تحلیل دقیق پاسخ-بسته ارتعاشات آزاد ورق های نسبتاً ضخیم مستطیلی ساخته شده از مواد هدفمند با لایه پیزوالکتریک

(1)

رﺎﺗ ﻳ ﻪﻟﺎﻘﻣ ﻪﭽﺨ :

رد ﻳ ﺖﻓﺎ 18 / 6 / 90

ﺬﭘ ﻳ شﺮ 10 / 7 / 90

ﺎﺳ رد ﻪﺋارا ﻳ ﺖ 11 / 7 / 90

ﻢﻴﺨﺿ

ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ

ﻖﻴﻗد ﻞﺣ ﻚﻳ ﻪﺋارا و ﺦﺳﺎﭘ

-

ياراد قرو ﻞﺑﺎﻘﻣ ي كﺮﺤﻣ ناﻮﻨﻋ ﻪﺑ ناﻮﺗ

ﻲ لﻮﺴﻛﺎﻣ

، ﺶﺷ

ﻐﺘﻣ يزﺎﺳاﺪﺟ شور زا ﺎﻫﺮﻴ

هدﺎﻔﺘﺳا

دﻮﺷ . ﻖﻴﻗد ﻞﺣ رد ﻂﻳاﺮﺷ ﺪﻨﻧﺎﻣ قرو

Exact Thick

Abstract-In this paper

of Mindlin plate theory. This structure has two opposite edges simply supported ( be used as a

properties vary continuously through the thickness of host equation, six complex coupled equations

potential and auxiliary functions as well as using separation of variables method verified by the available literature and

only the out of plane, but also the in boundary condition

Keywords: Mindlin theory

رﺎﺗ رد ﺬﭘ ﺎﺳ رد ﻪﺋارا

ًﺎﺘﺒﺴ ﻢﻴﺨﺿ

ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ

ناﺮﻬﺗ ،

ﻦﻴﻟﺪﻨﻴﻣ ﻖﻴﻗد ﻞﺣ ﻚﻳ ﻪﺋارا و

ﻪﺒﻟ ود ﻪﻛ ﺖﺳا نآ ﺮﺑ ضﺮﻓ ﻞﺣ ﻦﻳا رد ياراد قرو ﻞﺑﺎﻘﻣ ي

ﻲﻣ قرو ﻦﻳا زا ﻦﻳاﺮﺑﺎﻨﺑ ،ﺪﻧا كﺮﺤﻣ ناﻮﻨﻋ ﻪﺑ ناﻮﺗ

ﻜﻴﺗﺎﺘﺳاوﺮﺘﻜﻟا ﻪﻟدﺎﻌﻣ و نﻮﺘﻠﻴﻤﻫ ﻞﺻا زا هدﺎﻔﺘﺳا ﺎﺑ ﻲ

و ﻐﺘﻣ يزﺎﺳاﺪﺟ شور زا

وﺪﺤﻣ نﺎﻤﻟا ﻞﻴﻠﺤﺗ ﻲﻣ ﺪﻴﻳﺎﺗ د

دﻮﺷ

،ﻦﻴﻨﭽﻤﻫ .دﻮﺷ ﺎﻫﺮﺘﻣارﺎﭘ ﺮﻴﺛﺎﺗ

ي ﻂﻳاﺮﺷ ﺪﻨﻧﺎﻣ قرو

Exact closed

hick Rectangular

1- Assoc. Prof. of Mech 2- M.Sc. Student of Mech 3- P.h.d. Student of Mech

In this paper, free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis of Mindlin plate theory. This structure

has two opposite edges simply supported ( be used as an actuator. According to

properties vary continuously through the thickness of host six complex coupled equations

potential and auxiliary functions as well as using separation of variables method verified by the available literature and

only the out of plane, but also the in boundary conditions, gradient index

Mindlin theory,

قرو ﺎﻫ ي ﻧ

ًﺎﺘﺒﺴ

ﻪﻳﻻ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ

ناﺮﻳا ﺖﻌﻨﺻ و ﻢﻠﻋ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ هﺪﻜﺸﻧاد ،ﺪﺷرا ﻲﺳﺎﻨﺷرﺎﻛ يﻮﺠﺸﻧاد ناﺮﻬﺗ ،

ناﺮﻳا ﺖﻌﻨﺻ و ﻢﻠﻋ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ هﺪﻜﺸﻧاد ،اﺮﺘﻛد يﻮﺠﺸﻧاد ناﺮﻬﺗ ،

[email protected]

رﻮﺌﺗ سﺎﺳاﺮﺑ ﺪﻨﻤﺷﻮﻫ ي

ﻦﻴﻟﺪﻨﻴﻣ

ﻪﺒﻟ ود ﻪﻛ ﺖﺳا نآ ﺮﺑ ضﺮﻓ ﻞﺣ ﻦﻳا رد

هﺪﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧ رد ﻪﺘﺴﺑ راﺪﻣ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ﻲﻣ قرو ﻦﻳا زا ﻦﻳاﺮﺑﺎﻨﺑ ،ﺪﻧا

ﻜﻴﺗﺎﺘﺳاوﺮﺘﻜﻟا ﻪﻟدﺎﻌﻣ و نﻮﺘﻠﻴﻤﻫ ﻞﺻا زا هدﺎﻔﺘﺳا ﺎﺑ ﺪﻳﺪﺟ ﻞﻴﺴﻧﺎﺘﭘ و ﻓﺮﻌﻣ

ﻲ هﺪﺷ و ﻐﺘﻣ يزﺎﺳاﺪﺟ شور زا

ﻦﻴﻨﭽﻤﻫ وﺪﺤﻣ نﺎﻤﻟا ﻞﻴﻠﺤﺗ

ﻲﻣ ﺮ

،ﻦﻴﻨﭽﻤﻫ .دﻮﺷ راﺮﻗ ﻲﻣ دﺮﻴﮔ .

losed-form Rectangular

Bonded Piezoelectric Layers

Sh. Hosseini

Assoc. Prof. of Mech M.Sc. Student of Mech P.h.d. Student of Mech

*P.O.Box 16846

free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis of Mindlin plate theory. This structure

has two opposite edges simply supported ( actuator. According to

properties vary continuously through the thickness of host six complex coupled equations

potential and auxiliary functions as well as using separation of variables method verified by the available literature and the

only the out of plane, but also the in

gradient index and thickness of piezoelectric layers , Natural Frequency,

http://mechanics.journals.modares.ac.ir

57 - 74

دازآ قرو

ﺪﻨﻤﻓﺪﻫ داﻮﻣ زا هﺪﺷ ﻪﺘﺧﺎﺳ ﺎﺑ

ﻪﻳﻻ

ناﺮﻳا ﺖﻌﻨﺻ و ﻢﻠﻋ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ناﺮﻬﺗ ،

ناﺮﻳا ﺖﻌﻨﺻ و ﻢﻠﻋ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ هﺪﻜﺸﻧاد ،ﺪﺷرا ﻲﺳﺎﻨﺷرﺎﻛ يﻮﺠﺸﻧاد ناﺮﻳا ﺖﻌﻨﺻ و ﻢﻠﻋ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ هﺪﻜﺸﻧاد ،اﺮﺘﻛد يﻮﺠﺸﻧاد

16846 ، [email protected]

ﺎﺑ ﻪﻳﻻ يﺎﻫ رﻮﺌﺗ سﺎﺳاﺮﺑ ﺪﻨﻤﺷﻮﻫ

ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ﻣ

ﻲ ﺪﺷﺎﺑ . ﻪﺒﻟ ود ﻪﻛ ﺖﺳا نآ ﺮﺑ ضﺮﻓ ﻞﺣ ﻦﻳا رد

هﺪﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧ رد ﻪﺘﺴﺑ راﺪﻣ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ .دﻮﺷ

ﻜﻴﺗﺎﺘﺳاوﺮﺘﻜﻟا ﻪﻟدﺎﻌﻣ و نﻮﺘﻠﻴﻤﻫ ﻞﺻا زا هدﺎﻔﺘﺳا ﺎﺑ

ﻲﻜﻤﻛ ﻊﺑاﻮﺗ ﺪﻳﺪﺟ ﻞﻴﺴﻧﺎﺘﭘ و

و دﻮﺟﻮﻣ تﻻﺎﻘﻣ ﻦﻴﻨﭽﻤﻫ

ﻳﻻﺎﺑ رﺎﻴﺴﺑ ﺖﻗد ﺎﺑ ﻲ

ﻣ ﺴﻴ ﻲﻣ ﺮ

قرو ﺳرﺮﺑ درﻮﻣ ﻲ راﺮﻗ

form Free V

Rectangular Functionally Graded Bonded Piezoelectric Layers

Sh. Hosseini-Hashemi

Assoc. Prof. of Mech. Eng. Dept., Iran Univ M.Sc. Student of Mech. Eng. Dept., Iran Univ P.h.d. Student of Mech. Eng. Dept., Iran Univ P.O.Box 16846-13114, Tehran, Iran

free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis of Mindlin plate theory. This structure is composed of

has two opposite edges simply supported (i.e., Levy-type rectangular plates).

