c) and (d) indicates reinforcement cracking.

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throughout the whole composite, and so cannot give a proper description of damage localization. GSCM is based on continuum analysis and, therefore, cannot take into account the effects of reinforcement size on the composite properties. Apart from these limitations, GSCM united with the finite element analysis is still an effective method for the study of the strengthening and damage mechanisms of metal–matrix composites.

265

where ds c , ds m , d s f , d s i and ds d are, respectively, the incremental averaging stress tensors of the composite, matrix, residual reinforcement, residual interphase and ep cracked part, while d1 ep c , d 1 m , d 1 f , d 1 i and d 1 d are the corresponding incremental averaging strain tensors. Vm is the volume fraction of the matrix. Other definitions have been given in the text. According to the incremental constitutive relations of the composite and each of its phases, Eq. (A2) gives: ep ep S c ds c ˆ Vm S m ds m 1 …Vf 2 Vdf †Sf ds f 1 …Vi 2 Vdi †Si ds i

5. Conclusions

1 …Vdf 1 Vdi †Sd d s d 1. A transformation of the failure mechanism, from a mode of reinforcement fracture to a mode of void nucleation and growth, can be achieved by varying the interphase strength from a strong interphase bond to a weak one. 2. Weak interphase strength increases the composite ductility, but results in the strengthening effects of the brittle reinforcements decreasing unfavourably. 3. The void propagation and distribution in the case of weak and medium interphase strength tend to be homogeneous, but they are rather inhomogeneous for the case of high interphase strength. 4. For composites with high interphase strength or a perfect interphase, characterizing the composite stress–strain curves in terms of s0.2 fails to describe the initial workhardening rate, while for cases with medium and weak interphase strength there is a risk of losing insight into both the initial and succeeding work-hardening rates. 5. GSCM united with the finite element analysis is an effective method for the quantitative study of the strengthening and damage mechanisms of metal–matrix composites.

ep S c

(A3)

ep S m

and are, respectively, the incremental elasto– Here, plastic compliance tensors of the composite and matrix, while Sf, Si and Sd are the elastic compliance tensors of the reinforcement, interphase and cracked part, respectively. Combined with Eq. (A1), Eq. (A3) can be written as: ep ep S c ds c ˆ S m ‰ds c 2 …Vf 2 Vdf †ds f 2 …Vi 2 Vdi †d s i

2 …Vdf 1 Vdi †ds d Š 1 …Vf 2 Vdf †Sf ds f 1 …Vi 2 Vdi †Si ds i 1 …Vdf 1 Vdi †Sd ds d

(A4)

Then ep ep ep S c ds c ˆ S m d s c 1 …Vf 2 Vdf †…Sf 2 S m †d s f ep 1 …Vi 2 Vdi †…Si 2 S m †ds i

1 …Vdf 1 Vdi †…Sd 2 S m †d s d ep

(A5)

Using the inverse of ds c , Eq. (A5) gives: ep ep ep S c ˆ S m 1 …Vf 2 Vdf †…Sf 2 S m †ds f d s 21 c

1 …Vi 2 Vdi †…Si 2 S m †ds i ds 21 c ep

ep 1 …Vdf 1 Vdi †…Sd 2 S m †ds d ds 21 c

Acknowledgements The work was supported by the Key Project of the National Natural Science Foundation of China and the Cardiff University Advanced Chinese Engineering Centre.

The incremental averaging stress and strain tensors (with volume averaging after finite element analysis in each loading step), in the composite and in each of its phases, can be written in the form of mixture rules [34]: ds c ˆ Vm ds m 1 …Vf 2 Vdf †ds f 1 …Vi 2 Vdi †ds i (A1)

ep d 1 ep c ˆ Vm d 1 m 1 …Vf 2 Vdf †d 1 f 1 …Vi 2 Vdi †d 1 i

1 …Vdf 1 Vdi †d1 d

If    f ˆ ds f ds 21  i ds 21  d ds 21 (A7) A c ; Ai ˆ d s c and Ad ˆ d s c are defined as the average stress concentration tensors of the residual reinforcement, residual interphase and cracked part, respectively, Eq. (A6) can be expressed as:

Appendix A. Derivation of Eq. (18)

1 …Vdf 1Vdi †ds d

(A6)

ep ep ep  S c ˆ S m 1 …Vf 2 Vdf †…Sf 2 S m †A f ep   ep  1 …Vi 2 Vdi †…Si 2 S m †A i 1 …Vdf 1 Vdi †…Sd 2 Sm †Ad

(A8) When attention is confined to the elasto–plastic axial tensile properties of the composite, Eq. (A8) can be reduced to:  ep f 1 …Vi 2 Vdi † 1=Ecep ˆ 1=E ep m 1 …Vf 2 Vdf †…1=Ef 2 1=E m †a  i 1 …Vdf 1 Vdi †…1=Ed 2 1=E ep d  …1=Ei 2 1=E ep m †a m †a

(A2)

(A9)

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where a¯f, a¯i and a¯d are the components of the tensors defined in Eq. (A7).

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