analysis of mechanical behavior in cold-drawing deformation of polymers

Analysis of Mechanical Behavior in Cold-Drawing Deformation of Polymers

ZHU XlXlONG

Material Science & Mechanics Research Center, Ningbo University(315211), Zhejiang, China

SYNOPSIS

When stretched under uniaxial stress, ductile polymers usually exhibit unstable plastic deformation, which embodies two phases: ( a ) yielding with the formation of a neck and (b) cold-drawing with the propagation of necking shoulders. The mechanical state associated with this deformation behavior is analyzed. The discussion is divided into three parts. The first part is a general treatment of the constitutive function of flow stress in the plastic state, in which a series of relations among various characterizing parameters are formulated. The second part provides three mechanical criteria for necking deformation and propagation of necking shoulders: the condition of unstable plastic deformation requiring Dp = - (d In i /dc), < 0; the stabilizing deformation mode, which requires -yp = (dDJde) , > 0; and the obvious localization of unstable plastic deformation. The third part describes a mathematical model which can be used in calculations to fit the contour of the necking shoulder. This model is developed according to rational considerations for the relation of In i to E. Ex- perimental data on PE rod specimens are well fitted by this model. @ 1993 John Wiley & Sons, Inc. Keywords: unstable plastic deformation cold-drawing constitutive function mechanical criterion necking shoulder

INTRODUCTION

When a ductile polymer is stretched under uniaxial stress, the curve of load P versus elongation A1 will usually display the behavior shown in Figure 1. First there is unstable plastic yielding deformation at a local maximum, and a neck is formed. This is fol- lowed by unstable plastic deformation of cold-drawing, usually under nearly constant load over a large strain, with the propagation of necking shoulders along the specimen. Finally, when the necking shoulders have propagated through the entire length of the specimen, the deformation will exhibit the character of strain hardening, and the specimen will rupture at some higher stress.' The stresses corresponding to the yield point, the cold-drawing plateau, and the rupture point are known as the yield stress a,, cold-draw stress a d , and rupture stress uf,

Journal of Polymer Science: Part B: Polymer Physics, Vol. 31.1667-1675 (1993) 0 1993 John Wiley & Sons, Inc. CCC OSS7-6266/93/121667-09

respectively. The true stress a and true strain E will be used for our purpose.

Although yielding and cold-drawing have been utilized for a long time in polymer process technol- ogy, and have attracted more attention since the recent development of ultrahigh-modulus polymers, the mechanical behavior and relevant criteria are not well studied and there have been few quantitative and systematic discussion^.^-^ A simpler and clearer phenomenological analysis of the mechanical state of the deformation behavior according to the physical picture shown in Figure 1 is presented. Us- ing the constitutive function for plastic deformation, the mechanical criteria for neck formation and propagation of necking shoulders will be formulated. A mathematical model, which can be used to fit the contour of the necking shoulder in a specimen is provided.

Similar plastic deformation behavior can also appear in metals (i.e., the famous Luders bands); however, the morphology and the extent of deformation are not as clear and complete as in poly- m e r ~ ? . ~

1667

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Elogation, Al Figure 1. polymer in stretching.

Typical load vs. elongation curve of ductile

THE CONSTITUTIVE FUNCTION FOR PLASTIC DEFORMATION

Considerable attention has been given to the development of accurate constitutive equations of plasticity in recent years, and the general form of the equations for homogeneous plastic deformation is now well established." It is general opined that when elastic deformation can be neglected and the problem is limited to the case of uniaxial stress, the constitutive equations will include two groups. The first is the constitutive function of flow stress u, which will be a function of temperature T, plastic strain e, and some internal structure variables (internal variables) ~ i , characterizing the state of strain hardening or softening:

Since the plastic deformation process depends on the path of the deformation, the corresponding strain would also depend on the path. Thus, the equation for strain rate C characterizing the deformation process, and the kinetic evolution equations of the internal variables are also needed. Therefore, the second group of equations will be composed of the time rates of change of E and ~i as follows:

where f and gi are specific functions for each material.

