Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Analysis of Cracks in Piezoelectric Solids with Consideration of Electric Field and Strain Gradients

  • Jan Sladek
  • Vladimir Sladek
  • Michael WünscheEmail author
Living reference work entry
First Online:
DOI: https://doi.org/10.1007/978-3-662-53605-6_237-1
Download entry PDF
How to cite

Synonyms

Crack analysis; Electric field and strain gradient; FEM; J-integral; Size effect phenomenon

Definitions

Due to the progress in nanotechnology, the size of the electronic components and devices has been significantly reduced in recent years. If the dimension of the structure is of the same order of magnitude as the material length scale, the classical electromechanical coupling theory of piezoelectricity fails to describe the observed size-dependent phenomenon. Discrete atomistic methods such as molecular dynamics simulations can be utilized to analyze nano-sized structures. However, the required computational cost of such methods can be too high to investigate realistic engineering problems. An alternative and less expensive possibility is the formulation of an advanced continuum mechanics model which takes the size effect phenomenon into account, where the strain and electric field gradients are included in the constitutive equations.

Introduction

Smart piezoelectric materials offer certain important performance advantages over conventionally used metals due to the capability of converting electric energy into mechanical energy and vice versa. In recent years, micro- and nano-sized piezoelectric devices and components are getting increasing attention. The electric field and strain gradient effect is very strong for nano-sized dielectrics. The size-effect phenomenon of the piezoelectric solids has been investigated by several experiments (Buhlmann et al., 2002; Cross, 2006; Harden et al., 2006; Shvartsman et al., 2002; Baskaran et al., 2011; Catalan et al., 2011; Zhu et al., 2006). The classical continuum mechanics neglects the interaction of the material microstructure. The results are size-independent, and the classic theory cannot be applied to analyze nano-sized piezoelectric components. Therefore, an advanced and more appropriate size-dependent theory by considering the electric field and strain gradients is required. The rotation gradient effect in the framework of couple-stress theory is considered in piezoelectricity by Wang et al. (2004). Radi (2003) applied the couple-stress theory to investigate the stress singularity at the crack-tip under mode-I load condition. A more sophisticated theory for size-dependent piezoelectricity has been developed by Majdoub et al. (2008). A different approach to consider the size-effect has been presented by Hadjesfandiari (2013). There, the dielectric polarization is dependent on the mean curvature tensor. If the dielectric polarization is dependent on the strain gradient or curvature strain, a flexoelectric effect occurs (Kogan, 1964; Meyer, 1969; Sharma et al., 2006). The electric field gradient can also be included into the constitutive equations (Landau and Lifshitz, 1984; Yang et al., 2004). The duality between the theory of electric field gradient and the theory of flexoelectricity is given by Maugin (1980). A theory for nano-sized elastic dielectrics with the flexoelectric effects and the surface effects based on a variational principle has been introduced by Hu and Shen (2009).

Large strain gradients can occur also in macro-sized structures in zones of high stress concentration. This is, for example, the case in the vicinity of a crack-tip. The applications of the gradient theory to crack problems can be found in Aravas and Giannakopoulos (2009), Exadaktylos et al. (1996), Exadaktylos (1998), Fannjiang et al. (2002), Georgiadis and Grentzelou (2006), Shi et al. (2000), Vardoulakis et al. (1996), and Wei (2006). The near-tip fields for a crack in elastic or elastic-plastic materials have been considered by Huang et al. (1997). The boundary element method for 2D crack problems has been successfully developed by Karlis et al. (2007). The influence of the electric field gradient on an anti-plane crack has been analyzed by Yang (2004). The crack analysis of plane piezoelectric structures with the consideration of the strain gradient has been presented by Sladek et al. (2017).

In this entry, the size-dependent behavior of cracks in two-dimensional, linear piezoelectric materials is investigated. The size-effect phenomenon is considered by including the strain gradients and the electric field-strain gradient coupling in the constitutive equations. For the numerical computation of the corresponding boundary value problem, the finite element method (FEM) is used. As fracture parameter, the J-integral is formulated. Several numerical examples are presented and discussed to show the effects of the strain and electric field gradient on the J-integral, the crack opening displacements, and electric potentials.

Problem Statement

Let us consider a homogeneous linear piezoelectric solid with cracks of arbitrary shape as shown in Fig. 1.
Fig. 1

