AD-R168 749 IMPACT RESPONSE OF STRUCTURES--GLOBAL LOCAL APPROACH I.It(U) NORTHMESTERN UNIV EVANSTON IL L K KEER 81 DEC 84RFOSR-TR-86-8287 AFOSR-82-0338

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REPORT DOCUMENTATION PAGE

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!IT:L Ic %ACi RESPOrNSE OF -

-F* <1C 7-.-CTA T.CALAPROACH _____________________________

Leon M1. Keer

E:~ C i-- :.Z, I. 71M C VE~R 1 14 OA F Y'.-. ;.. _Ocy 5.PAC--Cc N T

C 4r COceEs I& SUS.'ECT TE siMS Cnti~ne oi reverig ii ngee±ary j.t~ zaenfy )y blocm nu ! -

z 4:~i sclu:' ons to p rctito ro 1lt ims of lr-o _-e ccntact bv a rigld

*r d, s. . Tlio 7,' or prorlefls t:it ,Te 0 j.e ere t~i _cllow.iq:

j1 IK: VWL? IMLATF1t L by :\oer and WOO. -he pr -,iilem o' tho cP.nimic-. ~-.lt ---.7.1 s rs e* C c~:cre tcta~WSsl . Th.ke i I,(ienter '- a

tO . ~~-,conikoC. CI - ., ind 3-rpl: iported cas;es Of

t r r Cul:

A'.~c'C3 -]F- *3

I?~~~~~~ - -- T:- ~C~ciI'

Cu~~~r AC: .,, ~l-,___________C_ 7 7 -S

d4

S.CURITY CLASSF ICATION OF THIS PACG

ABSTRACT (cont'd)

3. SMOOTH INDENTATION OF AN ISOTROPIC C\NTIL^EVER BE2t by Keer and Schonbera. 'elastic response of a cantilever beam of finite lenc.th to a frictionless cvlindrical

and flat indenter is studied. Solutions are obtained through a global-local techniqeis,wnich accounts for roth the local behavior near the indenter, as well as th lba] bea,behavior. Results are compared to Hertz theory and to beam theory solutions ind commtent ,

on the validity of the linear elastic analvsis are made.

SM!OOH INDENTATION OF A TRANSVERSELY ISOTROPIC CAN'TILEVER BEAMI bv Keer anr Schonbcr'The plane elastic response of a transversely isotropic beam of finite lentth to a

:rictionless cylindrical and flat indenter is studied. Local indenter stresses, as wLas displacements and rotations are computed for each case and plotted for ,arious ratsof contact width to beam length, and for various positions of the indenter. ac probcis solved using a nearl, isotropic and a highly anisotropic material. Results _re1:=,rrd to those o. the isotropic study, and to Hertz theory and beam theior" solution

0,

S.

.e .Q

- ~ ~ - - - -. ~ .

RESEARCH OBJECTIVES

The objective of the research is to attempt to develop systematic

I techniques for determining the response to dynamic impact, when the shape of

the impacting object is known. It is hoped that the dynamic response can be

obtained by the use of techniques that accurately model the stresses near the

point of impact as well as the stresses away from the point of impact. Tnis

technique is one tnat involves a global-local technique, where a more exact

description of the stress distribution is made near the contact (for example,

elasticity tneory) and a more approximate theory is used away from the point

of loan application (neam, plate or shell theory). An analysis should

incorporate tne fol.owin G features:

a. The elasti: an, Dr pl3stic stresses in the vicinity of the impacting

obe: s :,i :-1raely cnaracterized.

D. An 3C,: r.i:-, , tne material must be used, especially

w. wn n >.s' ie ' ' .: i.enear the impact.

1 c. A: po .nts :'t'n: om fie load the analysis must accurately model

nte Jynamiic response. Effects of support members, joints or other

restraints on tne response must be carefully modeled.

d. An evaluation of the potential towards damage, such as cracking,

plastic deformation, delamination, etc. must be made.

The research objectives include the study of both static and dynamic

solutions to indentation problems. The static ones are solved in order to

understand the elasticity result that will tend to form the basis for the

local solotion. This solution will be incorporated into a dynamic structural

.mechanics solution for a beam, plate or shell to obtain results of both a

global and local character. The first year of research concentratea on the

s],ution.s 3appropriate to bea, theory and to a) allY symr,]etric plate "hOry.

04

:i,(:-:

STATUS OF THE RESEARCH

The research is progressing along the lines of the original proposal.

The following are problems that involve the global-local approach that have

been solved for the static case:

1. SMOOTH INDENTATION OF AN ISOTROPIC CANTILEVER BEAM by L.M. Keer and

W. Schonberg. The elastic response of a cantilever beam of finite length

to a frictionless cylindrical and flat indenter is studied. Solutions

are obtained through a global-local technique, which accounts for both

the local behavior near the indenter, as well as the global beam

behavior. Results are compared to Hertz theory and to beam theory

solutions and comments on the validity of the linear elastic analysis are

made.

2. SMOOTH INDENTATION OF A TRANSVERSELY ISOTROPIC CANTILEVER BEAM by L.M.

Keer and W. Schonberg. The plane elastic response of a transversely

isotropic beam of finite length to a frictionless cylindrical and flat

indenter is studied. Local indenter stresses, as well as displacements

* and rotations are computed for each case and plotted for various ratios

of contact width to beam length, and for various positions of the

indenter. Each problem is solved using a nearly isotropic and a highly

anisotropic material. Results are compared to those of the isotropic

study, and to Hertz theory and beam theory solutions.

An extension of these problems to the dynamic case is presently under

. way. Possible titles are given in "PAPERS IN PREDARATION'

The following are tne dynamic problems that have oeen solved and are presently

"n press or ieinj considereG for publication.

. . V 1L JV I7 1PA..T ON A C i>CtJLAi PAT E by L .r rq k,.7. ,C2 .

,dyn : c iiip ct of a .......

elastic plate was solved. The indenter was assumed to have a radius of

curvature sufficiently large that areas of contact of the order of the

plate thickness could be considered. Clamped and simply supported cases

of plate impact were considered.

2. LOW VELOCITY IMPACT ON A TRANSVERSELY ISOTROPIC PLATE by L.M. Keer and

K.T. Woo. The problem of the dynamic impact of a rigid indenter striking

a transversely isotropic circular elastic plate was solved. The problem

geometry and formulation was similar to (1) except that the material

properties were for transverse isotropy. Three materials were chosen for

this case: cadmium, magnesium, and a laminated composite.

In addition to the above papers the dynamic impact of initially stressed0

- beams and cantilever beams are papers in preparation. They should be

submitted for publication relatively soon.

~A/03

.

LIST OF WRITTEN PUBLICATIONS (DURATION OF GRANT)

1. L.M. Keer and G.R. Miller, "Smooth Indentation of Finite Layer", ASCE

U Journal of Engineering Mechanics, Vol. 109, 1983, pp. 706-717.

2. L.M. Keer and R. Ballarini, "Smooth Contact between a Rigid Indenter and

an Initially Stressed Orthotropic Beam", AIAA Journal, Vol. 21, 1983 pp.

1035-1042.

S3. L.M. Kee and J.C. Lee, "Dynamic Impact of an Elastically Supported Beam-

--- Large Area Contact", International Journal of Engineering Science, in

press.

4. L.M. Keer and W. Schonberg, "Smooth Indentation of an Isotropic

Cantilever Beam", International Journal of Solids and Structures, in

press. (Included with report).

LIST OF PAPERS PUBLISHED IN PROCEEDINGS

1. L.M. Keer and R. Ballarini, "Smooth Contact between a Rigid Indenter and

Initially Stressed Orthotropic Beam", in Advances in Aerospace

Structures, Materials and Dynamics, ASME (1982).

2. L.M. Keer and J.C. Lee, "Dynamic Impact of an Elastically Supported Beam-

-Large Area Contact", Advances in Aerospace Structures, Materials and

Dynamics, ASME (1983).

3. L.M. Keer and K.T. Woo, "Low Velocity Impact on a Circular Plate"

Advances in Aerospace Structures, Materials and Dynamics, ASME (1934) in

press. (Included with report).

PAPERS SUBMITTED FOR PUBLICATION

. L.M. Keer and W. Schonberg, "Smooth Indentation of a Transversely

Isotropic Cantilever Beam".

- 2. L.3. Keer ald KT. WOoo, "Low Ve!ocity Impact on a Trafisvarsely is t r-e.

