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Effect of the Initial Stress State on the Effective Properties of Piezocomposite

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Mechanics of Composite Materials Aims and scope

A new numerical-analytical solution of the “effective modulus” problem of statistical mechanics of piezocomposites was obtained considering the presence of an initial stressed electromagnetic-elastic state of an irregular structure with ellipsoidal inhomogeneities. The Green function method for a homogeneous piezoelectromagnetic elastic medium was used. The new solution was validated by comparing with the known asymptotic solution for the case of an elastic laminate in the presence of an initial stress state in its layers. Results of numerical analysis of an influence of the initial stress state on the effective properties of elastic two-phase composites with layered, unidirectional-fibrous, and granular structures were presented. For a composite with spherical inclusions, the appearance of “induced” anisotropy at the macrolevel, owing to the presence of the initial stress state of the structure, was revealed. A numerical analysis of influence of the axisymmetric macrolevel initial stress state on the effective transversely isotropic electroelastic properties of a PZT-4/fluoroplastic composite with unidirectional piezoelectric fibers was presented. The effective characteristics of the piezocomposite significantly depending on the initial stress state were revealed.

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Authors and Affiliations

  1. Perm National Research Polytechnic University, Perm, 614990, Russia

    A. A. Pan’kov

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  1. A. A. Pan’kov
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Correspondence to A. A. Pan’kov.

Additional information

Translated from Mekhanika Kompozitnykh Materialov, Vol. 58, No. 5, pp. 1049-1068, September-October, 2022. Russian DOI: https://doi.org/10.22364/mkm.58.5.11.

Appendix

Appendix

Tensor components \( \overline{\textbf{A}}, \) \( \overline{\textbf{B}}, \) ⋯, and \( {\overline{\textbf{T}}}^{(2)} \) are the solutions of four independent systems of linear algebraic equations