A linear piezoelectric solid with cracks

With the consideration of the electric field-strain gradient coupling and a pure nonlocal elastic effect the nano-dielectric cracked solid satisfies the governing equations
$$\displaystyle \begin{aligned} \sigma_{ij,j}(\mathbf{x})-\tau_{ijk,jk}(\mathbf{x})=0, \end{aligned} $$
(1)
$$\displaystyle \begin{aligned} D_{i,i}(\mathbf{x})-Q_{ij,ji}(\mathbf{x})=0, \end{aligned} $$
(2)
and the constitutive equations (Hu and Shen (2009), Shen and Hu (2010))
$$\displaystyle \begin{aligned} \sigma_{ij}(\mathbf{x})=c_{ijkl} \varepsilon_{kl}(\mathbf{x})-e_{kij} E_{k}(\mathbf{x})+b_{klij} E_{k,l}(\mathbf{x}), \end{aligned} $$
(3)
$$\displaystyle \begin{aligned} D_{k}(\mathbf{x})=e_{kij} \varepsilon_{ij}(\mathbf{x})+\kappa_{kl} E_{l}(\mathbf{x})+f_{klmn} \eta_{lmn}(\mathbf{x}), \end{aligned} $$
(4)
$$\displaystyle \begin{aligned} \tau_{jkl}(\mathbf{x})=-f_{ijkl} E_{i}(\mathbf{x})+g_{jklmni} \eta_{nmi}(\mathbf{x}), \end{aligned} $$
(5)
$$\displaystyle \begin{aligned} Q_{ij}(\mathbf{x})=b_{ijkl} \varepsilon_{kl}(\mathbf{x})+h_{ijkl} E_{k,l}(\mathbf{x}). \end{aligned} $$
(6)
In Eqs. (1), (2), (3), (4), (5), and (6) σij, Dk, τjkl, and Qij are the stress tensor, the electric displacements, the higher-order stress, and electric quadrupole. The strain tensor εij, the electric field vector Ei, and the strain gradient tensor ηijk are defined by
$$\displaystyle \begin{aligned} \varepsilon_{ij}(\mathbf{x})=\frac{1}{2}\big[u_{i,j}(\mathbf{x})+u_{j,i}(\mathbf{x})\big], \end{aligned} $$
(7)
$$\displaystyle \begin{aligned} E_{j}(\mathbf{x})=-\phi_{,j}(\mathbf{x}), \end{aligned} $$
(8)
$$\displaystyle \begin{aligned} \eta_{ijk}(\mathbf{x})=\varepsilon_{ij,k}(\mathbf{x})=\frac{1}{2}\big[u_{i,jk}(\mathbf{x})+u_{j,ik}(\mathbf{x})\big], \end{aligned} $$
(9)
with ui and ϕ being the displacements and the electric potential, respectively. Further, cijkl, eijk, and κij are the elasticity tensor, the piezoelectric tensor, and the permittivity tensor; bijkl, fijkl, gijklmn, and hijkl are the quadrupole-strain coefficients, the electric field-strain gradient coupling coefficients, the purely nonlocal elastic effect coefficients, and the higher-order electric parameters.
The tensors possess the symmetry conditions
$$\displaystyle \begin{aligned} \sigma_{ij}=\sigma_{ji} \,\,,\varepsilon_{ij}=\varepsilon_{ji} \,\,, \tau_{ijk}=\tau_{jik} \,\,, \eta_{ijk}=\eta_{jik}, \end{aligned} $$
(10)
$$\displaystyle \begin{aligned} c_{ijkl} &= c_{ijlk}=c_{jikl} = c_{lkij} ,\\ e_{ijk} &= e_{jik} = e_{kij} \,\,, \kappa_{ij} = \kappa_{ji}, \end{aligned} $$
(11)
$$\displaystyle \begin{aligned} f_{ijkl} &= f_{ikjl} \,\,, g_{jklmni} = g_{kjlmni} = g_{jklnmi},\\ b_{ijkl} &= b_{ikjl} \,\,, h_{ijkl} = h_{ikjl}. \end{aligned} $$
(12)
As shown by Gitman et al. (2010) and Yaghoubi et al. (2017), the higher-order elastic parameters gijklmn are proportional to the conventional elastic stiffness coefficients cijkl by the internal length material parameter l
$$\displaystyle \begin{aligned} g_{jklmni}=l^{2}c_{jknm}\delta_{li}, \end{aligned} $$
(13)
with δli being the Kronecker delta. The electric field-strain gradient coupling coefficients fijkl and the quadrupole-strain coupling coefficients bijkl are considered as proportional to the piezoelectric coefficients eijk
$$\displaystyle \begin{aligned} f_{klmn}=M_{l} e_{kmn}, \quad M_{l}=(m^{2},m^{2})^{T}, \end{aligned} $$
(14)
$$\displaystyle \begin{aligned} b_{ijkl}=\varLambda_{j} e_{ikl}, \quad\varLambda_{j}=(\lambda^{2},\lambda^{2})^{T}, \end{aligned} $$
(15)
where m and λ are the scaling parameters. The higher-order electric parameters hijkl can be expressed by the dielectric constants κij and the scaling parameter q.
$$\displaystyle \begin{aligned} h_{ijkl}=q^{2}\kappa_{ik}\delta_{jl} \end{aligned} $$
(16)
The most practically relevant piezoelectric materials are transversally isotropic. For a poling in the positive x3-axis and x1 − x2 as isotropy plane, the general problem can be separated into a plane and anti-plane part. By applying the Voigt notation and assuming loadings only in the x1−x3 direction, the constitutive equations (3), (4), (5), and (6) may be reduced for the plane problem to
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \left[ \begin{array}{c} \sigma_{11} \\ \sigma_{33} \\ \sigma_{13} \end{array} \right] =&& \left[ \ \begin{array}{ccc} c_{11} & c_{13} & 0 \\ c_{13} & c_{33} & 0 \\ 0 & 0 & c_{44} \end{array} \right] \left[ \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{33} \\ 2\varepsilon_{13} \end{array} \right] - \left[ \begin{array}{cc} 0 & e_{31} \\ 0 & e_{33} \\ e_{15} & 0 \end{array} \right] \left[ \begin{array}{c} E_{1} \\ E_{3} \end{array} \right]\\ &&-\lambda^{2} \left[ \begin{array}{cccc} 0 & e_{31} & 0 & e_{31} \\ 0 & e_{33} & 0 & e_{33} \\ e_{15} & 0 & e_{15} & 0 \end{array} \right] \left[ \begin{array}{c} E_{1,1} \\ E_{3,1} \\ E_{1,3} \\ E_{3,3} \end{array} \right]\\ =&& \mathbf{C} \left[ \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{33} \\ 2\varepsilon_{13} \end{array} \right] - \mathbf{L} \left[ \begin{array}{c} E_{1} \\ E_{3} \end{array} \right] -\lambda^{2} \mathbf{M} \left[ \begin{array}{c} E_{1,1} \\ E_{3,1} \\ E_{1,3} \\ E_{3,3} \end{array} \right], \end{array} \end{aligned} $$
(17)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \left[ \begin{array}{c} D_{1} \\ D_{3} \end{array} \right] =&& \left[ \begin{array}{ccc} 0 & 0 & e_{15} \\ e_{31} & e_{33} & 0 \end{array} \right] \left[ \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{33} \\ 2\varepsilon_{13} \end{array} \right] + \left[ \begin{array}{cc} a_{11} & 0 \\ 0 & a_{33} \end{array} \right] \left[ \begin{array}{c} E_{1} \\ E_{3} \end{array} \right] \\ &&+m^{2} \left[ \begin{array}{cccccc} 0 & 0 & e_{15} & 0 & 0 & e_{15} \\ e_{31} & e_{33} & 0 & e_{31} & e_{33} & 0 \end{array} \right] \left[ \begin{array}{c} \eta_{111} \\ \eta_{331} \\ 2\eta_{131} \\ \eta_{113} \\ \eta_{333} \\ 2\eta_{133} \end{array} \right]\\ =&& {\mathbf{L}}^{T} \left[ \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{33} \\ 2\varepsilon_{13} \end{array} \right] + \mathbf{A} \left[ \begin{array}{c} E_{1} \\ E_{3} \end{array} \right] +m^{2} \mathbf{F} \left[ \begin{array}{c} \eta_{111} \\ \eta_{331} \\ 2\eta_{131} \\ \eta_{113} \\ \eta_{333} \\ 2\eta_{133} \end{array} \right] , \end{array} \end{aligned} $$
(18)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \left[ \begin{array}{c} \tau_{111} \\ \tau_{331} \\ \tau_{131} \\ \tau_{113} \\ \tau_{333} \\ \tau_{133} \end{array} \right] =&& -m^{2} \left[ \begin{array}{cccccc} 0 & e_{31} \\ 0 & e_{33} \\ e_{15} & 0 \\ 0 & e_{31} \\ 0 & e_{33} \\ e_{15} & 0 \end{array} \right] \left[ \begin{array}{c} E_{1} \\ E_{3} \end{array} \right] +l^{2} \left[ \begin{array}{cccccc} c_{11} & c_{13} & 0 & 0 & 0 & 0 \\ c_{13} & c_{33} & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{44} & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{11} & c_{13} & 0 \\ 0 & 0 & 0 & c_{13} & c_{33} & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{44} \end{array} \right] \left[ \begin{array}{c} \eta_{111} \\ \eta_{331} \\ 2\eta_{131} \\ \eta_{113} \\ \eta_{333} \\ 2\eta_{133} \end{array} \right]\\ =&& -m^{2} {\mathbf{F}}^{T} \left[ \begin{array}{c} E_{1} \\ E_{3} \end{array} \right] +l^{2} \mathbf{G} \left[ \begin{array}{c} \eta_{111} \\ \eta_{331} \\ 2\eta_{131} \\ \eta_{113} \\ \eta_{333} \\ 2\eta_{133} \end{array} \right], \end{array} \end{aligned} $$
(19)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \left[ \begin{array}{c} Q_{11} \\ Q_{31} \\ Q_{13} \\ Q_{33} \end{array} \right] =&& \lambda^{2} \left[ \begin{array}{ccc} 0 & 0 & e_{15} \\ e_{31} & e_{33} & 0 \\ 0 & 0 & e_{15} \\ e_{31} & e_{33} & 0 \end{array} \right] \left[ \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{33} \\ 2\varepsilon_{13} \end{array} \right] +q^{2} \left[ \begin{array}{cccc} a_{11} & 0 & 0 & 0 \\ 0 & a_{33} & 0 & 0 \\ 0 & 0 & a_{11} & 0 \\ 0 & 0 & 0 & a_{33} \end{array} \right] \left[ \begin{array}{c} E_{1,1} \\ E_{3,1} \\ E_{1,3} \\ E_{3,3} \end{array} \right]\\ =&& \lambda^{2} {\mathbf{M}}^{T} \left[ \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{33} \\ 2\varepsilon_{13} \end{array} \right] +q^{2} \mathbf{H} \left[ \begin{array}{c} E_{1,1} \\ E_{3,1} \\ E_{1,3} \\ E_{3,3} \end{array} \right]. \end{array} \end{aligned} $$
(20)
According to the Fig. 1, two kinds of boundary conditions are defined by:

  • Essential boundary conditions

    \(u_{i}(\mathbf {x})=\bar {u}_{i}(\mathbf {x})\) on Γu, Γu ⊂ Γ

    \(s_{i}(\mathbf {x})=\bar {s}_{i}(\mathbf {x})\) on Γs, Γs ⊂ Γ

    \(\phi (\mathbf {x})=\bar {\phi }(\mathbf {x})\) on Γϕ, Γϕ ⊂ Γ

    \(p(\mathbf {x})=\bar {p}(\mathbf {x})\) on Γp, Γp ⊂ Γ

    with
    $$\displaystyle \begin{aligned} s_{i}=\frac{\partial u_{i}}{\partial x_{j}}n_{j}, \quad p=\frac{\partial\phi}{\partial x_{j}}n_{j} \end{aligned} $$
    (21)

  • Natural boundary conditions

    \(t_{i}(\mathbf {x})=\bar {t}_{i}(\mathbf {x}) {\mathrm {on}} \varGamma _{t}, \,\varGamma _{t} \cup \varGamma _{u}=\varGamma , \,\varGamma _{t} \cap \varGamma _{u}=\oslash \)

    \(R_{i}(\mathbf {x})=\bar {R}_{i}(\mathbf {x})\) on ΓR, ΓR ∪ Γs = Γ, ΓR ∩ Γs = ⊘

    \(S(\mathbf {x}){=}\bar {S}(\mathbf {x})\) on ΓS, ΓS ∪ Γϕ=Γ, ΓS ∩ Γϕ= ⊘

    \(Z(\mathbf {x})=\bar {Z}(\mathbf {x})\) on ΓZ, ΓZ ∪ Γp = Γ, ΓZ ∩ Γp = ⊘

with
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} t_{i}&\displaystyle =&\displaystyle n_{j}\big(\sigma_{ij}-\tau_{ijk,k}\big)-\frac{\partial \rho_{i}}{\partial x_{j}}\varsigma_{j}\\ &\displaystyle &\displaystyle +\sum_{c}\|\rho_{i} ({\mathbf{x}}^{c})\|\delta(\mathbf{x}-{\mathbf{x}}^{c}) \end{array} \end{aligned} $$
(22)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} S&\displaystyle =&\displaystyle n_{k}\big(D_{k}-Q_{kj,j}\big)-\frac{\partial \alpha}{\partial x_{j}}\varsigma_{j}\\ &\displaystyle &\displaystyle +\sum_{c}\|\alpha ({\mathbf{x}}^{c})\|\delta(\mathbf{x}-{\mathbf{x}}^{c}) \end{array} \end{aligned} $$
(23)
$$\displaystyle \begin{aligned} R_{i}=n_{k}n_{j}\tau_{ijk}, \quad Z=n_{i}n_{j}Q_{ij} \end{aligned} $$
(24)
In Eqs. (22), (23), and (24) ni, is the outward unit normal vector, ςi is the unit tangential vector, and
$$\displaystyle \begin{aligned} \rho_{i}=n_{k}\varsigma_{j}\tau_{ijk}, \quad\alpha=n_{i}\varsigma_{j}Q_{ij}. \end{aligned} $$
(25)
The jump at a corner on the boundary Γ is defined as
$$\displaystyle \begin{aligned} \|\rho_{i} ({\mathbf{x}}^{c})\|=\rho_{i}({\mathbf{x}}^{c}+0)-\rho_{i}({\mathbf{x}}^{c}-0), \end{aligned} $$
(26)
$$\displaystyle \begin{aligned} \|\alpha ({\mathbf{x}}^{c})\|=\alpha({\mathbf{x}}^{c}+0)-\alpha({\mathbf{x}}^{c}-0). \end{aligned} $$
(27)
An important role in the crack analysis of piezoelectric solids plays the definition of crack-face boundary conditions. Since in the interior of the crack a medium with a limited electrical permittivity is present, cracks have to be treated in generally as semipermeable. By applying the capacitor analogy model introduced by Hao and Shen (1994), the electrical boundary condition may be written as
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} D_{n}^{+}(\textbf{x})&\displaystyle =&\displaystyle D_{n}^{-}(\textbf{x})=-\kappa_{c} \frac{\varphi^{+}(\textbf{x})-\varphi^{-}(\textbf{x})}{u_{n}^{+}(\textbf{x})-u_{n}^{-}(\textbf{x})}, \\ &\displaystyle &\displaystyle \textbf{x}\in\varGamma_{c^{\pm}}, \end{array} \end{aligned} $$
(28)
where κc = κrκ0 is the product of the relative permittivity of the considered crack medium κr and the permittivity of the vacuum κ0 = 8.854 ⋅ 10−12C∕(Vm). Dn and un are the normal components of the electric displacements and the mechanical displacements on the crack-faces. This crack-face boundary condition has been further improved by including electrostatic tractions, e.g., by Landis (2004) and Gellmann and Ricoeur (2012). For the limit cases of a crack medium with the electrical permittivity of zero, the substitution of κc = 0 into Eq. (28) leads to the known electrical impermeable crack-face boundary condition
$$\displaystyle \begin{aligned} D_{n}^{+}(\textbf{x})=D_{n}^{-}(\textbf{x})=0, \quad \textbf{x}\in\varGamma_{c^{\pm}}. \end{aligned} $$
(29)
This denotes in a physically sense that both crack-faces are free of electrical displacements. On the other side, the second limit cases of a medium in the interior of the crack with an infinite electrical permittivity result in the permeable electrical crack-face condition
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} D_{n}^{+}(\textbf{x})&\displaystyle &\displaystyle =D_{n}^{-}(\textbf{x}),\quad\varphi^{+}(\textbf{x})-\varphi^{-}(\textbf{x})=0, \\ &\displaystyle &\displaystyle \textbf{x}\in\varGamma_{c^{\pm}}. \end{array} \end{aligned} $$