.4

'a

PAPERS IN PREPARATION

1. L.M. Keer and W. Schonberg, "Dynamic Impact of an Elastically Supported

Cantilever Beam".

2. L.M. Keer and W. Schonberg, "Dynamic Impact of an Initially Stressed,

Orthotropic Beam".

.' PERSONNEL ASSOCIATED WITH RESEARCH EFFORT

1. W. Schonberg Research Assistant in Civil Engineering

2. K.T. Woo Research Assistant in Applied Mathematics (partiallysupported)

3. J.C. Lee Post Doctoral Fellow (partially supported)

0

U

0 O,

SMOOTH INDENTATION OF AN ISOTROPIC CANTILEVER REAM

by

Leon M. Keer and William P. Schonherg

Department of Civil Engineering

Northwestern University

Evanston, IL 60201

Abstract

The response of an isotropic cantilever beam of finite length under the

action of frictionless cylindrical and flat indenters is studied. Solutions

are obtained through a local-global technique, which accounts for hoth the

7ocal behavior near the indenter, as well as the global beam behavior. The

method of analysis superposes an infinite layer solution derived through the

use of integral transforms with a pure bending beam theory solution. Local

indenter stresses, as well as displacements and rotations are computed for

each case and plotted for various ratios of contact width to beam length, and

for various positions of the indenter. Where possible, the results are

compared to Hertz theory of contact stresses and to beam displacement and

rotation solutions.

1. INTRODUCTION

Consider the finite layer of length 2 and thickness shown in Fig. 1.

The layer is fixed at one end, free on the other, and loaded by an indenter

centered at a distance 9o from the fixed end. Although the theory Is

applicable for an arbitrary geometry, the types of indenters studied are

cylindrical (Fig. 1a) and flat (Fiq. 1b). Such problems are seen quite often

in mechanical applications and may also serve as models for impact phenomena

in gears and turbine blades. Usual methods which use a beam theory solution

to obtain an overall load-displacement relationship and then a Nertzian

-ontact solution to calculate local stresses under the indenter are fairly

limited, as will be shown in this paper. The method of solution used in this

paper is the superposition of an elasticity solution with a heam theory

solution. In this way, local contact stresses are represented by the

elasticity solution, while the global stresses are represented by the beam

theory solution.

A problem of the type considered here has already been solved by Keer and

*. Miller [1], and by Keer and Ballarini [2] for a beam that is clamped or simply

supported at both ends and loaded by a cylindrical indenter. Although not

treated numerically here, the intermediate case of a partially fixed end can

be solved for each type of indenter in a similar manner. Furt'nermore, the

met.ods used in this paper can be modified and expanded to study plate or

shell oroblems involving similar lo~dings rr.

The physical quantities of interest ie the stress dist-rhution under

-ach indenter, and the displacement and rotation of the beam under each of the

-t4- of "I V - -

-2-

r -*

ratio of indenter arm length to beam thickness, Zo/h, is treated as an

[dditional known parameter. However, in the flat indenter problem, thq

narameter ZI/h = Z0/h - c/h (i.e. the distance from the fixed Pnd of the heam

to the left-hand end of the indenter) is introduced as an additional known

parameter. In this manner, for a given 21/h, when the quantity c/h is varied

over a range of values the phenomenon known as "receding contact" can be

observed in the flat indenter problem. Ry varying the proper parameters as

required by the type of problem, then calculating deflections, rotations,

loads and stresses, an extensive study of the response of an isotropic

cantilever beam subjected to various loading conditions can he performed. The

results obtained are compared to beam theory solutions and Hertz contact

solutions for accuracy assessments.

U.

!O -3-

2, CASE I: CYLINDRICAL INDENTER - PROBLEM FORMULATION

The problem to he solved is that of an elastic layer of thickness h and

length Z indented by a cylindrical punch on its upper surface (Fig. la). The

conditions at the ends of the layer are those of a cantilever beam, clamped on

tne left and free on the right. The solution of the problem will he achieved

by a suitable superposition and matching of an elasticity solution and a heam

theory solution.

The boundary conditions for the elasticity problem can be written as

follows:

tyy(X,h) : 0I H < (2.1)* yy

T xyx,h) = 0 jxl < (2.2)

T xy(X,O) = 0 lxI < (2.3)

Syy (x,O) = 0 c < lxI < (2.4)

u (x,O) = 0- x2/_R < lXI < C. (2.5)y

The beam theory boundary conditions for the ends of the layer can be

Written as follows:

S= 0 x = -1o (2.6)

u, 0 x = - o 2.7)

= 0 x = Z (2.8)

V '10 (x.z

w where , 3 s-c the moment, shear and is the iverace v3lue nf t s e

-easv7 r t-se t'7 c.ness riven -v

h 3uYdy. (.0

0

.1A sui;tabe elastic-.ty solution that represents load-ing on the upper

surface of an i.sotrocpic elastic layer in plane str-ain with and no loading on

its lower surface is obtained using the techniques of Sneddon [4] and is given

* as follows:

-O E E()cos(x)+E (V'sin( x)rshJih(yyy I0 2S2 h23 F hc,

-[shza - 62+ y(a + ShaCha)]COSh( y)IdE (2.11)

* - 7 ~~~ES(E)sin(Ex)-EA( )cos( x) {s yB+s~h)sn(y

0 a2-sh2

+ Ey sh2a Cosh(,y')}d 21'

'0 E S( cos(ZIX)+EA ( )s5n(Ex) 'Fa~ + shI2 ys 2 ~sn~y

+x 0 sh2 -y shy ~l(2.13)

=7E 5 (I,)sin( x)-E.(E)cos(Ex)2*;u X (S2_h2 5) {f(1-?v)(a + shacha')- Ey sh2 7slnh( y)

-[ 2 + (1-2v')sh 23 - y(5 + sh~chS)7hosh(,iy)}d (2.1d)

- {f 2 -2(-vs2

- y(s + hc6)Sn yySl

I ~ 2(j>f~ ~Y '~ 2 's.~;y<. ~ 2.1F

S

where a = h, and u, v are the shear modulus and Poisson's Ratio of thie layer

material, respectively. The correspondinq equations for a layer In plan'e

.4 tres wth the same loadilng conditions are ohtained by replacing v with

v/(I~v). Furthermore, this suhstitution Is to he used throuqh the rest of

this paper to convert plane straln expressions to corresponding expressions in

plane stress.

It is seen that on y =h the normal and shear stresses vanish

automatically, and that on y =0, the shear stresses vanish. The normal

stress on y = 0 -is given as

T Y (x,O) IC f ~E(J)cos( x) + EA( )sin(Wx)d.:. (2.16)0

We let

ICEs) S f y~(t)JO(C'-t)dt (?.17)

0

CE A(r) = O(t )J, t )dt. 2~

Then Eqn. (2.16) becomes

c ii t d (t)dt .1S(X,0) f+ x f .(.9

YYX It2 -X 2 Xt /t 2 - 2

0 The moment and shea- due to the stresses as given by Eqns. (.1 23)are

g-Iven by

* '~l(X) dy E )~AisniY X2?l

SX

VE(x) f -r dy f j S(I,)s5flx) E- O OSWExy S A

00

We note that Eqns. (2.2n), (2.21) satisfy the condi.tion

- dM (2.22)

~TX

From Eqns. (2.10), (2.15) the average value of the slope is found to be

given by

E(X) f 1 a-(3-2v)shS FEs(4)sin (4x) - E (Wcos(WI)]d- (2.23)E 7-U (6-sna) A0

0 For the beam theory solution, the displacement is taken to be of the form

u x) ao + alx + az2+ a3X3. (.4

Assuming the hypotheses of Eule. -Bernoulli Beam Theory, the following results

* . are obtained:

u S(X) - y (y- h (2.25)

B T (2.26)

Using Eqns. (2.2d)-(2.26), the moment, shear and average slope are calculated

to be,

M ~' _) v x= ?( 2X) (2.27N3 -x X,

64 -7-

, -- ' - - - • . -

h

V (x) -r ,dx -6Da3

h 3u

-( x ) ax- dy a, + 2a2x + 3a3x

2

0

where 0 : h3/6(1-v). Thus, the solution sought will be a superposituon of

the elasticity solution given by Eqns. (2.11)-(2.16), (2.20)-(2.23) and the

beam theory solution given by Eqns. (2.24)-(2.29). The two solutions are

matched properly when the boundary conditions, Eqns. (2.1)-(2.9) are all

satisfied. This is achieved by solving for the constants a1 , a2 and a3 by

superposing the two solutions and applying the beam theory boundary

conditions. Thus, Eqns. (2.6), (2.8) and (2.q) become6

ME(Z ZO) + MB(Z-ZO) = 0 (P.30)

. VE(Z-Z o) + V8 .(Z-Zo0 = 0 (2.31)

(-Zo) + B(- O) = 0. (2.32)

Applying Eqns. (2.20)-(2.23) and (2.24)-(2.29) to Eqns. (2.3n)-(2.32) yields

the following:

1 2o(z-zo/2)*a 1 -U. f {E S FE(sinEYZ-zo) - E (VicoSV(2ZzO)J

0~£o

+ [ES( )cs(Z.z) + EA ()sinVYZ-ZO)I}d,

- 0 --(3-2,.*shS F ( S -o E (?,331

r4r

= 1 - .( \ . .-

-- -• ,-,~

....... . , , - . -. . ,- .. . . .

