Loading and unloading of an elastic-plastic fibre-reinforced cantilever

MECHANICS RESEARCH COMMUNICATIONS Vol. 19(4),333-340,1992. Printed in the USA 0093-6413/92 $5.00 + .00 Copyright (c) 1992 Pergamon Press Ltd.

LOADING AND UNLOADING OF AN ELASTIC-PLASTIC FIBRE-REINFORCED CANTILEVER

A. H. England and T. G. Rogers Department of Theoretical Mechanics, University of Nottingham, Nottingham, NG7 2RD, England

(Received 12 December 1991; accepted 31 December 1991)

1.

Introduction

This paper considers finite plane deformations of a cantilever beam composed of an elastic-plastic fibre-reinforced material in which the reinforcing fibres are aligned along the beam. The cantilever is deformed by a transverse point force applied to its upper surface as depicted in Fig. 1. It is assumed that the point force is a dead load. The effect of the fibres is to make the mechanical response of the composite material highly anisotropic with the local fibre-direction taking the role of a preferred direction. When the matrix material is metallic, finite deformations may result in plastic behaviour with a relatively small change in volume. Accordingly we assume that the composite satisfies the idealised kinematic conditions of incompressibility and that the composite is inextensible in the fibre-direction. We also assume that the fibres are continuously distributed throughout the material and deform with the composite. Pipkin, Rogers and Spencer, in a series of papers (1 to 4), have derived the properties of such ideal fibre-reinforced materials. In this paper we assume that the shearing response of the material is elastic-perfectly plastic with the uniaxial stress-strain relation for

S =

/

to,

loading

under simple shear defined by

Itol_
(1)

Itol > top,

where S and to denote the shear stress and amount of shear respectively, G is the elastic shear modulus, K is the yield stress and top is the critical shear angle defined by K = stress-strain relation for

s =

unloading

from a maximum shear tom is

{ r + 6~(to-tom),tom- ~ p -

K

Gtop. The corresponding

< to<_tom, (2)

to _< t o , . - 2 t o p .

If during unloading the amount of shear to at any point decreases to to,.--2top, then reverse yielding takes place, with the shear stress equal to --K. To the authors' knowledge there are few published 333

334

A.H.

E N G L A N D and T.G.

ROGERS

solutions which incorporate reverse plastic yield, and almost none when the material is anisotropic. Therefore it is interesting to observe that the solutions we investigate display this phenomenon. The constitutive relations (1) and (2) have been used to describe metal matrix composites at high temperatures (see Evans (5)) and obviously are the simplest relations to model elastic-plastic effects. In a recent paper Bradford, England and Rogers (6) have described finite plane strain deformations of a cantilever composed of an elastic ideal fibre-reinforced material when a point force is applied to its end face. Bradford (7) has also investigated other loading conditions on the cantilever and has extended consideration to an elastic-plastic ideal fibre-reinforced material. Papers covering this material are being prepared by the authors of (6), but unloading was not considered in (6) or (7). In Section 2 the deformation of the beam under a transverse point force applied to its upper surface is considered. In Section 3 the unloading of this system is investigated. The final completely unloaded deformation and residual stress field is described in the last section. We note that the solution also applies to the problem of three-point bending of a fibre-reinforced beam of the same thickness but twice the length of the beam shown in Fig. 1. D

I

J

H

F

D

/ J

C

J

A

C

I !

B G

2.

FIG 1

FIG 2

The Initial Configuration

The Finite Deformation

Point Force Applied to the

Q

B

UpperSurface

The physical problem is shown in Fig. 1. The cantilever beam ABCD has length L and depth h and is built-in along the edge AD. The displacement field on A D is taken to be zero. The inextensible fibres are assumed to be continuously distributed throughout the body and are parallel to the surfaces AB and DC. Pipkin and Rogers (2) have shown that when such materials undergo a finite plane-strain deformation the fibres form a family of parallel curves with straight normal lines. In addition, the deformation preserves the distance apart of any two fibres when measured along the normal lines. Hence the kinematics of the deformation field is relatively straightforward. The stress field in such materials has the dyadic representation

ELASTIC-PLASTIC FIBRE-REINFORCED CANTILEVER

=-p(L-aa)

+ T aa + s(a~+~a)

+ T33 /~ ~ .

