Torsion of a fiber reinforced hyperelastic cylinder for which the fibers can undergo dissolution and reassembly

International Journal of Engineering Science 48 (2010) 1179–1201

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Torsion of a fiber reinforced hyperelastic cylinder for which the fibers can undergo dissolution and reassembly Hasan Demirkoparan a, Thomas J. Pence b, Alan Wineman c,⇑ a

Carnegie Mellon University in Qatar, P.O. Box 24866, Doha, Qatar Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA c Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA b

a r t i c l e

i n f o

Article history: Available online 29 October 2010 Dedicated to K.R. Rajagopal on the occasion of his 60th birthday. His work serves as an inspiration to all who ply within the discipline of mechanics. Keywords: Hyperelasticity Fiber reinforcement Fiber dissolution Fiber reassembly

a b s t r a c t In previous work, the authors have developed a theory for treating microstructural changes in fiber reinforced hyperelastic materials. In this theory, fibers undergo dissolution as a result of increasing elongation and then reassemble in a direction defined as part of the model. Processes in which the fibers reassemble in the direction of maximum principal stretch of the matrix were specifically considered. This model was previously illustrated for various cases of homogeneous deformation. The present work studies the implications of the model during the non-homogeneous deformation of axial stretch and torsion of a circular solid cylinder composed of an isotropic matrix and families of helically wound fibers. It is shown that the process of fiber dissolution and reassembly produces complex morphological changes in the fibrous structure and hence, in the response of the cylinder. Such events can give rise to an outer layer of material in which the fibers have undergone dissolution and reassembly. The interface between this region and the as yet unaltered core material can then move radially inward as axial stretch and/or twist increase. Gradual reassembly of the fibers with increasing stretch and twist changes their contributions to the torque and axial force and their helical orientation. Different sequences of axial stretch and twist result in different morphologies in the fibrous structure. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction A variety of engineering materials and biological tissues consist of long chain filaments distributed in a softer matrix. Such materials confer significant mechanical benefits on structural behavior including stiffness in key directions, mechanical toughness as regards fracture resistance, and tailoring of properties so as to permit efficient joining to other materials. The large strain mechanical response of materials in which elastic fibers are embedded in an elastic matrix can be modeled in the context of anisotropic hyperelasticity where the fibrous structure is homogenized and represented by different properties in certain directions. To this end, various constitutive models in the context of anisotropic hyperelasticity have been developed as in, for example: [1–5]. The elastic response of such model materials can be considered as arising from a single microstructural mechanism during the entire range of response, namely the distortion of macromolecules in the matrix or in the molecular structure of the fibers. A number of recent publications consider new microstructural events that occur during the response of polymeric materials. Various approaches treat the Mullins effect ([6–12]) while others explore other aspects of damage in polymers ([13–15]). The microstructural event of interest here that occurs in polymeric materials is the scission of macromolecular ⇑ Corresponding author. Tel.: +1 734 936 0411; fax: +1 734 764 4256. E-mail address: [email protected] (A. Wineman). 0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2010.09.001

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network junctions and their re-crosslinking to form new macromolecular networks. This micromechanism was introduced by Tobolsky [16] to explain the results of extensive experiments on rubbery materials. These events have been modeled in a large deformation constitutive theory in a series of papers beginning with [17]. This theory can be used to describe changes in the matrix component of a fibrous composite material. It is also possible that the fibers themselves undergo microstructural change. Accordingly, this framework was extended in [18] so as to treat similar issues in the filamentary reinforcing networks in hyperelastic materials. In this theory, fibers undergo a process of dissolution as a result of increasing elongation followed by reassembly in a direction defined as part of the model. In [18] the fibers were assumed to reassemble in the direction of maximum principal stretch of the matrix. The implications of this model have been illustrated in detail for homogenous deformations of a material having fibers in the reference configuration that are all aligned in a common direction. In [18], a detailed study was carried out of the consequences of uniaxial stretching along the direction of the fibers. In that particular illustration, the reassembly process did not result in a change in fiber alignment. Later, Demirkoparan et al. [19] discussed increasing simple shear when the original fiber direction is perpendicular to the direction of shear. The resulting fiber elongation with increasing shear resulted in fiber dissolution over a constitutively determined interval of the amount of simple shear. Newly formed fibers became aligned in the current principal direction of maximum stretch, which is a direction that changes with the amount of simple shear. This resulted in the formation and structure of a fan-like fiber morphology at each material point. The purpose of the present work is to study the implications of the model when the deformation is non-homogeneous. In particular, we consider a circular solid cylinder consisting of an isotropic matrix and families of helically wound fibers. The cylinder is subjected to simultaneous axial stretch and twisting about its central axis. The case of a cylinder consisting of only isotropic elastomeric material undergoing microstructural change was previously considered in [20]. That work can be regarded as a study of microstructural change in the matrix component of a fiber reinforced composite. The present work is concerned with microstructural change in the fibrous component. Each fiber is stretched by an amount that depends on the radius, orientation, axial stretch and twist. In order to develop specific insights into the consequences of fiber dissolution and reassembly and to avoid complexity, we analyze the special case when there is a single family of fibers that are oriented in the axial direction and uniformly distributed along the radius. Moreover, it is assumed that the fibers undergo instantaneous dissolution and reassembly. Even with these simplifications, it is shown that the process of fiber dissolution and reassembly produces complex morphological changes in the fibrous structure. It is shown that, in general, there is an inner core of original material and an outer layer of material in which the fibers have undergone dissolution and reassembly. The interface between these regions can move radially inward as axial stretch and/or twist increase. Reassembly of the fibers in the outer layer changes their reference orientation from axial to helical with a radially varying angle. It is shown that different sequences of axial stretch and twist result in different morphologies in the fibrous structure. This paper is organized as follows. Section 2 presents an overview of the general constitutive framework introduced in [18] but generalized so as to consider multiple fiber families in the original reference configuration. Section 3 then considers the combined deformation of torsion and axial stretch. It contains a detailed discussion of the kinematics of deformation before and after dissolution and reassembly of the fibers, the issues involved in calculating the stresses and the satisfaction of the equations of equilibrium. Fiber dissolution and reassembly for the case of axially aligned fibers is examined in Section 4. It treats the influence of the sequence of axial stretch and twist on the radial motion of the interface between the inner core of original material and the outer layer of transformed material. Section 5 addresses how this sequence affects the morphological changes in fibrous structure. It is shown that different sequences connecting the reference configuration with the same deformed state produces different morphologies. Section 6 presents numerical examples for a specific choice of matrix and fiber properties and different axial stretch/twist sequence. Final comments are given in Section 7. 2. Constitutive theory Deformation of the material maps locations X in a reference configuration to locations x in the deformed configuration. The deformation gradient is F = @x/@X, and the right and left Cauchy Green deformation tensors are

C ¼ FT F;

B ¼ FFT

ð2:1Þ

with principal scalar invariants I1, I2, I3 and principal stretcheski (i = 1, 2, 3). The materials considered here are incompressible, so that

det F ¼

pffiffiffiffi I3 ¼ 1:

ð2:2Þ

Prior to microstructural change the materials are hyperelastic with a stored energy density function W(C) such that the Cauchy stress tensor is given by

T ¼ pI þ 2F

@W T F ; @C

ð2:3Þ

where p is the hydrostatic pressure arising from the constraint (2.2). The representative volume element in the continuum treatment contains both fibrous and matrix components, so that quantities of energy and stress are regarded as appropriate homogenized quantities. This is the standard view in anisotropic

H. Demirkoparan et al. / International Journal of Engineering Science 48 (2010) 1179–1201

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nonlinear elastic treatments of fiber composite materials. In addition, it shall be assumed that these homogenized quantities involve separately identifiable matrix and fiber contributions. This is also relatively common in hyperelastic treatments of fiber composite materials. Thus, prior to microstructural change, W in (2.3) is taken to be in the form W = Wm + Wf where Wm is the matrix energy storage density and Wf is the fiber energy storage density. The present treatment involves dissolution and reassembly of the fibrous component while the matrix in which the fibers are embedded remains fixed in its mechanical properties. The contribution of the matrix to the material stress is taken to be given as in conventional hyperelasticity. Thus, in general, the Cauchy stress is taken to be in the form