actuator. According to a power-law distribution of the volume fraction of the constituents, m properties vary continuously through the thickness of host

six complex coupled equations are introduced.

potential and auxiliary functions as well as using separation of variables method the finite element method

-plane modes of FG plate.

and thickness of piezoelectric layers requency, FGM, Power

http://mechanics.journals.modares.ac.ir ﻤﻠﻋ ﻪﻠﺠﻣ

ﻲ ﺸﻫوﮋﭘ ﻲ

هرﺎﻤﺷ 3 ﺰﻴﻳﺎﭘ 1390 ص ص 57

تﺎﺷﺎﻌﺗرا دازآ

ﺪﻨﻤﻓﺪﻫ داﻮﻣ زا هﺪﺷ ﻪﺘﺧﺎﺳ

*

ناﺮﻳا ﺖﻌﻨﺻ و ﻢﻠﻋ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ

ﻲﺘﺴﭘ قوﺪﻨﺻ ،ناﺮﻬﺗ 13114

- 16846 ،

ﻢﻴﺨﺿ ) ﺪﻨﻤﻓﺪﻫ ( FG ﺎﺑ

و ود ﻪﻳﻻ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ هﺪﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧ رد ﻪﺘﺴﺑ راﺪﻣ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ

ﻧاﻮﺗ ﻊﺑﺎﺗ لﺪﻣ ﻚ ﻲ

هدﺎﻔﺘﺳا ﻣ ﻲ .دﻮﺷ

،رﻮﻛﺬﻣ تﻻدﺎﻌﻣ ﻞﺣ رﻮﻈﻨﻣ ﻪﺑ ﻳ

ﺮﺴﻜ ي ﻲﻜﻤﻛ ﻊﺑاﻮﺗ

ﻪﺴﻳﺎﻘﻣ تﺎﻌﻟﺎﻄﻣ ﺎﺑ يا

و دﻮﺟﻮﻣ تﻻﺎﻘﻣ

- ﻪﺤﻔﺻ يا ﻳﻻﺎﺑ رﺎﻴﺴﺑ ﺖﻗد ﺎﺑ

ﺲﻧﺎﻛﺮﻓ يور يﺎﻫ

ﻲﻌﻴﺒﻃ قرو

ﻲﻧاﻮﺗ ﻊﺑﺎﺗ لﺪﻣ

.

Vibration

Functionally Graded Bonded Piezoelectric Layers

Hashemi^1*, H. Akhavan

Eng. Dept., Iran Univ.

Eng. Dept., Iran Univ Eng. Dept., Iran Univ 13114, Tehran, Iran

free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis is composed of a host FG plate and two bonded

type rectangular plates).

law distribution of the volume fraction of the constituents, m properties vary continuously through the thickness of host plate. Using H

are introduced. These equations potential and auxiliary functions as well as using separation of variables method

finite element method.

plane modes of FG plate.

and thickness of piezoelectric layers ower-Low Distribution

http://mechanics.journals.modares.ac.ir

هرود 11 هرﺎﻤﺷ

- ﻪﺘﺴﺑ تﺎﺷﺎﻌﺗرا

ﺪﻨﻤﻓﺪﻫ داﻮﻣ زا هﺪﺷ ﻪﺘﺧﺎﺳ

ﻲﺳﺪﻨﻬﻣ هﺪﻜﺸﻧاد ،رﺎﻴﺸﻧاد ناﺮﻳا ﺖﻌﻨﺻ و ﻢﻠﻋ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ

ﻲﺘﺴﭘ قوﺪﻨﺻ ،ناﺮﻬﺗ

ﻧ ﻲﻠﻴﻄﺘﺴﻣ قرو دازآ تﺎﺷﺎﻌﺗرا

ًﺎﺘﺒﺴ ﻢﻴﺨﺿ

زا ﻞﻜﺸﺘﻣ قرو ﻚﻳ وFG

.ﺪﻨﺷﺎﺑ ( ﻪﻳﻻ ﺎﻫ ي هﺪﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧ رد ﻪﺘﺴﺑ راﺪﻣ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ

زا ،FG ﻳ ﻧاﻮﺗ ﻊﺑﺎﺗ لﺪﻣ ﻚ

،رﻮﻛﺬﻣ تﻻدﺎﻌﻣ ﻞﺣ رﻮﻈﻨﻣ ﻪﺑ ﻳ

يﺮﺴﻜﻳ مﺎﺠﻧا ﻪﺴﻳﺎﻘﻣ تﺎﻌﻟﺎﻄﻣ

ﺮﺑ هوﻼﻋ يﺎﻫﺪﻣ جرﺎﺧ -

ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ﺮﺑ

ﺲﻧﺎﻛﺮﻓ يور

، FGM ﻲﻧاﻮﺗ ﻊﺑﺎﺗ لﺪﻣ

ibration Analysis of Functionally Graded

Bonded Piezoelectric Layers

, H. Akhavan²,M. Fadaee

of Science and Tech Eng. Dept., Iran Univ. of Science and Tech Eng. Dept., Iran Univ. of Science and Tech 13114, Tehran, Iran. [email protected]

free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis a host FG plate and two bonded

type rectangular plates). The closed circuit piezoelectric layers law distribution of the volume fraction of the constituents, m

plate. Using Hamilton's principle and These equations are

potential and auxiliary functions as well as using separation of variables method

The present exact solution can accurately predict not plane modes of FG plate. Finally, the effects of

and thickness of piezoelectric layers on the natural frequency istribution.

ﻖﻴﻗد ﻞ ﺦﺳﺎﭘ

-

ﺪﻨﻤﻓﺪﻫ داﻮﻣ زا هﺪﺷ ﻪﺘﺧﺎﺳ

ﻲﺳﺪﻨﻬﻣ هﺪﻜﺸﻧاد ،رﺎﻴﺸﻧاد ناﺮﻳا ﺖﻌﻨﺻ و ﻢﻠﻋ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ هﺪﻜﺸﻧاد ،ﺪﺷرا ﻲﺳﺎﻨﺷرﺎﻛ يﻮﺠﺸﻧاد ناﺮﻳا ﺖﻌﻨﺻ و ﻢﻠﻋ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ هﺪﻜﺸﻧاد ،اﺮﺘﻛد يﻮﺠﺸﻧاد

*

ﻲﺘﺴﭘ قوﺪﻨﺻ ،ناﺮﻬﺗ

ﻧ ﻲﻠﻴﻄﺘﺴﻣ قرو دازآ تﺎﺷﺎﻌﺗرا رﺎﺘﺧﺎﺳ ﻦﻳا زا ﻞﻜﺸﺘﻣ

ﻲﻠﻴﻄﺘﺴﻣ قرو ﻲﻨﻌﻳ) هدﺎﺳ يزﺮﻣ طﺮﺷ Levy-type

.ﺪﻨﺷﺎﺑ (

هدﺎﻣ رد صاﻮﺧ ﻊﻳزﻮﺗ FG

ﺪﻳآ . ،رﻮﻛﺬﻣ تﻻدﺎﻌﻣ ﻞﺣ رﻮﻈﻨﻣ ﻪﺑ

هﺪﻣآ ﺖﺳﺪﺑ ي ﺑ

ﻪ ﻪﻠﻴﺳو يﺮﺴﻜﻳ مﺎﺠﻧا

- ﻪﺤﻔﺻ يا ﺰﻴﻧ ﺮﺑ هوﻼﻋ

ﺖﻣﺎﺨﺿ ﺶﻳاﺰﻓا و ﻪﻳﻻ

ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ

ﺲﻧﺎﻛﺮﻓ ﻲﻌﻴﺒﻃ يﺎﻫ ،

FGM

nalysis of Functionally Graded Plate Bonded Piezoelectric Layers

M. Fadaee³

of Science and Tech.Tehran, of Science and Tech. Tehran of Science and Tech.Tehran

[email protected]

free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis a host FG plate and two bonded piezoelectric

The closed circuit piezoelectric layers law distribution of the volume fraction of the constituents, m

milton's principle and are exactly solved

potential and auxiliary functions as well as using separation of variables method. The accuracy of the frequencies is he present exact solution can accurately predict not

he effects of various on the natural frequency

ﻴﻠﺤﺗ ﻖﻴﻗد ﻞ

ﻠﻴﻄﺘﺴﻣ ﻲ

1 -

2 -

3 -

ﻦﻳا رد ﻘﻣ ،ﻪﻟﺎ ﻞﻴﻠﺤﺗ ﻧ ﻲﻠﻴﻄﺘﺴﻣ قرو دازآ تﺎﺷﺎﻌﺗرا

ﺖﺳا ﻪﺘﻓﺮﮔ راﺮﻗ .