$$ {\displaystyle \begin{array}{c}{a}_{ijk s}^{\left(1,1\right)}{\overline{A}}_{ksmn}+{a}_{ijk}^{\left(1,2\right)}{\overline{F}}_{kmn}^{(1)}+{a}_{ijk}^{\left(1,3\right)}{\overline{F}}_{kmn}^{(2)}={b}_{ijmn}^{(1)},\\ {}{a}_{ik s}^{\left(2,1\right)}{\overline{A}}_{ksmn}+{a}_{ik}^{\left(2,2\right)}{\overline{F}}_{kmn}^{(1)}+{a}_{ik}^{\left(2,3\right)}{\overline{F}}_{kmn}^{(2)}={b}_{imn}^{(2)},\\ {}{a}_{ik s}^{\left(3,1\right)}{\overline{A}}_{ksmn}+{a}_{ik}^{\left(3,2\right)}{\overline{F}}_{kmn}^{(1)}+{a}_{ik}^{\left(3,3\right)}{\overline{F}}_{kmn}^{(2)}={b}_{imn}^{(3)},\\ {}{a}_{ijk s}^{\left(1,1\right)}{\overline{B}}_{ksn}+{a}_{ik d}^{\left(1,2\right)}{\overline{H}}_{dn}^{(1)}+{a}_{ik d}^{\left(1,3\right)}{\overline{H}}_{dn}^{(2)}={c}_{ijn}^{(1)},\end{array}} $$
(A1)
$$ {\displaystyle \begin{array}{c}{a}_{ik s}^{\left(2,1\right)}{\overline{B}}_{ksn}+{a}_{ik}^{\left(2,2\right)}{\overline{H}}_{kn}^{(1)}+{a}_{ik}^{\left(2,3\right)}{\overline{H}}_{kn}^{(2)}={c}_{in}^{(2)},\\ {}{a}_{ik s}^{\left(3,1\right)}{\overline{B}}_{ksn}+{a}_{ik}^{\left(3,2\right)}{\overline{H}}_{kn}^{(1)}+{a}_{ik}^{\left(3,3\right)}{\overline{H}}_{kn}^{(2)}={c}_{in}^{(3)},\\ {}{a}_{ijks}^{\left(1,1\right)}{\overline{D}}_{ksn}+{a}_{ik d}^{\left(1,2\right)}{\overline{M}}_{dn}^{(1)}+{a}_{ik d}^{\left(1,3\right)}{\overline{M}}_{dn}^{(2)}={d}_{ijn}^{(1)},\end{array}} $$
(A2)
$$ {\displaystyle \begin{array}{c}{a}_{ik s}^{\left(2,1\right)}{\overline{D}}_{ks n}+{a}_{ik}^{\left(2,2\right)}{\overline{M}}_{kn}^{(1)}+{a}_{ik}^{\left(2,3\right)}{\overline{M}}_{kn}^{(2)}={d}_{in}^{(2)},\\ {}{a}_{ik s}^{\left(3,1\right)}{\overline{D}}_{ks n}+{a}_{ik}^{\left(3,2\right)}{\overline{M}}_{kn}^{(1)}+{a}_{ik}^{\left(3,3\right)}{\overline{M}}_{kn}^{(2)}={d}_{in}^{(3)},\\ {}{a}_{ij ks}^{\left(1,1\right)}{\overline{T}}_{ks}+{a}_{ij d}^{\left(1,2\right)}{\overline{T}}_d^{(1)}+{a}_{ij d}^{\left(1,3\right)}{\overline{T}}_d^{(2)}={f}_{ij}^{(1)},\end{array}} $$
(A3)
$$ {\displaystyle \begin{array}{c}{a}_{ik s}^{\left(2,1\right)}{\overline{T}}_{ks}+{a}_{ik}^{\left(2,2\right)}{\overline{T}}_k^{(1)}+{a}_{ik}^{\left(2,3\right)}{\overline{T}}_k^{(2)}={f}_i^{(2)},\\ {}{a}_{ik s}^{\left(3,1\right)}{\overline{T}}_{ks}+{a}_{ik}^{\left(3,2\right)}{\overline{T}}_k^{(1)}+{a}_{ik}^{\left(3,3\right)}{\overline{T}}_k^{(2)}={f}_i^{(3)},\end{array}} $$
(A4)