However, we shall take the conditions of constant temperature and only one internal variable K (the one-variable approximation). Since the mechanical behavior of polymers is sensitive to the strain rate, it is convenient to take the strain rate C to replace E as a state variable. Finally, we obtain the following constitutive function for the flow stress in much simplified form: ''-I3

U = U(k, K ) ( 3 )

Equation (3) can be also expressed in differential form as:

d In u d l n k ( 4 )

Thus under the simplified conditions mentioned above, the total or dynamic hardening d In u, i.e. the change of flow stress, can be decomposed into two parts: the first is the strain hardening ( a In u / d In ~)hd In K , and the second is the viscous hardening, ( d In u / d In C),d In 1.. Upon integration, we obtain:

Alnu=J(-) d In u dlnK+J(-) d In u dlnC dlnK a l n k

= A In US + A In uv ( 5 )

In general, the strain in the specimen will be a function of the coordinate position x and time t, i.e. E = E ( x , t ) . If consideration is limited to the change of u in a given section element, t may be replaced by E , because the later will be a unique function of t . This leads to the following evolution equation for the variation of u with c at fixed position (we are not concerned with a time-dependent internal variable K ) .

d In u

We may define two characteristic parameters as follows:

strain rate sensitivity (7)

H = ( - In ") ( - a ") strain hardening rate (8) dlnK X

MECHANICAL BEHAVIOR IN POLYMERS 1669

Although V is a unique function of state variables ( K , k ) , and in general takes a value larger than 0, H is path-dependent. For an unambiguous definition of H, therefore, the deformation path must be specified. For instance, (1) if k is constant,

(2) if P is constant,

(3) if u is constant,

We can also introduce the dynamic hardening rate as follows:

S = ( F ) X

S is also path-dependent, and must be specified for different deformation paths, e.g., (1) if u is constant

( 2 ) if P is constant,

If the constitutive function a( K , k ) is rewritten in terms of strain rate k( K , a), then the evolution equation for strain rate may be obtained

d In 1.

We introduce a new characteristic parameter D = - ( a In k /d&) , , which represents the dependence of the strain rate on the strain, and will play a de-

cisive role in the following discussions. From these definitions we can obtain the relation

It should be noted that D is also path-dependent. By specifying the deformation path and comparing this equation with eqs. ( 10) - ( 14), we obtain:

(1) if u is constant,

(2) if P is constant,

For example, in a creep test of uniform deformation, D, and Dp will represent the decelerations of the deformation rate a t constant u or P, respectively.

Unstable Plastic Deformation of Specimen

The yielding and cold-drawing deformations shown in Figure 1 contain two types of nonuniform deformation processes: one is the initiation of local necking, and the other is the propagation of necking shoulders along the specimen. Although both arise from the local instability of deformation, they are different in behavior. The former belongs to the geometrical instability of deformation described in the classical work of Considere (1885) .' If the specimen contains geometrical inhomogeneities, or local defects (local weak points), the plastic yield deformation will preferentially develop in these areas. Once the local plastic deformation begins, succeed- ing plastic deformation will also be concentrated in this local region by virtue of intrinsic strain softening of the polymeric material, Finally necking will develop, The critical point from stable to unstable deformation is the local maximum yield point in the P versus A1 curve shown in Figure 1; the corresponding criterion can be expressed as (du ldc ) , = u. With the constitutive relations suggested above, it can be also represented as

H p = 1, D p = O (19)

The practical situation is such a case. For example, when V > 0, D is positive as long as Hp > 1. This means that the fluctuation in k will decrease

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with increasing e, and the deformation will be sta- bilized by this deceleration. On the other hand, when Hp < 1 and Dp < 0, any fluctuation in E will increase with increasing e. This is the case with unstable deformation, and necking will develop in a certain area of the specimen. If we consider the change in H p , two modes of deformation transition can appear. If Hp decreases with increasing e and changes from > 1 to < 1, a transition from stable to unstable behavior will occur at the point Hp = 1, and necking will gradually develop. We call such behavior a destabilizing deformation mode, and the critical strain with Hp = 1 the stability limit e s ~ . If Hp increases with increasing e and changes from < 1 to > 1, corresponding to a transition from unstable to stable behavior, i. will decrease gradually and approach zero. The necking and the corresponding plastic deformation in this local region will not develop further, but its fronts, i.e. the necking shoulders comprising the in- terfaces of necking and non-necking regions, will propagate along the specimen. This is the state of cold-drawing plastic deformation. We call such behavior a stabilizing deformation mode and the critical strain with Hp = 1 the instability limit, EIL. The criterion to distinguish these two modes is given by:

If rp > 0, the stabilizing mode and continuous cold- drawing plastic deformation will appear; whereas if

rp < 0, the destabilizing mode and only the local necking deformation can appear until rupture of the specimen.

