Dynamic behavior of structures composed of strain and workhardening visco-plastic materials

DYNAMIC BEHAVIOR OF STRUCTURES COMPOSED OF STRAINANDWORKHARDENING VISCO-PLASTIC MATERIALS A. R. S. PONTER Department

of Engineering. Leicester University.

Leicester. England

(Received 17 May 1979; in revised form 5 h-ember

1979)

Abstract-A dynamic convergence thereon is proven for a class of visco-plastic constitutive equations involving internal state variables, which provides an extension of a result due to Martin[l]. The class of constitutive equation, which includes the Malvem material when effective plastic strain is adopted as the state variable, is expressed in terms of a Row potential. For a simple model structure both convergence and divergence is demonstrated. The example also demonstrates that the theorem provides a ttdkiint but not a necessary condition for convergence, as convergence can occur even when the conditions of the theorem are not satisfied.

I. INTRODUCTION

During the last ten years approximate techniques have been developed by Martin, Symonds and others[lJ] which provide solutions to the problem of a structure subjected to an initial impulse. The technique employed, the “mode”-solution method, derives from an observation by Martin[I], that structures subjected to differing initial distributions of velocity have convergent velocity fields during the deformation process. Under some circumstances the velocities may converge to a “mode” solution which is provided by the solution for a particular distribution of initial velocities, and in other circumstances the “mode” solutions are defined by the extremal of a functional of the velocity distribution[3, 61.The technique has been developed for large deformation of structures[5), and provides a simple direct method of assessing the most significant features of the deformation of impulsively loaded structures and obviates the need for a complete analysis. The theory has, however, been developed only for constitutive relationships for which either the inelastic strain or strain rate is given in terms of the instantaneous stress. Although a wide class of material models fall within this category (elastic, perfectly plastic, deformation theory plastic, visco-rigid plastic and viscous materials),the important class of work and strain hardening viscoplastic models are excluded. The purpose of the paper is to present extensions of the convergence proof of Martin[ l] to a class of state variable equations which may be expressed in terms of a dissipation potential. This class has been discussed by Ponter[7] for quasi-static loading of a body. For uniaxial stress u with resulting elastic strain e and inelastic strain t), the constitutive relationship take the form i=Gc+

i=i+t;,

(1) (2)

where e denotes the total strain, G the elastic compliance and 0 a potential function of stress and a state variable S. By choosing a particular state variable, eqn (2) and (3).yield a specific functional form for the potential R. For example, we may choose

s = l(0)

(4)

where I is some arbitrary function, and restrict stress histories to these for which u > 0 and u 193

794

increases equation

A. R. S. PONTER

in time. Eliminating

d

between (2) and (3) and using (4) results in the differential

(5) The general solution of (5) is given

by

aa,S) = mr-g(u))

(6)

where g(u)= Subsitituting

I

P2du

(7)

this solution into (2) yields if = R’(u - g(o))

(81

which may be recognized as Malvern’s[8]relationship where g(u) is the static yield stress corresponding to monotonic flow to inelastic strain v, and u-g(u) is the dynamic overstress. Hence the eqn (2) and (3) provided a limited but important class of constitutive relationship, and we propose to show that this class can possess convergence properties for dynamically loaded body if $I is a convex function of its arguments. In Section 2 some results from 171are reviewed and the inversion of the constitutive relationship is discussed. In Section 3 three p~icul~ cases are described, isotropic and kinematic strain hardening and work hardening visco-plastic models. In Section 4 the convergence theorem is formally derived for the general class. This is followed by Section 5 where a simple structure is analyzed for situations where sufficient conditions for the convergence theorem are both satisfiedand violated. This is achieved by varying the materialparameters in a simple strain hardening visco-plasticconstitutive relationship. In particular, the conditions are violated for materials whose static plastic stress strain curve has increasing slope with strain. For su~ciently pronounced strain h~dening characteristics nonconvergenceis exhibited by the model although we also find cases where convergence occurs. 2. A CLASS OFCONSTITUTIVE

RELATIONS

We are concerned with the small strain behavior of a material with inelastic strain rates which are expressed in terms of a simple potential function fI(cij, S,) where uii is the current stress and S, is the current value of a set of state variables.The total strain
eij+ Vij