(30)

Numerical Solution Algorithm

The prescribed boundary value problem is solved with the finite element method (FEM). The required weak form of the boundary value problem in the electric field gradient theory can be derived from the principle of virtual work as
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle \int\limits_{V}\big(\sigma_{ij} \delta u_{i,j} + \tau_{ijk} \delta u_{i,jk} - D_{k} \delta \phi_{,k}\\&\displaystyle &\displaystyle \quad- Q_{ij} \delta \phi_{,ij} \big)\mathrm{d}\varOmega =\int\limits_{\varGamma_{t}}\bar{t}_{i}\delta u_{i}\mathrm{d}\varGamma + \int\limits_{\varGamma_{R}}\bar{R}_{i}\delta s_{i}\mathrm{d}\varGamma\\&\displaystyle &\displaystyle \quad+ \int\limits_{\varGamma_{S}}\bar{S}\delta \phi\mathrm{d}\varGamma + \int\limits_{\varGamma_{Z}}\bar{Z}\delta p\mathrm{d}\varGamma. \end{array} \end{aligned} $$
(31)
After spatial discretization, the variational statement (31) is leading to a system of linear algebraic equations
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle \sum_{e_{k}}\sum_{c=1}^{n}\int\limits_{-1}^{1}\int\limits_{-1}^{1}\Bigg\{ \Big[B_{\varepsilon}^{e_{k}a_{k}}\Big]^{T} \bigg ([C] \Big[B_{\varepsilon}^{e_{k}c}\Big]-[L] \Big[B_{E}^{e_{k}c}\Big] -\lambda^{2} [M] \Big[B_{EG}^{e_{k}c}\Big]\bigg)\\ &\displaystyle &\displaystyle +\Big[B_{\eta}^{e_{k}a_{k}}\Big]^{T} \bigg(-m^{2}[F]^{T} \Big[B_{E}^{e_{k}c}\Big] +l^{2} [G]\Big[B_{\eta}^{e_{k}c}\Big]\bigg)\\ &\displaystyle &\displaystyle -\Big[B_{EG}^{e_{k}a}\Big]^{T} \bigg(\lambda^{2}[M]^{T} \Big[B_{\varepsilon}^{e_{k}c}\Big] +q^{2} [H]\Big[B_{EG}^{e_{k}c}\Big]\bigg)\\ &\displaystyle &\displaystyle -\Big[B_{E}^{e_{k}a_{k}}\Big]^{T} \bigg ([L]^{T} \Big[B_{\varepsilon}^{e_{k}c}\Big]+[A] \Big[B_{E}^{e_{k}c}\Big] +m^{2} [F] \Big[B_{\eta}^{e_{k}c}\Big]\bigg) \Bigg\}\Big\{q^{(e_{k}c)}\Big\} \Big|\mathrm{det}\big[J^{e_{k}}\big]\Big|\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\\ &\displaystyle &\displaystyle =\sum_{e_{k}}\Bigg(\int\limits_{V_{t}^{e_{k}}}\{T\}N^{a_{k}}\mathrm{d}\varGamma + \int\limits_{V_{R}^{e_{k}}}\{R\}n_{j}b_{j}^{e_{k}a_{k}}\mathrm{d}\varGamma + \int\limits_{V_{S}^{e_{k}}}\{S\}N^{a_{k}}\mathrm{d}\varGamma + \int\limits_{V_{Z}^{e_{k}}}\{Z\}n_{j}b_{j}^{e_{k}a_{k}}\mathrm{d}\varGamma \Bigg),\\ &\displaystyle &\displaystyle (k=1,2,\ldots,K), \end{array} \end{aligned} $$
(32)
where Na is the shape function of the a-th node of the e-th element; Je is Jacobian; the material matrices [C], [L], [M], [F], [G], [H] and [A] are defined in Eqs. (17), (18), (19), and (20); and
$$\displaystyle \begin{aligned} \begin{array}{rcl} \{T\}&\displaystyle =&\displaystyle (\bar{t}_{1} \,\,\,\, \bar{t}_{2} \,\,\,\,\bar{t}_{3}), \quad\{R\}=(\bar{R}_{1} \,\,\,\, \bar{R}_{2} \,\,\,\,0), \\ \{S\}&\displaystyle =&\displaystyle (0 \,\,\,\, 0 \,\,\,\,\bar{S}), \quad\{Z\}=(0 \,\,\,\, 0 \,\,\,\,\bar{Z}), \end{array} \end{aligned} $$
(33)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &&\Big[B_{E}^{ea}\Big]= -\left( \begin{array}{ccc} 0 & 0 & b_{1}^{ea} \\ 0 & 0 & b_{3}^{ea} \end{array} \right),\quad\Big[B_{\varepsilon}^{ea}\Big]= -\left( \begin{array}{ccc} b_{1}^{ea} & 0 & 0 \\ 0 & b_{3}^{ea} & 0 \\ b_{3}^{ea} & b_{1}^{ea} & 0 \end{array} \right),\quad\Big\{q^{(ec)}\Big\}= -\left( \begin{array}{c} u_{1}^{(ea)} \\ u_{3}^{(ea)} \\ \phi^{(ea)} \end{array} \right),\\ &&\Big[B_{\eta}^{ea}\Big]= -\left( \begin{array}{ccc} b_{11}^{ea} & 0 & 0 \\ 0 & b_{13}^{ea} & 0 \\ b_{13}^{ea} & b_{11}^{ea} & 0 \\ b_{13}^{ea} & 0 & 0 \\ 0 & b_{33}^{ea} & 0 \\ b_{33}^{ea} & b_{13}^{ea} & 0 \end{array} \right),\quad\Big[B_{EG}^{ea}\Big]= -\left( \begin{array}{ccc} 0 & 0 & b_{11}^{ea} \\ 0 & 0 & b_{13}^{ea} \\ 0 & 0 & b_{13}^{ea} \\ 0 & 0 & b_{33}^{ea} \end{array} \right). \end{array} \end{aligned} $$
(34)
The coefficients \(b_{i}^{ea}\) and \(b_{ij}^{ea}\) follow from the spatial derivations
$$\displaystyle \begin{aligned} \hat{b}_{i}=\frac{\partial}{\partial x_{i}}, \quad\hat{b}_{ij}=\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}, \end{aligned} $$
(35)
as defined in Sladek et al. (2017).

