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Piezoelectric material has been emerging as a popular building block in MEMS devices owing to its unique mechanical and electrical material properties. However, the reliability of MEMS devices under buckling deformation environments remains elusive and needs to be further explored. Based on the Talreja's tensor valued internal state damage variables as well as the Helmhotlz free energy of piezoelectric material, a constitutive model of piezoelectric materials with damage is presented. The Kachanvo damage evolution law under in-plane compressive loads is employed. The model is applied to the specific case of the postbuckling analysis of the piezoelectric plate with damage. Then, adopting von Karman's plate theory, the nonlinear governing equations of the piezoelectric plates with initial geometric deflection including damage effects under in-plane compressive loads are established. By using the finite difference method and the Newmark scheme, the damage evolution for damage accumulation is developed and the finite difference procedure for postbuckling equilibrium path is simultaneously employed. Numerical results show the postbuckling behaviors of initial flat and deflected piezoelectric plates with damage or no damage under different sets of electrical loading conditions. The effects of applied voltage, aspect ratio of plate, thick-span ratio of plate, damage as well as initial geometric deflections on the postbuckling behaviors of the piezoelectric plate are discussed.

The use of piezoelectric materials in intelligent structures has received considerable attention in recent years due to the intrinsic direct and converse piezoelectric effects. Piezoelectric materials have been used as sensors or actuators for the control of the active shape or vibration of structures. Defects such as microcracks, voids, dislocations and delamination are introduced in piezoelectric materials during the manufacturing and poling process. The existence of these defects greatly affects the electric, dielectric, elastic, mechanical and piezoelectric properties of the piezoelectric materials, especially the service life of piezoelectric structures. When subjected to mechanical and electrical loads, these defects may grow in size and cracks may propagate leading to premature mechanical or electrical fatigue failure. Therefore, it is important to understand the growth of these defects, the damage accumulation and the overall effect of these defects on the average mechanical and electrical properties of piezoelectric structures.

Damage in fiber-reinforced composite materials has been extensively investigated, and many theories have been established and used to predict the life of composite structures. Based on the framework of irreversible thermodynamics with internal state variables, Talreja [

Modeling and analysis of multilayer piezoelectric beams and plates have reached a relative maturity as attested by the numerous papers. Mindlin [

In the present study, a new constitutive model for piezoelectric materials using the Talreja's tensor valued internal state damage variables and the Helmhotlz free energy of piezoelectric material is presented. This model is then applied to a specific case of postbuckling analysis of piezoelectric plates under in-plane compressive loads. By adopting von Karman's plate theory and using the finite difference and the Newmark scheme, the damage evolution for damage accumulation is developed and the finite difference procedure for postbuckling equilibrium path is simultaneously employed. In the numerical examples, the effects of variation in the load parameters, damage influences and geometric parameters of the plate on postbuckling equilibrium paths are discussed.

Consider a representative volume element of a piezoelectric solid with a multitude of damage entities in the form of microcracks, as shown in _{ij}

Now if there are _{r}_{k}_{ij}

Physically, the damage tensor

For the case of damaged piezoelectric material without temperature effect where the damage is represented by internal state variables, the Helmholtz free energy of piezoelectric material can be written as a function of the transformed elastic strains, the electric field vector and damage internal variables, that is:

The transformed stress components _{ij}_{i}

When the damage induced by the cracks in the piezoelectric material has the orthotropic property, the irreducible integrity bases for a scalar polynomial function of two symmetric second rank tensors can be expressed as [_{13} = _{23} =0 and applying Voigt notation to describe strains and damage variables, the bases of invariants can be further written as:

Using the above stated irreducible integrity bases, the Helmholtz free energy of piezoelectric materials can be expressed as a quadratic expression of the strains or the electric field intensity, a mixture quadratic expression of strains and electric field intensity and a linear expression of damage variables [_{0} is a constant, _{1} is a linear function of strains, _{2} is a linear function of damage variables and _{3} is a linear function of the electric field intensity. Then the stresses and the electric displacements can be expressed as:

Assuming that there is only one damage mode in the representative volume element, the relations of the strains, the stresses, the electric field intensity and the electric displacements in

In present study, consider that the matrix cracks in the piezoelectric plate are parallel to the coordinate plane 2–3, all damage variables except _{1} are zero, then the coefficient matrixes in

Due to the fact the cracks are parallel to the coordinate plane 2 – 3, the effect of the damage on the stiffness of the plate in this coordinate plane 2 – 3 can be neglected, which means the component

Letting _{3}=0 based on plane-stress assumption and using

In the present research, the Kachanvo damage evolution law [_{eq}_{f}

The relations between the electric fields _{x}_{y}_{z}

For the piezoelectric plate, only thickness direction electric field _{z}_{e}

Now, consider a thin piezoelectric plate with transverse cracks subjected to uniformly distributed in-plane compressive load

Setting _{x}_{y}_{xy}

Suppose the damage variable remains constant through the thickness of plate. Denoting _{x}_{y}_{xy}_{x}_{y}_{xy}

Using _{ij}_{ij}

The resultants and couples due to the piezoelectric effect can be written as:

Introducing the following dimensionless parameters:

By using

Suppose the boundary of the piezoelectric plate is simply movably supported, the dimensionless boundary conditions can be expressed as:

The dimensionless damage evolution equation of the piezoelectric plate subjected to the uniformly in-plane compressive load can be written respectively as follows:

Taking the mid-surface normal stress of the piezoelectric plate as the equivalent stress _{eq}

Suppose the dimensionless initial geometric deflection is taken as:

To seek the approximate solutions of the governing _{Jp}

Moreover, using the Newmark scheme, the inertia in

For every time step, the iteration lasts until the difference of the present value and the former is smaller than 0.1%, then continue the calculation of the next step.