T ¼ pI þ 2F

@W m T F þ Tfib : @C

ð2:4Þ

Prior to dissolution and reassembly, Tfib is given by

Tfib ¼ 2F

@W f T F @C

ðbefore microstructural changeÞ:

ð2:5Þ

It is assumed that Wm depends only on the deformation of the matrix while Wf depends upon the deformation of all of the fiber families. It is further assumed that the matrix is isotropic in its reference configuration so that Wm = Wm(I1, I2). It is presumed that there are Nf families of fibers in the representative volume element corresponding to the material point. Let M(j) with 1 6 j 6 Nf be a unit vector that gives the orientation of fiber family j in the representative volume element. In the present modeling, it is assumed that Wf is the sum of separately identifiable contributions from each fiber family, meaning that

Wf ¼

Nf X

W f ðjÞ ;

W f ðjÞ ¼ W f ðjÞ ðK ðjÞ ; K ðjÞ Þ;

ð2:6Þ

j¼1

where K(j) and K ðjÞ are notations for

K ðjÞ ¼ MðjÞ  CMðjÞ ¼ FMðjÞ  FMðjÞ ;

K ðjÞ ¼ MðjÞ  C2 MðjÞ ¼ CMðjÞ  CMðjÞ :

(j)

ð2:7Þ

ðjÞ

Here K is the square of the fiber stretch, while K accounts for fiber shearing effects. When there is only a single family of fibers, K(j) is often written as I4, and K ðjÞ is often written as I5. In what follows we shall further assume that Wm only depends upon I1 and that W f ðjÞ only depends upon K(j). This simplification is not necessary for the development of the theory, but it does allow for simplification in the overall exposition. Similarly, while the mechanical description of fiber dissolution and reassembly would generally depend upon quantities analogous to both K(j) and K ðjÞ , the development here will be in terms of quantities analogous to only the fiber stretch quantity K(j). By virtue of these assumptions it follows that (2.4) and (2.5) give a Cauchy stress in the form N

T ¼ pI þ 2

f X dW m ðjÞ Bþ Tfib dI1 j¼1

ð2:8Þ

with ðjÞ

Tfib ¼ 2

dW f ðjÞ dK

ðjÞ

FMðjÞ  FMðjÞ

ðbefore microstructural changeÞ:

ð2:9Þ

We now turn to the consideration of fiber dissolution and reassembly. It is assumed that the j fiber family begins to dissolve (j) if K(j) reaches the activation value K ðjÞ increases past K ðjÞ s . If K s there will then be additional fiber dissolution. The amount of ðjÞ dissolution is presumed to increase as K(j) increases. If K(j) reaches the value K f > K ðjÞ s then it is presumed that the original fibers in the j family have completely dissolved. Here the subscripts s and f stand for start and finish, and it is presumed that ðjÞ the constants K ðjÞ s and K f are material properties. It is possible to modify the development presented here in order to permit some portion of original fibers to be immune to ðjÞ dissolution. In this case, when K(j) exceeds K f , there are both some remnant portion of the original fibers and another portion of the reassembled fibers. This possibility leads to additional complexity and is not considered here. ðjÞ The dissolution as a function of K(j) causes Tfib to no longer be given by (2.9). Specifically, dissolution by itself causes the fiber contribution to the stress field to be diminished. In addition, reassembly gives rise to additional material stiffening that ðjÞ is described by additional stresses. Thus, once microstructural change has occurred in the j fiber family, the stress Tfib is taken to have the form ðjÞ

Tfib ¼ 2bðjÞ

dW f ðjÞ dK

ðjÞ

ðjÞ

FMðjÞ  FMðjÞ þ Treassembly :

ð2:10Þ

The first of the terms on the right side of (2.10) gives the contribution due to any undissolved original fibers. Here the value b(j) decreases from an initial value of one as the original fibers dissolve. Specifically, the value b(j) is a function of the maximum value that K(j) has ever reached. In particular, it is not a function of K(j) itself, because decrease in K(j) after it has ex(j) ceeded K ðjÞ over all past times up to and including s does not result in any type of fiber healing. The maximum value of K

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ðjÞ ðjÞ (j) the current time shall be denoted by K ðjÞ  . The  function b decreases from one to zero over the interval K s 6 K max 6 K f . In max the simplest modeling, the function bðjÞ K ðjÞ is regarded as known, and simple constitutive specifications for this function max can be used for modeling purposes. ðjÞ The second of the two terms on the right side of (2.10), namely Treassembly , gives the contribution of the reassembled fibers. ðjÞ The specification of Treassembly will generally require a detailed consideration of the physical processes associated with the fiber reassembly. Alternative processes will require very different mathematical models. For our purposes here we follow the framework discussed by us in [18,19]. Specifically, we here consider models for processes in which all of the dissolved fibers immediately reform into new fibers that have similar mechanical properties to the original fibers. The similarity in properties must account for the different underlying matrix state at the time of reassembly. Namely, the deformed state of the matrix at the instant of reassembly becomes the natural reference state for the reassembled fibers. This is consistent with the broader notion of multiple reference states which is discussed in detail in the context of isotropic interpenetrating networks in [17]. In addition, the model must not only specify the mechanical properties of the reassembled fibers, but also the direction of the reassembled fibers. Here we continue to follow our previous work by considering a model for which the direction of newly formed fiber is taken to be the direction of maximum principal stretch in the matrix at the instant of reassembly. Reassembly in the present modeling is tied directly to dissolution and so also correlates directly with K ðjÞ max on the interval ðjÞ

ðjÞ

ðjÞ

ðjÞ ðjÞ and K ðjÞ s 6 K max 6 K f . In order to describe the stress contribution Treassembly using a common formula for both K max < K f ðjÞ

ðjÞ

ðjÞ

ðjÞ ðjÞ K ðjÞmax P K f it is useful to introduce K ðjÞ  ¼ MinfK max ; K f g. Thus if K max < K f , meaning that some of the original fibers still re-

main, then

K ðjÞ 

¼

K ðjÞ max .

Alternatively, if

K ðjÞ max

>

ðjÞ Kf ,

ðjÞ

meaning that all of the original fibers have dissolved, then K ðjÞ  ¼ K f . Dis-

solution, and hence reassembly, takes place at all past instances of the deformation during which K(j) was equal to K ðjÞmax while _

ðjÞ

ðjÞ simultaneously increasing through the interval K ðjÞ 6 K f . We use superscript ðÞ to indicate quantities associated with s 6 K _

ðjÞ these instances. Using this notation, we say that dissolution and reassembly has taken place for K ðjÞ 6 K ðjÞ s 6 K  . While this notation may be somewhat cumbersome, it allows for the consideration of processes for which dissolution may or may not have gone to completion. It also allows for processes in which K(j) exhibits episodes of both increase and decrease during the dissolution process, in which case only the periods of increase beyond the previous maximum give rise to dissolution. By _

ðjÞ

this means, the reassembly stress Treassembly is expressed as an integral with respect to K ðjÞ as follows: ðjÞ

Treassembly ¼ 2

Z

ðjÞ

K

ðjÞ Ks

_

aðjÞ ðK ðjÞ Þ

dW f ðjÞ dK

ðjÞ

^ _

^ _

_

jbðjÞ F MðjÞ  F MðjÞ dK ðjÞ K

ð2:11Þ

and it is now necessary to explain the numerous notations in (2.11). Here it is to be noted that, in addition to the previously _ ^ motivated ðÞ notation, there is now an additional ðÞ notation. We now discuss the terms in (2.11) that are associated with these two notations. _