with the coefficients

$$ {\displaystyle \begin{array}{c}{a}_{ik s}^{\left(1,1\right)}={\delta}_{ik}{\delta}_{js}-{U}_{ijd b}^s\left[{\overset{\sim }{C}}_{dbks}+{\delta}_{dk}{\sigma}_{bs}^{\ast 0}+\left(1-2{v}_1\right)\left({\overline{C}}_{dbks}+{\delta}_{dk}{\overline{\sigma}}_{bs}^0\right)\right]\\ {}-{U}_{ijd}^{s(1)}\left[\tilde{e}_{dk s}+{\delta}_{dk}{D}_s^{\ast 0}+\left(1-2{v}_1\right)\left({\overline{e}}_{dk s}+{\delta}_{dk}{\overline{D}}_s^0\right)\right]\\ {}-{U}_{ijd}^{s(2)}\left[\tilde{h}_{dk s}+{\delta}_{dk}{B}_s^{\ast 0}+\left(1-2{v}_1\right)\left({\overline{h}}_{dk s}+{\delta}_{dk}{\overline{B}}_s^0\right)\right],\\ {}{a}_{ijk}^{\left(1,2\right)}={U}_{ijd b}^s\left[\tilde{e}_{kdb}+\left(1-2{v}_1\right){\overline{e}}_{kdb}\right]-{U}_{ijd}^{s(1)}\left[{\overset{\sim }{\lambda}}_{dk}+\left(1-2{v}_1\right){\overline{\lambda}}_{dk}\right],\\ {}{a}_{ijk}^{\left(1,3\right)}={U}_{ijd b}^s\left[\tilde{h}_{kdb}+\left(1-2{v}_1\right){\overline{h}}_{kdb}\right]-{U}_{ijd}^{s(2)}\left[{\overset{\sim }{\mu}}_{dk}+\left(1-2{v}_1\right){\overline{\mu}}_{dk}\right],\\ {}{a}_{ik s}^{\left(2,1\right)}=-{\varPhi}_{id b}^s\left[{\overset{\sim }{C}}_{dbks}+{\delta}_{dk}{\sigma}_{bs}^{\ast 0}+\left(1-2{v}_1\right)\left({\overline{C}}_{dbks}+{\delta}_{dk}{\overline{\sigma}}_{bs}^0\right)\right]\\ {}-{\varPhi}_{id}^{s(1)}\left[\tilde{e}_{dk s}+{\delta}_{dk}{D}_s^{\ast 0}+\left(1-2{v}_1\right)\left({\overline{e}}_{dk s}+{\delta}_{dk}{\overline{D}}_s^0\right)\right]\\ {}-{\varPhi}_{id}^{s(2)}\left[\tilde{h}_{dk s}+{\delta}_{dk}{B}_s^{\ast 0}+\left(1-2{v}_1\right)\left({\overline{h}}_{dk s}+{\delta}_{dk}{\overline{B}}_s^0\right)\right],\\ {}{a}_{ik}^{\left(2,2\right)}=-{\delta}_{ik}+{\varPhi}_{id b}^s\left[\tilde{e}_{kdb}+\left(1-2{v}_1\right){\overline{e}}_{kdb}\right]-{\varPhi}_{id}^{s(1)}\left[{\overset{\sim }{\lambda}}_{dk}+\left(1-2{v}_1\right){\overline{\lambda}}_{dk}\right],\\ {}{a}_{ik}^{\left(2,3\right)}={\varPhi}_{id b}^s\left[\tilde{h}_{kdb}+\left(1-2{v}_1\right){\overline{h}}_{kdb}\right]-{\varPhi}_{id}^{s(2)}\left[{\overset{\sim }{\mu}}_{dk}+\left(1-2{v}_1\right){\overline{\mu}}_{dk}\right],\\ {}{a}_{ik s}^{\left(3,1\right)}=-{\varPsi}_{id b}^s\left[{\overset{\sim }{C}}_{dbks}+{\delta}_{dk}{\sigma}_{bs}^{\ast 0}+\left(1-2{v}_1\right)\left({\overline{C}}_{dbks}+{\delta}_{dk}{\overline{\sigma}}_{bs}^0\right)\right],\\ {}-{\varPsi}_{id}^{s(1)}\left[\tilde{e}_{dk s}+{\delta}_{dk}{D}_s^{\ast 0}+\left(1-2{v}_1\right)\left({\overline{e}}_{dk s}+{\delta}_{dk}{\overline{D}}_s^0\right)\right]\\ {}-{\varPsi}_{id}^{s(2)}\left[\tilde{h}_{dk s}+{\delta}_{dk}{B}_s^{\ast 0}+\left(1-2{v}_1\right)\left({\overline{h}}_{dk s}+{\delta}_{dk}{\overline{B}}_s^0\right)\right],\\ {}{a}_{ik}^{\left(3,2\right)}={\varPsi}_{id b}^s\left[\tilde{e}_{kdb}+\left(1-2{v}_1\right){\overline{e}}_{kdb}\right]-{\varPsi}_{id}^{s(1)}\left[{\overset{\sim }{\lambda}}_{dk}+\left(1-2{v}_1\right){\overline{\lambda}}_{dk}\right],\\ {}{a}_{ik}^{\left(3,3\right)}=-{\delta}_{ik}+{\varPsi}_{id b}^s\left[\tilde{h}_{kdb}+\left(1-2{v}_1\right){\overline{h}}_{kdb}\right]-{\varPsi}_{id}^{s(2)}\left[{\overset{\sim }{\mu}}_{dk}+\left(1-2{v}_1\right){\overline{\mu}}_{dk}\right],\end{array}} $$

the right parts of the 1st system of equations (A1)