رﺎﺘﺧﺎﺳ ﻦﻳا

ﻲﻠﻴﻄﺘﺴﻣ قرو ﻲﻨﻌﻳ) هدﺎﺳ يزﺮﻣ طﺮﺷ ﺑﻴ

هﻮﺤﻧ نﺎ هدﺎﻣ رد صاﻮﺧ ﻊﻳزﻮﺗ

ﻪﺘﺴﺑاو ﻞ ﻣ ﺖﺳﺪﺑ ﻲ ﺪﻳآ

ﺲﻧﺎﻛﺮﻓ ﺖﻗد ﺎﻫ

هﺪﻣآ ﺖﺳﺪﺑ ي ﻲﻨﻴﺑ نورد يﺎﻫﺪﻣ -

ﻧﺎﻳداﺮﮔ ﺲﻳﺪﻧا ، ﻲ

ﺖﻣﺎﺨﺿ ﺶﻳاﺰﻓا و ﻦﻴﻟﺪﻨﻴﻣ يرﻮﺌﺗ

، ﺲﻧﺎﻛﺮﻓ

nalysis of Moderately Plates with

Iran Tehran, Iran .Tehran, Iran

free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis piezoelectric layers

The closed circuit piezoelectric layers law distribution of the volume fraction of the constituents, m

milton's principle and Maxwell electrostatic solved introducing the new e accuracy of the frequencies is he present exact solution can accurately predict not

various parame on the natural frequency are investigated.

ﻴﻠﺤﺗ ﻠﻴﻄﺘﺴﻣ

هﺪﻴﻜﭼ - ﻦﻳا رد

،ﻪﺘﺴﺑ ﺳرﺮﺑ درﻮﻣ ﻲ

ﻲﻠﻴﻄﺘﺴﻣ قرو ﻲﻨﻌﻳ) هدﺎﺳ يزﺮﻣ طﺮﺷ .دﺮﻛ هدﺎﻔﺘﺳا اﺮﺑ

ي ﺑ

ﻪﻟدﺎﻌﻣ د ﻳ ﺴﻧاﺮﻔ ﻴ ﻪﺘﺴﺑاو ﻞ

ﻣ ﻲ .دﻮﺷ ﺲﻧﺎﻛﺮﻓ ﺖﻗد

ﺮﺿﺎﺣ ﺶﻴﭘ نﺎﻜﻣا ﻲﻨﻴﺑ

زﺮﻣ ي ﻧﺎﻳداﺮﮔ ﺲﻳﺪﻧا ،

ﺪﻴﻠﻛ :نﺎﮔژاو ﻦﻴﻟﺪﻨﻴﻣ يرﻮﺌﺗ

oderately ith Two

free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis layers. The plate The closed circuit piezoelectric layers can law distribution of the volume fraction of the constituents, material Maxwell electrostatic introducing the new e accuracy of the frequencies is he present exact solution can accurately predict not parameters such as are investigated.

هﺪﻴﻜﭼ ،ﻪﺘﺴﺑ ﻲﻠﻴﻄﺘﺴﻣ قرو ﻲﻨﻌﻳ) هدﺎﺳ يزﺮﻣ طﺮﺷ .دﺮﻛ هدﺎﻔﺘﺳا ﻪﻟدﺎﻌﻣ ﻣ ﺮﺿﺎﺣ زﺮﻣ ﺪﻴﻠﻛ

free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis he plate can aterial Maxwell electrostatic introducing the new e accuracy of the frequencies is he present exact solution can accurately predict not such as

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(3)

ﺦﺳﺎﭘ ﻖﻴﻗد ﻞﻴﻠﺤﺗ -

ﻪﺘﺴﺑ تﺎﺷﺎﻌﺗرا دازآ قرو

ًﺎﺘﺒﺴﻧ يﺎﻫ ...

ﻲﻤﺷﺎﻫ ﻲﻨﻴﺴﺣ خﺮﻫﺎﺷ نارﺎﻜﻤﻫ و

58

1 - ﻪﻣﺪﻘﻣ

داﻮﻣ ﻫ ﻨﻤﻓﺪ

1ﺪ ًﺎﺣﻼﻄـﺻا ﻪـﻛ ﻲـﻣ هﺪـﻴﻣﺎﻧFGM

ﺪﻧﻮـﺷ ﻞـﺴﻧ

ﺪﻳﺪﺟ ي ﺎﻫرﺎﺘﺧﺎﺳ زا ي

ﺘﻳزﻮﭙﻣﺎﻛ ﻲ ﻂﺳﻮﺗ رﺎﺑ ﻦﻴﻟوا ﻪﻛ ﺪﻨﺘﺴﻫ

ناﺪﻨﻤﺸﻧاد زا هوﺮﮔ ﻚﻳ ﻨﭘاژ

ﻲ لﺎﺳ رد 1984 ـﻓﺮﻌﻣ ﻲ .ﺪﻧﺪـﺷ

داﻮــﻣ داﻮــﻣ ،FGM

ي ﻦﮕﻤﻫﺎــﻧ ــﻟو2

ﻲ .ﺪﻨﺘــﺴﻫ پوﺮــﺗوﺰﻳا

ﺑ داﻮﻣ ﻦﻳا رﺎﺘﺧﺎﺳﺰﻳر ﻪ

ﻪﻧﻮﮔ ا ي ـﺑ ﻪـﻛ ﺖﺳا ﻪ

و ﻪﺘـﺳﻮﻴﭘ ترﻮـﺻ

ﺠﻳرﺪﺗ ﻲ ﺎﺘـﺳار رد هدﺎـﻣ ﻚﻳ زا ي

ﻪـﺑ ﺺﺨـﺸﻣ هدﺎـﻣ

ا ي ًﻼﻣﺎـﻛ

ﻣ ﻞﻴﻣ توﺎﻔﺘﻣ ﻲ

ﻪـﺴﻳﺎﻘﻣ رد صاﻮﺧ ﻪﺘﺳﻮﻴﭘ تاﺮﻴﻴﻐﺗ ﻦﻳا .ﺪﻨﻛ

صاﻮﺧ ﻪﺘﺴﺴﮔ تاﺮﻴﻴﻐﺗ ﺎﺑ ﻧﺎـﻬﮔﺎﻧ ﺮﻴﻴﻐﺗ ﻞﻜﺸﻣ ،ﺐﻛﺮﻣ داﻮﻣ

ﻲ

ﻊـﻳزﻮﺗ و هدﺮـﻛ فﺮـﻃﺮﺑ ار نﻮﮔﺎـﻧﻮﮔ هدﺎـﻣ ود سﺎﻤﺗ ﺢﻄﺳ رد ﻣ ﺐﺒﺳ ار ﻪﺘﺳﻮﻴﭘ ﺶﻨﺗ ﻲ

دﻮﺷ 2] 1, .[

ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ داﻮـﻣ ﺎـﻫ

ي اراد ﻪـﻛ ﺪﻨﺘـﺴﻫ ي

ـﺻاﻮﺧ ﻲ ﻦﻴـﺑ ،

ﻜﻳﺮﺘﻜﻟا صاﻮﺧ ﻲ

ﻜﻴﺘﺳﻻا و ﻲ

ﻲﻣ هدﺎﻔﺘـﺳا ﻪـﺑ ﻪـﺟﻮﺗ ﺎﺑ .ﺪﻨﺷﺎﺑ

هدﺮﺘﺴﮔ ا ي ﺎﻫﺮﮕـﺴﺣ رد داﻮـﻣ ﻦـﻳا ﻪﻛ كﺮـﺤﻣ و3

4 ﺎـﻫ ،ﺪـﻧراد

ور ﺮﺑ ﻪﻌﻟﺎﻄﻣ ي

قرو اراد ﻪﻛ ﻲﻳﺎﻫ ي

ﻪﻳﻻ ﻣ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ﻲ

ﻨﺷﺎﺑ ﺪ

ﻪﺟﻮﺗ رﺎﻴﺴﺑ ي ﻪﺑ ار ﻦﻴﻘﻘﺤﻣ زا .ﺖﺳا ﻪﺘﺧﺎﺳ فﻮﻄﻌﻣ دﻮﺧ

ﺖـﻳزﻮﭙﻣﺎﻛ ﺎﺑ هاﺮﻤﻫ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ داﻮﻣ زا هدﺎﻔﺘﺳا هزوﺮﻣا ﺎـﻫ