(9)

eij = cijhkuhk

and the inelastic strain rate liiiand the state rates S, are expressed in terms of CIas: (1la)

an

St=-%

(1ItO

The quantities S, may form a set of scalar quantities or the components of tensor quantities with respect to some fixed axis. The eqn (1I) defines a restricted class of consitutive relationships which certainly contain some well known forms [7], but excludes others. We are

Dynamic behavior of viscoplasiic materials

19s

exploiting the particular strong properties of (I 1) which allow extension of known results to a wider, but incomplete, class of constitutive relationships. Before discussing particular forms of eqn (1l), we review properties of R derived elsewhere. In (71 it was shown that R is a convex function of both ajj and S, provided the following inequalities hold: at constant state a small change in dcij results in a corres~ndi~ change d& so that

dajjdtijjz0,

Sj=O;

(12)

and for constant Uij,a small change in state dSt produces a change in $ such that ‘c- 0, ci, = 0. dSfdS[

(13)

From (12)and (13)the convexity of R may be derived, (141 Reversing the roles of (& S/) and (af, $1 in (14) and adding the resulting inequality to (14) results in the inequality;

Equality in (14)and (15) may occur for stresses and states at which equality occurs in (12)and (13)for non-zero duil and dSj. In inequality (12)such conditions would occur in the rigid region for material with an initial yield surface and in (13)for states of stress and states for which the state is frozen. The sign of in~u~i~y (13)is in~ue~~~ by the notion of strain h~deni~; the rate of increase of the state quantities tends to decrease as the state variables increase. If the inequality were reversed then Q would be convex in ~6 and concave in St. The eqn (11) may be inverted without difficulty.We define dual potentials by

so that

(17) From (17) we see that

and hence from (ll), (16) and (I@,

s;=($$

11W

By analogous arguments to those used for deriving the convexity of 0 we conclude that fi is convex in &jbut concave in S,.

A. R. S. PONTER

796

It is worth notingthat if inequality(13)was reversed then fi wouldbe convex in t&and S,.Hencethe reversalon inequality(13)interchangesthe convexity propertiesof Q and fi and it is clearthat both R and fi cannot be convex together.

3. PARTICULAR

FORMS

OF

THE CONSTITUTIVE

RELATIONSHIP

Specificationsof the state variable S, gives further restriction on the constitutive relationship (1 I) ingests in the form of the dependence of Q on the deviatoric stress C$ and S,. We consider three special cases, involving a single state variable, corresponding to isotropic and kinematic strain hardening and work hardening. (a) hofropic strain-hardening We define a single state variable as some function of an effective strain 5: s = I( -u)

(21)

where (22) where Jt is a homogeneous scalar function of degree one. Further Ctis assumed to be a function of q5(oti)where # is a homogeneous of degree one. Substituting these conditions into the constitutive relationships (11) and eliminating Uiiyields,

-=-f$, a4

Aan

A=@$).

(23) 81

We term conjugate those $ and 4 for which A is a constant, which we may take as unity without loss of generality. For example

form conjugate pairs. The general solution of (23)is given by fi=il(#-g(C)),

g(a)=&PdR

S=&)=/(g’)‘“da

(26)

and the constitutive relationship (I la) and (1lb) reduce to:

This general form corresponds to a generalization of a Malvern material 181,and g may be interpreted as a measure of the history dependent “static” flow stress which increases with increasing strain. The dual potential fi may be most easily derived by first assuming v’= 0, and hence It(#) + R*(Jr)= Cjjtiij= 4&

(2%

191

Dynamic behavior of viscoplastic materials

Substituting (# -8) for 4 in (29)and noting (17)we obtain fi((Ljii, lj)=R*(JI)+#(tS)II/.

(30)

The sufficient conditions for the convexity of Q, inequalities (12) and ( 13),may be translated into restrictions on the general function Q and g: (31) (32) where use has been made of eqns (21), (22) and (28). Hence sufficient conditions for the convexity of R are given by