It is interesting to compare the strain hardening behavior of two modes at P = const, as inferred from the above considerations. Obviously, the dynamic hardening for both modes can be represented by the integration of Sp = ( 8 In a/&) = 1, with the result:

A In u = In u - In ui = e - ei (21)

where ui and ei are the initial stress and strain on entering the deformation state with P = const. In coordinates In u versus e, A In u for brhh modes are represented by straight lines, as shr wn in Figure 2; but their components (i.e., st. a;*i hardening and viscous hardening) are quite difierent: ( 1 ) destabilizing mode

A In US = 1: Hp de > A In u,

A l n u v = - i : V D p d e < O

(2) stabilizing mode

Figure 2. Schematic plots of flow stress vs. strain and decomposition into contributions from strain hardening and viscous hardening: (a) Destabilizing mode: urn, em, local maximum stress and strain; ui , ei , initial stress and strain; a,, saturation stress. (b) Stabilizing mode: ui, e i , stress and strain at front of necking shoulder; e,, strain at center of shoulder; e,,

strain at end of shoulder.


Typical examples of these components are also shown in Figure 2.

By supposing V = const in the first-order approximation, we can also obtain from eq. ( 18) :

1 V Dp d e = - A In uv (24)

This gives, at least qualitatively, a dependence of In i on c, which is analogous with the dependence of In uv on E, as shown in Figure 3. On the other hand, when I.( c ) according to eq. (18) is known, the localized strain c( t ) is given by the inverse function:

The whole unstable plastic deformation from the local maximum to the constant load plateau, as shown in Figure 1, essentially consists of two parts (i.e., the yielding and cold-drawing deformations as discussed in this section). Moreover these two parts are continuous with each other. Although the practical process is not at constant load, and the situation may be more complicated, the basic scheme is still analogous with the constant-load process described above. For example, the connecting point between the two parts is at the inflection point of the curve.

A Mathematical Model of Cold-Drawing Deformation

In order to treat unstable plastic deformation theo- retically, the roles of two independent factors (i.e.,

plastic instability and flow localization) have to be clarified. The former is a problem of the mechanical state only, specified by the criteria of eqs. (19) and (20), according to the constitutive relations discussed above. On the other hand, the flow localization involves spatial differences in strain and strain rate, and will occur if gradients in strain or strain rate are accentuated. Thus, the localization criteria have to be derived from a consideration of the kinetics of deformation in space and time. ln the case of stationary propagation of a necking shoulder, the strain distribution in space at a given time would be in equivalent to the change with time for a fixed coordinate position. For this reason, we can give an unified treatment by combining two variables x and t into a new variable z by the relation z = x + u t , where u is the propagation speed of the necking shoulder. When z = 0, x = ut , this indicates that the center of the necking shoulder is at position x at timet.Bysettingc(x,t) = e ( z ) , e ’ ( z ) = d e ( z ) / d z , eq. ( 2 5 ) may be rewritten as:

where e, is the strain at the center of the necking shoulder ( z = 0).

According to the preceding discussion of unstable plastic deformation at constant load, a quantitative mathematical model, which may be used to analyze the stationary propagation of the necking shoulder is developed in this section. Specifically, two power functions, connected continuously, are used to fit

Figure 3. stabilizing mode.

Typical curves of In E vs. E according to eq. (24): (a) destabilizing mode; (b)

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the curve of In 1. versus c as shown in Figure 3b. This can be expressed in differential form as follows:

Upon integration and exchange z and e for t and c, we obtain

where the limits ef (corresponding to ci ) and e, are the corresponding strains at the front and back of the necking shoulder, respectively. e, also corre- sponds to the strain, for which e' decreases to ei again (corresponding to 1.i ) ; thus it also represents the characteristic strain produced when the necking shoulder is passed. The integration constant e: = (de/dz), is the maximum gradient of strain at the center of the necking shoulder. It should be noted that this is only the change of strain or strain rate produced by passing the necking shoulder. The strain rate will at first increase to the maximum e:, and then gradually decrease to the plateau value. As a result, the specimen will experience a large cold- drawing strain: ed = er - e,. When the specimen is stretched at constant load, creep deformation will also occur; and for this reason, the total strain will be equal to the sum of the cold-drawing strain ed and the creep strain eg.

