Some homogeneous elastic materials are capable of sustaining

finite equilibrium deformations with discontinuous

strains. For materials of this kind, the energetics of isothermal,

quasi-static motions may differ from those conventionally

associated with elastic behavior. When equilibrium states

involving strain jumps occur during such motions, the rate of

increase of stored energy in a portion of the body may no

longer coincide with the rate of work of the external forces

present. In general, energy balance now includes an additional

effect due to the presence of moving strain discontinuities. As

a consequence, the macroscopic response of the body may be

dissipative. This fact makes it possible to model certain types

of inelastic behavior in solids with the help of such "unstable"

elastic materials; see, for example, Abeyaratne and Knowles

(1987a,b,c)

Abeyaratne, R.

Knowles, J. K.

Caltech Authors - Main

Lin. P. P . and Datseris, P., 1986, "Development of a Position and Force Sen· 
sor for Robotic Applications," Proceedings of 1986 /£££International Con-
ference on Robotics and Automation, Vol. 3, Apr. 7-10, pp. 1798-1805. 
Nadai. A., 1915, "Die Formanderungen und die Spannungen \'On 
Rechteckigen Elastichen Pauern," Forsch A. D. Gebiete D. lngenieurwesens, 
Berlin, Nos. 170 and 171. 
Szilard. R., 1974, Theory and A nalysis of Plates, New Jersey, Prentice Hall, 
PP· 36-42. 
Timoshenko, S. and Woinowsky-Krieger, S., 1959, Theory of Plates and 
Shells. S«ond Edition, McGra.,-Hill, New York, pp. 113-11-1. 
Unstable Elastic Materials and the 
Viscoelastic Response of Bars in Tension5 
R. Abeyaratne6 and J. K. Knowles7 
Some homogeneous elastic materials are capable of sustain-
ing finite equilibrium deformations with discontinuous 
strains. For materials of this kind, the energetics of isother-
mal , quasi-static motions may differ from those conventional-
ly associated with elastic behavior. When equilibrium states 
involving strain jumps occur during such motions, the rate of 
increase of stored energy in a portion of the body may no 
longer coincide with the rate of work of the external forces 
present. In general, energy balance now includes an additional 
effect due to the presence of moving strain discontinuities. As 
a consequence, the macroscopic response of the body may be 
dissipative. This fact makes it possible to model certain types 
of inelastic behavior in solids with the help of such "unstable" 
elastic materials; see, for example, Abeyaratne and Knowles 
(1987a,b,c). 
The purpose of the present note is to illustrate behavior of 
this kind with the help of an especially simple example involv-
ing the extensional deformations of a bar treated as a one-
dimensional continuum. The bar is composed of an unstable 
elastic material of the type considered by Ericksen (1975). We 
show that the force-elongation relation (or "macroscopic 
response") of the bar during a quasi-static motion may be 
viscoelastic when a moving strain jump is present, even though 
the underlying constitutive law is elastic in the sense that, at 
each particle, the present stress is determined by the present 
strain. 
Consider an elastic bar which, in the reference configura-
tion, occupies the interval [0,L) of the x-axis and has constant 
cross-sectional area A. In a deformation, the particle at x is 
carried to y =x+ u(x), where u is the displacement. We 
assume that u is continuous on [O,L], and that u ' (x) exists and 
is continuous for O::sx::ss and s::sx::sL, but u ' (s+) and 
u '(s- ) may differ; here O<s<L. If x?!s, the strain at xis 
E (x) = u ' (x). It is required that f(x) > - I for x?!s, so that the 
mapping x - y is invertible. We take the left end of the bar to 
be fixed, so that u(O) = 0. 
Suppose that the material is elastic with strain energy per 
unit reference volume W(E). The nominal stress response 
function is then 
a(E) = W '(E), (I) 
so that the nominal stress in the bar at particle x is 
u(x) = a(f(x)). We shall be concerned with the special 
"trilinear" stress response function given by 
5
The results reported here were obtained in the course of an in,estigation sup-
ported in pan by the U.S. Office of Naval Research under Contract 
N~l4-87-~-0l l 7. 
Department of Mechanical Engineering, Massachuseus lnstitute of 
T~hno_J~gy, Cambridge, MA 02139. 
D1v1s1on of Engineering and Applied Science, California lnstitute of 
T cchnology, Pasadena, CA 91 125. 
Manuscript received by ASME Applied Mechanics Division ovember 10 
1987. • • 