ﺎﻫدﺮﺑرﺎﻛ ي ﻔﻠﺘﺨﻣ ﻲ ﺎـﻫدﺮﺑرﺎﻛ ﻪﻠﻤﺟ زا .ﺪﻧراد ي

رد داﻮـﻣ ﻦـﻳا

ﺎﻫرﺎﺘﺧﺎــﺳ و تﺎــﺷﺎﻌﺗرا لﺮــﺘﻨﻛ ي

ﺪﻨﻤــﺷﻮﻫ ــﻣ5

ﻲ ﺪــﺷﺎﺑ 3] .[

كﺮﺤﻣ هزﺎﺳ ﻪﺑ ﺎﻫﺮﮕﺴﺣ و ﺎﻫ ﺎﻫ

ي ﻠﺻا ﻲ ﻪﻛ

ًﺎﺣﻼﻄﺻا نآ ﻪﺑ

ﺎـﻫ

ﻣ ﻪﺘﻔﮔhost ﻲ ﻣ ﻞﺼﺘﻣ ،دﻮﺷ ﻲ

نآ لﺎﺼﺗا و ﺪﻧﻮﺷ ﻞﻜﺷ ﻪﺑ ﺎﻫ

ﺎﻫ

مﺮﻓ و ﺎﻫ ي ﻔﻠﺘﺨﻣ ﻲ ﻣ ترﻮﺻ ﻲ ـﻄﻴﺤﻣ ﻂﻳاﺮـﺷ ﻪـﺑ ﻪﻛ دﺮﻴﮔ ﻲ

و

ﺎﻫزﺎﻴﻧ ي ﺗﺎﻴﻠﻤﻋ ﻲ ﮕﺘﺴﺑ ﻒﻠﺘﺨﻣ ﻲ

دراد 4] .[

رﻮﺌﺗ ي قرو ﻚﻴﺳﻼﻛ ـﺷﺮﺑ ﻞﻜـﺷ ﺮﻴﻴﻐﺗ ﺮﻴﺛﺎﺗ6

ﻲ ـﺳﺮﻨﻳا و ﻲ

ﻧارود ﻲ ﻤﻧ ﺮﻈﻧ رد ار ﻲ

اﺮـﺑ ﻂـﻘﻓ نآ زا ﻞـﺻﺎﺣ ﺞﻳﺎـﺘﻧ و دﺮﻴﮔ ي

قرو ﺎﻫ ي ندﺮـﻛ فﺮـﻃﺮﺑ رﻮـﻈﻨﻣ ﻪـﺑ .ﺖـﺳا لﻮـﺒﻗ ﻞﺑﺎﻗ كزﺎﻧ

ﺘﺳﺎﻛ ﻲ ﺎﻫ ي رﻮﺌﺗ ﻪﺑ طﻮﺑﺮﻣ ي

اﺮﺑ ﻚﻴﺳﻼﻛ ي

،ﻢﻴﺨﺿ ًﺎﺘﺒﺴﻧ قرو

رﻮﺌﺗ ي لوا ﻪﺒﺗﺮﻣ ﻲﺷﺮﺑ ﻞﻜﺷ ﺮﻴﻴﻐﺗ )7

ﺮﻴﻴﻐﺗ ﺮﻴﺛﺎﺗ ﻪﻛ (FSDT

1. Functionally Graded Materials 2. Inhomogeneous

3. Sensors 4. Actuators 5. Smart

6. Classical plate Thoery

7. First-Order shear Deformation Plate Theory

ﺷﺮﺑ ﻞﻜﺷ ﻲ

ﺳﺮﻨﻳا و ﻲ ﻧارود ﻲ ﻣ ﺮﻈﻧ رد ﺰﻴﻧ ار ﻲ

دﺮﻴﮔ ﻪﻠﻴـﺳﻮﺑ ،

ﻦﻴﻟﺪﻨﻴﻣ ﺪﺷ دﺎﻬﻨﺸﻴﭘ ﺶﻧارﺎﻜﻤﻫ و8

5] . [ رﻮﺌﺗ ﺎﺑ ﻪﺴﻳﺎﻘﻣ رد ي

-

ﺎﻫ ي رﻮﺌﺗ ﺮﺗﻻﺎﺑ ﻪﺒﺗﺮﻣ ي

ﮔﺪﻴﭽﻴﭘ و ﻢﺠﺣ ﻦﻴﻟﺪﻨﻴﻣ ﻲ

ﺗﺎﺒﺳﺎﺤﻣ ﻲ

ﺦـﺳﺎﭘ لﺎـﺣ ﻦﻴﻋ رد و دراد يﺮﺘﻤﻛ رﺎﻴﺴﺑ ﺎـﻫ

ي نآ زا ﻞـﺻﺎﺣ

ﻲﻣ ﻦﻴﻣﺎﺗ ار زﺎﻴﻧ درﻮﻣ ﺖﻗد .ﺪﻨﻨﻛ

ﻲﺳرﺮﺑ نﻮﻨﻛﺎﺗ يﺎﻫ

قرو يور يدﺪﻌﺘﻣ ﻲﻠﻴﻄﺘﺴﻣ يﺎﻫ

،FG

،ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ

9FGPM

قرو و ﺎﺑ FG

ﻪﻳﻻ ﻚـﻳﺮﺘﻜﻟاوﺰﻴﭘ يﺎـﻫ

ﺖﺳا هﺪﺷ مﺎﺠﻧا .

اﺮﻳﺮﻓ نارﺎﻜﻤﻫ و10

6] ،[ ﺲﻧﺎﻛﺮﻓ ﺎﻫ ي ـﻌﻴﺒﻃ ﻲ

ﻌﺑﺮﻣ قرو ﻲ ار FG اﺮﺑ ي زﺮﻣ ﻂﻳاﺮﺷ ي

.ﺪـﻧدروآ ﺖﺳﺪﺑ ﻒﻠﺘﺨﻣ

ﻞﻴﻠﺤﺗ ﻌﺑﺮﻣ قرو دازآ تﺎﺷﺎﻌﺗرا ﻲ

ﺎـﺑ ﻂﻳاﺮـﺷ يزﺮـﻣ ﺎـﺑ هدﺎـﺳ

ﻻﺎﺑ ﻪﺒﺗﺮﻣ دوﺪﺤﻣ نﺎﻤﻟا شور ﻚﻳ زا هدﺎﻔﺘﺳا و ﺎﻨﻣﺎﻳداﺮﭘ ﻂﺳﻮﺗ

ﺎﻳدﺎﭘﻮﻳﺪــﻧﺎﺑ

11ي 7] .ﺖــﻓﺮﮔ راﺮــﻗ ﻪــﻌﻟﺎﻄﻣ درﻮــﻣ[ ﻨﻴــﺴﺣ

ﻲ

ﻲﻤــﺷﺎﻫ نارﺎــﻜﻤﻫ و12

9] 8,

، [ ﻞــﻴﻠﺤﺗ ﻖــﻴﻗد ﺦــﺳﺎﭘ - ﻪﺘــﺴﺑ

ﻧ قرو دازآ تﺎﺷﺎﻌﺗرا

ًﺎﺘﺒﺴ ﻠﻴﻄﺘـﺴﻣ ﻢﻴﺨـﺿ ﻲ

ﺮﺑ ارFG ﺎـﻨﺒﻣ ي

رﻮﺌﺗ ي ﻦﻴﻟﺪﻨﻴﻣ در و

13ي اﺮﺑ ي عﻮﻧ ﺶﺷ زﺮﻣ طﺮﺷ

ي ﻒـﻠﺘﺨﻣ

.ﺪـﻧداد راﺮـﻗ ﻲـﺳرﺮﺑ درﻮﻣ اﺮﺗﺎـﺑ و ﮓـﻨﭼ

]14

10 و ﺶﻧﺎـﻤﻛ ،[

راﺪـﻳﺎﭘ ﺖﻟﺎﺣ تﺎﺷﺎﻌﺗرا قرو15

ﻌﻠـﺿ ﺪـﻨﭼFG

16ﻲ ﺎـﺑ ﻂﻳاﺮـﺷ

يزﺮﻣ هدﺎﺳ ار ﻪﺋارا .ﺪﻧداد ﺪـﻌﺑ ﻪﺳ ﻖﻴﻗد ﻞﻴﻠﺤﺗ ي

ﺲﻧﺎـﻛﺮﻓ -

ﺎﻫ ي ﻌﻴﺒﻃ ﻲ ﺎﻫدﻮﻣ ﻞﻜﺷ و ي

ﻠﻴﻄﺘـﺴﻣ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ قرو ﻲ

ﺎـﺑ

يزﺮﻣ ﻂﻳاﺮﺷ هدﺎﺳ

اﺮﺑ ي عﻮﻧ ود زﺮﻣ طﺮﺷ ي ﻜﻳﺮﺘﻜﻟا ﻲ زﺎﺑ راﺪﻣ

ﻪﺘﺴﺑ راﺪﻣ و لﺎﻴﭘﻮﻴﻛ ﻂﺳﻮﺗ

]17

11 ﺳرﺮﺑ درﻮﻣ[ ﻲ

راﺮﻗ ﺖـﻓﺮﮔ .