~‘30,

~“30,

g’,O and g”dO,

(33)

and are equivalent to the conditon that ~~) shall be convex and g(6) shall be concave functions of their arguments. In fact, the convexity of Q may be derived directly from these conditons and a direct and simple proof is given in the appendix. The meaning of these inequalities in terms of uniaxiai behavior of the model may be seen from Figs. 1and 2. Stress strain curves at constant strain rates must have positive slope (g’3 0) but must decrease with increasing strain (g CO). This condition is usually observed in metals and alloys, but mild steel provides a notable exception. On the other hand, stress-strain rate curves at constant strain must have positive slope (0” 3 0) but the slope may either increase or decrease with increasing strain rate. Hence the inequalities(33)place a more signikant restriction on the strain h~de~ng characteristics than on the strain rate sensitivity of the material. A ~~icul~Iy simple form of f2 which contains a sufkient number of parameters to allow the description of many materials over, at least, some range of stress-strain and strain rate is given by;

=o

(34)

9

which gives rise to the uniaxial reiations~p; a,,>uo+c7

Fii. 1.Forconvexa, stress-strain cwwsptconstant strainratemustbpvepositive increasing strain u, ,.

[

2

1. I/n

dope, decreasing with

(35)

A. R. S. PONTER

798

u11

V,, constant v,, = 0

~

Fig 2. For convex fi, stress-strain rate curves at constant strain must have positive slope, but may either increase or decrease with increasing strain rate li,,.

*

.

The “static” stress-strain curve (titI = 0) is Qven by Qll -= -WI--@0 m

vo

5

[

I

(36)

’

and a > 0, m > 1 are suiTicientcondition for the convexity of 0. A simple re~tionship, without an initial yield stress is provided by

-0

i.e.

!h&9&!-[!!iJJ;p>o,

q>o.

(38)

Fire 3 shows the correlation of both (35) and (38) with data for 304 stainless steel at 7OT, from the data of Hause@]. Clearly eqn (39 provides the better result, but the more restricted form (38)provides a quite satisfactory fit to the data. For both eqns (34)and (38)the state variable S may be derived from (26):

where m = q and 6 = a0 for eqn (38).

* Ff.

bc-’ 1

3. Comparison Of eqns (34) and (38) with test data for 3@t !%illk66

SW

(Hauser

[91).

199

Dynamic behavior of viscoplaslic materials (b)

~ine~ufi~-sfr~i~ dampening

Kinematic behavior may be generated by assuming S, are the components tensor quantity Sip The simplest form occurs when

of a

second order

Sij = C Ujj

where C denotes a constant. The state equations then yield that fI = R(@ij- C Uij)

(39)

and Szis convex, provided fI” 2 0 and C > 0. Constitutive equations in which the state is given by the plastic work have been discussed by Perzena[lO] and Bodner and Partom[ 1I]. If in eqn (1I) we choose S = l(W),

W=

’

i0

Uii&jdt+

(401

and assume that 0 = ~(#(~ij~, S), then eqns (11) yield, on eliminating 6,

which possess the general solution; Q = Q(de-c(w’)where g(W) = j (!‘)2dW.

(421

The strain rate equation is now given by (43) This lotion is of a similar form to that suggested by Elodner and Par-tom [II], but their equation cannot be expressed in terms of a potential of the form (42).As (42)and (43)appear to be essentially new, their properties will not be pursued further in any detail. It is worth noting, however, that sufficientconditions for convexity of Q are given by the restriction that fI shall be a convex function of (Band a concave function of S. The resulting restriction becomes a combined restriction on the form of Q and g(w) and will not be pursued here. lTHECONVERGENCETHEOREh4

Consider a body of volume V surface S subjected to an impulse at time f = 0 which gives rise to an initial velocity distribution 4(O). Surface traction #&act over part of Y: .Yr and body force F,(f) act within V. Over the remainder of Y: which we denote by Yap,, displacements Ui remain zero. The theorem concerns two such bodies subjected to identical conditions in all respects except the conditions at time t = 0. One body has initial stress ~b, velocity ri’ and state S/, whereas the second body has differing initial value Q& tii, S?. We are concerned with the relationship between the subsequent histories in the two bodies. We first note the followi~ relationship, which foltow from the ~ons~utive relationship (9) and (IO)and the principle of virtual velocities:

A. R. S. POFCER

800

where A”[/ii’- cif]= v;p(hi’-rif)(&-$)dV I

(45)

and A,[trfi- a$] = Hence Am denotes the total kinetic energy associated with velocity dis~bution 4 and A,(q) denotes the total strain energy associated with stress dis~bution @ii Combini~ (44)with the convexity condition (15)yields dA Tiiso

(47)

A=A&ij’ -rii2)+A~,(qe--q:)+Al(S~-S:),

(48

where

and A, =

; (4’ - S&(SJ -S&V.