J-Integral

The virtual extension of a crack along its plane leads to a change in the potential energy Π of a cracked body Ω
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} -J&\displaystyle &\displaystyle =\frac{\mathrm{d}\varPi}{\mathrm{d}a}=\int\limits_{\varOmega}\frac{\mathrm{d}U}{\mathrm{d}a}\mathrm{d}\varOmega -\int\limits_{\partial\varOmega_{t}}\bar{t}_{i}\frac{\mathrm{d}u_{i}}{\mathrm{d}a}\mathrm{d}\varGamma -\int\limits_{\partial\varOmega_{R}}\bar{R}_{i}\frac{\mathrm{d}s_{i}}{\mathrm{d}a}\mathrm{d}\varGamma -\int\limits_{\partial\varOmega_{S}}\bar{S}\frac{\mathrm{d}\phi}{\mathrm{d}a}\mathrm{d}\varGamma -\int\limits_{\partial\varOmega_{Z}}\bar{Z}\frac{\mathrm{d}p}{\mathrm{d}a}\mathrm{d}\varGamma\\ &\displaystyle &\displaystyle =\int\limits_{\varOmega}\frac{\mathrm{d}U}{\mathrm{d}a}\mathrm{d}\varOmega -\int\limits_{\partial\varOmega}\bigg(t_{i}\frac{\mathrm{d}u_{i}}{\mathrm{d}a}+R_{i}\frac{\mathrm{d}s_{i}}{\mathrm{d}a}+S\frac{\mathrm{d}\phi}{\mathrm{d}a}+Z\frac{\mathrm{d}p}{\mathrm{d}a}\bigg)\mathrm{d}\varGamma. \end{array} \end{aligned} $$
(36)
Following the procedure prescribed in Sladek et al. (2017), the J-integral may be expressed by
$$\displaystyle \begin{aligned} J=J_{0}+J_{c^{+}}+J_{c^{-}}-J_{\varepsilon}=0, \end{aligned} $$
(37)
with
$$\displaystyle \begin{aligned} \begin{array}{rcl} J_{0}&\displaystyle =&\displaystyle \int\limits_{\varGamma}\bigg(U n_{1}-t_{i}\frac{\partial u_{i}}{\partial x_{1}}-R_{i}\frac{\partial s_{i}}{\partial x_{1}}\\&\displaystyle &\displaystyle -S\frac{\partial \phi}{\partial x_{1}}-Z\frac{\partial p}{\partial x_{1}} \bigg)\mathbf{d}\varGamma, \end{array} \end{aligned} $$
(38)
$$\displaystyle \begin{aligned} \begin{array}{rcl} J_{c^{+}}&\displaystyle =&\displaystyle \int\limits_{\varGamma_{c^{+}}}\bigg(U n_{1}-t_{i}\frac{\partial u_{i}}{\partial x_{1}}-R_{i}\frac{\partial s_{i}}{\partial x_{1}}\\&\displaystyle &\displaystyle -S\frac{\partial \phi}{\partial x_{1}}-Z\frac{\partial p}{\partial x_{1}} \bigg)\mathbf{d}\varGamma, \end{array} \end{aligned} $$
(39)
$$\displaystyle \begin{aligned} \begin{array}{rcl} J_{c^{-}}&\displaystyle =&\displaystyle \int\limits_{\varGamma_{c^{-}}}\bigg(U n_{1}-t_{i}\frac{\partial u_{i}}{\partial x_{1}}-R_{i}\frac{\partial s_{i}}{\partial x_{1}}\\&\displaystyle &\displaystyle -S\frac{\partial \phi}{\partial x_{1}}-Z\frac{\partial p}{\partial x_{1}} \bigg)\mathbf{d}\varGamma, \end{array} \end{aligned} $$
(40)
$$\displaystyle \begin{aligned} \begin{array}{rcl} J_{\varepsilon}&\displaystyle =&\displaystyle \int\limits_{\varGamma_{\varepsilon}}\bigg(U n_{1}-t_{i}\frac{\partial u_{i}}{\partial x_{1}}-R_{i}\frac{\partial s_{i}}{\partial x_{1}}\\&\displaystyle &\displaystyle -S\frac{\partial \phi}{\partial x_{1}}-Z\frac{\partial p}{\partial x_{1}} \bigg)\mathbf{d}\varGamma. \end{array} \end{aligned} $$
(41)
Here, ni denotes the outward unit normal vector of the contour as shown in Fig. 2.
Fig. 2