_

The ðÞ notation in (2.11) appears in the previously mentioned K ðjÞ which allows for an integration over all past instances _ of dissolution. The function aðjÞ ðK ðjÞ Þ represents the density of the newly formed fibers during the reassembly process and so _ ðjÞ ðjÞ is only defined on the interval K ðjÞ 6 K f . In order to ensure that the reassembled fibers account for all of the just diss 6 K solved fibers, the function a(j) is related to the previously introduced function b(j) as follows:

Z

ðjÞ

K max

ðjÞ Ks

_

_

ðjÞ aðjÞ ðK ðjÞ ÞdK ðjÞ ¼ 1  bðjÞ ðK max Þ; )

aðjÞ ¼ 

_

dbðjÞ ðjÞ

dK max

ð2:12Þ

:

_

The other use of the notation ðÞ in (2.11) occurs in MðjÞ which is a unit vector giving the orientation of the reassembled fibers in _

the configuration corresponding to the instant of reassembly. As indicated previously, the direction given by MðjÞ is taken to be _

the direction of maximum principal stretch in the matrix at the instant of reassembly. Introducing F as the deformation gradient _

_

_ _

at the instant of reassembly, it follows that MðjÞ is the unit eigenvector of B ¼ F FT associated with the maximum eigenvalue. ^

^

We now turn to consider the ðÞ notation in (2.11). In general ðÞ quantities correspond to the portion of the deformation between the instant of reassembly and the current time. The overall deformation gradient F can be decomposed as ^ _

F¼FF ^

ð2:13Þ ^

_

so that F in (2.11) is given by F ¼ FF1 . An even fuller notational scheme would make this even more explicit in two additional respects: first it would identify the particular fiber family under consideration and, second, it would identify a specific _

_

ðjÞ

_

^

_

^

ðjÞ K ðjÞ value on the dissolution interval K ðjÞ 6 K f . The first of these would replace F and F with, say, FðjÞ and FðjÞ , while the s 6 K _

_

second would make each of these a function of K ðjÞ . In order to not encumber our notation even more, we simply write F and ^

F in what follows. b ðjÞ corresponds to the square of the stretch experienced by the reassembled fiber due to the deformation The variable K that takes place after reassembly, i.e.,

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H. Demirkoparan et al. / International Journal of Engineering Science 48 (2010) 1179–1201 ^

^ _

^ _

_

^ _

K ðjÞ ¼ F MðjÞ  F MðjÞ ¼ MðjÞ  C MðjÞ :

ð2:14Þ

A comparison of (2.11) with (2.9) shows that the reassembled fiber is assumed to have the same strain energy density as the original fiber. That is, the reassembled fibers in the modeling presented here have the same mechanical properties as the ^

original fibers. The ðÞ notation indicates that stretch in the reassembled fibers is measured relative to the natural reference state of the reassembled fibers. More detailed discussion of the model, along with an examination of useful connections be_

^

tween ðÞ quantities and ðÞ quantities, is found in [18,19]. Many aspects of the model can be illustrated for the special case in which the dissolution and reassembly process occurs at a single value of K(j) instead of over an interval of K(j) values. This means that there is instantaneous dissolution of all original fibers and their instantaneous reassembly into new fibers with a new reference configuration. Formally, this means ðjÞ ðjÞ (j) that K f ¼ K ðjÞ and b(j) appearing in (2.12) can operationally be regarded as, respectively, a Dirac s ¼ K ACT , say. In this case a delta function and a unit step-down function (from 1 to 0). Under these circumstances, the Cauchy stress continues to be given by (2.8), but now (2.10) and (2.11) consolidates into a simple rule. In order to express the result in a concise fashion, introduce standard notation

W 0f ðjÞ ¼

dW f ðjÞ dK

ð2:15Þ

ðjÞ

ðjÞ

for differentiation with respect to the argument. Then there is complete and abrupt microstructural change at K ðjÞ ¼ K ACT , and ðjÞ it follows that Tfib is simply given by ðjÞ Tfib

¼

8 < 2W 0f ðjÞ ðK ðjÞ ÞFMðjÞ  FMðjÞ ; :

_

ðjÞ

if K ðjÞ max < K ACT ;

ð2:16Þ

_

ðjÞ b ðjÞ Þ b 2W 0f ðjÞ ð K FMðjÞ  b FMðjÞ ; if K ðjÞ max > K ACT : ðjÞ

ðjÞ

According to (2.16), the discontinuity in the stress Tfib at K ðjÞ max ¼ K ACT arises from two causes. The first of these is due to the _

(j)

change in direction of the fiber from that of M in the original reference configuration, to that of MðjÞ in the configuration at the time of reassembly (which is the natural reference configuration for the newly formed fiber). The second concerns the ^

^

change from F to F (and hence from K(j) to K ðjÞ Þ. Here it is useful to note that at the instant of microstructural change we have _

^

F ¼ F and F ¼ I (viz. (2.13)). ðjÞ ðjÞ Consequences of the special case K f ¼ K ðjÞ s ¼ K ACT are immediately observable in homogeneous deformation. As discussed for uniaxial extension in [18], this can lead to a discontinuity in the stress-stretch behavior due to what can be interpreted as a sudden drop in load carrying capacity. As regards the structure of the reassembled fibers in simple shear, discussed in [19], ðjÞ whereas the case K f > K ðjÞ s can generally lead to a fibrous fan morphology at each material point (due to the range of direc_

ðjÞ

ðjÞ

_

ðjÞ so that the type of fan morphologies tions in MðjÞ in the integral (2.11)), the case K f ¼ K ðjÞ s ¼ K ACT involves only a single M discussed in [19] consolidate into a single direction for the reassembled fiber.

3. The combined deformation of torsion and axial stretch The object of this paper is to demonstrate the consequences of this type of modeling in the context of inhomogeneous states of deformation that require consideration of the equations of equilibrium. In order to demonstrate some of the main issues that arise in this setting we, at this point in the development, restrict attention to the special case in which there is only a single family of fibers (i.e., j = 1) and for which the dissolution and reassembly process for these fibers takes place at a single value of K. As suggested by the discussion at the end of Section 2, the value of K at which dissolution and reassembly takes place will be denoted by KACT, and the Cauchy stress reduces by virtue of (2.8) and (2.16) to

T ¼ pI þ 2W 0m ðI1 ÞB þ Tfib

ð3:1Þ

with

Tfib ¼

8 < 2W 0f ðKÞFM  FM;

if K max < K ACT ;

_ _ : 2W 0 ð K b Þb F M b F M; if K max > K ACT : f

ð3:2Þ

The issues under consideration will be demonstrated in the context of the combined deformation of torsion and axial stretch. 3.1. Kinematics of deformation We consider a solid circular cylinder whose undeformed configuration is given by fðR; H; ZÞ : 0 6 R 6 R0 ; 0 6 H < 2p; L 6 Z 6 Lg. Here R = 0 provides the coordinates for a material point in the reference configuration and (r, h, z) provides its coordinates in the current configuration. We also let {eR, eH, eZ} and {er, eh, ez} be unit basis vectors in the reference and deformed configurations. The deformation of interest is a volume preserving combination of torsion and axial stretch in the form

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R r ¼ pffiffiffiffiffi ; kz

h ¼ H þ wZ;

z ¼ kz Z;

ð3:3Þ

where kz and w are constants giving the axial stretch and the twist per unit length in the reference configuration. The lateral surface R = Ro is required to be traction free and the nature of the end conditions at Z = L and Z = L will be discussed later in this development. The material properties are assumed to be independent of H and Z, which will have the effect of allowing all subsequent processes of dissolution and reassembly to also be independent of H and Z. These processes, however, will exhibit substantial dependence on R as described more fully in what follows. The deformation gradient tensor F for the deformation (3.3) is given by

1 wR F ¼ pffiffiffiffiffi ðer  eR þ eh  eH Þ þ pffiffiffiffiffi eh  eZ þ kz ez  eZ : kz kz