- r FE(VsinV(zQ) E EW~COW~Z-Z0)l (2.34)0

a rE (r)sin(Z-zu) E <W)co Z(_Zo)l d (2.35)

The two remaining boundary conditions given by Eqns. (2.5) and (2.7)

.4 still need to he satisfied. First, consider Eqn. (2.5):

UE(x,O) + uB(xO) = A - x2/2R (2.36)Uy Y ,

Differentiating with respect to x:

3uE auB

y (x,O) + Y(x,O) x (2.37)

Separating Eqn. (2.37) into a symmetric and an antisymmetric part with respect

to x yields

3U Ey xay (x,O) + 2a2x = - (2.3)* ax A

EauE

Y (x,O) + a,+ 3a~x = 0. (2.39)ax S

Consider first Eqn. (2.38). Substituting for U E (x,O) according to Eqn.y

(2.15), comhining terms in ES(g) and EA(), and making use of Fqns. k2.17),

(2.18) yields

c X(Z-20)Jc -ht f I'shgchS' s .* x2cs;-° " sn (-Z ,(t gt

+ -2 ,",'(x +a x cos-Zo') +sin ~ZZ 0 ~~~0 o -sh 2

0---(z-z -,

- -

04 -9 -

d4

An asymptotic evaluation of the kernels in Eqn. (2.40) shows that they

Uare all convergent at the lower li1mit of integration. However, at the uOper

limi th fist term in the first kernel is divergent. This is adjusted by

adding and subtracting the ter-n

0 0

in Eqn. (2.40) as follows:

h3C c C [h'A(+sh~cha- r I p (t) f sin( x)i0( t)d~dt + J p(t) f ['6- ±1)sin(2

*+ 2Losyz...z0 ) + x(2.-.)]J( td0)

+ f;,(t) f u -sn (z-z0) - cos (Z-zc)]Jj( t)d 'dt - (2.41)

After simplifcaio (i.e. through use of the Weber-SchafheItlin Intear-il

[5]), Eqn. (d1) reduces to

h3 C C Ox ((x) - )I -4(tK1 (x,t)dt - f (t'K2(x,t)dt (2 ..42)0 0

where

00

* ~+~ ~- (2,43)

K2(X ,t) = tx. 2.4

Oet-jtn~ g t (2.39), PrnfOrrming sV-.ii'ar m-anim)u1al"cnS v/-.e'-S

-10C-

d4C ,3 +~ao

3 3 -2Anf I(t h' { ~sha.c1 cos( X +

0 0

+ sin (Z-Z 0 ) - - cosr(Z-Z) r)~t~~ (.5

An asymptotic analysis reveals that all thP terns are bounded as 0, but as

+ the first term of the fi rst kernel 41s (14vergant. This is corrected by

adding and subtracting the term

0Gh3 C

0 0

,n Eqn. (2.45) as follows:

h3 C c(t 7h3 ,'34sh~,ch3Ii-6- f O(t) f cos( x)jl( t)d~dt + f t c)

0 0 0 0 2Sp

+ h3 ~-3-2'))shB co(£ 0 0(1-ZO/2)T2TT v T a(5-sha)c o) COSCZ-ZO)

-- s-n (Z-.)--- cosY(Z-.O)}JI(Zt)d~dt

c CO h3 6-(3-2v)sh~ ___________112 sn(XZo+ f pt)f I T Tn}-v (a-s h5)Jsn(£)-____

0 0

ZO X2- cos (Z.-ZO) + 4.sinr(Z-ZO)IJo(rt)d~dt =0. (2.46)

Afte sirnlifiatin Eq. (2~' ecomes

494

h3 C co~ (x) + f (t)K3(X,t)dt - (t)K,(x,t)dt =0 (?.47)

1 00

kj(X,t) = X2(2.48)

h3j~t (as3~a+j~xj ~j ~&

ins. f2.42) and (2.47) are the two coupled integral equations for the unknown

'I~ary functions 4 (x), O(x). These equations are solved numerically. Once

I x), ;(x) are obtained, all necessary physical quantities may be calculated.

Stresses may he calculated using Eqn. (2.19):

T (x,O) = c dt (2.+19) (t dyyf

7 i resultant load due to the symmetr~c stresses is ohtained as follows:

c cP -f T dx =-rf (t)dt. (2.50)

-c 0

The resultant moment due to the antisyrnmetr-Ic stresses is obtained as follows:

c Cm r xT dx f - t~t (2.51)

C 0

i n order' to valuate the deflection under the indenter, A, the constant an

* -us, I,.? iie. Suerir~ cns 2 and 22' y-e'Ids

. . . . . . . . . . . . .

u (x,0) L!. f a + shachB FI *s (X + E (sin( xY d

. + ao + ax a2x2 + a3x 3 . (2.52)

Applying Eqn. (2.7) yields

ao= . _sh_ acha E ( )cos(rZo) _ EA( )sin( o-L- + a1Zo _ a2Z a34

(2.53)Uwhere a,, a2 , a3 are given by Fqns. (2.33)-(2.35). Substituting Eqns. (2.33)-

(2.35), (2.53) into Eqn. (2.52) and recalling that u y(0,0) = A yieldsIy

c O - + ,B h I-cos EZo + -'t -(3- L.o)sh B sin( ZO)

f f 11 p 2-sh2a E g T - shaJ

"i " 2 - 2 } Jo( t)d 4 - - - 0 3ZZajd

C 0

+ f (t)[ f {-v a+sh~chB sin L° +o 6-(3-2v)sha cos(,Zo)}di( t)d,p -U

* a ~ 0 a2 -sh2S (as~

t ldt. (2.54)

An asymptotic analysis reveals that both kernels are bounded as + 0,

but the first two terms in each kernel are divergent as + -. This is

adjusted by adding and subtracting the appropriate terms to yield

c cf'= (t)KS(t)dt + f (t)K6(t)dt (2.55)0 0

01

04 -13-

I-( ,1 sh~ h 1-cos zo Zo sin( £O)K5 (t) - { + I30 2-shB-

31 cos (z-zo- Jij( t)dr - cs --

+ i 3-2v O - it XJ(3-£40) (2.56)

sin( Zo) Zo cos( Zo)K6 (t) 1-v r ra+shachl + I) ' CJ5(ctZd)

0h 1J2-sh2 d* -~0

1-V t t Zj. (2.57)

A quantity of interest in this problem, as well as in the subsequent

study of the dynamic case, is the rotation of the beam under the indenter.

More specifically, the rotation of the beam at the point under the indenter

ahout which the moment produced by the antisymmetric stresses is zero (i.e.

the "eccentricity" of the load) is of special interest. This point is

obttained from statics simply as

x = -e = M/P. (2.58)

Superimposing Eqns. (2.23) and (2.29) and making use of Eqns. (2.17), (2.1,)

ind (2.33)-(2.35) yields

, )~ h -r (3-2v)Sha snx~i o

s 0

sin (Z-ZO) cos ,(-zo )

* ', ZL/)-'-,x/ (yx

C(t) r h3 a-(3-2v)sha (cos xcOS z0 )U f O(t) 1- T2(I-7) ((s-sh )00

cos (Z-ZO) sin (Z-Zu)""- " +[ZO(,Z-ZO/2)+x(1'Zo)-X2/2] (2"0 +x) 2.Jlt dit

(2 59)

Note that the boundary condition S(-10) = 0 is automatically satisfied by

Eqn. (2.59). An asymptotic analysis of the kernels in Fqn. (2.59) reveals

that they are bounded as + 0, but that as Z + -, the first term in each is

divergent. This can be adjusted as before to become the following:

c C h3 sin~x+sin~o, cosV(Z-Z o ).*] (x) =- f (t)If I- a- - sh j -&x+o)J

0 0Tr 0 2 3-'v !