Journal of Applied Mechanics 
·BRIEF NOTES 
f 
µ.E , - I <E ~EM, 
µ.EM 
a(E) = ( Em +EM -2E), 
Em-EM 
l µ.(E -Eo). Em ~E<OO. 
(2) 
The graph of a ( E) is shown in Fig. I , where the meanings of 
the constants µ., EM , Em and Eo =EM+ Em are made clear. One 
may think of the material as exhibiting three phases: the first 
phase is represented by the rising branch of the stress-strain 
curve through the origin, the third phase corresponds to the 
final rising branch, and the declining portion of the curve 
represents an unstable intervening second phase. In the pres-
ent case, the first and third phases are both associated with the 
same modulus µ. . 
In the absence of body force, the bar will be in equilibrium 
if the nominal stress a(x) satisfies 
a(x} = u= constant, 0 ~ x ~ L. (3) 
A deformation with a strain discontinuity of the kind 
described above will correspond to an equilibrium state if it is 
of the form 
( EtX, O~x~s, 
u(x) = ~ 
LE 2x+ (E t -E2)s, s~x~L, 
where the strains E 1 and E2 are constant and such that 
a(E1) = 0(€2) = a. 
(4) 
(5) 
We shall be concerned here only with those deformations of 
the form (4) in which 
(6) 
so that only phases one and three are represented. For an 
equilibrium state of this kind, the total energy stored in the bar 
is 
(7) 
Now suppose that the bar is in a quasi-static motion during 
which, at each instant 1, the displacement u (x,t) is of the form 
(4), with s=s(I), E1 =E1 (t} and E2 =e2(r). The restrictions (5) 
and (6) are to hold for all t, with a = a(I). Assume that s(t), 
E 1(t) and E2 (t} are continuous and piecewise continuously dif-
ferentiable int. During such a motion, the energy E(t) is given 
by (7) with E1, e2 , and s now functions of time. A direct 
calculation gives 
E(I) =FU>6U> + A[ -f(t)Js(I), 
where F(t) =a(t)A is the force, 
Fig. 1 Stress response function 
(8) 
JUNE 1988, Vol. 55 / 491 
BRIEF NOTES 
o(t) =u(L,t)=t1{t)s(t) + t 2(t)(L-s(t)) (9) 
is the elongation of the bar at time t, and f( t) is given by 
f(t) = W(t2(t))- W(t1(t))-a(f)(t2(t)- t1 (f)). (10) 
The first term on the right in (8) is the rate of work of the force 
acting on the end x = L of the bar. The second term may be in-
terpreted as the rate of work done by a fictitious " traction" 
f ( t) in moving the strain discontinuity at x=s(t). For the 
stress-strain relation (2), one finds using (1) , (6) and (10) that 
f(t) = - t 0a(t) . (1 1) 
If the motion takes place isothermally, the second law of 
thermodynamics requires that E-Ffi be nonnegative; thus the 
motion must be such that 
f(t)s(t) ~ o. (12) 
Because of (5) and (6), specifying the history of the stress 
acting on the bar determines t 1 (t) and t 2(t) , but leaves undeter-
mined the location s (t) of the strain jump, and therefore the 
elongation history o (/) of (9) as well. 
In formulas (7) and (9), one may regard s as an " internal 
variable" akin to those arising in microstructural theories of 
plasticity; see, for example, Rice (197 1). This suggests that , by 
analogy with such theories, the constitutive description of the 
material should be augmented by relating t he fictitious trac-
tionf(f) to the velocity of the moving discontinuity s(l). One 
form that such a "kinetic relation" might take is 
s (t) = V(f(t) ), (13) 
where Vis a function determined by the material. Using (1 1) , 
one infers from (13) that 
s (t) = V( -t0a (l)). (14) 
Note from (12) that every admissible V must be such that 
V(f)f~ 0. 
Let ")' (f) =o (t) IL be the relative elongation of the bar. The 
relation between 'Y and F - the macroscopic response o f the 
bar - may now be determined as follows. Using (2) and (6) in 
(9) yields 
o(t) = t 0[L-s(t)] + a(t)LIµ, (15) 
from which, with the help of (14), one finds the nonlinear 
viscoelastic relation 
)' (l) =Fc t ) /µA-e0 V( -e0F(t) I A)I L. (16) 
If in particular the kinetic response function V is specified 
through V(f) =f/v, where vis a positive constant , (13) pro-
vides a linearly " viscous" kinetic relation between f and s. 
The macroscopic response relation (16) specializes to 
)' (f) =F(t) l µA + el,F(t) l vLA. (17) 
This is precisely the form of the response relation 
characteristic of the so-called "Maxwell" spring-dashpot 
model of elementary viscoelasticity. Other forms of 
macroscopic response can be obtained by replacing (13) by a 
more general kinetic relation . 
A more extensive discussion of the one-dimensional theory 
of quasi-static motions of bars composed of unstable elastic 
materials may be found in Abeyaratne and Knowles (l 987c). 