ð3:4Þ

The left Cauchy–Green deformation tensor B = FFT appearing in (2.8) is given by

B¼

pffiffiffiffiffi 1 1 þ w2 R2 er  er þ eh  eh þ kz wRðeh  ez þ ez  eh Þ þ k2z ez  ez kz kz

ð3:5Þ

I1 ¼

2 þ w2 R2 þ k2z : kz

ð3:6Þ

and

The fibers are assumed to wind around the cylinder with orientation in the reference configuration prior to dissolution and reassembly given by the unit vector

M ¼ sin XeH þ cos XeZ ;

ð3:7Þ

where the orientation angle may vary with radial position, i.e., X = X(R). The function X(R) may be discontinuous. The deformation maps the unit vectors M into

FM ¼

  1 wR pffiffiffiffiffi sin X þ pffiffiffiffiffi cos X eh þ kz cos Xez : kz kz

ð3:8Þ

Thus FM  FM, which appears in the first of (3.2), is given by

FM  FM ¼

pffiffiffiffiffi 1 ðsin X þ wR cos XÞ2 eh  eh þ kz cos Xðsin X þ wR cos XÞðeh  ez þ ez  eh Þ þ k2z cos2 Xez  ez : kz

ð3:9Þ

Consequently the square of the fiber stretch before dissolution and reassembly is given by

1 ðsin X þ wR cos XÞ2 þ k2z cos2 X: kz

K¼

ð3:10Þ

A fiber undergoes instantaneous dissolution and reassembly when the condition K = KACT is met at the spatial location of that fiber. Note for all w – 0 that K as given by (3.10) varies with R. Thus for the case in which KACT is a material constant independent of R, it follows that the condition K = KACT, if it is met at all, will only be met at special values of R. For any fixed value of R, let td be the time at which dissolution and reassembly occurs. Then the second of (3.2) is to be used for the determination of Tfib at this R when t P td. The following discussion concerns the determination of the quantities ^

^

_

_

F, K and M appearing in (3.2)2. The tensor F is the deformation gradient F at time td and so is given by (viz. (3.4)) _

 wR _ _ _ _ 1 _ F ¼ qffiffiffiffiffi er  eR þ eh  eH þ qffiffiffiffiffi eh  ez þ kz ez  ez ; _

_

_

kz

kz

ð3:11Þ

n_ _ _ o _ _ where kz is the axial stretch per unit length at time td and w is the twist per unit length at time td. In addition, er ; eh ; ez are ^ _

unit basis vectors in the deformed configuration at time td. For t P td the deformation gradient F is decomposed as F ¼ F F, ^

whereupon it follows that the tensor F appearing in (3.2)2 is given by ^

F¼

  _ ^ ww R   w ^ _ _ _ _ _ _ _ _ kz 1  R z er  er þeh  eh þ _ eh  ez þ _ ez  ez ¼ qffiffiffiffiffi er  er þeh  eh þ qffiffiffiffiffi eh  ez þkz ez  ez : pffiffiffiffiffi ^ ^ kz kz kz kz kz kz

vffiffiffiffiffi u_ u  tk

ð3:12Þ

For the second equality in (3.12) we have defined new quantities _

^

kz ¼

kz _

^

;

w¼

ðw  wÞ _

ð3:13Þ

;

ðkz Þ3=2

kz ^

_

in order to express F in the same format as F in (3.4) and F in (3.11).

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H. Demirkoparan et al. / International Journal of Engineering Science 48 (2010) 1179–1201 _

_ _

As discussed previously, the fiber direction at the time of fiber reassembly is given by the eigenvector of B ¼ F FT that is asso_

_

ciated with the maximum eigenvalue. Let k2max > 1 be this maximal eigenvalue and M be the associated unit eigenvector. It is _

_

_

_

immediate from (3.5) at time td that 1=kz is an eigenvalue of B with eigenvector er . The other two eigenvalues of B have eigen_

vectors orthogonal to er . These other two eigenvalues can be calculated explicitly, and the larger of these two is given by _

k3z

_

ffi  _ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ 2 _ _ _ 2 4 þ 1þ wR þ k3z  1 þ 2 k3z þ 1 w2 R þ w4 R

f

ð3:14Þ

:

_

2kz It is immediate from (3.14) that _

_

fP

_

k3z þ 1 þ ðw RÞ2 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ ðk3z  1Þ2

_

P

2kz

1

ð3:15Þ

:

_

kz _

_

The second inequality in (3.15) is strict unless both w ¼ 0 and kz < 1. Thus it may be concluded that

qffiffiffi

_

_

kmax ¼

ð3:16Þ

f: _

_

_

The w ¼ 0, kz < 1 corresponds to simple uniaxial compression, in which case k max is a repeated eigenvalue qffiffifficase of   _ special _ _ _ k max ¼ f ¼ 1=kz . With the exception of this special case, the direction of reassembly M is given by _

_ _

_ _

M ¼ sin X eh þ cos X ez ; where _

k3z

_

ð3:17Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  2   _ _ _ _ 2 4 3 3 2 4 þ1þw R þ kz  1 þ 2 kz þ 1 w R þ w R _

2 2

tan X ¼

_

3=2

2kz

ð3:18Þ

:

_

wR

_

_

_

_

In contrast, for simple uniaxial compression, any vector orthogonal to ez is an eigenvector of B with eigenvalue kmax ¼ 1=kz . This case will not be considered further in what follows. _ The deformation that takes place between the time of reassembly and any later time maps the unit vectors M as given by (3.17) into

0

1 _ _ ^   _ ^ Bsin X w R cos XC F M ¼ @ qffiffiffiffiffi þ qffiffiffiffiffi Aeh þ kz cos X ez : ^ ^ kz kz ^ _

^

_

^

ð3:19Þ

_

Then F M  F M, which appears in the second of (3.2), is given by ^ _

^ _

F MF M ¼

1 ^

qffiffiffi       _ _ 2 _ _ _ _ 2 ^ ^ ^ ^ sin X þ w R cos X eh  eh þ kz cos X sin X þ w R cos X ðeh  ez þ ez  eh Þ þ kz cos X ez  ez

kz ð3:20Þ ^

and the stretch quantity K as given by (2.14) becomes ^

K¼

1 ^



 _ _ 2 _ ^ ^ sin X þ w R cos X þ k2z cos2 X :

ð3:21Þ

kz _

_

_

The vector M in (3.17) can also be mapped to the original reference configuration giving a vector in the direction F1 M. Then _

_

_

_

Mref ¼ kmax F1 M is a unit vector in this direction. Thus, with this notation: M in (3.7) gives the unit vector that is aligned with _

the fibers in the original reference configuration prior to dissolution and reassembly, and Mref is a unit vector giving the orientation of the fibers, also in the original reference configuration, after dissolution and reassembly. In this way, the physical processes being modeled are such that the orientation of the fibers in the original reference configuration change their ori_

_

_

_

_

_

_

_

entation from M to Mref at time td. It further follows for t > td that Mref ¼ kmax F1 M is the unit eigenvector of C ¼ FT F that is _

_

associated with the maximum eigenvalue kmax . For t > td this Mref is also expressible in the general form of (3.7) provided that _

the angle X is replaced by Xref with

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_

k3z

_

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi _ 2 _ _ _ 2 4 3 3 2 4 þ1w R þ kz  1 þ 2 kz þ 1 w R þ w R _

2 2

tan Xref ¼

_

:

ð3:22Þ

2wR 3.2. Stresses It now follows from (3.1), (3.2), (3.5), (3.9) and (3.18) that the Cauchy stress T is of the form