- [ZO(XZO/2)+x(Z-Zo)-x2/2] + -2- d

[ h3 COS X-COS, 7

+ f (t){ 7 [ ( a. shax-O o(.t)]d - q(Zo+x)}dt0 0

+ sgn(x) ( -)[Tf td +1 c (t in1(x~tHc

Ix

Thus, we can easily evaluate I (x -e).

0;

3. PROBLEM SOLUTION

4" Tn order to be ahle to solve for the auxil ry functions "x, (X,

Ss '?_.?'-{2. , (2."17)-(2.dS) must be n7n-esnaize . Ts i

04 -15-

accomplished by the use of the following non-dimensional parameters:

S = c/h e e/ h -. (x) -cy VY) (3.1a,b), (3.2)Rh 3

a = X0/h y = /h(x -cy y. (3.lc,d, (.3)Rh

3

u : t/c y : x/c (3.1e,f)

Once the non-dimensionalized auxiliary functions '(y), D(y) are obtained,

Eqns. (2.19), (2.50), (2.51), (2.55)-(2.57) and ( 2 .0) can he used to

calculate non-dimenslonalzed stresses, loads, moments, deflections and

rotations, respectively. These non-dimensional quantities may be transformed

back to real quantities via the following relationships:

yy P yy/C A = hIA/R (3.4), (3.7)Y y

P = h2 .P/(!-v)R = h ;/R. (3.5), (3.8)

*M = h ljM/(l-v)R (3.6)

In order to assess the accuracy of the results for displacement and

rotat;on under the indenter, the following beam theory solutions are use 1 :

A I 1 Z + 1 m (3.9)B 77 p C .

pt'Z_,- ) + M(zf) ,I

0t sncfli e ,t tat, 'Ithough the for'-,,,t I of t-e equ. trs -"

.1 _eor- ~r . . e 1a.e p esrs - l~ t~

the use of beam theory equations in comparison studies dictates that

conditions of plane stress should be assumed in the evaluatinn of Epis. (2.55

and (2.60). This assumption would be consistent with the use of a thin hpam

in an experimental verification of the results presented here. ror the

numerical evaluation of Eqns. (2.55) and (2.60) under conditions of plane

stress, Poisson's Ratio is taken to be equal to 0.35 (this corresponds to a

Poisson's Ratio of 0.26 if Eqns (2.55), (2.60) were to be evaluated under

conditions of plane strain).

4. OBSERVATIONS AND CONCULSIONS

Solutions to the problem were obtained for 6 = 0.2, 0.5 and 1.0, y

5.0, 10.0, 15.0 and 20.0, and for each y, a = 0.25y, 0.5y, 0.75y. It was

found that the stress distributions for the different values of Z/h remained

virtually identical. This is illustrated in Tables I and 2, which compare

peak symmetric and total stresses for different values of c/h and Z0 /h. A

typical symmetric stress distrihution is shown in Fia. 2; that of the total

stress under the indenter is shown in Fig. 3. As can be seen in Fig. 2, for

small values of c/h the Hertzian distribution approximates the stress under

the indenter quite well. As c/h increases, the distribution changes

signi ficantly. in Fig. 3 we see that as c/h increases, the location of Peak

st-ess slifts to the left and the distribution becomes less and less

oaraholic. This is because for small values of c/h, the indenter acts like a

roint load. Pence, for small c/h, we have a small region of concentrated

stress. _,owever, as c/h increases, the shace of the indenter hecomes more and

s ,s such, t e c irvature o the em is -i-h re orone ne,

* -17-

Tables 3 and 4 show a comparlson hetween the el. st~rity solution

developed here and the classical heam theory so'jt4nr. The elasticity

solution is seen to agree well with the beam theory solution. Yowever, as r/h

app-oaches Z0 /h the two solutions differ to - mucn larger extent. This

behavior is due to the fact that the elasticity solution ts valid only for

c/h << Z0/h.

It is important to note that the rotations in Table 4 have no mean-ng

unless they are convertep to real values. Since the elasticity and beam

theory solutions were derived using small angle approximations, it must he

ensured .hat 7 < ;,, where .0 is a preretermnea aqle below which rotations

are considered "smal I." Thus, it is possible to place a lowpr bound on R, the

radius of curvature of the indenter:

However, 4 t ust also be ensured that no yielding occurs anywhe-e ;n the beam,

..e., that "max < where

max= h/21c, 0 h3/12 (42) 4.3max max

and y is the yield stress of the material. The maximum moment occurs at they

fixed end of tne team and is g'vee by

la : O

O' = c';-; r 0,< '/ S l~r n, hcu 'dH o D.

_a*<

20!-'3dJr~ ) z45

f % = ?0.5 (0.3' rad , = 55,000 ksi and (-I,5? k, then Fqns. !.1)

ao (a.5) may be combined into the following:

R/h > max(2.865 ,.0(p4)L 46

Judg4r'g from the values of , P and M, it is evident that ?.R656 1.60F(ao - M)

for each cise considered. Thus, En. (A.6) reduces to

R/h > 2.265_;. (4.7)

Under these criteria, Table 5 shows the minimum allowable values nf R/h

to ensure small deflections and no yielding. It is seen that for large c/h

ai'(r for lona beams ,with c/h > 0.5, the indenter is practically flat. It is,

therefore, of some interest to solve the problem of a flat indenter. This

solution -s found in the next section.

5. C EF II: FLAT IDENTEP - PRORLEM FORMULATION

The problem to he solved is that of an elastic layer of thickness h and-1-In , hy ts uo P'r h) .h T! T e

._ . ntn. , hv a flat punch on ts upoer sur'ace (Fic. . Th

'i.<a.i C t r:ns 'or the elasi ity proeM and h3m t r rqhlem rr rfo

- , •- upooAi'' _p~'~j-r

(,h) = 0 <5.1

yy l<

t (x,h) = 0 x < w 5.2)xy

U (x,o) = 0 x(.xy I<

T ry (x,O) = 0 c < xl < 54)['yu Y (X,0) = L 0 < XI < c (5.5)

S0 x = - o (5.6)

uy =0 ( (5.7)

M 0 x= -9.o (5.P)

V =0 x Z- 0 (5.Q)

[t is further imposed that the stresses be non-slngular at x = C:

I y (C,0) < i " (5.10)

Ths condition ensures a smooth deflection of the beam for x > 0 and results

-- n a "receding contact" F61; here, as the load, increases, the contact length

* decreases with load.

Ths suitable elasticity solution for this problem is given by Eqns.

Olt (2.11) through (2.15). Using this formulation, the normal stress is gven by

(-yy , .f -ES ()coscx + EA( )sin.xd . (5.11)• 0

Por a flat pinch, to haie singularities at x q c, we ,t

c

-- 20-

C

EA(.) jic) + ((t)J1( t)d . (5.13)A0

i Substituting into Eqn. (5.11) yields

xC C- H(C-X) + f P(t)dt + x f p(t)dt (5.14)

Yc2_x 2 X t2_x2 t t2 _x2

The first term in Eqn. (5.14) is singular at x : ± c. For the singularity to

vanish at x = c, it rmist he true that A + R = 0. Then Eqn. (5.14) becomesI

c xc *(~t c tTy (xO) - - c - x H(c-x) + f(t)dt x f (t)dt5.5)

VI t t2-x

The moment, shear, and average slope are given by Eqns. (2.20), (2.21) and

(2.23), respectively. As before, the beam theory solution is taken to be

U B (x) - ao + alx + a2x2 + a3x

3 (5.16)

resulting in expressions for moment, shear and average slope given by Eqns.

(2.27), (2.28) and (2.29), respectively. The constants a a 2 , a are solved

for as before, with identical outcome. Eqn. (5.5) is treated in a similar

manner. Differentiating with respect to x and separating into symmetric and

antisymmetric components yields

au E

Y (x,O) + 2a2x = 0 (5.17)A

- (xO) + a, + 3a3x2 = 0. (5.18

,'nsiderinq eat e4uationr r a:i sa

. -21 -

as before resuIts in the fol owing equatons for A, R, p(t and (tl"

AK1 (x,c' h3 C#x :..