ﻦــــﭼ ﻫ و18

ﻤ ﺶﻧارﺎــــﻜ 12]

قرو ﻚــــﻳ دازآ تﺎــــﺷﺎﻌﺗرا ،[

ﻠﻴﻄﺘﺴﻣ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ﻲ

ﺪﻨﻤﻓﺪﻫ ًﺎﺣﻼﻄﺻا ﻪﻛ هﺪـﻴﻣﺎﻧFGPM

ﻲﻣ دﻮﺷ ﺎﺑ ار ﺎﻀﻓ شور زا هدﺎﻔﺘﺳا ي

19ﺖﻟﺎﺣ راﺮﻗ ﻪﻌﻟﺎﻄﻣ درﻮﻣ

قرو و ﺪﻧداد ـﺒﻳﺮﻘﺗ رﻮﻄﺑ ارFGPM

ﻲ ترﻮـﺼﺑ رﺎﺘﺧﺎـﺳ ﻚـﻳ

8. Mindlin

9. Functionally Graded Piezoelectric Materials 10. Ferreira

11. Pradyumna and Bandyopadhyay 12. Hosseini-Hashemi

13. Reddy 14. Cheng and Batra 15. Steady state vibrations 16. Polygonal

17. Cupial 17. Chen 18. State space 19. Zhong and Yu

(4)

59

ﻪﻳﻻ ا ي اﺮﺑ ار ﻖﻴﻗد ﻞﺣ و ﺪﻧدﺮﻛ لﺪﻣ ي

ﻒـﻠﺘﺨﻣ يزﺮﻣ ﻂﻳاﺮﺷ

.ﺪﻧدروآ ﺖﺳﺪﺑ ﻮﻳ و ﮓﻧاژ

]1

13 اﺮـﺑ يﺪـﻌﺑ ﻪـﺳ ﻖﻴﻗد ﻞﺣ ،[ ي

رﺎــﺒﺟا و دازآ تﺎــﺷﺎﻌﺗرا ﻞــﻴﻠﺤﺗ ي

ﻚــﻳﺮﺘﻜﻟاوﺰﻴﭘ قرو ﻚــﻳ

ﻠﻴﻄﺘﺴﻣFGPM ﻲ ﺎـﺑ ﻂﻳاﺮـﺷ يزﺮـﻣ هدﺎـﺳ .ﺪـﻧدﺮﻛ ﻪـﺋارا ار

ـﻜﻳﺮﺘﻜﻟا ﻞﻴﺴﻧﺎﺘﭘ و ﻲﻳﺎﺠﺑﺎﺟ ﻲ

قرو ار ﻪـﺑ ﺮـﺳ ﻞﻜـﺷ ي

ﻪـﻳرﻮﻓ

.ﺪﻧدﺮﻛ جاﺮﺨﺘﺳا ار ﺖﻟﺎﺣ تﻻدﺎﻌﻣ ﺲﭙﺳ و هداد ﻂﺴﺑ

در ي ﮓﻨﭼ و 14]

،[ ﺪﻨﻤﺷﻮﻫ قرو ﻚﻳ ﻞﻣﺎﺷFG

قرو ﻚﻳ

داﻮﻣ زا هﺪﺷ ﻪﺘﺧﺎﺳ ﺪﻨﻤﻓﺪﻫ

، ﻪﻛ ﻪﻳﻻ ﻪـﺑ ﻚـﻳﺮﺘﻜﻟاوﺰﻴﭘ ناﻮـﻨﻋ

كﺮﺤﻣ ﻂﻘﻓ ور ي ﻨﻏ ﺢﻄﺳ ﻲ ﺰﻠﻓ زا هﺪﺷ نآ2

هﺪﺷ ﺐﺼﻧ ﺖـﺳا

ﺪﻧداد راﺮﻗ ﻪﻌﻟﺎﻄﻣ درﻮﻣ ار ﻪـﻛ ﻲﻣﺎـﮕﻨﻫ .

ـﻨﻏ ﺢﻄـﺳ ﻲ

زا هﺪـﺷ

راﺬــﮔرﺎﺑ ضﺮــﻌﻣ رد ﻚﻴﻣاﺮــﺳ ي

ﻜﻴﻧﺎــﻜﻣﻮﻣﺮﺗ ﻲ

راﺮــﻗ ﺖــﻓﺮﮔ

،

ﺶﻨﺗ و ﻲﻳﺎﺠﺑﺎﺟ ار

دروآرد لﺮﺘﻨﻛ ﺖﺤﺗ ﺪﻧ

ار تﺎﺷﺎﻌﺗرا ﻪﻨﻣاد و

ـﻜﻳﺮﺘﻜﻟا ﻊـﺒﻨﻣ ﺎـﺑ كﺮﺤﻣ ﻪﻠﻴﺳﻮﺑ ﻲ

ـﺜﻨﺧ ﻲ .ﺪـﻧدﺮﻛ و يرﺎـﺨﻓ

نارﺎﻜﻤﻫ 15]

، [ تﺎﺷﺎﻌﺗرا ﻲﻄﺧﺮﻴﻏ

يرﺎﺒﺟا و دازآ قرو

ﺎﺑFG

ﻚــﻳﺮﺘﻜﻟاوﺰﻴﭘ ﻪــﻳﻻ ود و ﻲﻜﻴﻧﺎــﻜﻣ ،ﻲــﺗراﺮﺣ يﺎــﻫرﺎﺑ ﺖــﺤﺗ

ﻲﻜﻳﺮﺘﻜﻟا ار

ﺶﻧﺮﻛ ﻂﺑاور زا هدﺎﻔﺘﺳا ﺎﺑ -

ﻲـﻄﺧ ﺮـﻴﻏ ﻲﻳﺎـﺠﺑﺎﺟ

ﻦﻣرﺎﻛ نﻮﻓ دوﺪﺤﻣ نﺎﻤﻟا شور و3

.ﺪﻧداد راﺮﻗ ﻲﺳرﺮﺑ درﻮﻣ

ﻪﻟﺎﻘﻣ ﻦﻳا رد دازآ تﺎﺷﺎﻌﺗرا ﻞﻴﻠﺤﺗ ياﺮﺑ ﻖﻴﻗد شور ﻚﻳ

ﻲﻠﻴﻄﺘﺴﻣ ﻦﻴﻟﺪﻨﻴﻣ قرو ﺎﺑFG

ﻪﻳﻻ ﺎﻫ ي ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ياﺮـﺑ

يﻮﻟ ًﺎﺣﻼﻄﺻا ﻪﻛ ﻒﻠﺘﺨﻣ يزﺮﻣ طﺮﺷ عﻮﻧ ﺶﺷ -

4ﻪﻧﻮﮔ ﻪﺘﻔﮔ

ﻣ ﻲ

،دﻮﺷ ﺪﺷ ﻪﺋارا ﺖﺳا ه . ﻦﻴﻨﭽﻤﻫ ﻞﻜـﺷ ﺮـﻴﻴﻐﺗ ﺮﻴﺛﺎـﺗ

يﺎـﻫ

ـﻧارود ﻲـﺳﺮﻨﻳا ﺮـﺛا و ﻲﺷﺮﺑ ﻲ

.ﺖـﺳا هﺪـﺷ ﻪـﺘﻓﺮﮔ ﺮـﻈﻧ رد

ﻪﻳﻻ ﺎﻫ ي ﺪـﺷ ﻪـﺘﻓﺮﮔ ﺮـﻈﻧ رد ﻪﺘﺴﺑ راﺪﻣ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ه

ﺪـﻧا ،

ـﻣ ﻞـﻤﻋ كﺮـﺤﻣ ﻚﻳ ناﻮﻨﻋ ﻪﺑ رﺎﺘﺧﺎﺳ ﻦﻳا ﻦﻳاﺮﺑﺎﻨﺑ ﻲ

.ﺪـﻨﻛ

ﻜﻴﺗﺎﺘـﺳاوﺮﺘﻜﻟا تﻻدﺎـﻌﻣ و نﻮﺘﻠﻴﻤﻫ ﻞﺻا ﻲ

اﺮـﺑ لﻮـﺴﻛﺎﻣ ي

تﻻدﺎﻌﻣ جاﺮﺨﺘﺳا ﻪﻠﺌﺴﻣﺮﺑ ﻢﻛﺎﺣ

، ﺪﺷ ﻪﺘﻓﺮﮔ رﺎﻜﺑ ﺖﺳا ه

. رد

،يزﺮــﻣ ﻂﻳاﺮــﺷ ﺮــﻴﻈﻧ قرو يﺎــﻫﺮﺘﻣارﺎﭘ ﺮﻴﺛﺎــﺗ ﺰــﻴﻧ نﺎــﻳﺎﭘ ﺖﻣﺎﺨﺿ ﻧﺎﻳداﺮﮔ ﺲﻳﺪﻧا و ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ﻪﻳﻻ