T ¼ T rr er  er þ T hh eh  eh þ T zh ðeh  ez þ ez  eh Þ þ T zz ez  ez

ð3:23Þ

with normal stresses

2 0 W ðI1 Þ; kz m 2 ¼ p þ ð1 þ w2 R2 ÞW 0m ðI1 Þ þ T fib hh ; kz

T rr ¼ p þ

ð3:24Þ

T hh

ð3:25Þ

T zz ¼ p þ 2k2z W 0m ðI1 Þ þ T fib zz

ð3:26Þ

and shear stress

pffiffiffiffiffi T zh ¼ 2 kz wRW 0m ðI1 Þ þ T fib zh ;

ð3:27Þ

where I1 is given by (3.6). At those locations where dissolution and reassembly has not occurred, the fiber stresses are given by

T fib hh ¼

2 ðsin X þ wR cos XÞ2 W 0f ðKÞ; kz

0 2 T fib zz ¼ 2kz cos XW f ðKÞ; ffiffiffiffi ffi p T fib kz cos Xðsin X þ wR cos XÞW 0f ðKÞ: zh ¼ 2

ð3:28Þ ð3:29Þ ð3:30Þ

Conversely, at those locations where dissolution and reassembly has already taken place, the fiber stresses are given by

T fib hh ¼

2 ^

  _ _ 2 ^ ^ sin X þ w R cos X W 0f ðK Þ;

ð3:31Þ

kz ^

_

^

0 2 2 T fib zz ¼ 2kz cos X W f ðK Þ; qffiffiffiffiffi   _ _ _ ^ ^ ^ T fib W 0f ðK Þ: ¼ 2 k X sin X þ w R cos X cos z zh

ð3:32Þ ð3:33Þ

By virtue of (3.23) the requirement of vanishing tractions on the lateral surfaces becomes simply

T rr ðRo Þ ¼ 0:

ð3:34Þ

Equilibrium in the absence of body forces requires that div T = 0. It then follows from the equations of equilibrium in the axial and circumferential directions that the pressure only depends upon r. The equation of equilibrium in the radial direction reduces to

dT rr 1 þ ðT rr  T hh Þ ¼ 0; r dr

ð3:35Þ

which, by virtue of (3.3), may be written as

dT rr 1 þ ðT rr  T hh Þ ¼ 0: R dR

ð3:36Þ

For the deformation (3.3), dependence upon r is equivalent to dependence upon R and so we write p = p(R). Together (3.24), (3.36) and the boundary condition Trr(Ro) = 0 give the following integral expression for the pressure

p¼

2W 0m ðI1 Þ  kz

Z R

Ro

1 ðT rr ðnÞ  T hh ðnÞÞdn: n

At those radial values where dissolution and reassembly has not occurred, (3.24), (3.25) and (3.28) provide

ð3:37Þ

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T rr  T hh ¼ 

2w2 R2 0 2 W m ðI1 Þ  ðsin X þ wR cos XÞ2 W 0f ðKÞ; kz kz

ð3:38Þ

while at those radial values where dissolution and reassembly has already occurred, (3.24), (3.25) and (3.31) provide

T rr  T hh ¼ 

  _ _ 2 ^ ^ 2w2 R2 0 2 W m ðI1 Þ  ^ sin X þ w R cos X W 0f ðK Þ: kz kz

ð3:39Þ

Thus, in general, the interval of integration in (3.37) is partitioned into subintervals corresponding to alternating radial zones where dissolution and reassembly has or has not taken place. At the radial locations separating such zones, it will generally be the case that the Cauchy stress components Thh, Tzz, Tzh are discontinuous. In contrast, traction continuity requires that Trr is continuous at such radial locations. The expressions (3.24) and (3.37) give that Trr may be expressed in the form

T rr ¼

Z

Ro

1 ðT rr ðnÞ  T hh ðnÞÞdn; n

R

ð3:40Þ

which, with the aid of (3.38) and (3.39), shows that Trr as given by the present development is indeed continuous as a function of R. It is also to be remarked that this development is consistent with an initial fiber orientation that varies with R and with energy expressions Wm(I1) and Wf(K)that may contain constitutive parameters that vary with R. This radial variation in material properties may include discontinuity in material properties at particular radial locations. The development as given above, and in particular the stress expressions (3.24)–(3.33) with pressure given by (3.37), is consistent with such material property discontinuities. This is because (3.40) with (3.38) and (3.39) ensures that Trr is continuous at any such radial locations of material property discontinuity. We now turn to consider conditions on the ends Z = L and Z = L. The twisting moment Q and axial force P on these ends are given by

Q ¼ 2p P ¼ 2p

Z

Z

ro

T zh r 2 dr ¼

0 ro

T zz r dr ¼

0

Z

2p k3=2 z

2p kz

Z

Ro

T zh ðRÞR2 dR;

ð3:41Þ

0 Ro

T zz ðRÞR dR;

ð3:42Þ

0

where Tzh is given by (3.27) and (3.30) where dissolution and reassembly has not occurred, and by (3.27) and (3.33) where dissolution and reassembly has occurred. Similarly Tzz is given by (3.26) and (3.29) where dissolution and reassembly has not occurred, and by (3.26), (3.32) where dissolution and reassembly has occurred. A variety of problems can be considered for the solid cylinder, all conforming to boundary conditions (3.34):    

(S1): (S2): (S3): (S4):

specified specified specified specified

w and kz whereupon the stresses, Q and P can be immediately calculated. w and P whereupon Eq. (3.42) is to be solved for kz prior to the determination of Q. kz and Q whereupon Eq. (3.41) is to be solved for w prior to the determination of P. Q and P whereupon Eqs. (3.41) and (3.42) are to be solved for w and kz.

4. Fiber dissolution for the case in which the original fibers are axially aligned For the rest of this paper we focus attention on the problem (S1) with X = 0 in (3.7). This means that prior to deformation the fibers are aligned with the cylinder axis. It is assumed that the originally undeformed cylinder is stress free. This means that the original hyperelastic response prior to microstructural change is such that all stress components vanish when w = 0 and kz = 1. It follows from (3.24)–(3.30), (3.37), and 3.38 that this will be the case if Wf(K) is such that W 0f ð1Þ ¼ 0. We wish to consider the effect of deformation by varying w and kz. This causes variation in K where it follows from (3.10) with X = 0 that

K¼

w2 R2 þ k2z : kz

ð4:1Þ

We shall further assume that all material properties are independent of R. In particular this gives the same activation value KACT throughout the cylinder. Since dissolution and reassembly is taken to occur only if the fibers are stretched beyond their undeformed state it follows that KACT > 1. 4.1. Deformation paths in the (kz, w2) parameter plane The deformation (3.3) combining axial stretch and twist is given in terms of kz and w, and these two parameters may vary independently. In what follows it is convenient to describe this by means of deformation paths in a parameter plane of (kz, w2)-values. Here the use of w2 is motivated by the way in which w appears in (4.1). In general, this path is taken as beginning at the undeformed configuration values (kz, w2) = (1, 0).