335

(3)

Here p and T are the reaction stress components introduced by the constraints of incompressibility and inextensibility respectively. The current fibre and normal directions are denoted by the unit vectors and tz and ~ = a X ~. The reaction stress component T33 in the ~ direction simply maintains the plane strain configuration. The shear stress S has already been defined in (1) and (2) in terms of the shear strain ~b. When the fibres are initially straight it may be shown (3) that at any material point the amount of shear is given by the angle between the normal line through that point in the deformed configuration and in the reference configuration. Thus in the present case the shear strain ~b is the angle of inclination of the normal ~ to the line AD. It is also shown in (2) that the reaction stresses are sufficiently general to satisfy the equilibrium equations so that any kinematically admissible solution is also statically admissible. However not every statically admissible deformation of an elastic material corresponds to a state of minimum potential energy. This criterion has been used in (6) to determine the appropriate elastic deformation fields. The elastic-plastic deformations considered here are natural extensions of the elastic fields found in (6). The deformed configuration shown in Fig. 2 is the two-fan solution of (6). If the applied dead-load F is sufficiently large then the fan angle o% will exceed the plastic-shear angle ~bp and the beam will deform plastically in the region AEGPHI. The fan angle o% may be determined by examining the overall equilibrium of the part of the cantilever to the right of a normal line NN' in this plastic region. Hence, along the direction n,

Fcosolo=hK=hGd# p,

Oto>q~p .

(4)

Here we have assumed that the beam is sufficiently long, and the point P of application of the force F sufficiently far from D, that the deformed beam actually has a uniformly sheared section AEPH. Within the fan region DAI the material deforms elastically with the shear stress defined by (1) l . Similarly, within the fan region IAH, at all points along the normal line AJ with shear angle ~bm the shear stress is K = G ~bp for ~bp < ~m < °to" Identical results hold for the regions GPQ, GPE under the point force. The remainder of the stress field may be found by using the methods described in (2) and (6). We note that the region PQBC is completely undeformed and stress free. In the context of conventional elastic-plastic beam theory, the region ADH corresponds to a plastic hinge; the present analysis shows the structure of such a hinge for a highly anisotropic elastic-perfectly plastic cantilever. In the following analysis, for reasons to be discussed later, we shall assume that the loading on the beam is sufficiently large to ensure that the fan angle ~t0 is greater than 2.6266 Op. This is not an unduly restrictive assumption since K is usually much less than (7 so that the plastic shear angle ~bp is normally quite small.

3.

Kinematics

It is possible to follow the deformation as the point force in Fig. 2 is gradually reduced to zero. Several different regions appear which are discussed later. In the completely unloaded configuration a

336

A.H.

ENGLAND and T.G. ROGERS

possible kinematically admissible deformation is shown in Fig. 3. D

I

j

D

iI

H

i

Y

P

C

A

g

FIG 3 A Kinematically Admissible Deformation

FIG 4 The End Region of the Unloaded Beam

There is a curved section AWHD connecting the end AD with the straight section WXYH of the beam. There will be an image of the curved region adjacent to the point force. The set of material points AI which were just at yield under the load F (see Fig. 2) becomes the dashed line AI in Fig. 3 and will be referred to as the 6p--line. Similarly the material points on the line AH in Fig. 2 lie along the dashed line AH in Fig. 3 and will be referred to as the ot0--1ine. We denote by r0(6 ) the radius of curvature of the lower edge fibre AW at the point where the shear angle of the normal is 6. Then the arclength along the edge from A to this general point is

6 s=

f0

r0 ( 6 ' ) d 6 ' = I ( 6 ) .

(5)

Hence, once I (6) has been found, (5) is the intrinsic equation of AW. For any curved section of the beam it is convenient (see 1) to describe the position of any material point M in terms of its quasi-polar coordinates ~, 6 where ~ is the distance of the point M from the lower boundary measured along the normal line through M and 6 is the angle of inclination of that normal. Consider a material point M lying within the fan region DAH of Fig. 2 with coordinates and 6m at the onset of unloading. Let us denote the shear angle at M by 6 in the unloaded configuration Fig. 3. The inextensibility of the fibre passing through M implies that the arclength along this fibre from the end AD to the point M satisfies

~6m=f60 {ro(qY)+~}d6'=~qS+I(6). Hence for the material point M the current shear angle 6 is related to its maximum value 6 m through

6 m=6+ sly,

(6)

where s = I (6)For the particular point H on the upper surface we have ~ = h with the shear angle 6m = ~0 in Fig. 2. But if in the unloaded configuration the shear angle at H is ~, then (6) gives

ELASTIC-PLASTIC FIBRE-REINFORCED CANTILEVER

337

a o = a+sw/h,

(7)

where s w is the arclength AW along the lower surface to the base of the normal line HW. The kinematic results (6) and (7) hold true even if there is a fan region located within the curved section AWHD of the beam provided the centre of the fan lies on the edge fibre AW.