Further A,O.

(49)

Equality in (47)occurs only when ai = af and S,’= S? for strictly convex 0 and when ril = d2. Equality occurs in (49)under the same conditions. Hence a$, uf and S,’ approach ai, u$ and S2 in the sense that A must reduce in magnitude. It is clear that either A, or A, or A, may increase in value during the deformation process. Indeed, if S,’= St at t = 0, i~~l~y A, will certainly increase but less rapidly than the combined decrease of A, and 4, Our principal interest will be in the case where the electric term A, is negligible and initially S,’= S/. At t = 0 A, must desrease in time and will continue to do so until A, has become of a similar magnitude to A,,. The convergence may be expected to be initially dominated by q’ approaching uf and eventually $’ approaching Sp when A, is sutkiently small. How this behavior manifests itself in the behavior of a simple structure is demostrated in the section. 5. AN EXAMPLE

Consider the structural model shown in Fii. 4. Two masses are attached to weightiess beam of tength I which is simply supported at each end. The ~~rnat~s of the stru&ure may be expressed in terms of the vertical ~p~cements

Pii, 4. Simpk structural model considered as an example.

Dynamic behavior of viscoplaslic materials

801

Udt) and U*(t) of the two masses. Deformation is assumed to be confined to points A and B where hinges form which rotate with angles 4, and tJzgiven by

&=f(3U,-UJ,

$*=5(3Lj2-U,).

(50)

The moments applied to these hinges are given by N=$3P,+P*)*

M*=&(3P,+P,)

(51)

where PI and PZ are the D‘Alembertforces applied by the weights to the beam and hence ml?,=-PI,

ml&=-P,.

(52)

For the relationship between 4, and M, we adopt a simple relationship of the form of eqn (35): (53) where t+Ji = Jd I&ldf. Equations (50)-(53)may be combined into a pair of second order differential equations for Ui. For 4 > 0 and $ > 0, the equation takes the form 4 = ~&l[3(3lil

& = &l[(3li,

- tiz)“”- (3& - ti,)“”+ 8”“(2+ 38, - &)]

(54)

- d*)“”- 3(3&- I#” + 8”“(- 2 + /3,- 3831

(55)

and similar equations hold for differing sign combinations of $i. The equations are expressed in terms of nondimensional variables: Ii, =

.= pi1 163 &$ pt 16 M,’

and @,= A’lrn[I,‘j3li, - &Idr’]lim, & = A’lrn [I’ 131i2 - ri,Jdr’]“n, 0 7 = 256.M,‘, and A = m11,212 256.8M,,I&,~ NJ* In terms of these nondimensional quantities the solution is governed by I&at t = 0, and the material parameter m, n and A. The quantity A governs the degree of strain hardening in the model as A = 0 corresponds to purely viscous behavior. We may compute values of A which have some resemblences to reality by the following approximate argument. We may identify rotation rate for the special case u, = ti2.The initial kinetic energy of the structure is then given by

&=2 1m(ti,*+ i2*) = $ &*.

A. R. S. PONTER

802

With & = 0 and (li = & then Mi = 2 M,

where M, is the static initial yield moment. Hence this choice of initial velocities corresponds to a doubling of the initial yield stress. We may now write; A =&

where

W =2 MO&,.