ﻲ درﻮﻣ ﻲـﺳرﺮﺑ

ﺪﺷ هداد راﺮﻗ ﺖﺳا ه

. رد ،ﺮـﺿﺎﺣ ﻪـﻟﺎﻘﻣ ﺖﺒﺜﻣ تﺎﻜﻧ ﻪﻠﻤﺟ زا

1. Metal-rich 2. Ceramic-rich 3. Von karman 4. Levy-type

ﻦﺘﻓﺮﮔ ﺮﻈﻧ ﻲﮕﺘـﺴﺑاو

نورد تﺎـﺷﺎﻌﺗرا ﻦﻴـﺑ5

- ﻪﺤﻔـﺻ ا

6ي و

جرﺎﺧ - ﻪﺤﻔﺻ ا

7ي ـﻣ ﻲ ﺲﻧﺎـﻛﺮﻓ ﻦﻳاﺮﺑﺎـﻨﺑ ،ﺪـﺷﺎﺑ نورد يﺎـﻫ

-

ﻪﺤﻔﺻ هزاﺪﻧا ﻞﺑﺎﻗ ﺰﻴﻧ يا ﻲﻣ يﺮﻴﮔ

ﺳرﺮﺑ ﻦﻴﻨﭽﻤﻫ .ﺪﻨﺷﺎﺑ ﻲ

ﺎﻫ

هداد نﺎﺸﻧ ﻦـﻳا ﻦﺘﻓﺮـﮕﻧ ﺮﻈﻧ رد ﻪﻛ ﺪﻧا

ﻲﮕﺘـﺴﺑاو ﺎـﻫﺎﻄﺧ

ي

ﻢﺸﭼ ﺮﻴﮔ ي ﻣ دﺎﺠﻳا ﻲ .ﺪﻨﻛ ﻞـﺣ ﺞﻳﺎـﺘﻧ ندروآ ﺖـﺳﺪﺑ ياﺮـﺑ

ﻖﻴﻗد

، راﺰﻓا مﺮﻧ رد ﻲﺴﻳﻮﻧﺪﻛ Mathematica

و هﺪـﺷ مﺎﺠﻧا

ﺖﺳﺪﺑ ﺞﻳﺎﺘﻧ آ

راﺰـﻓا مﺮﻧ زا ﻞﺻﺎﺣ ﺞﻳﺎﺘﻧ ﺎﺑ هﺪﻣ ABAQUS

ﺖﺳا هﺪﺷ ﻪﺴﻳﺎﻘﻣ و

و ﺖﻗد ﺖﺤﺻ ﺮﺿﺎﺣ ﻞﺣ ﺪـﻴﻳﺎﺗ درﻮـﻣ

ﺖﺳا هﺪﺷ ﻪﺘﻓﺮﮔ راﺮﻗ .

2 - ﻪﻠﺌﺴﻣ ﺮﺑ ﻢﻛﺎﺣ تﻻدﺎﻌﻣ

2 - 1 - ﺳﺪﻨﻫ ﺶﻳارآ ﻲ

قرو ﻚﻳ ﻠﻴﻄﺘﺴﻣFG

ﻲ

ًﺎﺘﺒﺴﻧ و ﻢﻴﺨﺿ ﺢﻄﺴﻣ

لﻮـﻃ ﻪـﺑ وa

ضﺮﻋ ﻪﻛ ،b ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ﻪﻳﻻ ود نﺎﺴﻜﻳ

و ﻻﺎـﺑ رد ﻦﻴﻳﺎـﭘ

ﻪـﺑ

.ﺪﻳﺮﻴﮕﺑ ﺮﻈﻧ رد ار هﺪﺷ ﻞﺼﺘﻣ نآ ﻞﻜـﺷ رد ﻪـﻛ رﻮﻄﻧﺎﻤﻫ

) 1 (

،هﺪــﺷ هداد نﺎــﺸﻧ قرو ﺖﻣﺎﺨــﺿ

،FG 2h ﻪــﻳﻻ ﺖﻣﺎﺨــﺿ و

ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ hp

ﻲﻣ .ﺪﺷﺎﺑ ﻪﺒﻟ ود ي داﺪـﺘﻣا رد ﻪـﻛ قرو ﻞﺑﺎﻘﻣ

رﻮﺤﻣ x2

ﻨﻌﻳ) دراد راﺮﻗ ﻲ

ﻪـﺒﻟ داﺪـﺘﻣا رد ﺎـﻫ

ي x1=0 و x1=a (

اراد هراﻮﻤﻫ ي

ﻂﻳاﺮﺷ يزﺮﻣ ـﻣ هدﺎﺳ ﻲ ـﺷﺎﺑ ﻨ ،ﺪ ﻟﺎـﺣرد ﻲ ود ﻪـﻛ

ﻪﺒﻟ ي ﻲﻣ ﺮﮕﻳد ﻧاﻮﺗ ﻨ

،دازآ ﺪ ﻢﺘـﺴﻴﺳ .ﺪـﺷﺎﺑ رادﺮـﻴﮔ ﺎـﻳ و هدﺎﺳ

) ﻦﻳﺰﺗرﺎﻛ تﺎﺼﺘﺨﻣ x1,x2,x3

( اﺮﺑ ي ـﺿﺎﻳر تﻻدﺎﻌﻣ جاﺮﺨﺘﺳا ﻲ

ﺑ ،ﺖﺳا هﺪﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧ رد ﻪ

ﻪﻜﻳرﻮﻃ ﻚـﻳ رد تﺎـﺼﺘﺨﻣ اﺪـﺒﻣ

و هﺪﺷ ﻊﻗاو قرو ﻪﺷﻮﮔ يﺎـﻫرﻮﺤﻣ

x₁,x₂ ﻪﺤﻔـﺻ رد ي

ﻧﺎـﻴﻣ ﻲ

هﺪﺷ ﻊﻗاو قرو ﺪﻧا

. و ﻲﻳﻻﺎﺑ ﺢﻄﺳ ﻨﻴﻳﺎﭘ

ﻲ ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ﻪﻳﻻ ﺮﻫ

ﺑ ﻪ هﺎﺗﻮﻛ لﺎﺼﺗا يﺎﻫدوﺮﺘﻜﻟا زا ﻞﻣﺎﻛ رﻮﻃ هﺪﻧﺎﺷﻮﭘ

ﺖـﺳا هﺪـﺷ .

ﻠﻴﺧ ﺎﻫدوﺮﺘﻜﻟا ﺖﻣﺎﺨﺿ ﻲ

قرو ﺖﻣﺎﺨﺿ زا ﺮﺘﻜﭼﻮﻛ FG

ضﺮﻓ

.ﺖﺳا هﺪﺷ رد ﻦﻳاﺮﺑﺎﻨﺑ

ﻄﺑاور ﻲ ﺮﻴﺛﺎﺗ زا ،هﺪﺷ هدروآ ﻪﻣادا رد ﻪﻛ

ﻪـﻳﻻ ود ﺮـﻫ .ﺖـﺳا هﺪـﺷ ﻲـﺷﻮﭘ ﻢﺸﭼ ﺎﻫدوﺮﺘﻜﻟا ﻲﻜﻴﻧﺎﻜﻣ ي

ﻧﺎﻴﻣ ﻪﺤﻔﺻ ﺮﺑ دﻮﻤﻋ داﺪﺘﻣا رد ﻚﻳﺮﺘﻜﻟاوﺰﻴﭘ ﻲ

ﺖـﺒﺜﻣ ﺖﻬﺟ رد

ﺒﻄﻗ ﺖﻣﺎﺨﺿ ﻲ

هﺪﺷ .ﺪﻧا

5. Coupling 6. In-plane 7. Out of plane

(5)

ﻲﻤﺷﺎﻫ ﻲﻨﻴﺴﺣ خﺮﻫﺎﺷ نارﺎﻜﻤﻫ و

لوا ﻪــﺒﺗﺮﻣ ﻲــﺷﺮﺑ ﻞﻜــﺷ ﺮــﻴﻴﻐﺗ ﻦﻴﻟﺪــﻨﻴﻣ

،

زا ﻲـﻄﺧ ﻊﺑﺎـﺗ ناﻮﻨﻋ ﺖﻣﺎﺨـﺿ

قرو ﺖﻣﺎﺨﺿ داﺪﺘﻣا رد ﻪـﻣادا رد ﻪـﻜﻳرﻮﻃ

1 2 3 1 2 3 1 1 2

1 2 3 1 2 3 2 1 2

1 2 3 1 2

( , , , ) ( , , ) ( , , )

( , , , ) ( , , )

U x x x t u x x t x x x t V x x x t x x t x x x t W x x x t w x x t

ﻪﺤﻔﺻ رد ي

ﻧﺎﻴﻣ ﻲ ،قرو

ﻲﻧﺎـﻴﻣ تﺎﺤﻔـﺻ .ﺪﻨﻛ

:ﺖﺳا هﺪﺷ هداد ﺮﻳز ترﻮﺻ

11 ,1 1,1

22 ,2 2,2

33

12 ,2 ,1 1,2 2,1

13 1 ,1

23 2 ,2

0 u z

v z

u z

w w

ε ψ

ε

γ ν ψ ψ

γ ψ γ ψ

= +

=

= + + +

= +

ﻖﺘـﺸﻣ ﺐـﻴﺗﺮﺗ نﺎـﺸﻧ ﺮﻃﺎﺧ ﺪﻳﺎﺑ

.ﺖﺳا هﺪﺷ ضﺮﻓ ﺮﻔﺻ سﺎﺳاﺮﺑ

ﻪﺑ ترﻮﺻ :

11 22

12 12

13 23

( ) 1

h h h h h

E x σ

σ

σ γ

σ ν σ

 