4.

The Completely Unloaded Configuration

The shape of the beam is conveniently determined by examining the equilibrium of the part of the body to the right of a normal line. A careful inspection of the curved region AWHD of Fig. 3 indicates that as the applied load is gradually decreased several distinct regions appear until, when the beam is completely unloaded, the end region of the final configuration is as shown in Fig. 4. This solution holds provided the initial beam angle ot0 exceeds 2.62660p. The material in the hatched region AVU has reverse yielded. There is an identical end region adjacent to the point P at which the force was applied. The region R 3, namely WXYH, is the straight section of the beam. A material point in this section originally lay in the region AEPH of Fig. 2 and experienced the shear strain ot0 at the maximum load F. If the current shear angle is a then on resolving along the normal line NN' we find

hG[Op+

or--a0] = 0

(8)

since there is now no applied load. Hence the beam angle becomes

ot = oto - Op ,

(9)

when the beam is completely unloaded. We also see that the arclength A W is specified in (7) to be

s w = hop.

(10)

Hence the arclength A W depends only on the plastic shear angle 0 p and is independent of the magnitude of the finite deformation (provided a 0 is sufficiently large). To determine the shape of the adjacent region R e we need to resolve along a general normal line with shear angle ~b lying in VWHJ. In the part of the region below the line A l l the material points had a maximum shear angle ao, whereas for those in the upper part the maximum shear was

0m

defined in

(6). Hence for equilibrium h

f~c~G(dpp+dP-~o) d~+ f~

G(dpp+dp--qJm)d~=O,

(11)

0t

where Got is defined from (6) as the distance along the normal line from A W to the point of intersection with A l I (on which O m = °to) so that

/(~) ~ a = a0_~b "

(12)

In addition when ~ exceeds ~a then Cbrn satisfies (6), and hence on substituting in (11) we find

h

s

338

A.H.

ENGLAND

and T.G.

ROGERS

which may be solved for 0 to give

= ~ _ ± exd-~e 0

h

"~

- 1}

(13)

s

In (13) s = I(0) is the arclength along A W

from A to the base of the normal line in region R 2 with

shear angle 0. Thus (13) is the intrinsicequation of the edge fibre A W

in the region R 2.

W e note that at the point W, ~b = ot and s = sw = hop from (7) so that (13) is identically satisfiedat this point. Equation (11) has been derived under the assumption that the material has not reverse yielded. Reverse yieldingwill firsttake place at a point when Op + dp--dpm is equal to --q~p. Hence in region R 2 this will firstoccur when 0 takes its least value (along VJ) and 0 m has its greatest value which is ot0 in the lower part of the region. Accordingly, reverse yieldingtakes place when ~b = Oto--2t#p = ~bv .

(14)

Hence the normal line VJ which forms the boundary of the region of reverse yielding has the shear angle ~to - 2~bp which we shah denote by Or. In addition the arclength A V is (from 13) defined by

,v h --

exp

\

Sv

--

1

)

=

(15)

20p,

s v = 0.3733h ~bp.

which yields

(16)

To the left of the normal line VJ reverse yielding is possible since 0 < Ov. Along any normal line with shear angle q~, the shear stress is computed from the angles

Op+dp--et o

for ~ < ~a

/

Op + 0 -- Om

for ~ > ~a

/

(17)

where ~a is defined in (12) as the distance along the normal to the point of intersection with the material line AII and dPrn is defined in (6). In (171), the angle is less than - - % (since 0 < a0--2q~p) and the material is in reverse yield. In (172) reverse yielding takes place for ~ _
(18)

and we see that, in general, ~a < ~R' We note that along the normal line VJ, ~a and ~R are equal, so defining the position of the point U on Fig. 4. Hence the region AVU of reverse yielding contains the region under the a0-1ine and is shown hatched in Fig. 4. The set of material points AI were just at yield with the shear angle dpp in the finite deformation shown on Fig. 2. These points form the curved q~p--line AI in Fig. 4 and points above this line have not suffered a plastic deformation. From (6), the normal line in the region R 0 with the shear angle 0 will intersect AI at

i(¢) = ~P -- q~p--0" It is clear physically that ~p must be greater than ~R and this is obviously true mathematically.