A regular domain around the crack-tip

For a local crack-tip coordinate system with n1 = 0 as well as vanishing natural boundary conditions on the crack-faces \(\varGamma _{c^{\pm }}\), the terms \(J_{c^{+}}\), \(J_{c^{-}}\) are zero, and Eq. (37) reduces to
$$\displaystyle \begin{aligned} \begin{array}{rcl} J&\displaystyle =&\displaystyle \int\limits_{\varGamma}\bigg(U n_{1}-t_{i}\frac{\partial u_{i}}{\partial x_{1}}-R_{i}\frac{\partial s_{i}}{\partial x_{1}}\\&\displaystyle &\displaystyle -S\frac{\partial \phi}{\partial x_{1}}-Z\frac{\partial p}{\partial x_{1}} \bigg)\mathbf{d}\varGamma\\ &\displaystyle =&\displaystyle \int\limits_{\varGamma_{\varepsilon}}\bigg(U n_{1}-t_{i}\frac{\partial u_{i}}{\partial x_{1}}-R_{i}\frac{\partial s_{i}}{\partial x_{1}}\\&\displaystyle &\displaystyle -S\frac{\partial \phi}{\partial x_{1}}-Z\frac{\partial p}{\partial x_{1}} \bigg)\mathbf{d}\varGamma, \end{array} \end{aligned} $$
(42)
with ti, Ri, S, and Z defined in Eqs. (22), (23), (24) and
$$\displaystyle \begin{aligned} U=\frac{1}{2}\sigma_{ij}\varepsilon_{ij}+\frac{1}{2}\tau_{ijk}\eta_{ijk}-\frac{1}{2}D_{i}E_{i}-\frac{1}{2}Q_{ij}E_{i,j}. \end{aligned} $$
(43)

Using the asymptotic expansions of the fields in the vicinity of the crack-tip, the second integral may be evaluated in closed form. In a numerical solution algorithm, it is more accurate to compute the first integral along a contour away from the crack-tip.

Numerical Examples

In this section, several numerical examples are presented and discussed to indicate the influences of the electric field and strain gradients on the J-integral and the field quantities. In all examples, the piezoelectric material PZT-5H is applied which has the linear constants
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle C_{11}=126.0\,\mathrm{GPa}, \quad C_{13}=84.1\,\mathrm{GPa}, \\ &\displaystyle &\displaystyle C_{33}=117.0\,\mathrm{GPa}, \quad C_{44}=23.0\,\mathrm{GPa},\\ &\displaystyle &\displaystyle e_{31}=-6.5\,\mathrm{C/m}^{2}, \quad e_{33}=23.3\,\mathrm{C/m}^{2}, \\ &\displaystyle &\displaystyle e_{15}=17.0\,\mathrm{C/m}^{2},\quad\kappa_{11}=15.04\,\mathrm{C/(GVm)}, \\ &\displaystyle &\displaystyle \kappa_{22}=13.0\,\mathrm{C/(GVm)}. \end{array} \end{aligned} $$
(44)
In order to assess the effects of the strain and electric field gradients, the size factors l, m, λ, and q according to the Eqs. (17), (18), (19), and (20) are defined by
$$\displaystyle \begin{aligned} \begin{array}{rcl} l^{2}&\displaystyle =&\displaystyle \alpha l_{0}^{2}, \quad m^{2}=\alpha m_{0}^{2}, \\ \lambda^{2}&\displaystyle =&\displaystyle \beta \lambda_{0}^{2}, \quad q^{2}=\beta q_{0}^{2}. \end{array} \end{aligned} $$
(45)
The scaling parameters are chosen as l0 = 5 ⋅ 10−9m, m0 = 2.5 ⋅ 10−6m, λ0 = 2.5 ⋅ 10−6m and q0 = 5 ⋅ 10−10m.

All numerical computations have been performed with the FEM program COMSOL Multiphysics by implementing the governing equations in the weak form (31). A higher-order Argyris-type element with continuity is used (Argyris et al., 1968).

A Square Plate with a Central Crack

In the first example, as shown in Fig. 3, a square plate with a central crack and the geometrical parameters a = 1.0 ⋅ 10−7 m, w = 5a is investigated.
Fig. 3

A square plate with a central crack under electromechanical loading

The top and the bottom boundaries of the plate are subjected to a combined electromechanical loading. The mechanical loading in all computations is taken as t3 = 1.17 ⋅ 106 MPa. The tractions and the electrical displacements on the left and right lateral sides of the plate are zero. The crack-faces are free of mechanical tractions and electrical displacements. The poling direction (P.) of the material is taken to be parallel to the x3-axis.

The computed J-integral for the strain and electric field gradient size factors \(\alpha \, = \,\beta \, = 0 - 4\) are presented in Fig. 4 for different electrical loadings in the range − 5 ⋅ 10−4C∕m2 ≤ D3 ≤ 1.2 ⋅ 10−3 C∕m2. The crack is assumed as electrically impermeable.
Fig. 4

J-integral for different strain and electric field gradient size factors α = β

It can be observed that the J-integral decreases with increasing strain and electric field gradient parameter α in the whole investigated interval of D3. The pick values of the curves are shifted to smaller electric displacement D3 if the gradient parameter α increases.