U -" -- + , ,(t)Kix,t'dt" + RK,2 (x,C + (tlK 2 ('x,t Ht = ,5.0

0oCC

AKI(x,c) + v(t)k3 (x,t)dt - R K(xd) + K--(x) - j ,(t)K.(x,t)dt n (5.20)

0 0

where

x {h3(, +shichk + j> Jo(,x) + x cos (Z _Zo);J o( )d + ×0 02-sh2 7

~5.?1)KAI

K (),n) = nx (5.22)

K3(x,n) = X2 (5.23)

j 0 +shichl<,( ,n = h -- --- - 1(x )J ( q d 5.24)

0 o 2sh

and r = c or t. Eqns. (5.19), (5.20) are two equations in four unknowns, A

. third equation was obtained earlier by enforcing non-sinaularity of normal

- "stress at x = C:

A + = O. (5.25)

• The fourth is found by enforcing the boundary ccndition given by Eqn. (5.51.

Af.er solvr fnr the constont ao as before (with M-entical results), qns.

05.5), (?.15), (2.21), (5.12), (5.13) and (5.25) may be used in a manner

O 4 .2"_

C C-I (t)Ks(t)dt -f (t)K 6 (t~dt

0 0

K6(c) -K 5('c)

* where

1-cos~zo si n Zo 3 cosE2.-Zo)K5(t) ***V~f(a+shacha + 1)(----J-£ J~ndo s2-sh2 h3 E

-----c +. X 0 -.Z.(3-o (5. 27)

os - ){~s~h

K6 (n) Z En )d E ___-

o 62-Sh2a 0- -ha J(nd

1 V _ _9__ _ - U ( 5 . 2 8 )

2.0+ T, 2

and q c or t. Making use of Eqns. (5.25), (5.26) in Eqns. (5.19), (5.20)

vy el ds

h3 C Kl(x,c)-K2(X,C)6 (x - f (t){Kj(x,t) + K6(c) - K5(c)TK()d

c 0KI(x,c)-K2(X,C) K(~d-f p(t)' K2 (X ,t) + K6 c -K() K(t)}dt

-Kj(x,c)+K 2(X,C)K6(c) - K5(C) (.9

0h 3 c K3(X,c)+4K4(X,C).0 ~x + f -(t)(K 3(x,t) + K6 (C -5T F5 c K5 (t)ldt

0

c K3(x,c)-KL4(X,C)-~tYK(x,t) -7K 6 c) - K5 (c)

-23-

Thus, Eqns. (5.29) , (5.30) are those used to solve for the unknown auxiliary

functions -J(×), ;(x). These equations are solved numerically. Once

y(x), '(x' are ohta. ned, all necessary physical quantities may then be

cal cul ated.

Stresses may be calculated using Eqn. (5.15):

c cT (x,O) - - x H(c-x) + ____td + x I ,(t)dt (5.31)yy

where B is given hy Eqn. (5.?6).

The resultant load due to the symmetric stresses is found to he given by

CP 7B - T f (t)dt. (5.32)

0U

The resultant moment due to the antisymretric stresses is found to be given by

C

M - cB - I t$(t)dt. (5.33)

0

The rotation of the beam at any point x > c is found to he given by

3- 2,) + c* g(x) B[K 3 (x,c)-K 7 (x,c)] - p-#-h + [ ,(t)K 7 (X,t)dt + f (t)K8(x,t)dt

o 0

(5.34)

0

whe re

x 3" S- -I -- 7 " -S q'

-TS :-,2 --

-24-

7T + T 3-2v[Zo(Z-Zo/2)+x(Z-Z°)-x 2/2] + 7 3 (5.35)

K8(xn) 1 3'COS- j - sn) dh - (Zo+x)n (5.36)0

and n c or t.

6. DROBLEM SOLUTION

Following a scheme similar to that of Sec. 3, we define

a (X) = -ACy VY(y) (6.1), (6.2)h5

-(x) : D (y) (6.3)

Non-dimensionalized stresses, loads, moments and rotations are calculated and

may be transformed back to real quantities through

- =y : / C ( 6 .4 )

P =vA/1v (6.5)

M ( (6.6)

(6.7)

0 -In order to assess the accuracy of the solution ohtained, two tests are

performed. Fi rst, the rotation of the beam just beyond x c is compared t

'a* q ie4 hv e ..he standard -;eam theory solution

*, -25-

where E c+. Second, from the iatu-e of a receding contact problem, as the

contact length decreases to zero, the load P increases to a certain limit

value. ?ecause the moment , goes to 7ero as the contact length goes to zPro,

the beam theory solution for displacement tells us that the load DE must

approach the beam theory load PBT for a cantilever beam prohlem:

"-I FT= 30 /Z (6.9)

The limit load um PE is compared to the value of PBT for each casec/h-EO

considered.

Once again, the use of beam theory equations in subsequent comparison

studies dictates that conditions of plane stress should he assumed in the

solution of Eqns. (5.29) and (5.30). As before, Poisson's ratio is taken to

be equal to 0.35 for the numerical solution of these equations.

7. OBSERVATIONS AND CONCLI)SIONS

* Solutions to tha problem were obtained for 6 = 0.25, 0.5 and 1.0, y =

10.0 and 20.0, and for each Y, Z = 0.25y, 0.375y, 0.5y and 0.75y. It was

found that the stress distrhutions for the different values of 2/h were

v"rtuall./ identical. A typical distribution of the total stress under the

-enter, nor'alized by a factor of C-X2, 4s shown -n F .c. 9, An

examinit2.n of the plots of the total stress undrer the indente revpaiS that

Z. s i~o- -, s; z]l - at x = - t, hut zero at x = c. 7 T -Ore , . . .r

O 7 s

results from a more pronounced effect of the antisymmetric component in the

solution for cases where c/h is large.

Tables 6 and 3 show a comparison between the solution developed here and

the classical beam theory solution. As can he seen, the elasticity solution

agrees quite well with the beam theory solution. Again, it is noted that the

1 rotations in Table 6 have no meaning unless they are converted to real

values. Following the same procedure as before, an upper bound can be placed

on the ratio of displacement under the indenter to beam thickness:

A/h < mi j;O/;, (1-v)a /6v(aP-M)j (7.1)

Assuming the same values of ;0, ay and u, Eqn. (7.1) reduces to^y

,I/h < 0.349/u. (7.2)U

Under these criteria, Table 7 shows the maximum allowable values of A/h to

ensure small rotations and no yielding.

The phenomenon of receding contact in this problem is borne out hy Fig.

7. As c/h goes to zero, the non-dimensionalized load parameter approaches a

limit value greater than zero. In Table 3, the limit load of the elasticity0I'

solution is seen to agree quite well with the limit load of the beam theory

solution. Furthermore, the limit load is highest in those cases where the

i9denter is close to the wall.

In Fi. 7 tnp relationships hetween the loads and contact widths for the

V~; 3 r~ Cises C~nsidred a- e3 to vi.late the condit-on imposed on receding

01i :-- -- .-eb - e~s;:re te e!iSAece and u-queness of t'e- solutions

' , '1- 7-

the applied loads on that heam. The contact length in a recedinq contact

problem will be independent of the level of loading only if the ratio of tho

resultant moment to the applied load is constant 'o, all llvels of loall-n.

In the problems solved through the course of this study, tH ratio nf moment

to load changes as the applied loading changes; thus, the contact length will

change as well, the imposed conditions are not violated, and the problems

studied are indeed receding contact problems.

S. ACKNOWLEDGEMENT

The authors are grateful for support from the AFOSR (Grant AFOSP-22-003301.

0!

04 _-

*1

References

2. Keer, L.M. and Miiller, G.R., "Smooth Indentation of a Finite Layer,"Journal of the Enqineerinq Mechanics r)ivis-on, ASC-, Vol. IOQ, IQ3, po.706-717.

2. Keer, L.M. and Ballarini, R., "Smooth Contact hetween a Rigid indenterand an Initially Stressed Orthotropic Ream," AIAA Journal, Vol. 21, 1983,pp. 1035-1042.

3. Keer, L.M. and Miller, G.R., "Contact between an Elastically SupportedCircular Plate and a Rigid Indenter," International Journal ofEngineering Science, Vol. 21, 1983, pp. 681-6q0.

4. Sneddon, 1.N., courier Transforms, McGraw-Hill Rook Co., Inc., New York,N.Y., IQ51.

5. Watson, G.N., A Treatise on the Theory of Ressel Functions, 2nd Ed.,Cambridge University Press, Cambridge, Great Rritain, Iq66.