(19)

ELASTIC-PLASTIC

FIBRE-REINFORCED

CANTILEVER

339

Hence along a normal line in the region R o with a shear angle O, a material point in the region 0 < ~ <_~R has undergone reverse yielding, a point in ~R -< ~ -< ~p has undergone plastic unloading to the final shear stress value G (Op + O-05ra) when 0 m is defined by (6), and a material point in the region _> ~p was originally in the elastic fan region DAI of Fig. 2 so that its current shear stress depends only on its current shear strain 05. Hence for equilibrium

~R

~p

:o

h :o

,,0,

Using (18), (19) and (6) this yields 05 _±in( h ~051,- 0 5 / -

(21)

in the region R.

When s = 0 then 05 = 0, so there is no fan region at the edge AD. The upper limit is defined by the normal line IV through the point I at the end of the 05p--line. Let this have the shear angle 050; then from (6) we see the arclength AV must have the value s o = h (05p-- 05o)'

(22)

and on substituting for 05o in terms of s o in (21) we find

_ So[1 +

05~ -~-

~ s o /j"

On comparing this equation with (15) we see s o = sv

-

0.3733h05p

confirming that the normal line from I intersects the edge A W at the point V. Hence, from (22), 05o = 0 . 6 2 6 6 % .

(23)

Thus the region R 1 in Fig. 4 is a fan region with the fan angle 0.6266 05p < $ < ~t0 - 2 % .

(24)

Clearly this solution only exists ff the finite deformation is sufficiently large so that c~0 > 2.626605p. It may be confirmed that the radius of curvature d s / d $ of the edge fibre AV and VW is positive so that s is a monotonically increasing function of 05 in AV and VW. It is also possible to confirm that the radius of curvature is infinite at the point W where the curved region joins the uniformly sheared section of the beam. The shear stress along a normal line with shear angle 05 in the region R o has the form shown on Fig. 5. We note that the resultant force, the area under the curve, must be zero. A similar curve describes the shear stress in the regions R 1 and R 2. In the straight section R 3 of the beam the shear stress is zero. The edge fibres AB and CD must carry finite forces since there is a discontinuity in the shear stress across the edge fibres. Integrating along AV we find T = T A + Ks,

(25)

A.H.

340

c~

ENGLAND and T.G. ROGERS

•/i

p FIG 5 The Shear Stress Along a Normal Line in the Region R 0 where T A is the tension at the point A and s is the arclength. In the range VW we find

ga

ds T = T A + K s v -- f gay G(gap -- a o + ga) - ~ dga = T A - G (gap --a o + ga)s + G J(ga) ,

(26)

where s = I(ga) is the arclength from A and

ga J(ga) = f

I(ga) alga,

ao--2ga p _< ga < Oto--~p .

(27)

a0--2gap We note that J(ga) is a positive increasing function of ga on the interval VW. In the uniformly sheared section of the beam, the tension in the edge fibre becomes the constant force T A + G J ( a o - - O p) •

(28)

On the remaining sections of the lower fibre the shear stress is reversed in sign and hence at the point Q, directly under the point of application of the force, the tension must equal T A. However the tension at this point must be zero. Hence the tension in the lower edge fibre is defined by (25), (26) and (28) with T A = 0. The function J(ga), where the arclength s = I(ga) is related to the angle ga by (13), requires numerical evaluation. The upper fibre must carry an equal but opposite force to that in the lower fibre so that its straight section must carry the compressive force -- G J (0% -- gap).

REFERENCES [1] [2] [3] [4] [5] [6] [7]

J.F. Mulhern, T. G. Rogers and AJ.M. Spencer. Proc. Roy. Soc. A 3 0 1 , 4 3 7 (1967). A . C . Pipkin and T. G. Rogers. J. Appl. Mech., 3 8 , 6 3 4 (1971). T.G. Rogers and A. C. Pipkin. Q. Appl. Math. 29, 151 (1971). A.J.M. Spencer. Deformations of Fibre-reinforced Material~, Clarendon Press, Oxford, 1972. J.T. Evans. Scripta Met. 22, 1223 (1988). I.D.R. Bradford, A. H. England and T. G. Rogers, Acta Mechanica, forthcoming. I.D.R. Bradford. Ph.D. thesis, University of Nottingham, 1991.