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Let this path be denoted by P = P(s) where s is an appropriate path parameter (say, arc length in the (kz, w2)-plane). That is P(s) = {(kz(s), w2(s))js: 0 ? sfinal} for some final value sfinal. In keeping with the assumption that initially (kz, w2) = (1, 0) it follows that kzjs=0 = 1 and w2js=0 = 0. At this initial point (4.1) then gives that K = 1 for all R. Starting at (kz, w2) = (1, 0) the deformation path is initially such that K < KACT for all R. On this initial portion of P(s), the material behavior is that of a hyperelastic solid. One then finds that elementary manipulations using (3.27) and (3.30) in (3.41) and (3.6), (3.29), (3.37), (3.38) in (3.42) gives

Z  4p Ro  0 W m ðI1 Þ þ W 0f ðKÞ wR3 dR; kz 0 ! ! ! Z Ro 2 þ w2 R2 w2 R2 0 0 W m ðI1 Þ þ 2kz  2 W f ðKÞ R dR; 2kz  P ¼ 2p k2z kz 0

Q¼

ð4:2Þ ð4:3Þ

where K is again given by (4.1), and I1 is given by (3.6). In arriving at the expression (4.3) for P it is necessary to reverse the order of integration in the double integral that one obtains after substituting from (3.38) and (3.37) into (3.42). The value of K given by (4.1) is the same for all R if and only if w = 0. If w = 0 then (3.3) is a homogeneous deformation, in pffiffiffiffiffiffiffiffiffiffi which case dissolution and reassembly takes place simultaneously at all values of R when kz ¼ K ACT . This situation is discussed at length in [18]. More generally, if w – 0 then K is increasing in R so that its maximum value occurs at R = Ro. In this case the condition K = KACT is first met at R = Ro; whereupon dissolution and reassembly initially occurs only at the outer radius. Both situations are associated with the deformation path P(s) encountering the locus of points in the (kz, w2) plane satisfying w2 R2o þ k3z  kz K ACT ¼ 0. In particular, there is no dissolution so long as w2 R2o þ k3z  kz K ACT < 0. We now consider the dissolution of the fibers if the deformation path enters into the region for which w2 R2o þ k3z  kz K ACT > 0. 4.2. The transformation front R = RACT The activation condition for the fiber at radius R is given by (4.1) with K = KACT, which together is restated as

w2 R2 þ k3z  kz K ACT ¼ 0:

ð4:4Þ

This gives the combinations of kz and w2 that cause dissolution and reassembly of the fiber at radius R. Alternatively, one may say that the fiber at radial value R is activated for the process of dissolution and reassembly when (4.4) is first met. Clearly, for a given (kz, w2) there is at most one positive value of R satisfying (4.4). Here we also recall that we have previously

Fig. 1. Graph of a generic deformation path in (kz, w2) plane with fiber dissolution and reassembly occurring in the blue portions. This path encounters the activation curve for R = Ro at the point (a) and the location of fiber dissolution and reassembly moves inward within the cylinder up to point (b). Between (b) and (d) there is no fiber dissolution and reassembly. Fiber dissolution and reassembly restarts at (d) and concludes at the point (e). At point (e) all of the fibers have dissolved and reassembled.

H. Demirkoparan et al. / International Journal of Engineering Science 48 (2010) 1179–1201 _

1189

_

introduced kz and w in order to indicate such values at the instant of dissolution and reassembly. It thus follows from (4.4) _

_

that kz and w are specific functions of R once the deformation path is specified. Curves in the (kz, w2)-plane that follow from (4.4) upon fixing the value of R will be referred to as activation curves. Fig. 1 shows these curves for radii R = 0.1Ro, 0.2Ro, . . . , 0.9Ro, Ro. The lower thick black curve corresponds to radius Ro which is the previously described locus where dissolution and reassembly first occurs. The curve for R = 0.9Ro is then the next lowest curve, while R = 0.1Ro gives the highest curve inpFig. ffiffiffiffiffiffiffiffiffiffi1. For this figure we take KACT = 1.69, which corresponds to dissolution and reassembly in pure stretching when kz ¼ 1:69 ¼ 1:3. Note that all of the activation curves pass through the points pffiffiffiffiffiffiffiffiffiffi kz = 0, w2 = 0 and kz ¼ K ACT , w2 = 0. The first of these points is indicated with an open circle, since the stretch kz must be positive. The slope of any such activation curve at kz = 0 is KACT/R2 > 0 and so approaches infinity as R approaches zero. pffiffiffiffiffiffiffiffiffiffi 2 The slope of the activation curves at kz ¼ K ACT is 2K ACT/Rffi < 0 and so approaches negative infinity as R approaches zero. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Each activation curve has a local maximum at kz ¼ K ACT =3, where

w2 R2 ¼

2 pffiffiffi 3ðK ACT Þ3=2 : 9

ð4:5Þ

Hence the maximum value of w2 is monotonically increasing to infinity as the activation value R approaches zero. Thus the pffiffiffiffiffiffiffiffiffiffi vertical lines at kz = 0 and kz ¼ K ACT are asymptotes to the activation curves corresponding to radii R that approach zero. Consider the curved deformation path P(s) with w2 > 0 as shown in Fig. 1. This path encounters the activation curve for R = Ro at the point labeled (a) in Fig. 1. This heralds the onset of fiber dissolution and reassembly, and only the location R = Ro undergoes this process at point (a) on the deformation path. Now consider the path P(s) as it continues into the region

w2 R2o þ k3z  kz K ACT > 0;

ð4:6Þ

as shown in Fig. 1. In addition to point (a), there are four additional points: (b)–(e) on the curved deformation path that are of special significance for the ensuing discussion. We first indicate their geometrical characterization with respect to the activation curves in Fig. 1. We then follow this with an explanation of the significance of these points as it relates to dissolution and reassembly. After entering into the region (4.6), each location (kz, w2) on the deformation path intersects a particular activation curve. Points (b) and (c) are the only locations where this intersection is tangential in our depiction. Starting at point (a) and moving to point (b) the deformation path progressively encounters activation curves for ever smaller R. However, starting at point (b) and moving to point (c) the opposite is true, namely the deformation path now progressively encounters activation curves for ever larger R. Each of the activation curves encountered between (b) and (c) had previously been encountered between (a) and (b), albeit at a different (kz, w2) since the deformation path between (a) and (b) has no common points with the deformation path between (b) and (c). Starting at point (c) and moving forward on the deformation path through (d)–(e) the deformation path once again encounters activation curves for progressively smaller R. The geometrical characterization of point (d) is that this is the point on the deformation path where it encounters the same activation curve as at location (b), that is (b) and (d) are different points on the same activation curve. The geometrical characterization of point (e) is that pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi kz ¼ K ACT at this location, where we note from the discussion after (4.5) that the locus kz ¼ K ACT has an asymptotic interpretation as the activation curve for R = 0. With the Fig. 1 geometrical characterization of the locations (a)–(e) established, we now turn to consider the fiber dissolution process on the deformation path between (a) and (e). Here we recall that the present modeling posits that reassembly occurs immediately after dissolution. However, for the remaining discussion of the curved deformation path containing (a)– (e) we shall only refer to the dissolution process, as this depends upon the condition (4.4) irrespective of any subsequent reassembly. Proceeding from (a) to (b) on the curved deformation path, the location R satisfying (4.4) decreases from R = Ro as determined by the intersection of the deformation path with the activation curves. Thus the location of dissolution moves inward. Denote this location by R = RACT. Thus, for any fixed (kz, w2) between (a) and (b), RACT is the unique value of R where the activation condition is satisfied. This radius separates untransformed material 0 < R < RACT from transformed material RACT < R < Ro. Proceeding from (b) to (c) on the curved deformation path, the location R satisfying (4.4) now increases from its value at (b). Thus the fibers at location R satisfying (4.4) are already dissolved. Therefore the location RACT separating transformed from untransformed material remains fixed as one proceeds from (b) to (c) on the deformation path. Proceeding from (c) to (d) on the deformation path it is still the case that the location R satisfying (4.4) is a location where the fibers are already dissolved, even though the value of K is once again increasing at all locations in the cylinder. In particular K evaluated at RACT is still less than KACT between (c) and (d). Thus the location RACT remains at rest during this portion of the deformation path as well. The location (d) on the deformation path once again gives K = KACT at R = RACT. Proceeding from (d) to (e) on the deformation path it is now again the case that RACT decreases because the deformation path encounters activation curves with ever pffiffiffiffiffiffiffiffiffiffi smaller values of R. At point (e) the deformation path reaches the vertical line kz ¼ K ACT . In approaching the vertical line the path intersects all the remaining small R activation curves. At point (e) the location RACT reaches the centerline R = 0. This can also be seen from (4.4) which, upon setting R = RACT and rearranging, gives

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H. Demirkoparan et al. / International Journal of Engineering Science 48 (2010) 1179–1201

RACT ¼

pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi kz K ACT  k2z ! 0 as kz ! K ACT : jwj