5. 9undurs, J. and Stippes, M., "Role of Elastic Constants in Certain* Contact Problems," Journal of Applied Mechanics, Vol. 37, 1970, pp. Q65-

970.

U

0 -2

.h=10.0 Z/h=20.0

Sc/h 2o/h=2.5 Z /h=5.0 Z /h=7.5 Zo/h=5.0 20/h=1.0 Z /h=15.'rS_ 0 0. 5 0 0 0

0.5 0.622 0.622 0.617 0.635 0.635 0.622

0.2 0.632 00625 0.670.25052 0. 632

1.0 0.560 0.561 0.561 0.560 0.564 0.562

Table 1: Maxlmum Symmetric Stresses (Case I

*I. iZ/h=10.0 X/h=20 .0

c/h /h=2.5 . /h=5.0 z /h=7.5 Z. /h=5. 2,o/h=10.0 2. /h=15.n0 0 0 0 0

0.2 0.636 0,636 0.636 0.636 0.636 0.636

0.5 0.6a 0.649 0.645 0, 50 0.649 0.640

3.0 0.932 0.983 0.981 0.982 0.998 0083

Table 2: Max-mum Total Stresses (Case T)

0

04

2 /h=2.5 2 /h=5.0 Z /h=7.500 0

Ream lstiit eam ReamElasticity Theory TheoryTheory

I I

2.0 2.0' 16.1 16.1 54.3 54.3 0.2

15.5 15.3 127.3 127.5 438.6 A39.0 0.5

376.2 383.5 3487.0 3498.0 8705.0 8716.0 1.0

z o/h=5.0 £ /h=10.0 z /h=15.00 0 0

ElastlaityiBea, Rean BeamTheory Theory Elasticity Theory

16.2 16.2 127.1 127.6 435.7 435.7 0.2

123.5 128.8 1000.0 1001.0 3466.0 3467.0 0.5

3327.0 3338.0 35890.0 35900.0 101540.0 101570.0 1.0

a

Table 3: Displacement Comparison (Case I)

z /h=2.5 Zo/h=5.0 Xo/h=7.5

*ati Beam Elasticity R eam BeamSElastci Theory Theory Elasticity Theory c/h

*186.4 215.9 48.8 15.1630 1704.0 10

2. /h=5.0 2. /h=10.0 2./h=15.00 0 0

*Iasticity Beam Elasticity Rem Elasticity Rem c/hSTheo ' y Theory eory

a1.0 1 9I 1d35 43.6 0.2R 1.9.3 1 3.0 385.0 38.6 0.5

* - .o/h5 E2o/h~ 2'Kllfll.O fl/h 1

II

1~' 'eam jem1aa

ii ~~ ~ D Elsict Ihoy Eatcit hor EaTity Ter c/

z / h= 10.0 //=1?0.9

C 'n1 L. /h2.5 ./=. . Ih=7.5 2. /h=5.0 2 /h10O.O 2. /hiq n0 0 0 0 0' "0"

0.2 4.0 14.0 31.0 14.0 55, 125

0.5 .6.0 108.0 250.0 1 lr .0 4?.3 92R.0

1.0 535.0 2S25.0 42.l .0 2695.fl 5080.0 2R690 .0

Tat1e 5: Minimum A 11 nv e iValues of R/h (Cas e T

• h=2.5 ZL/M=5 .Z /h=7.5

Rastic Ream I Beam Ream /ElastcryyElasticity heory Elasticity I c/h

T hoTheoheoy Theory

S4.08 0.54 0.280 M85 0.192 0.193 0.2

S 0.457 0.463 0.263 0.264 0.183 0.183 0.5

0.403 0.303 0.2.39 0219 0.172 0.166 oO

, i5.0 i/h=10.0 2.i/h=15.0

Elasticity Beam Elasticity Beam Elasticity Ream c/hTheory Theory Theory

0.2?0 0.286 0.145 n. 141i 0.098 o.0qg 0.?

0.263 0.26A 0.110 0.140 0. 9 5 0.095 0.5

.229 0.21Q 0.133 0,]30 0.0,2 0.09 1

.4Tb e P- in merS ts [

! Z/h~O.O Zh=20.n

S &2h.5 Z/h=5.O ,lh=7.5 ./h=5.1 in/h= q., Z/h=.9

0.. .58 1.24 I1R, 1.2 2.39 3.54

0.5 0.76 1.32 1.90 1.32 2.d3 3.63

1.0 0.36 1.46 2.02 1.46 2.62 3.82

Table 7: !Maximum Allowable Values of (/h (Case 1-)

2Z/h lir PE (xlO- 2) PBT (xi0 2 )

2.5 3.104 3.200

7.5 n.118 0.119

j 5.0 0.39 o.no

10.0 0 .fQ9 O.n5O

15.0 0.015 0.015

Table A: Limit Lcad Cora -ison (Case IT2

04

eg

U

I

-V/

C

1-- I

0

-c

0

0~~

.4

I _.

i0

*X

-- c

__ _

0N

0-

.

1 1.000

.800

c/h 0.5 c/ h 0.2 a Hertz

0-.600-

.400-

.200O

0.000 .200 .400 .600 .800 1.000

Fizure

c /h 1.0 C y/ 1.-000 Cy

.00

.40

.20

-1.000 -.500 0.000 .500 I.000

0 x/G

F 7u r 3

20.000-

16.000- 10 /h 5.0

CY /h =15.0

I1 0 /h =10.018.000

4.000-

0.000 I

0.000 .200 .400 .600 .800 1.000

0 c/h

UI

20.000

18.00010/h 10.0

16.000-

14.000

12.000 10 /h 5.0

rZ 10.000

8.000 10/h 15.0

* 6.000

4.000

2.000

0.000 I*

0.000 2.400 4.800 7.200 9.600 12.000SR A/h 2 ( x 10')

Figure 5

. --

6'

04

1.000 Cyy~YiX IC)2

6p

.800

c/h =0.2

.200h05

[-~

100 .I50 .l00 0 50 1.000

ii)x /C

04

5.000

4.000it/ =.

'0

--- 3.000

2.000/

1.00

0.000 = .20 .0600 .800 1.000c/h

LOW VELOCITY IMPACT ON A CIRCULAR PLATE

L. M. Kee- and T. K. Woo

Department of C-*vi1 Eng-neer~ngThe rechnolog4,Ca1 Institute

Northwestern UniversityEvanston, 11]4no:s

04

An tvest' ,it'on is inade on tee low veloc4-y :impact on a c4rcular p13te

C y a i indent er . The ;)ate response to 'mTpact shnows; .ome of Lit not all

of the s4 4Iilr problem for a beam-. The local c-ontact stresses were developed

from an e1 asttyteoryv, ,qnile tne global stresses were obtained from plate

Tne pronlem of tne low v elOCi.t'y Thdentation of an isotrop 4c ci*rcular

9h, t-c plate by a 'Mgtd puncn has neen i;nvestigated. This studjy uses the

assumption that an elasIt tneory may be used for calculating the local

effects and a Berncuilli - Euler plate theory can be uised for calculating the

overall1 dynamic. response.

n tni4s p r ole7 a spnhenic al1 ind en t er w"Itn. a ve ry lar,4e ra iuijs o f

curvature strikes the plate with a low i mpact vel1ocity. Since both the0 2~~~Lre553 re loa! and contact area are unknown and depen uo tm, hepobe

s formul3ted ano reud to souin o f ,i otr r nerleutn

wo t.ypes of solujtions are superposed. Pr :s tee- static solutten

analao's to thne one developed z y Keer anr '4Ai1>r I;teother is a standard

dYnam plt solution3 whiten can '-e obtai-ned fromentll- Eu ler plate

3-3li vs se e eto. iraft 2>'%

*~~~~~~~~~ n' rsr sroto sasue o~ Tcwe >C

0-4

Snie time ,-4se pressi re d4st r4 'u 4on 4 s cal culIatepd. 'in example f;,r the Case )f

plane strai4ns 1 velocity ias been given i.n an earlier paper rny 'leer a-d Lee

PWOBLEM FJRML'LATION

The formulation uses tnie assumption that elastodynamic effects are

negl~l 4 ie wnen compared with the effects due to the static deformation.

~ rdynamic effects are incorporated i nto the plate bending sal uti on.