The coefficients k,, kd, rn, and the integration constant e: may be determined in general by fitting the theoretical formula to the experimental data (e.g., Marquez-Lucero et al.'s recent series of experiments on the stationary propagation of the tensile necking shoulder in rod and membrane specimens of polyethylene) .14 The data reported, which are accurate and comprehensive, can be used to fitting calculations according to the mathematical model suggested above. By fitting eq. (28) to the experimental data for a rod specimen of PE, we obtain the following results:

ln(e') = 0.004784

12.3257(e - ec)4 rising section

10.0953 (e - e,) (29)

descending section

with e: = 1.0490 and e, = 0.7000 at the center of the necking shoulder. Comparisons of theoretical and experimental curves are shown in Figure 4.

0 -

h

-b I 1- 3 Y E

3 rA

W

-9 2 -

3 I I I -0 .5 0 0. 5

1.

B 5 - 0.

\

8

Ralative strain, (e-G)

(a)

0

=exp[-12.3257(e--+

0 0. 5 1. 0 1. 5 Strain, e

Figure 4. Comparisons of theoretical curves obtained by fitting calculations with experimental curves: (a) In ( -e l ) versus ( e - e,.); (b) e'/e: versus e .

By introducing e = ef and e = e, into eq. (28) and setting e) = e:, we obtain:

In order to obtain a relation for the contour of the necking shoulder in the state of stationary propagation, we have to do another integration of eq. (28). By introducing the integral function:


and integrating eq. (28) , we obtain

where represents the inverse function of eq. (31 ) , and zf and z, are the values of z at the front and back of the necking shoulder. This is a func- tional expression for the contour of the necking shoulder in terms of the relationship of e with z . The characteristic width of the necking shoulder may be defined by w = ed/e: = ( e , - ef ) / e : . By using the experimental data reported by Marquez-Lucero et al. to evaluate the coefficients and integration constants, good agreement is obtained between the theory and experiments, as shown in Figure 5.

It is apparent from eq. (32) that the contour of necking shoulder in a phenomenological description is determined by two integration constants ec and e:. But they cannot be reduced to two phenomenological constants. They are probably related to the microscopic process occurring in the necking region.

Analysis of Deformation Localization

As noted in the preceding sections, the process of plastic deformation appearing in Figure 1 consists of two parts, i.e. the yielding deformation with the formation of the neck, and the cold-drawing deformation with the propagation of the necking shoulder along the specimen. These two regimes are continuously connected. We can consider that the former

would provide the necessary conditions for the development of the latter: i.e. would serve as the "nucleation phase" for the whole process of cold-drawing deformation. Obviously, the main point is the localization of plastic deformation. We turn now to a more detailed discussion of this problem.

Although the deformation path is not specified in the nucleation step, we can visualize eq. (28) as a quantitative demonstration of the localization condition. It is highly probable that the nucleation takes place at a certain weakened or stress-concentrated region in the specimen where the yielding stress has a local minimum. Let us consider a tensile test of a specimen. The nominal stress rate, uN( uN = P / A , , where A, is the initial section area) can be expressed by machine equation:

where Em is the effective modulus of stretching sys- tem, urn is the cross-head speed of the test machine, and 1 is the length of the specimen. The nucleation phase will appear only if the local deformation rate is so high that the yield stress in this local region, ( (J:)LW, is smaller than the average yield stress of the specimen The chance for (u:)Loc < ( U:)AV will be the greater, if (E,)LOC, which is the corresponding local yielding strain, is smaller. The yield stress will be reached when iN = 0, i.e. when u = E dx. The corresponding (c , )LOC may be estimated approximately as follows. Assuming that the nucleation has been initiated, then in the local yielding area we have:

0

Coordinate parameter, 2

Figure 5. Plot of e vs. z , illustrating contour shape of necking shoulder under stationary propagation condition: comparison of theoretical curve obtained by fitting calculations with experimental curve.