ð4:7Þ

In summary, RACT decreases its value on the curved deformation path segments: (a)–(b), and (d)–(e). The transformation front may be then be regarded to be in quasi-static motion. Parameterizing the deformation path with arclength variable s we may then regard RACT ¼ RACT ðsÞ. The motion of the transformation front R = RACT then correlates with the change in location of the point (kz, w2) = (kz(s), w2(s)) on the deformation path. Let k_ z and w_ give the derivative of the deformation variables kz and w with respect to the path parameter s. Then, at fixed R let K_ give the derivative of K with respect to (kz, w2). Eq. (4.1) gives

! 2 2 2 _K ¼ 2wR w_ þ 2kz  w R k_ z : kz k2z

ð4:8Þ

Then (4.8) evaluated at R = RACT(s) gives the rate of change of K at the activation radius. The activation radius at path parameter s satisfies the condition

K ACT ¼

w2 ðsÞR2ACT ðsÞ þ k2z ðsÞ: kz ðsÞ

ð4:9Þ

Differentiate (4.9) with respect to path parameter s and let the result be combined with (4.8) evaluated at R = RACT(s). This gives

! 2R2ACT _ R2ACT 2k2z _ _RACT ¼  kz Kj _  2 kz : wþ RACT ¼  w kz w2 w

ð4:10Þ

The transformation front R = RACT is in motion at any instant of the deformation if and only if K = KACT with K_ > 0 at R = RACT. These conditions are met on portions (a)–(b) and (d)–(e) of the deformation path in Fig. 1, whereupon the first of (4.10) gives the expected result R_ ACT < 0 confirming that the transformation front moves in the direction of decreasing radius. In contrast, on the portion (b)–(d) of the transformation path it is the case that K < KACT at R = RACT so that the transformation front remains stalled on this portion of the deformation path. 4.3. Twisting moment and axial load For any deformation path prior to the start of dissolution and reassembly, i.e., for w2 R2o þ k3z  kz K ACT < 0, the twisting moment Q and axial force P continue to be given by (4.2) and (4.3). However for w2 R2o þ k3z  kz K ACT > 0 the effect of dissolution and reassembly gives rise to different expressions for the stresses in the transformed part of the cylinder RACT < R 6 Ro. If the pffiffiffiffiffiffiffiffiffiffi deformation path has not crossed the vertical line kz ¼ K ACT then RACT > 0. This in turn leads to new expressions for Q and P in the form

4p Q¼ kz

P ¼ 2p

Z

RACT

0

Z

vffiffiffiffiffi 0 1 u^   Z Ro u   _ _ _ ^ ^ t 4 p k z B C 0 cos X sin X þR w cos X W 0f ðK ÞAR2 dR; W 0m ðI1 Þ þ W 0f ðKÞ wR3 dR þ @RwW m ðI1 Þ þ kz RACT kz

RACT

2kz 

0

þ 2p

Ro

RACT

k2z

! W 0m ðI1 Þ

þ 2kz 

w2 R2 k2z

!

!

W 0f ðKÞ

R dR

1 0   1 _ _ 2 ^ ^ ! sin X þR w cos X 2 2 2 C C B B _ 2þw R C 0 ^ C B B2 kz 0 2 W W ðI Þ þ cos X  ðK Þ CR dR: C B 2kz  B 1 m f ^ A A @ @ kz k2z kz kz 0

Z

2 þ w2 R2

ð4:11Þ

ð4:12Þ

Here, as in (4.3), the original double integral in the original expression for P is evaluated into a single integral after reversing the order of integration. pffiffiffiffiffiffiffiffiffiffi Conversely, if the transformation path has (ever) passed into the region kz P K ACT then the fibers at all radial values have dissolved and reassembled. For such a fully transformed cylinder, the expressions for Q and P follow by formally setting RACT = 0 in (4.11) and (4.12). _ _ When evaluating (4.11) and (4.12), it is useful to keep in mind kz and w have been used to indicate the stretch and twist, respectively, at the instant of dissolution and reassembly at a radius R. Thus, each value of the arclength variable s on the deformation path P(s) from (a) to (b) or (d) to (e) corresponds to ! all of the following: a specific radius R, a specific instant of dissolution and reassembly, and a specific pair _

_

_

_

w ¼ wðRÞ; kz ¼ kz ðRÞ on a specific deformation path.

_

_

kz ; w2

_

_

_

related by w2 R2 þ kz K ACT  k3z ¼ 0. This implies that

H. Demirkoparan et al. / International Journal of Engineering Science 48 (2010) 1179–1201

1191

As a more quantitative example, consider a deformation of pure torsion (kz = 1) with w increasing from its initial value of 0. Then dissolution begins at R = Ro when w2 attains the value w2 ¼ ðK ACT  1Þ=R2o . According to (4.10), R_ ACT ¼ ð2R2ACT =wÞw_ so that continued increase in w causes the location of dissolution and reassembly to move inward. At any value of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2 > ðK ACT  1Þ=R2o the current location of the transformation front is simply RACT ¼ K ACT  1=w. Here it is to be noted that RACT can never reach the centerline R = 0 on this kz = 1 deformation path, although it does approach the centerline asymptotically as w ? 1. Of special interest is the twisting moment Q which, prior to dissolution and reassembly, is given by (4.2) with kz = 1, namely

Q ¼ QðwÞ ¼ 4pw

Z

  W 0m ð3 þ w2 R2 Þ þ W 0f ð1 þ w2 R2 Þ R3 dR

Ro

0

ð4:13Þ

After the onset of dissolution and reassembly, that is for w2 > ðK ACT  1Þ=R2o , the twisting moment is given by (4.11) with _ ^ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kz = 1 and RACT ¼ RACT ðwÞ ¼ ðK ACT  1Þ=w. For this deformation it follows from (3.13) that kz ¼ kz ¼ 1, so that (3.18) gives _

tan X ¼

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ACT þ 3 þ K ACT  1 ; 2

ð4:14Þ

whence _ 2 cos X ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi ; 4 þ K ACT  1 þ K ACT þ 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ K ACT  1 þ K ACT þ 3 sin X ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi ; 4 þ K ACT  1 þ K ACT þ 3 _

which are independent of R. In addition, w ¼ ^

w¼w

ð4:15Þ

ð4:16Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ACT  1=R so that (3.13) gives

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ACT  1 : R

ð4:17Þ

Then (3.21) in conjunction with (4.15)–(4.17) yields, after algebraic simplification, ^

K¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 þ 2K ACT  2 K ACT  1 K ACT þ 3 þ 4w2 R2 þ 4wR K ACT þ 3  K ACT  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 6 þ 2K ACT þ 2 K ACT  1 K ACT þ 3

ð4:18Þ

Hence (4.11) for this deformation can be written

Z Q ¼ 4pw þ 4p

Z

Ro

0 Ro RACT

W 0m ð3 þ w2 R2 ÞR3 dR þ 

^ W 0f ðK Þ ^

^

Z 0

RACT

 W 0f ð1 þ w2 R2 ÞR3 dR

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ cos X sin X þ wR  K ACT  1 cos2 X R2 dR; _