If Jli= 0 and J;i= Jlo then again M = 2 M,, a degree of strain hardening which is similar in m~nitude to that exhibited by the data of Fig‘2. Hence, for this hypothetical case, assuming Jto is the final rotation of the hinges the moments Mibegin and end at the same value, 2 MO.If we assume that on average Miremain close to this value, the total energy dissipated in the hinges is given by 4 MO&and equals the initial kinetic energy K.Hence, A = l/32 and this value may be expected to yield an upper limit on a reasonable range of A. With & = & = &,, then the nondimensional velocities at 7 = 0 are li, = liz= 4. In the following numerical examples, we choose initial velocities which yield the same values of the nondimensional kinetic energy as cii= ti2= 4. The eqns (54) and (55) were solved numerically by treating them as lirst order equations in iii and integratingto compute pi*The time interval was reduced until the solution showed no appreciable change in the solution. Cases were run for values of m for which the theorem indicated convergence occured fm 2 1) and also for cases where the theorem was not valid (m < 1) to see if non-convergencecould be observed. Two combinations of the exponents n and m were studied. In all cases the value n = 5 was chosen and solutions were computed for a range of values of A for m = 3 and m = l/3. In the case m = 3, the conditions of the theorem were satisfied and convergence was expected, whereas for m = l/3 the conditions of the theorem were not satiafiedand convergence may or may not occur. In fact, we discovered that in this later case, convergence occurred for sufficientlysmall values of A but divergence occured for larger values of A. The theorem was tested by evaluating the two solutions corresponding to ti2= 0 and 4, = IiJ3, such that their initial kinetic energy were the same. For m = 3 the solutions for increasing A are very similar in form and closely follow the A = 0 solutions. In Fii. 5 the ~spl~ements (ui, u3 are shown and we see that all histories follow similar paths, except that the finaldisplacements are markedly affected by the value of A. For A = 0.3 the finaldeflections are approximately one-half the values for A = 0. The velocity trajectories are shown in Fig. 6. Strain hardening appears to have little effect on either the velocity ratio ti,/& or the displacement shape, but causes the structure to come to rest more quickly and with a smaller deflection. The values of the convergence quantity A/K0with 7 is shown in Fig. 7 for the extreme cases A = 0 and A = 0.5. For A = 0, A = A, decreases rapidly until 7 = 1.5when it continues to decline very slowly. For A = 0.5, Au decreases even more quickly and, in fact, decreases throughout

Fii.

5. Noa~ime~ioMi

displacements (u,, rr3 for II = 5 and HI= 3 for a range of values of the non-dimensional material strain-hardening parameter A.

803

Dynamic behaviorof viscoplastic materials

Fii. 6. Nondimensional velocity paths for n = 5, m = 3 and a rangeof values of A.

%e 10 9 6 7 6 5 5 3 2 1

I

2

4

3

5

7

b

9

Fig. 7. Variationof convergance quantity A = A, + A, for II= 5 and m = 3.

1

3

2

Fig. 8. Nondimensional veioci!y par&s for tl

=

b

5

6

t

%

3, m = i/3. Note wide divergencefrom mode soiution for A -0.1.

A. R. S. PONTER

804

the history of the structure. ASincreases to a maximum value of less than 10%of A(0)before decreasing in value. Hence AS remains very small and A is dominated by A,..It should be emphasized that in this example, the effect of strain hardening is sufficiently great to reduce the final displacement by a factor of two and to reduce the deformation time by a similar order of magnitude. These results imply that for moderate amounts of strain hardening the displacement shape and velocity distribution are relatively insensitive to the amount of strain hardeni~, but the total de~e~tion and response time are markedly affected, This phenomenon may be explained by the notion of mode solutions which will be discussed in a forthcomi~ paper. For the case of m = t/3, we find that convergence occurs for su~ciently small values of A as the structure is brou~t to rest before the strain hiders terms are of significant~gnitude. The velocity histories for A = 0.01 and 0.1 are shown in Fig. 8 and the displacements and A’s are shown in Fiis. 9 and 10. For A = 0.01 the solutions are very close to A = 0 solutions. For A = 0.1, however, the solutions show non-convergent properties and have distinctly differing properties from those for M = 3. in Fii. 8 it can he seen that the velocity paths of both solutions initially follow the parts corresponding to A = 0 and subsesuently move over to and remain close to the line & = 34 (i.e. 9, = 0). This behaviour is reflected in the displacement paths shown in Fii. 9 and imply that the total history consists of two parts. During the initial part & remains small and during the final part I& remains small. This type of behaviour contrasts strongly with mode solutions where the displacement form remains constant in time. The fact that convergence has not occurred can be clearly seen in Fig. 10 where A,, A, and A both increase and decrease during the course of the deformation history.

i

i Fig.

9. Non-dimensional

i

i

lb

ri

i4

lb

ra

io

2i

2i

displacements (ul, Q) for n = 3 and m = l/3 for a range of values of the non-dimensional material strain hardening parameter A.

06

Fig. 10. Va~at~n

of convergence quantity A = A, + ASfor n = 5, m = 113. Note noa~onvergence

for

7 > I.