1674 XIXIONG

u, = li dx = s,” i dx = wi,exp[-k,(e - eC,)”]

(34)

In the integration, we have supposed that the deformation is localized in a narrow region of width w and, as a crude approximation, has constant 1.. By substituting eq. ( 30) , we finally obtain

In the preceding derivation, we have recovered variables c and t, because the discussion is formulated in an appropriate way. Since the logarithmic terms appearing in the preceding equation can only produce small variations, the order of magnitude of (c,)LW is determined mainly by the parameter kr. Thus, requiring a smaller ( e , ) L m is equivalent to requiring a larger coefficient kr. For example, from the experimental dataI4 on a rod specimen of PE, we have obtained k = 12.3257 by fitting calculations. This is the localization condition required to produce the cold-drawing deformation, and can be represented as

Finally we discuss the estimation of the strain which can be measured in experiments. The measured value of a strain is usually the average strain ct in a certain gauge length of the specimen. This strain includes two parts: the cold-drawing strain &d

as discussed above and the creep strain e, produced in the period of propagation of the necking shoulder across the gauge length of the specimen.* There are two methods to estimate the total strain e,. One method is to estimate the macroscopic strain. Sup- pose that lo is the initial gauge length of the specimen, the coordinate position of the center of the necking shoulder is initially a t X I = -6, but the final position is at xp = 0. As the necking shoulder moves, the gauge length changes from lo to 1, so that the propagation time is given by t = E/v and the initial position x1 is shifted to x1 = -1. If the characteristic width of the necking shoulder is very small (i.e. w -4 1) the strain cl at x = -1 is very close to the ex-

* In the strict sense, the total strain should also contain contributions furnished by the nonstationary processes of initiating and terminating the propagation of the necking shoulder, but we shall ignore these in the first approximation.

perimental value et. Introducing et instead of cl into the well-known relation, 1 = &exp(ct), and taking u, = ct( u + ug) _N q u where v, (6v) is the constant creep rate, we obtain

1 ef dc t = ; = s , E(E)

(37) ’Jm e, = -exp(-c,) lo

This is essentially an estimation by using the inverse function of eq. (25).

Another method is to make separate estimations. By taking cg = ugt, &d = c, - ef = ( e , - c f ) + taken z = -1, we can finally obtain an estimating formula:

[-ki/m e,loexp(et)]/kA/m, ’ where we have already

We anticipate that the results estimated from the two methods will be comparable.

CONCLUSIONS

The mechanical state and behavior for the local plastic deformations involved in yielding with the initiation of necking in a stretched specimen and subsequent cold-drawing with propagation of necking shoulders along the specimen are analyzed in detail.

According to the practical conditions of deformation in uniaxial tension of a polymer, a suitable, simplified constitutive function for the flow stress in the plastic state is suggested. The relations among various characteristic parameters, including the dynamic hardening rate S , the strain hardening rate H, the strain-rate sensitivity V, and the deceleration in strain rate D, as well as the behavior character- istics of these parameters are expressed and discussed. The content of this section provides the necessary theoretical basis for the subsequent analysis.

According to the above discussion, three criteria must be satisfied in order to arrive at the mechanical state of cold-drawing deformation:

1. The condition of initial unstable plastic deformation: Dp = - (d In i / d ~ ) ~ < 0;


2. The stabilizing deformation mode: y,,

3. The localization condition: k, B 1. = (dDJdc) , > 0;

From the rational consideration of the constitutive function, a quantitative mathematical model, which may be used to fit the contour of the necking shoulder under the condition of stationary propagation, is developed. Specifically, two continuously connected power functions define the rational curve of In .i versus E. When this is used to fit experimental data for a rod specimen of PE, the agreement is rea- sonable.

The author thanks Mr. Huang Xusheng for calculations to fit the experimental data of Marquez-Lucero et al.’*

REFERENCES A N D NOTES

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Received October 17, 1991 Revised May 3, 1992 Accepted December 31, 1992

analysis of mechanical behavior in cold-drawing deformation of polymers

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