 

   

=

   

  −  

   

 

2. Transverse deflection 3. Slope rotations

ﻲﻤﺷﺎﻫ ﻲﻨﻴﺴﺣ خﺮﻫﺎﺷ

لوا ﻪــﺒﺗﺮﻣ ﻲــﺷﺮﺑ ﻞﻜــﺷ ﺮــﻴﻴﻐﺗ زا ﻲـﻄﺧ ﻊﺑﺎـﺗ ناﻮﻨﻋ

2ﻲﺿﺮﻋ قرو ﺖﻣﺎﺨﺿ داﺪﺘﻣا رد

ـﺑ ﻲﻳﺎـﺠﺑﺎﺟ ناﺪﻴﻣ . ﻪ

ﻪـﻣادا رد ﻪـﻜﻳرﻮﻃ

16 :[

1 2 3 1 2 3 1 1 2

1 2 3 1 2 3 2 1 2

1 2 3 1 2

( , , , ) ( , , ) ( , , )

( , , , ) ( , , )

ν ψ

= +

=

ﻪﺤﻔﺻ ا ي ﻪﺤﻔﺻ رد

ﺶﺧﺮﭼ يﺎـﻫ ﻲﻧﺎـﻴﻣ تﺎﺤﻔـﺻ3

ﻲﻣ ﺺﺨﺸﻣ ار نﺎﻣز .ﺪﻨﻛ

11 ,1 1,1

22 ,2 2,2

12 ,2 ,1 1,2 2,1

13 1 ,1

23 2 ,2

( )

u z v z

u z

w w

ε ψ

γ = +ν + ψ +ψ

2 و 3 ﺑ دراد راﺮﻗ ﻪ

ﻖﺘـﺸﻣ ﺐـﻴﺗﺮﺗ

ﻣ نﺎﺸﻧ ار ﻲ .ﺪﻫد نﺎـﺸﻧ ﺮﻃﺎﺧ ﺪﻳﺎﺑ

.ﺖﺳا هﺪﺷ ضﺮﻓ ﺮﻔﺻ

اﺮﺑ ﻲﻳﺎﺠﺑﺎﺟ ي

ﻲﻧﺎﻴﻣ قرو ﻪﺑ

3

12 2 12

1 0 0 0

0 0 1 0 0

( ) 2

0 0 0 0

E x ν ν

σ γ

ν

 

 

 − 

 

   

 

   

−  

   

 

   

 

   

 

 

Transverse deflection rotations

ﻲﻤﺷﺎﻫ ﻲﻨﻴﺴﺣ خﺮﻫﺎﺷ

ﻲﻳﺎﺠﺑﺎﺟ ناﺪﻴﻣ

لوا ﻪــﺒﺗﺮﻣ ﻲــﺷﺮﺑ ﻞﻜــﺷ ﺮــﻴﻴﻐﺗ - ﻪﺤﻔﺻ ا ي ﺑ ﻪ زا ﻲـﻄﺧ ﻊﺑﺎـﺗ ناﻮﻨﻋ

هﺪﺷ هداد ﻂﺴﺑ و ﺪﻧا

ﺶﻤﺧ ﻲﺿﺮﻋ

ﺖﺳا ـﺑ ﻲﻳﺎـﺠﺑﺎﺟ ناﺪﻴﻣ .

ﺖﺳا هﺪﺷ هدﺮﺑ رﺎﻛ 16]

1 2 3 1 2 3 1 1 2

1 2 3 1 2 3 2 1 2

1 2 3 1 2

( , , , ) ( , , ) ( , , )

( , , , ) ( , , )

ψ

ν ψ

= +

ﺎﻫ ي نورد - ﻪﺤﻔﺻ

ﻲﺿﺮﻋ ﻲﻳﺎﺠﺑﺎﺟ

1و وψ ψ2

ﺶﺧﺮﭼ

x1, و ﻲﻣ ﺺﺨﺸﻣ ار نﺎﻣزt

ﺶﻧﺮﻛ و ﺶﻨﺗ ﻂﺑاور

ﺑ ﻲﻳﺎﺠﺑﺎﺟ ﻪ

12 ,2 ,1 ( 1,2 2,1)

نآ لﺎﺒﻧد ﻪﺑ ﻪﻛ ﺎﻣﺎﻛ ﺖﻣﻼﻋ 1

، 2

x1

، x2

و x3

ﻣ نﺎﺸﻧ ار

دﻮﻤﻋ ﻲﺿﺮﻋ ﺶﻨﺗ ﻪﻛ ﻢﻴﻨﻛ

33 ي σ

ﺶﻨﺗ ﻂﺑاور ،كﻮﻫ نﻮﻧﺎﻗ -

اﺮﺑ ﻲﻳﺎﺠﺑﺎﺟ

12 12

1 0 0 0

0 0 1 0 0

2

0 0 0 1 0

2

0 0 0 0 1

2 ν

σ γ

ν

 

 

 − 

 

   

 

   

−

 

   

 

   

 

   

 − 

 

 

3 - ﻲﻳﺎﺠﺑﺎﺟ ناﺪﻴﻣ

ﺎــﻨﺒﻣﺮﺑ ي رﻮــﺌﺗ ي لوا ﻪــﺒﺗﺮﻣ ﻲــﺷﺮﺑ ﻞﻜــﺷ ﺮــﻴﻴﻐﺗ

ﻲﻳﺎﺠﺑﺎﺟ يﺎﻫ نورد -

هﺪﺷ هداد ﻂﺴﺑ ي ﻘﻣ راﺪ ﻲﺘﺑﺎﺛ ﺖﺳا

ﺑ ،هﺪﺷ نﺎﻴﺑ ﻪ

ﺖﺳا هﺪﺷ هدﺮﺑ رﺎﻛ

1 2 3 1 2 3 1 1 2

1 2 3 1 2 3 2 1 2

( , , , ) ( , , ) ( , , )

U x x x t u x x t x x x t V x x x t x x t x x x t

وu ﻲﻳﺎﺠﺑﺎﺟv ﺎﻫ

ﻲﺿﺮﻋ ﻲﻳﺎﺠﺑﺎﺟ قرو رﻮﺤﻣ لﻮﺣ ,x2

4 - ﺶﻧﺮﻛ و ﺶﻨﺗ ﻂﺑاور

ﺶﻧﺮﻛ ﻂﺑاور -

ﺑ ﻲﻳﺎﺠﺑﺎﺟ

نآ لﺎﺒﻧد ﻪﺑ ﻪﻛ ﺎﻣﺎﻛ ﺖﻣﻼﻋ تﺎﺼﺘﺨﻣ ﻪﺑ ﺖﺒﺴﻧ

1

دﻮﻤﻋ ﻲﺿﺮﻋ ﺶﻨﺗ ﻪﻛ ﻢﻴﻨﻛ ﺶﻨﺗ ﻂﺑاور ،كﻮﻫ نﻮﻧﺎﻗ

(

11 22

12 12

13 23

1 0 0 0

0 0 0 0

1 2

ε ε

σ γ

γ ν γ

 

 

 

 

 

 

 

 

   

 

   

 

   

 

   

 

   −  

 

 

60

و ﻻﺎـﺑ رد ﻚـﻳﺮﺘﻜﻟاوﺰﻴﭘ

صاﻮــﺧ ﺮـﺴﻛ

قرو ﺖﻣﺎﺨـﺿ داﺪـﺘﻣا رد ًﻻﻮـﻤﻌﻣ ،هﺪﻨﻫد ﻞﻴﻜﺸﺗ داﻮﻣ قرو صاﻮﺧ ﻪﻛ ﺖﺳا نآ ﺮﺑ ضﺮﻓ ،ﻪﻌﻟﺎﻄﻣ ﻦﻳا رد

ﻲﻳﻻﺎﺑ ﺢﻄﺳ .ﺪﻨﻛ قرو

لﻮـﻃ رد

ﻲﻟﺎـﮕﭼ و ﮓـﻧﺎﻳ لوﺪـﻣ .ﺖـﺳا هﺪـﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧرد ﺖﺑﺎﺛ نﺎـﻴﺑ ترﻮـﺻ

( ) ( ) ( )

E x E E V x E

ρ ρ ρ ρ

نﺎـﺸﻧ ار

Vf x

ﻂـﻘﻓ و ﺖـﺳا

1. Volume fraction

2 - 3 ﺎــﻨﺒﻣﺮﺑ ﻲﻳﺎﺠﺑﺎﺟ قرو اراد ي ﺑ ،هﺪﺷ نﺎﻴﺑ

) 3 (

u ﻲﺿﺮﻋ ﻲﻳﺎﺠﺑﺎﺟw

قرو

2 - 4 ﺶﻧﺮﻛ ﻂﺑاور

) 4 (

نآ لﺎﺒﻧد ﻪﺑ ﻪﻛ ﺎﻣﺎﻛ ﺖﻣﻼﻋ تﺎﺼﺘﺨﻣ ﻪﺑ ﺖﺒﺴﻧ دﻮﻤﻋ ﻲﺿﺮﻋ ﺶﻨﺗ ﻪﻛ ﻢﻴﻨﻛ ﺶﻨﺗ ﻂﺑاور ،كﻮﻫ نﻮﻧﺎﻗ

) 5 (

ًﺎﺘﺒﺴﻧ يﺎﻫ ...