$$ {\displaystyle \begin{array}{c}{b}_{ijmn}^{(1)}={U}_{ijk s}^s{\overline{C}}_{ksmn}+{U}_{ijk}^{s(1)}\left({\overline{e}}_{km n}+{\delta}_{km}{\overline{D}}_n^0\right)+{U}_{ijk}^{s(1)}\left({\overline{h}}_{km n}+{\delta}_{km}{\overline{B}}_n^0\right),\\ {}{b}_{imn}^{(2)}={\varPhi}_{ik s}^s{\overline{C}}_{ksmn}+{\varPhi}_{ik}^{s(1)}\left({\overline{e}}_{km n}+{\delta}_{km}{\overline{D}}_n^0\right)+{\varPhi}_{ik}^{s(2)}\left({\overline{h}}_{km n}+{\delta}_{km}{\overline{B}}_n^0\right),\\ {}{b}_{imn}^{(3)}={\varPsi}_{ik s}^s{\overline{C}}_{ksmn}+{\varPsi}_{ik}^{s(1)}\left({\overline{e}}_{km n}+{\delta}_{km}{\overline{D}}_n^0\right)+{\varPsi}_{ik}^{s(2)}\left({\overline{h}}_{km n}+{\delta}_{km}{\overline{B}}_n^0\right)\end{array}} $$

for the 2nd and 3rd systems of equations (A2), (A3)

$$ {\displaystyle \begin{array}{c}{c}_{ijn}^{(1)}=-{U}_{ijk s}^s{\overline{e}}_{ksmn}+{U}_{ijk}^{s(1)}{\overline{\lambda}}_{kn},\kern1em {c}_{in}^{(2)}=-{\varPhi}_{ijk s}^s{\overline{e}}_{nks}+{\varPhi}_{ik}^{s(1)}{\overline{\lambda}}_{kn},\\ {}{c}_{in}^{(3)}=-{\varPsi}_{ik s}^s{\overline{e}}_{nks}+{\varPsi}_{ik}^{s(1)}{\overline{\lambda}}_{kn},\kern1em {d}_{ijn}^{(1)}=-{U}_{ijk s}^s{\overline{h}}_{nks}+{U}_{ijk}^{s(2)}{\overline{\mu}}_{kn},\\ {}{d}_{in}^{(2)}=-{\varPhi}_{ik s}^s{\overline{h}}_{nks}+{\varPhi}_{ik}^{s(2)}{\overline{\mu}}_{kn},\kern1em {d}_{in}^{(3)}=-{\varPsi}_{ik s}^s{\overline{h}}_{nks}+{\varPsi}_{ik}^{s(2)}{\overline{\mu}}_{kn}\end{array}} $$

and for the 4th system of equations (A4)

$$ {\displaystyle \begin{array}{c}{f}_{ij}^{(1)}=-{U}_{ij k s}^s{\overline{\beta}}_{ks}+{U}_{ij k}^{s(1)}{\overline{\pi}}_k+{U}_{ij k}^{s(2)}\overline{\vartheta},\\ {}{f}_i^{(2)}=-{\varPhi}_{ik s}^s{\overline{\beta}}_{ks}+{\varPhi}_{ik}^{s(1)}{\overline{\pi}}_k+{\varPhi}_{ik}^{s(2)}{\overline{\vartheta}}_k,\\ {}{f}_i^{(3)}=-{\varPsi}_{ik s}^s{\overline{\beta}}_{ks}+{\varPsi}_{ik}^{s(1)}{\overline{\pi}}_k+{\varPsi}_{ik}^{s(2)}{\overline{\vartheta}}_k\end{array}} $$

to calculate the sought-for values \( \overline{\textbf{A}}, \) \( {\overline{\textbf{F}}}^{(1)}, \) ⋯, and \( {\overline{\textbf{T}}}^{(2)} \) taking into account the designations of singular components Gs (see Eqs. 13)), difference tensors \( \tilde{\textbf{C}}, \) \( \overline{\textbf{C}}, \) \( {\overline{\sigma}}^0, \) ⋯,(see Eqs. (10) and (18)).

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Pan’kov, A.A. Effect of the Initial Stress State on the Effective Properties of Piezocomposite. Mech Compos Mater 58, 733–746 (2022). https://doi.org/10.1007/s11029-022-10063-w

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