To ensure the accuracy and effectiveness of the present method, a test example was calculated for postbuckling analysis of isotropic rectangular plate with initial geometric deflection. Comparison of postbuckling response curves for isotropic rectangular plate with initial geometric deflection is shown in _{0} denotes the center deflection of the plate. The close agreements between the present results and those of reference [

To study the piezo-effects and damage effects on the postbuckling behavior of the plates, several numerical examples were solved for initial flat and deflected plate. A piezoelectric plate consisting of the PZT-5A including initial damage is considered for postbuckling analysis. The material properties of PZT-5A are given as follows:

When the effect of damage is omitted and the linear strain-displacement relations are adopted, the dimensionless governing equation corresponding to

The corresponding dimensionless boundary conditions of the simply movable supported plate can be written as:

Considering a harmonic displacement solution for this buckling problem

Substituting _{mn}

The least critical buckling load _{cr}

When the geometric parameters are given as _{1} = _{2} =0.1, _{0} =0.1, the critical buckling load _{cr}_{0} of the piezoelectric plate and the in-plane compressive load

A parametric study has been carried out and typical results are shown in _{0}, _{e}_{cr}_{cr}_{1} = _{2} = 0.01.

When the damage effect is in consideration, the material parameters related to damage in all examples are taken as:

_{f}^{−2},

_{1} = _{2} = 0.1. Three electrical load conditions, referred as 1, 2 and 3, are considered. It can be seen that the negative control voltage results in the increase of the buckling load and the decrease of postbuckled deflection at the same compressive loads. In contrast, the positive control voltage decreases the buckling load and induces larger postbuckled deflections. It can be concluded that the positive control voltage acting upon the piezoelectric plate is equivalent to a compressive piezoelectric force acting in the in-plane direction of the plate to some certain extent, which leads to the smaller buckling loads.

_{1} = _{2} = 0.1. It can be seen that the larger the initial deflections of the plate, the larger the postbuckled deflection of the plate under the same compressive load, and that the postbuckled deflections of the plate under different initial deflections will reach the same value with the increase of postbuckling loads.

_{e}

_{e}

This paper presents an approach to investigate the postbuckling analysis of piezoelectric plates including damage effects using Talreja's tensor valued internal state damage variables and the Kachanvo damage evolution equation. The effects of applied voltage, plate aspect ratio, thick-span ratio, damage as well as initial geometric deflections on the postbuckling behaviors of the piezoelectric plate are investigated. Numerical results show that the nonlinearity of structure has a great influence on the postbuckling paths of the piezoelectric plate. The negative control voltage results in the increase of the buckling loads and the decrease of postbuckled deflections under the same in-plane compressive loads, whereas the positive control voltage decreases the buckling loads and induces larger postbuckled deflections. The buckling loads increase with the increase of the thick-span ratio of the plate, and the control voltage has a small effect on the postbuckling behaviors of the plate with lower thick-span ratio. When the damage and damage evolution are considered, the postbuckled deflection of the plate will gradually grow with the increase of the time until the damage reaches a characteristic damage state. The external in-plane compressive loads and the applied control voltage have great effects on the postbuckled deflections of the plate and the damage development. The negative control voltage can decrease the degradation rate of the stiffness of the piezoelectric structures and will provide a control mean for the damaged smart structures.

Partial support for this project provided by National Natural Science Foundation of China (51075204); Aeronautical Science Foundation of China (2012ZB52026); the Fundamental Research Funds for the Central Universities (NS2014024) is gratefully acknowledged.

Zhigang Sun carried out the theory and calculation. Xianqiao Wang provided the thought. Zhigang Sun wrote the paper. Xianqiao Wang reviewed and edited the manuscript. All authors read and approved the manuscript

The authors declare no conflict of interest.

Representative volume element with internal damage variables for piezoelectric materials.

Geometric configuration of a piezoelectric plate with transverse cracks under the uniform compressive in-plane loads.

Comparison of postbuckling response curves for isotropic rectangular plate with initial deflection (_{0} =0.1,

Response curves of the centre deflection

Comparisons of postbuckling response curves for initially flat and deflected piezoelectric plate without damage under different electrical loads.

Effect of electrical loads on postbuckling response curves of piezoelectric plate without damage under two different initial deflections.

Effect of initial geometric deflections on the postbuckling response curves of piezoelectric plate without damage.

Effect of thick-span ratio on postbuckling response curves of piezoelectric plate without damage under different electrical loads (the inset figure is a zoom-in snapshot of the region around orgin point to depict the difference of three cases).

Effect of aspect ratio on postbuckling response curves of piezoelectric plate without damage under different electrical loads.

Effect of external loads on postbuckling response curves of piezoelectric plate with damage and initial deflection.

Effect of initial deflections on the postbuckling response curves of piezoelectric plate with damage.

Effect of electrical loads on postbuckling response curves of piezoelectric plate with damage and initial deflection.

Effect of aspect ratio on postbuckling response curves of piezoelectric plate with damage and initial deflection.

Effect of thick-span ratio on postbuckling response curves of piezoelectric plate with damage and initial deflection.