Multiple Neutral Axes in Bending of a Multiple-Layer Beam With Extremely Different Elastic Propertiesby Yan Shi, John A. Rogers, Cunfa Gao, Yonggang Huang

Journal of Applied Mechanics

About

Year
2014
DOI
10.1115/1.4028465
Subject
Mechanics of Materials / Computational Mechanics

Text

Multiple Neutral Axes in Bending of a Multiple-Layer Beam With

Extremely Different Elastic Properties normal after deformation does not hold any more. The energy method is used to establish a simple analytic model for multi-lay­ ered structures with extremely different elastic properties. The analytic model is validated by finite element analysis. [DOI: 10.1115/1.4028465]

Yan Shi

State Key Laboratory of Mechanics and Control of Mechanical Structures,

Nanjing University of Aeronautics & Astronautics,

Nanjing 210016, China;

Department of Civil and Environmental Engineering,

Northwestern University,

Evanston, IL 60208;

Department of Mechanical Engineering,

Northwestern University,

Evanston, IL 60208;

Center for Engineering and Health,

Northwestern University,

Evanston, IL 60208;

Skin Disease Research Center,

Northwestern University,

Evanston, IL 60208

Development of stretchable and flexible electronics involves both hard (stiff) and soft (compliant) materials [1-3], such as sili­ con and PDMS, which have the elastic moduli different by five orders of magnitude or more. These extremely different materials often appear in a multilayer structure [4—8], as illustrated in

Fig. 1.

For the multilayer stmcture that consists of extremely different materials subjected to bending, the Kirchhoff assumption [9] that the straight lines normal to the midsurface remain normal after de­ formation does not hold anymore, as shown by the FEA [10], The purpose of this paper is to establish a simple, analytic model for the multilayer structure with extremely different elastic properties and to validate the model by FEA.

Figure 1 shows a three-layer stmcture with the thickness hbottom, ^m iddle, and hlop, and Young’s modulus £bottom, £ middie, and £ top, respectively, where the middle layer is much softer (more compliant) than the top and bottom layers, i.e., £ middle « £ bottom and £ top. The bottom and top layers have thickness much smaller than the length and therefore can be modeled as beams. For the bend curvature k applied to the stmcture, the strain in each beam can be written as

John A, Rogers

Department of Materials Science and Engineering,

Frederick Seitz Materials Research Laboratory,

University of Illinois at Urbana-Champaign,

Urbana, IL 61801

Cunfa Gao

State Key Laboratory of Mechanics and Control of Mechanical Structures,

Nanjing University of Aeronautics & Astronautics,

Nanjing 210016, China

Yonggang Huang1

Department of Civil and Environmental Engineering,

Northwestern University,

Evanston, IL 60208;

Department of Mechanical Engineering,

Northwestern University,

Evanston, IL 60208;

Center for Engineering and Health,

Northwestern University,

Evanston, IL 60208;

Skin Disease Research Center,

Northwestern University,

Evanston, IL 60208 e-mail: y-huang@northwestem.edu

The multi-layer beam consisted of materials with extremely differ­ ent elastic properties, such as silicon and Polydimethylsiloxane (PDMS), has many important applications. The Kirchhoff assump­ tion that the straight lines normal to the midsurface remain ’Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the

Journal of A pplied M echanics. Manuscript received July 4, 2014; final manuscript received August 25, 2014; accepted manuscript posted September 2, 2014; published online September 17, 2014. Assoc. Editor: Arun Shukla.

T ^bottom for 0 < y < , ( ^bottom \ e =K[y - — J £ = — 1^ bottom — ^middle ^top ^bottom + ^middle — y — ^bottom + ^middle ~F htop U) where the coordinate y denotes the distance from the bottom surface (Fig. 1), ebottom and etop are the membrane strains of the bottom and top layers, respectively, and are to be determined.

Continuity of strain across the layer interfaces y — hbottom and y ^bottom + ^ m iddle gives the strain in the middle layer as llbottom + htop\ y middle ^bottom ^top “1“ K . ^bottom “1“ £top . ^bottom ^top + ------- 2------- + K-------- 4 ~ f o r ^bottom 5^ y ^ ^bottom "t- ^middle ^middle (2)

For pure bending, the net force in each cross section of the struc‘^ bottom ~b ^ middle "b h to prhture is zero, J0 or £ top in the bottom, middle, and top layers, respectively.

Eedy = 0, where £ = £ bottom, £ middie, y

V h EU t o p ' l~tOp • ^ m id d le ' ^ m id d le ^ b o t t o m ' ^ b o t to mH i 0 x

Fig. 1 Schematic illustration of a multilayer beam composited of materials with extremely different elastic properties

Journal of Applied Mechanics Copyright © 2014 by ASME NOVEMBER 2014, Vol. 81 / 114501-1

Substitution o f Eqs. (1) and (2) into the above integral gives one equation for bottom and £top ^bottom ^ bottom "b ^middle^middle \ ^bottom + Etop t^op “b ^m iddle ^ middle \ o ^ » ^bottom ^top2 I ^top — ^m iddle "middle X ^ (3)

T he other equation for bottom and £top can be obtained by m inim izing the total strain energy l E s 2dy in the structure w ith respect to bottom (or £toP) for a given curvature k , which, together with Eq. (3), gives ^bottom ^ bottom "h ^middle ^ middle \

I ^bottom

Etop t^op T ^rniddle ^ middle “ 6 ) t^op — ^middle ^ middle X ^bottom “b h\ 12 top (4)

Fig. 2 C om parison o f th e strain d istribution from FEA [10] and present theory

The m em brane strains £bottom and £top are then obtained analytically ^bottom ^middle^middle

Ebottom ^ bottom (4/Zbottom 2/?top) “b (^middle ^ middle)

E bottom h bottom^top^top ^bottom r. . 0 ^m iddle^middle 0 ^m iddle^m iddle . ^ZSmiddie/Zmjddie) 2 4 + o —-------- ;---------- r o --- —— ---------- h

Ebottom ^ bottom ^middle ^ middle ^top^top Ebottom ^ bottom Ftop ^ top

Stop K~

E t^op t^op (4/i top 2/?bottom) “b (^rniddle ^ middle )

E bottom h bottom ^ top ^ top ‘top 24 + 8 ^middle ^ middle ^bottom ^ bottom

Q ^middle^middle 8 ----F---h---------£ top "top 2 (^middle ^ middle )

E bottom h bottom ^ top ^ top (5)

For the lim it Fmiddle^middle ^bottom^bottom and ^-top^top’ SUCh as the m iddle layer is not m uch th icker than the top and bottom layers, the above solution is sim plified to ^middle ^ middle ^ ^^bottom ^top