,e~ solution of this problem is obtai;ned by superpositi1on of the static

solutofl with a dyna3mic plate solution.

The boundary conditilon for an elastically supported layer loaded by a

s-nootn rid i 4, ndenter are as follows:

ZZ(r,h) = 0 0 <r < (1

* tzr (r ,h) = 0 04r < - (2)

r,o" = 0 0 r~ ( 3)zr

z(r,o) =-p(r) 0 r C(5)

wnere 4s tne radius of the contact area.

.n e onundary conditi*ons at the ends of the plate are:

=iz 0 r a (6)M =0 r =a

*~~-i ', ~ a~ so ne plate, 0 4s the moment, K is spring cnstant,

-e -r, j-1 ) n 4 C~ t c<,e S

or

le e stic! tj solution that represents loading on the upper surface of

y e.st': lyer s jiven in tne following for by ise of integral transforms:

2ow, ;) 1-2v)(S + sh3ch3) , jZSM2 3>Qi~z)r 0

_ (1-2vlsh2 -r 32 _Z(3 + shIch3)ch(Sz)}

i(9)

2;u - , 2(i-v)sn 23 - 32 + -Zz 0

(3 + sh3ch3]sh(jz) - 2(l-v)(3 shchO )

- ,zsh 26ch(jz)J (,r)d (10)0

I~~~ ~ j Tr Ei)5 - shich;Ish(.iz)

32 + - -z(3 + shch3)sch(Zz)Jo(,r)d

--F-o

1 r E P32 + (1 - 2v)sh2 3 - i

r + shachB)]ch(,z) ," [zsh 23

* - (1-2v) (S + shSchS)sh(z))}J (,r)dj (11)

z , o E( ) {-32 - ,z(3 + shjch3)]sh(iz)

+ zsh 2 IChZ))j,(IrdF I

+ +

- .~ l 1c, : I J .(-:r'-d*- 1

6 . ."' n ,n : s : s" ,

,- ~ - ,--- - . -. 7

s4"e i,; -P< us, resp-ac-4ve1 s.

rc 3 ,.

s- n ) Ve r r q , 7

7

0

7~n tn i-neaa~ o s ci'A, e~ f, st d r vmothen -io uTie nd asea' slp~e

laate stlitt-*o 4n sieen! anc tha'tht e

077

0*

S rj - 0 *:: - - - 'flr0~ -

J' aS {'I, q1 ) ;s'. d1 ) A ' e t e r iii n e d t o h e

ara

I + ('ch - sn

s

s V 3 3 c.a n del-erined -" 1 a S4 Term1 ar oa y

- , ,-,-snchF

-2 -:dh (21)• , ,,- ,SS h - S 2: .

'-_ m',D y " s\ ln

.00 - nCK

* ,- -

[ r o n q Fron eqations 115 , I1, (16), (20, and (22)

I -c ° h3 3 + shen3z o 7 I -shr

(Ja 2 - r2 -Jo(,a)• ×~~( (o a ) - J o( r ,) 7 + v0

1 32ch3- sh 23

-7 1 + 2v ( 2 shT J I a2

+ h2 ( - (3 - 2v)sh3)j (- )lld7 23)+ 2(I - v)a 3- sh")d3()

As 0 0, an asymptotic analysis shows the terms in equation (23) to be

bouided. However, as , -, the terms are divergent. By adding and

s-Dtract ng

S- r p(r )dr -- (Jo( a) - (r))d (24)

(3-2v) a - h2 a rc r'p(r')

J( ,r)J( a)d~dr (25)

nte difficulty can be removed.

3YNAMIZC PLATE THEORY SOLUTION

A solut4on can be obtained by using a normal mode expansion together w'ito

Laplace tLansform. The equation governing the normal ,node expans;on 4s

S

r' re tne -,ice sn apes anc (i s tne n: na rmin c )perat r Tne

":o1 c-an e wr4tten 3S Y y +~ wniere Y, Y2 sa Lsfy reSpeC,'?v

± ~2y~ Q 2y-,2y +'~l 0 (21)

~vnr 4 s the Lapl ace operator. Because of ax-*al symmetry, we ontai n

rn

* ~Cf as tne so0b1t;on

J< r) + 3YO(2 r) (28)

* - Iv 32yr n

,,~n-*~ nias tne Soljt~on

n C (r) + DK 3 n r (29)

nliT

0 ~VA SC1 ~t>D ~r)

S4ne~tn' j nave j Si ngUjarity at r J , they i%-t strjze.

a .-js, h sol'ution i.s reduced to

Y" r) ,AJO(,r + CIO(3 r (31)

-or tne clamped Case

3Y (a)Y (a) 0 (32)

Su -js 4 i ttng int o equ aton (31) we get the f requ ency equ a tion

a-, -'v n, ~

Ia{~a)J~(3d) - J(na).a)=(3

Y (r) r) r (3a)

nns nh sa0(3o ns redce n

r For the simply supported case, the boundary conditions are

0

y~a n at r =a (35)

Uat r =a (36)

3'- 0 32)

04

n n

Yn JO( n r) - Jo(3n) n. (r) ~38)

To derive tie dynamically loaded solution, we take the Laplace transform

)f the governinlg equation,

r-3r + -h -~~-=p~r't) (39)

wrie re

G2 =

Sh

The Laclace? t-3n-sfor of 39) -s

2-- + - ) 2 7(rS) + s2 7(r,S) = f(r,s),ph (O*r r 6r

and tne t-ransformi, 7(r,s), 5(r,s) may )e expanded 'n a series of normal

.lodes Y \'r, as

7(r~s) a(s~Y (r)n'I n. n

Sr,S n n9. ~"~ (rK

44

s = (as ) + (dJ2 ( aR ).

S r3 (r,s)i f_,'r ,r 2C)

where Yn r is the sol ttOn of

n7Yn(K - 3n r) = 0 (43)

32( + ±12 )2 9P a Y (r S2 fl- aY (r)'5 r r- r n=l n n n(rl ann

1 (44)- 7 1bnY (r);n ntt1 n

U

12 32 4a y (r) + S2 Y (r)

(45)

- b 1nb Y ((45

J (3 4 2 S2)a Y r) 1 w b (r)

n n n -n -- ,1 n6

I fl -n n - Y (

,i n

0 , i i " - " " " I " i - | : : : " " - . . ' " " - ' I • -t.

'~r~s) ph a- (J(a n J',(aBn) n- !- +

n. n

a y(r,s)Y (r)cr (48)U n

,. Snqce,.2

=(s G234) s'n GB 2 t (49)n n

Yn(r)1 1 n r

o G3,a' n ) +pn G n nzL o,,

.a r'Y (r) t p(u,) s4n G3 2 t - r)d~ir (50n n

['- ' ., L ,T .J' F R ,!L TI

*he :'tegra' eqiiaton - o e 3o ye-s I s a VoJ Ier ra inteqra -<;oton oyen

-2.;

0-

or

¥ . *m. b .. * ,

-- r

'i+" 1 go, , a - J3(ar')' + J6(a - .

0+

hth2

" ] +2v)( " h 5 ) J, (,,a).,-~~~ as3l22v( snZ3

k]J d( Ia)

h 2a O- (3 - 2v )sh ) J ( a )] }d~ d r -

1 2(l - t)a ( sh3

1 + 1 y Yn r

rY (r') p(u,,) sin G32 (t - )drdr' (51)n n

A e contact rel'cn we have

(Irt) 6(t) - 0 < r < c(t) (52)

wnere 2(t) is the relative approach and R is the radius of curvature of the

stri er.

* From Newton's equation of motion

<t) = -2M tc)-t 2M , (t - 0 c rp(r, )drd (53)

:, 7 is the velocity of the striker as it hits the plate and M is the ratio

7 - plate t a tat of the str Ker The method t' determIne the

, .,.ct a . :s ,ive , , , et . ,

04/

N A S~ Yi AD RES' TS

sn s q t P i tg ral equ ation (,5 2) s omie techniques have neen usedl tL.

3a. care ofto:e s nctular4 ty and speed of convergence. Ti rd order

dulinoamals are used to approx-,nate tne uinknown finction, since increasing the

order of polynomiJC to higher order does not mean that better accuracy w'il 5e

acui eved . Filon's formula 4s used to 4inprove the accuracy of the

tri gnometrical parts of the kernal . The overestimated contact area and

Tieifco starting valuje of the Volterra integral, equati- 4on help to ensure the

aL cy of thie nurieri cal results.