و ﻻﺎـﺑ رد ﻚـﻳﺮﺘﻜﻟاوﺰﻴﭘ

ﻪــﻛ ﺪﻨﺘــﺴﻫ ﻲــﺒﻛﺮﻣ داﻮــﻣ ( صاﻮــﺧ

تاﺮـﻴﻴﻐﺗ ﻞـﻴﻟد ﺠﻳرﺪـﺗ

ﻲ ﺮـﺴﻛ

ﻲ ﻲﻳﻻﺎﺑ ﺢﻄﺳ .ﺪﻨﻛ

ﻨﻴﻳﺎــﭘ ﺢﻄــﺳ ﻪــﻜﻴﻟﺎﺣ رد ﻲ

قرو

نﻮﺳآﻮﭘ ﺐﻳﺮﺿ لﻮـﻃ ردν

ﻲﻟﺎـﮕﭼ و ﮓـﻧﺎﻳ لوﺪـﻣ .ﺖـﺳا هﺪـﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧرد ﺖﺑﺎﺛ ـﺑ قرو ﺖﻣﺎﺨـﺿ داﺪﺘﻣا رد ﻪﺘﺳﻮﻴﭘ رﻮﻃ ﻪ

نﺎـﻴﺑ ترﻮـﺻ

3 3

( ) ( ) ( )

c m f m

E x E E V x E

x V x

ρ ρ ρ ρ

= − +

ﻜﻴﻣاﺮـﺳ ﻲ نﺎـﺸﻧ ار

3 3

( ) ( 1)

2 2

f

V x x

= h+

ﻧﺎـﻳداﺮﮔ ﺲﻳﺪـﻧا ﻲ

ﻂـﻘﻓ و ﺖـﺳا

. Volume fraction

تﺎﺷﺎﻌﺗرا دازآ قرو

ًﺎﺘﺒﺴﻧ يﺎﻫ

ود ﺎـﺑ ﻪـﻳﻻ و ﻻﺎـﺑ رد ﻚـﻳﺮﺘﻜﻟاوﺰﻴﭘ

ﻪــﻛ ﺪﻨﺘــﺴﻫ ﻲــﺒﻛﺮﻣ داﻮــﻣ (

ﻪ تاﺮـﻴﻴﻐﺗ ﻞـﻴﻟد

ﻲﻧاﻮﺗ ﻣ ﺮﻴﻴﻐﺗ ﻲ

، ﻨﻴﻳﺎــﭘ ﺢﻄــﺳ ﻪــﻜﻴﻟﺎﺣ رد

راﺪﻘﻣ نﻮﺳآﻮﭘ ﺐﻳﺮﺿ

ﻲﻟﺎـﮕﭼ و ﮓـﻧﺎﻳ لوﺪـﻣ .ﺖـﺳا هﺪـﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧرد ﺖﺑﺎﺛ ـﺑ قرو ﺖﻣﺎﺨـﺿ داﺪﺘﻣا رد ﻪﺘﺳﻮﻴﭘ رﻮﻃ

3 3

( ) ( ) ( )

c m f m

E x E E V x E

x V x

ρ ρ ρ ρ

= − +

و يﺰـﻠﻓ اﺰـﺟا ﺐﻴﺗﺮﺗ ﻜﻴﻣاﺮـﺳ

ترﻮﺻ :

( ) ( 1)

2 2

= + g

ﻧﺎـﻳداﺮﮔ ﺲﻳﺪـﻧاg

ﺦﺳﺎﭘ ﻖﻴﻗد ﻞﻴﻠﺤﺗ -

ﻪﺘﺴﺑ تﺎﺷﺎﻌﺗرا

ﻢﻴﺨـﺿ ًﺎﺘﺒﺴﻧ قرو FG

نآ

داﻮﻣ صاﻮﺧ ﺪﻨﻤﻓﺪﻫ

ﺪــﻨﻤﻓﺪﻫ )

ﻪــﻛ ﺪﻨﺘــﺴﻫ ﻲــﺒﻛﺮﻣ داﻮــﻣ (FGMs

ﺑ ﺎﻫ ﻪ رﻮﻃ ﺑ ﻪﺘﺳﻮﻴﭘ ﻪ

ﺎﺑ ﺖﻣﺎﺨﺿ داﺪﺘﻣا رد ﻊﺑﺎﺗ لﺪﻣ

ًﻼﻣﺎــﻛ ﻚﻴﻣاﺮــﺳ

،

ﻲﻣ ﺰﻠﻓ ًﻼﻣﺎﻛ .ﺪﺷﺎﺑ

راﺪﻘﻣ

ﻲﻟﺎـﮕﭼ و ﮓـﻧﺎﻳ لوﺪـﻣ .ﺖـﺳا هﺪـﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧرد ﺖﺑﺎﺛ ـﺑ قرو ﺖﻣﺎﺨـﺿ داﺪﺘﻣا رد ﻪﺘﺳﻮﻴﭘ رﻮﻃ

ﻲﻣ ﺮﻴﻴﻐﺗ ﻪﻣادا رد هﺪﺷ ﺪﻨﻛ

:

c m f m

E x E E V x E

ρ ρ ρ ρ

و ﺑc ﻪ و يﺰـﻠﻓ اﺰـﺟا ﺐﻴﺗﺮﺗ

ﻲﻤﺠﺣ ﺮﺴﻛ و ﺪﻫد Vf

ﺑ ﻪ ترﻮﺻ

ﺖﺳا هﺪﺷ ﻪﺘﻓﺮ .

g

ﻣ ار ﺖﺒﺜﻣ ﺮﻳدﺎﻘﻣ ﻲ

.دﺮﻳﺬﭘ

ﺦﺳﺎﭘ ﻖﻴﻗد ﻞﻴﻠﺤﺗ

ﻞﻜﺷ ﻢﻴﺨـﺿ ًﺎﺘﺒﺴﻧ قرو1

ﻦﻴﻳﺎﭘ نآ

2 - 2 - داﻮﻣ صاﻮﺧ

داﻮــﻣ ﺪــﻨﻤﻓﺪﻫ

نآ ﻲﻜﻴﻧﺎﻜﻣ ﺑ ﺎﻫ

1ﻲﻤﺠﺣ قرو ﺖﻣﺎﺨـﺿ داﺪـﺘﻣا رد ًﻻﻮـﻤﻌﻣ ،هﺪﻨﻫد ﻞﻴﻜﺸﺗ داﻮﻣ

ﺮﻴﻴﻐﺗ ﻲﻣ .ﺪﻨﻛ قرو صاﻮﺧ ﻪﻛ ﺖﺳا نآ ﺮﺑ ضﺮﻓ ،ﻪﻌﻟﺎﻄﻣ ﻦﻳا رد

ﺎﺑ ﺖﻣﺎﺨﺿ داﺪﺘﻣا رد قرو ) x3=h (

ًﻼﻣﺎــﻛ

) x3=-h ( ﻲﻣ ﺰﻠﻓ ًﻼﻣﺎﻛ

ﻞﻴﻠﺤﺗ ﻲﻟﺎـﮕﭼ و ﮓـﻧﺎﻳ لوﺪـﻣ .ﺖـﺳا هﺪـﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧرد ﺖﺑﺎﺛ

ﺑ ﻲﻣﺮﺟ ﻪ ـﺑ قرو ﺖﻣﺎﺨـﺿ داﺪﺘﻣا رد ﻪﺘﺳﻮﻴﭘ رﻮﻃ

ﻲﻣ ﺮﻴﻴﻐﺗ ﻪﻣادا رد هﺪﺷ

) 1 (

ﺲﻳﻮﻧﺮﻳز m

ﻲﻣ ﻲﻤﺠﺣ ﺮﺴﻛ و ﺪﻫد

) 2 (

ﮔ ﺮﻈﻧ رد ﺖﺳا هﺪﺷ ﻪﺘﻓﺮ

ﻣ ار ﺖﺒﺜﻣ ﺮﻳدﺎﻘﻣ ﻞﻜﺷ

2 داﻮــﻣ نآ ﻲﻜﻴﻧﺎﻜﻣ ﻲﻤﺠﺣ ﺮﻴﻴﻐﺗ ﺎﺑ ﺖﻣﺎﺨﺿ داﺪﺘﻣا رد قرو ) ﻞﻴﻠﺤﺗ ﺑ ﻲﻣﺮﺟ ﻲﻣ ﺮﻴﻴﻐﺗ ﻪﻣادا رد هﺪﺷ )

ﻲﻣ

)

ﻣ ار ﺖﺒﺜﻣ ﺮﻳدﺎﻘﻣ