References 
Abeyaratne, R. and Knowles, J . K., 1987a, "Non-elliptic Elastic Materials 
and the Modeling of Elastic-plastic Behavior for Finite Deformation," Journal 
of the Mechanics and Physics of Solids, Vol. 35, pp. 343-365. 
Abeyaratne, R. and Knowles, J . K., J987b, "Non-elliptic Elastic Materials 
and the Modeling of Dissipative Mechanical Behavior: An Example," Journal 
of Elasticity, Vol. 18, pp. 227- 278. 
Abeyarame, R. and Knowles, J . K. , 1987c, " On the Dissipative Response due 
to Discontinuous Strains in Bars of Unstable Elastic Material," Technical 
Repon No. I , ONR Contract NOOOl4-87-K-0117, California Institute of 
Technology, September, 1987; to appear in International Journal of Solids and 
Structures. 
492 / Vol. 55, JUNE 1988 
Ericksen , J . L., 1975, "Equilibrium of Bars," Journal of Elasticity, Vol. 5, 
pp. 191-20 1. 
Rice, J . R. , 1971, " Inelastic Constitutive Relations fo r Solids: An Internal 
Variable Theory and its Application to Metal Plasticity," Journal of the 
Mechanics and Physics of Solids, Vol. 19, pp. 433-455. 
Diffusion of Self-Equilibrating End Loads in 
Elastic Solids 
D. Durban8 and W . J . Stronge9 
Introduction 
The classical Papkovich-Fadle eigenfunction analysis for 
axial diffusion of self-equilibrating end loads, in plane strain 
linear elasticity, predicts exponential decay of the form exp 
( - kz/ h) where z ~ 0 is the axial coordinate, h-half 
thickness, and k is any root of the associated eigenvalue equa-
tion . The lowest rate of axial decay for an elastic solid at small 
strains is governed by the first symmetric eigenfunction with 
k 1 = 2.1061 + i 1.1254. The next term, in order of increasing 
magnitude of Re [ k), is given by the fi rst antisymmetric eigen-
function with k2 = 3.7488 + i 1.3843. These values of k, as 
well as the higher roots, are determined by the transcendental 
equations sin 2k ± 2k = 0 where the ( + ) and ( - ) signs cor-
respond to symmetric and antisymmetric eigenfunctions, 
respectively. 
A recent paper by Abeyaratne, Horgan and Chung (1985) 
has addressed the plane strain end effect problem within the 
framewo rk of finite strain elasticity. The authors consider an 
incrementally self-equilibrating end disturbance imposed on a 
uniformly stretched strip. The axial decay rate is again ex-
ponential but the eigenvalue k is now dependent on the 
magnitude of initial stretch . Furthermore, the variation of k 
with the initial stretch displays considerable sensitivity to the 
form of the strain energy function. Unfortunately, that study 
has been restricted to the subspace of symmetric eigenfunc-
tions so it reveals only part of the complete picture. 
The main purpose of the present note is to demonstrate that 
the smallest decay rate (which provides a bound on the validity 
of Saint-Venant's principle) can be associated with the first 
antisymmetric eigenfunction. It is necessary, therefore, to ex-
amine the complete space of eigenfunctions (i.e. , symmetric 
and antisymmetric) allowing for competition among the cor-
responding eigenvalues over the entire range of prestrain and 
material parameters . 
The basic setting is similar to that of Abeyarame, Horgan 
and Chung (1985), though we differ in the solution method 
and follow the Hill and Hutchinson (1975) stream function 
formulation. The material is assumed to be hyperelastic and 
incompressible. Specific examples are illustrated for the 
Mooney-Rivlin model and for a power-law solid. A few 
related topics are discussed and some general results are 
presented. 
Formulation of the Problem 
A semi-infinite strip of rectangular cross section (Fig. 1) is 
uniformly stretched under plane strain conditions. The current 
axial stress is a, = a and the principal stretches are >.., = t.. and 
t..x = }.. - 1 • The deformed thickness is denoted by 2h and ~e 
faces x = ± h are stress free. Note that h = H It.. where H 15 
the original undeformed thickness. 
8 Department of Aeronautical Engineering, Technion, Haifa 32000, Israel. 
9 Dcpartment of Engineering, University of Cambridge, Trumpington Street, 
Cambridge CB2 I PZ, England. 
87 Manuscript received by ASME Applied Mechanics Division, August 25, 19 · 
Transactions of the ASME 


English

Unstable Elastic Materials and the Viscoelastic Response of Bars in Tension

https://authors.library.caltech.edu/51241/1/386803.pdf

Unstable Elastic Materials and the Viscoelastic Response of Bars in Tension

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main