The crack opening displacements \(u_{3}^{+}\) and the electrical potentials ϕ+ of the upper crack-face \(\varGamma _{c^{+}}\) obtained for the electrical loadings \(D_{3}^{1}=-5.0\cdot 10^{-4}\,\mathrm {C/m^{2}}\), \(D_{3}^{2}=0\) and \(D_{3}^{3}=1.0\cdot 10^{-3}\,\mathrm {C/m^{2}}\) are shown in Fig. 5 for the strain and electric field gradient size factors α = β = 0, 2 and 4.
Fig. 5

Crack opening displacements \(u_{3}^{+}\) and electrical potentials ϕ+ of the upper crack-face \(\varGamma _{c}^{+}\) for α = β = 0, 2, 4

The Fig. 5 indicates that the electrical loading has a significant influence on the crack opening displacement \(u_{3}^{+}\) and the electrical potential ϕ+ for all considered strain and electric field gradient size factors α. The analyzed cracked plate is symmetric with respect to the horizontal and vertical midplanes. Therefore, the crack opening displacement \(u_{3}^{+}\) and the electrical potential ϕ+ are also symmetric. Since the piezoelectric material PZT-5H is transversally isotropic, no shear stress components are induced by the applied loadings, and the discontinuity in the tangential component of the crack-displacement \(\varDelta u_{1}=u_{1}^{+}-u_{1}^{-}\) is zero. The presence of the strain and electric field gradients decreases the crack opening displacement \(u_{3}^{+}\) and the electrical potential ϕ+. In the classical piezoelectric crack model with α = 0, both the crack opening displacements and the electrical potential exhibit the square root behavior near the crack-tip. The application of the strain and electric field gradient theory (α > 0) changes the asymptotic crack-tip behavior of the crack opening displacements and electrical potential as discussed in Sladek et al. (2017).

A Rectangular Plate with an Edge Crack

In the next example, an edge crack in a rectangular plate is investigated. According to the Fig. 6, the geometrical data a = 1.0 ⋅ 10−7 m and w = 5a are chosen for the numerical computation.
Fig. 6

A rectangular plate with an edge crack

On the left and right lateral sides of the plate, a mechanical tensile loading t3 = 1.17 ⋅ 106 MPa is applied. The mechanical stresses are zero on the top and bottom boundaries of the plate. The electrical displacements are zero on all external boundaries.

The J-integral obtained for the strain and electric field gradient size factors α = β = 0, 2, 4 using the electrically impermeable (ip) and permeable (p) crack-face boundary conditions are given in Table 1. A comparison of the corresponding crack opening displacements \(u_{3}^{+}\) and the electrical potentials ϕ+ is shown in Fig. 7
Fig. 7

Crack opening displacements \(u_{3}^{+}\) and electrical potentials ϕ+ of the upper crack-face \(\varGamma _{c}^{+}\) for α = β = 0, 2, 4 and different crack-face boundary conditions

Table 1

J-integral for different crack-face boundary conditions and strain and electric field gradient size factors α = β

 

J [J/m2 ]

α

0

2

4

impermeable (ip)

7.1 ⋅ 10−6

6.2 ⋅ 10−6

5.9 ⋅ 10−6

permeable (p)

7.9 ⋅ 10−6

6.9 ⋅ 10−6

6.6 ⋅ 10−6

Figure 7 indicates an important influence of the electrical crack-face boundary condition on the crack opening displacements and the electrical potential for both the classical (α = 0) and the strain and electric field gradient model (α = 2, 4). The jump of the electrical potential Δϕ vanishes for the permeable crack-face boundary condition. Further, this crack model leads in the considered example to larger crack opening displacements. The rectangular plate is symmetric with respect to the midplane in the x1-direction. As a consequence, the electrical potential ϕ+ is zero for the case of the transversally isotropic material with poling in the x3-direction and under the applied tensile loading. As expected, the J-integral decreases with increasing strain and electric field gradient parameter α.

Conclusion

In this entry, general 2D boundary value problems of piezoelectric nano-sized structures with cracks under electromechanical loadings are investigated. The size-effect phenomenon is described by the strain and electric field gradients in the constitutive equations. The FEM formulation has been developed for the solution of the corresponding boundary value problem. Several numerical examples are presented and discussed to show the effects of the combined electromechanical loading, the crack-face boundary conditions and the strain and electric field gradients on the J-integral, the crack opening displacements and the electric potentials. It is shown that the strain and electric field gradient model produces smaller values of the J-integral than those obtained with the classical piezoelectric model for the same geometry and loading conditions.

Notes

Acknowledgements

The authors acknowledge the support by the Slovak Science and Technology Assistance Agency registered under number APVV-14-0216, VEGA 1/0145/17 and the Slovak Academy of Sciences Project (SASPRO) 0106/01/01.