Dynamic behavior of viscoplastic materials

805

6. CONCLUSION A dynamic convergence theorem has been proven for a class of viscoplastic constitutive equations involving internal state variables. The constitutive equations are expressed in terms of a scalar potential function and particular forms are derived by assuming that the state may be calculated from either the plastic strain, the effective plastic strain or the plastic work. For the case of effective plastic strain the form becomes the familiar Nalvern material. For a particular simple model, structure dynamic solutions are generated for cases when the convergence proof holds, and the resulting solutions follow closely the mode-type of solution of a visco-plastic solid. For circumstances when the convergence proof does not hold, nonconvergence can be demonstrated, although it is clear that for weak strain hardening (corresponding to small values of the non~imensionaf parameter A) convergence can occur as the theorem is not a necessary, but only a sufhcient condition for convergence.

Ac&nowlrdgcmenfs-The work for this paper was carried out at Brown University, Providence, Rhode Island. and the support of the Division of Engineering is gratefully acknowledged. The author would like to express his thanks to Paul Symonds. Sol Bodner and John Martin for helpful discussions during the course of the work. The c&&ions invotved in the example were carried out by Helmut Bez at Leicestcr Un~ve~jty, and his assistance is ~tefuily ack~l~~.

REFERENCES I. J. B. Martin, A note on uniqueness of solutions for dynamically loaded rigid-plastic and rigid-viscoplastic continua. J. Appl. Mech. Tmns. ASME X3,207-209 (1966). 2. J. B. Martin and P. S. Symonds. Mode approximatjons for impulsively loaded rigid-plastic structures. J. Engxg. Me& Div. ASCE 91, EMS, 43-66 (1966). 3. 1. B. Martin and L. S. S, Lee, Approximate solutions for impulsively loaded elastic-plastic beams. I. A@. Me&, Trans. ASME 35, W-809 (1!%8). 4. P. S. Symonds, Approximation techniques for impulsively loaded structures of rate sensitive plastic behavior. I. S.I.A.M. 2S(3), 462473 (Nov. 1973). 5. P. S. Symonds and C. T. Chon. Approximation techniques for impulsive loading of structures of time dependent plastic behavior with finite deflections. Proceedings, Institarfe of Physics CdnJcrmce, on fhr Mechunicul Pmperfics of Mu~e~als at High Srmin Rates, S&i No. 21.29%316 (Dec. 1974). 5. P. S. Symonds and T. W~zbicki, On an extermal principle for mode form solutions in plastic structurat dynamics. f. Appl. h&h., Tmns. ASME 42,63%440 (1975). 7. A. R. S. Ponter. Convexity and associated continuum properties of a class of constitutive relationships. Journal de Micanique 15(4), (1976). i. L. E. Malvern. The propagation of longitudinal waves of plastic deformation in a bar exhibiting a strain rate effect. J. APP. Me&., Trans. ASME 18,203 (1951). 9. F. E. Hauser, Techniques for measuring stress-strain relations at high strain rates. Expff. Me&. 6,395 (H66~. 10. P. Perxena and W. Wojno, ~~y~~s of rate sensitive plastic materials. A&r. M&I. SOS. 20,499 (I%@. II. S. R. Bodner and Y. Partom, Consitutive equations for elastic-viscoplastic strain-hardening materials. L Appt. Mech. Tmns. AWE 42,385-389 (1975).

APPENDIX The inequaiity (14) may be derived in the case f&r,.

s) = f&#(q)

-g(u‘)).

(Al)

and

g(6) =

1’*dB * /(fi)=s

.

(AZ)

directly from the conditions (33) n,ZO

w

R"Z0

(A4)

g'i?O

(A-V

g”s0

(A@

and the further condition that d(oii) shall be convex. Transforming g(p) = g(s) where for A2 (A7)

A. R. S. Potam

806

and

fA8) lncqualitiis (A33) and (A4)imply IIS convexityof f2 R(d’)-ll(cb3-n’($3(
fA9t

Simiiarity(AT)mid (AS)imply tie concavityof B(S) -(8(s’)-fffs”~)+5q~~s’-~~zco.

(AlOf

If 4’ and cb*are replacedby 4’ - $0’) and d- U(A in (A91and inequality(AlO)mWpiicd by fk‘(#) * 0 is added to the resultinginequality,we obtain

On further noting that

and that

and

then (Al I) becomes

as required.