Calclations are based on a variety of geometr-*cal and material

2~'m, eer s. They are the fol1ow-'ng:

Ra.d1ius of StriKer: 40 cm

Radius of DpIte: 10 cm

Elasticity Constant: 200 Gpa

7h4 c-ness of Plate: I cm

.1~s uisn s Raio .25

Density of Plate: 7.8335 g/cm3

- r-s 2-3 snow the contact area and contact force for cases of

~eetve Iociti es and mas s rat s . As tne mass ratio is increased the

~ cc o -pact becomes shorter; however, changing the initial velocity does

-o Jf~ the per'od of -Iinpact. By comparing witm eer and Lee

?~~ npc fr r pae sshorter t"hun tnat of the e:.

4~ e*'o rij ccde to t e stror ne be y e2ffect i e t i re-

- -

- - - - - - - - - - - - - - -- - - - I ~-- rL

- rDqeqt as [I]. Tne pressure distribution is shown in Fig. 4-7, whi:h

cor -spond to several values of the mass ratio. The non-Hertzian solution

ne is to apper in g. 4, and the wrapping effect, Keer and Miller [I],

necones more important. The pressure distribution begins to concentrate more

closely to the edge of the contact than the center.

Sigures 3 - 10 show the pressure distribution for different velocities.

t is seen that the ratio of the peak forces tends to be proportional to the

ratio ar velocities. Ths shape of tne force nistory anc contact area are very

s' nilar for these cases.

Fi gure 11 snows the pressure distri bution for the impact of a clamped

plate by a striker with velocity 20 m/s and a mass ratio 1. As can be seen

the rate of increase of the force is the same as in the case of the simple

s,,:port plate. Non-Hertzian solution occurs during a portion of the impact

c:' . Te period of impact is noticeably shorter since the clamped plate is

Figure 12 snows the force history and contact length when the thickness

of the plate is 0.125 cm. As the plate becomes less stiff, we observe a

double striKe phenomenon, siilar to that noted by Keer and Lee [4] for the

related beam proolem. The velocity of the striker fortties case has been

reduced to 2 m's to make elastic behavior assumption valued as the thickness

,2 tne plate 4s reduced. Figures 13, to 14 show the pressure distribution for

tqs re:uced tncness case. At the timewise point near detachment, the plate

* :~: -,s r ,hr h-gh stress.

-i e•

: "et j v _a+ ver . t +e _ ~i e stre phenoiienon cn still occ jr t t'e l

e-s 'yt zcnt st Re:. ela cd o-serv :onS to ')ur results are calcjl I

i? s i -14 - '-:,or , a ol oel at ve y thi ck )l 3tes t.ne re! 3t '. n

•orce an ime 4s no'.ei to oe nearly constant for a portion ot tne t' 'no

jraraineters such as these may prove to be useful for the design of an impact

.es. -n wh4cn ne force 4s kept relatively constant dur,ng a prescribed period

.re au.nors are grateful for support from the A4 Force Pff;ce o:

* Sc~entf'c Research Grant AFOSP 32-0330

001

•0.4

Ue . e- , L.%I. and '4 1 len, G.R., "Contact Between an Elastically Supportedcilar Plate and a Ri gi indenter," International Journal of

Enjneeriny Science, Vol . 21, No. 6, 1983.

2. r dratf, K.F., Wave Motion in Elastic Salids, Ohio State University Press,-1975.

,. n;'ai , N., Keer, L.M., and Mura, T., "Non-Hertzian Stress Analysis--'iormal and Sliding Contact," International Journal of Sol ids and

* Struct-res, Vol. 19, 1983, pp. 357-373.

. Keer, L.M. and Lee, J.C., "Dynamic Impact of an Elastically Supported:'ea--L3are Area Contact", International Journal of Engineering Science,

2ess.

.,ds;iith , W., Impact, The Theory and Physical Behavior of CollidingSolias, Edward Arnold Ltd., London, 1960.

0

U1

•0

0I

S

'p

,S

7gj re Capt i)ns

I -q. 1: Geometry for Circular Plate Loaded by ?lgid Indenter

Fi. 2: Contact Lenth Vs. Time for Simply Supported Plate Subjected toimpact (v = 20 m/s)

Fi. 3: Force History for Simply Supported Plate Subjected to Impact8 (v = 20 mls)

1g. : Pressure Distribution for Simply Supported Plate (M = 0.75,v = 20 m/s)

.5: Pressure Distribution for Simply Supported Plate (M = 1.0,Uv = 20 m/s)

* ig. 6: Pressure Distribution for Simply Supported Plate (M = 1.5,

v = 20 m/s)

* Fig. 7: Pressure Distribution for Simply Supported Plate (M = 2.0,v = 20 m/'s)

Fig. 3: Pressure Distribution for Simply Supported Plate ( M 1 .0,v = 15 m/s)

Fig. 9a): Force History for Simply Supported Plate Subjected to Impact

• . i Contact Length s. Time for Simply uppur tu Plate JU Uj Utad oImpact (M = 1.0)

g.l 1d: 0 ressure Distribution for Clamped Plate (v = 20 m/s,M = 1 .0)

Fig ]1 a Contact Lengtn vs. Time for Clamped Plate Subjected toTransverse Impact (v 20 m/s, M = 1.0)

* :,' -orce History for Clamped Plate Subjected to Transver-Impact (v = 20 m/s, m = 1.0)

-'g 12 . force History for Simply Supported Plate Subjected to impact (V2 m/s. Th'ckness of Plate = 0.125 cm)

2 ~; ,- Cn acr Lengtn vs Time for Si;np ly Supported Plate S!,Jec, Pd.1 I = ? fl/s. Tnickness of Plate = 0.125 :,0)

* . r;.j .str on fr Simply S(ppor'ed Plate ' _ . V

i 4's. T> -ess of Plate 0.125 CM

.-t, e~ : ater SI.l - '_

0'.

eq

a

'1

A

U I c~ ~

*

C

C

01

@4

247

20T M 0.75

'D T

12-

00

oT, SECON (10-4

U 150- M=0.7 5

- 120

rj 1.5S60-

30 2

00

0 3 6 9 12

T, SECOND (*104)

04

0

.4

1.0 G~a

(M -m o ., s

... 4

1.2 GPa

Li

5:Pnrssure Distr4'-,;±c: "-r

V ., -2 m's)

1.0 GPa

F,.6: pressure D-lstributCf for

Sirnm-3 Surcortsd Plate(M~~~2 m I.5 -S)

1.0 GPa

4 see

* . :Pressure D-tibton for

Si~nply Supported Kiate=.0, v; -0 ::O s)

I. GPa

771 14S-, secm'

00

t

DS

1 1201

go-

00

00

0 .36 9 12

T, SECOND 10-4~)

6 24-

20 -

S16-

Sv= 20 m

E 12-

z/

V 15 m

4-

0 3 6 9 12T. SECOND (*10-4

0n r c ' T*:,-

( 1.2 GPa

- ~ sec

ClarmDed Pla tcI~~ ~~ ( 0 -m I0'~s, !-

20

I TI--I-

, " 16 /

z 12

z8-)

4-,•c

4

0 1 2 3 4 5 6 7

T, SECOND (* 10 - 4)

er

F-4 . 1 , C o n t a t L e n ' t' v s T im e o rCied ? ate Subjected to7rans':er e D,3c t,( 0

n, " , Y = 1 .)

eq-

U 2751-250-

225-

200--

175-

150-

S 125--

1001

251

0 1 2 3 4 5 6 7

T, SECOND (*10~

Fi.11(b): Force Pso--:rCaPlate Sulcjcc:e i t

-.erse T±7c2Ct (V .:/s

L -

--

|- -//

-" /I I

o /' / /,

O I M l '1 I

0 6 12 1'8 24 30 3'8 42 48 5'4 60

T, SCOND(x1O -4)

- *, " *,,.<- . • " -

16-

o II

~/

12-

6 ,/o\

z M=1.O

E--

M=1.5

z

4

0 6 12 18 24 30 36 42 48 54 60T, SECOND (x 10 - 4)

.-0 . , : : t : t Z. :, . ,. _ - ! -. ,,

0 ..

0"z " : -

0~

U

.46Ga

0

U

0

zr~s~3.~* rtx-1:

01

SI

.46G~a

Uig.IL:Pressur szmii.c

1V =.5, V 2 -

Thick-:ess cf in0 0. 1,5 C6

Kmoo

o 7