References

  1. Aravas N, Giannakopoulos AE (2009) Plane asymptotic crack-tip solutions in gradient elasticity. Int J Solids Struct 46:4478–4503CrossRefGoogle Scholar
  2. Argyris JH, Fried I, Scharpf DW (1968) The tuba family of plate elements for the matrix displacement method. Aeronaut J 72:701–709CrossRefGoogle Scholar
  3. Baskaran S, He X, Chen Q, Fu JF (2011) Experimental studies on the direct flexoelectric effect in alpha-phase polyvinylidene fluoride films. Appl Phys Lett 98:242901CrossRefGoogle Scholar
  4. Buhlmann S, Dwir B, Baborowski J, Muralt P (2002) Size effects in mesiscopic epitaxial ferroelectric structures: increase of piezoelectric response with decreasing feature-size. Appl Phys Lett 80:3195–3197CrossRefGoogle Scholar
  5. Catalan G, Lubk A, Vlooswijk AHG, Snoeck E, Magen C, Janssens A, Rispens G, Rijnders G, Blank DHA, Noheda B (2011) Flexoelectric rotation of polarization in ferroelectric thin films. Nat Mater 10:963–967CrossRefGoogle Scholar
  6. Cross LE (2006) Flexoelectric effects: charge separation in insulating solids subjected to elastic strain gradients. J Mater Sci 41:53–63CrossRefGoogle Scholar
  7. Exadaktylos G (1998) Gradient elasticity with surface energy: mode-I crack problem. Int J Solids Struct 35:421–456MathSciNetCrossRefGoogle Scholar
  8. Exadaktylos G, Vardoulakis I, Aifantis E (1996) Cracks in gradient elastic bodies with surface energy. Int J Fract 79: 107–119CrossRefGoogle Scholar
  9. Fannjiang AC, Chan YS, Paulino GH (2002) Strain gradient elasticity for antiplane shear cracks: a hypersingular integrodifferential equation approach. SIAM J Appl Math 62:1066–1091MathSciNetCrossRefGoogle Scholar
  10. Gellmann R, Ricoeur A (2012) Some new aspects of boundary conditions at cracks in piezoelectrics. Arch Appl Mech 82:841–852CrossRefGoogle Scholar
  11. Georgiadis HG, Grentzelou CG (2006) Energy theorems and the J integral in dipolar gradient elasticity. Int J Solids Struct 43:5690–5712MathSciNetCrossRefGoogle Scholar
  12. Gitman I, Askes H, Kuhl E, Aifantis E (2010) Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity. Int J Solids Struct 47:1099–1107CrossRefGoogle Scholar
  13. Hadjesfandiari AR (2013) Size-dependent piezoelectricity. Int J Solids Struct 50:2781–2791CrossRefGoogle Scholar
  14. Hao TH, Shen ZY (1994) A new electric boundary condition of electric fracture mechanics and its applications. Eng Fract Mech 47:793–802CrossRefGoogle Scholar
  15. Harden J, Mbanga B, Eber N, Fodor-Csorba K, Sprunt S, Gleeson JT, Jakli A (2006) Giant flexoelectricity of bent-core nematic liquid crystals. Phys Rev Lett 97:157802CrossRefGoogle Scholar
  16. Hu SL, Shen SP (2009) Electric field gradient theory with surface effect for nano-dielectrics. CMC Comput Mater Continua 13:63–87Google Scholar
  17. Huang Y, Zhang L, Guo TF, Hwang KC (1997) Mixed mode near-tip fields for cracks in materials with strain-gradient effects. J Mech Phys Solids 45:439–465CrossRefGoogle Scholar
  18. Karlis GF, Tsinopoulos SV, Polyzos D, Beskos DE (2007) Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity. Comput Methods Appl Mech Eng 196:5092–5103CrossRefGoogle Scholar
  19. Kogan ShM (1964) Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals. Sov Phys Solid State 5:2069–2070Google Scholar
  20. Landau LD, Lifshitz EM (1984) Electrodynamics of continuous media, 2nd edn. Butterworth-Heinemann, Oxford, pp 358–371CrossRefGoogle Scholar
  21. Landis CM (2004) Energetically consistent boundary conditions for electromechanical fracture. Int J Solids Struct 41:6291–6315CrossRefGoogle Scholar
  22. Majdoub MS, Sharma P, Cagin T (2008) Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys Rev B 77:1–9CrossRefGoogle Scholar
  23. Maugin GA (1980) The method of virtual power in continuum mechanics: applications to coupled fields. Acta Mech 35:1–80MathSciNetCrossRefGoogle Scholar
  24. Meyer RB (1969) Piezoelectric effects in liquid crystals. Phys Rev Lett 22:918–921CrossRefGoogle Scholar
  25. Radi E (2003) Strain gradient effects on steady-state crack growth in linear hardening materials. J Mech Phys Solids 51:543–573MathSciNetCrossRefGoogle Scholar
  26. Sharma P, Maranganti R, Sharma ND (2006) Electromechanical coupling in nanopiezoelectric materials due to nanoscale nonlocal size effects: green function solution and embedded inclustions. Phys Rev B 74:014110CrossRefGoogle Scholar
  27. Shen SP, Hu SL (2010) A theory of flexoelectricity with surface effect for elastic dielectrics. J Mech Phys Solids 58:665–677MathSciNetCrossRefGoogle Scholar
  28. Shi MX, Huang Y, Hwang KC (2000) Fracture in a higher-order elastic continuum. J Mech Phys Solids 48: 2513–2538CrossRefGoogle Scholar
  29. Shvartsman VV, Emelyanov AY, Kholkin AL, Safari A (2002) Local hysteresis and grain size effects in Pb(Mg1/3Nb2/3)O-SbTiO3. Appl Phys Lett 81: 117–119CrossRefGoogle Scholar
  30. Sladek J, Sladek V, Stanak P, Zhang Ch, Wünsche M (2011) An interaction integral method for computing fracture parameters in functionally graded magnetoelectroelastic composites. CMC Comput Mater Continua 586:1–34Google Scholar
  31. Sladek J, Sladek V, Stanak P, Zhang Ch, Tan CL (2017) Fracture mechanics analysis of size-dependent piezoelectric solids. Int J Solids Struct 113:1–9CrossRefGoogle Scholar
  32. Vardoulakis I, Exadaktylos G, Aifantis E (1996) Gradient elasticity with surface energy: mode-III crack problem. Int J Solids Struct 33:4531–4559CrossRefGoogle Scholar
  33. Wang GF, Yu SW, Feng XQ (2004) A piezoelectric constitutive theory with rotation gradient effects. Eur J Mech A Solids 23:455–466CrossRefGoogle Scholar
  34. Wei Y (2006) A new finite element method for strain gradient theories and applications to fracture analyses. Eur J Mech A Solids 25:897–913MathSciNetCrossRefGoogle Scholar
  35. Yaghoubi ST, Mousavi SM, Paavola J (2017) Buckling of centrosymmetric anisotropic beam structures within strain gradient elasticity. Int J Solids Struct 109:84–92CrossRefGoogle Scholar
  36. Yang XM, Hu YT, Yang JS (2004) Electric field gradient effects in anti-plane problems of polarized ceramics. Int J Solids Struct 41:6801–6811CrossRefGoogle Scholar
  37. Yang J (2004) Effects of electric field gradient on an antiplane crack in piezoelectric ceramics. Int J Fract 127:L111–L116CrossRefGoogle Scholar
  38. Zhu W, Fu JY, Li N, Cross LE (2006) Piezoelectric composite based on the enhanced flexoelectric effects. Appl Phys Lett 89:192904CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Construction and Architecture, Slovak Academy of SciencesBratislavaSlovakia

Section editors and affiliations

  • Eduard-Marius Craciun
    • 1
  1. 1.Faculty of Mechanical, Industrial and Maritime Engineering"Ovidius" University of ConstantaConstantaRomania