_

ð4:19Þ

_

where RACT = RACT(w), K ¼ K ðwRÞ, and X are as given above. One also finds that the orientation of the reassembled fibers are _ independent of R and w when mapped back to the original reference configuration. This is because (3.22) with kz ¼ 1 gives _

tan Xref ¼

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ACT þ 3  K ACT  1 : 2

ð4:20Þ

5. Deformation path dependent morphological changes The development of the previous section indicates how dissolution in the (kz, w2)-region takes place when the deformapffiffiffiffiffiffiffiffiffiffi tion path P(s) is within the subregion bounded by the two curves w2 R2o þ k3z  kz K ACT ¼ 0 and kz ¼ K ACT . That development also showed how the first of these curves is the RACT = Ro activation curve, and the second of these curves is (asymptotically) the RACT = 0 activation curve. We shall refer to the region between these two curves as the active transformation subregion (active subregion for short). Similarly, the initially untransformed subregion (untransformed subregion for short) consists of points (kz, w2) obeying w2 R2o þ k3z  kz K ACT < 0. Finally, the fully transformed subregion consists of those points (kz, w2) obeying pffiffiffiffiffiffiffiffiffiffi kz > K ACT . The untransformed subregion contains the initial point (kz, w2) = (1, 0) of the deformation path P(s). If P(s) remains in this subregion, then there is no transformation and the material behaves like a hyperelastic solid as described by the original material specification. If the path P(s) enters the active subregion then, so long as it either stays in this subregion or reenters into the untransformed subregion, the cylinder will contain a transformation radius R = RACT that separates the transformed outer zone from the untransformed core. The location R = RACT can only move inward, and may or may not be undergoing inward motion as a function of s. If the path P(s) enters the fully transformed subregion then microstructural change is complete, and the material remains fixed in its mechanical properties. The mechanical response is then again that of an unchanging fiber reinforced hyperelastic solid, where now both the orientation of the fibers and the stress free configuration for the

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fibers will vary with radial location due to the now completed process of dissolution and reassembly. In this section we consider the morphology of the reassembled fibers. 5.1. Deformation paths with a common endpoint in the fully transformed subregion The morphology of the reassembled fibers is highly dependent on the deformation path within the active subregion. Consider six different paths as shown in Fig. 2 and labeled P1(s), P2(s), P3 (s), P4(s), P5(s), P6(s). As in Fig. 1 we again take pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi K ACT ¼ 1:3. All six deformation paths begin at (kz, w2) = (1, 0) and end at (kz, w2) = (1.5, (p/3)2). Since 1:5 > 1:3 ¼ K ACT the ending point is in the fully transformed region. Each of the four paths P2(s), P3(s), P4(s), P5(s) consists of three line segments which we now proceed to describe. The first line segment in each path P2(s), P3(s), P4(s), P5(s) connects (kz, w2) = (1, 0) to (kz, w2) = (1, q(p/3)2) where q= 0.2, 0.4, 0.6, 0.8 respectively for P2(s), P3(s), P4(s), P5(s). These first line segments each have kz = 1 and so are vertical in the (kz, w2) region; we shall denote them by P2a(s), P3a(s), P4a(s), P5a(s), respectively. The second line segment in each path connects (kz, w2) = (1, q(p/3)2) to (kz, w2) = (1.5, q(p/3)2) for q as just given. These segments have constant w2 and so are horizontal in the (kz, w2) region; we shall denote them by P2b(s), P3b(s), P4b(s), P5b(s), respectively. The third and final line segment in each path connects (kz, w2) = (1.5, q(p/3)2) to (kz, w2) = (1.5, (p/3)2). These segments each have kz = 1.5 and so are vertical in the (kz, w2) region; we shall denote them by P2c(s), P3c(s), P4c(s), P5c(s). The remaining two paths P1(s) and P6(s) each involves only two line segments. For P1(s) the first line segment connects (kz, w2) = (1, 0) to (kz, w2) = (1.5, 0), while the second line segment connects (kz, w2) = (1.5, 0) to (kz, w2) = (1.5, (p/3)2). Thus P1(s) corresponds to taking q = 0 in the previous three segment construction, and so does not have the initial vertical segment. We shall denote the two segments of P1(s) as P1b(s) and P1c(s). For P6(s) the first line segment connects (kz, w2) = (1, 0) to (kz, w2) = (1, (p/3)2), while the second line segment connects (kz, w2) = (1, (p/3)2) to (kz, w2) = (1.5, (p/3)2). Thus P6(s) corresponds to taking q = 1 in the previous three segment construction, and so does not have the final vertical segment. We shall denote the two segments of P6(s) as P6a(s) and P6b(s). Each path ends in the fully transformed region so that all of the original fibers have dissolved and reassembled. Even though each path ends at the same point (kz, w2) = (1.5, (p/3)2) in the fully transformed zone, the new fiber morphology is very different in each case. It is to be noted that path P1(s) consists of uniaxial extension on segment P1b(s) such that all of the fibers dissolve and pffiffiffiffiffiffiffiffiffiffi reassemble when ðkz ; w2 Þ ¼ ð K ACT ; 0Þ. In this case the z direction is maximally stretched, and one finds from (3.18) that _

the orientation angle of reassembly is X ¼ 0 in the current configuration. Turning to the remaining paths P2(s)  P6(s) we note that dissolution and reassembly is completed on each path during its respective horizontal path segment P2b(s)–P6b(s). This completion takes place at the location on this segment where pffiffiffiffiffiffiffiffiffiffi kz ¼ K ACT . As regards the onset of dissolution and reassembly, paths P2(s)–P4(s) all have initial segments confined to the untransformed subregion so that, for these paths, the transformation onset is also on the horizontal path segments

Fig. 2. Graphs of the six deformation paths in (kz, w2) plane for a material with KACT = 1.69. Fiber dissolution and reassembly occurs on the blue portions of each path. Each path starts at (kz, w2) = (1, 0) and ends at (kz, w2) = (1.5, (p/3)2).

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P2b(s)–P4b(s) respectively. In contrast, for paths P5(s) and P6(s) one finds that the initial kz = 1 segments encounter the active subregion at w2 ¼ 0:69=R2o so that for these paths the dissolution and reassembly begins on P5a(s) and P6a(s). In the active subregion, the transformation front advances inward so long as the arclength variable s is also increasing with time. If, at some instant of time, the deformation is halted then RACT remains fixed. If during some interval of time, the deformation is reversed, meaning that the point (kz, w2) retraces the path in the reverse direction, then the transformation front remains fixed during the reversal. The transformation front only resumes its motion once the point (kz, w2) has returned back to its original point of reversal. Other than temporarily halting the inward advance of the transformation front R = RACT, such reversals have no effect on the final outcome of the dissolution and reassembly process since we are here requiring that the deformation point (kz, w2) always remains on the given deformation path.

5.2. Path dependent morphology of the reassembled fibers _

During reassembly, the newly created fibers are oriented at an angle X with respect to the axial direction as determined from (3.18). Since subsequent deformation causes line segments to rotate, deformation after reassembly changes the orientation of the reassembled fibers. Since fibers at different radial locations R reassemble at different times, the orientation an_

gle X as a function of R is in fact with respect to different configurations. It is convenient to represent orientation as a function of R with respect to a common configuration. When the original reference configuration is chosen as a common ref_

erence, one obtains the angle Xref as given by (3.22). pffiffiffiffiffiffiffiffiffiffi Dissolution and reassembly is complete once the deformation path reaches the boundary kz ¼ K ACT of the fully trans_

formed subregion. Fig. 3 shows the orientation angle Xref of the newly formed fibers as a function of R for each of the six _

paths. For deformation path P1(s) the graph in Fig. 3 is simply Xref ¼ 0 for all R since reassembly on P1(s) takes place in uni_

axial extension. The graphs for P5(s) and P6(s) involve constant Xref given by (4.20) for the values of R associated with the _

kz = 1 portions (i.e., P5a(s) and P6a(s)). In order to correlate Fig. 3 with the discussion in Section 4.2, recall that Xref in the ref_

erence configuration is the image of X in the configuration where reassembly occurs. Also, each R in Fig. 3, is associated with an instant of dissolution and reassembly when R = RACT. At initial activation, RACT = Ro corresponding to R = Ro in Fig. 3. Then, as s increases along deformation paths P2(s), . . . , P6(s), the activation radius RACT moves inward, during which, as shown in _

Fig. 3, Xref is decreasing. points There are slope discontinuities in the Fig. 3 graphs for P5(s) and P6(s) at the R values associated with the corner _ where portions P5a(s) and P6a(s) meet portions P5b(s) and P6b(s). In addition, all six graphs in Fig. 3 meet at R = 0, Xref ¼ 0.

_

Fig. 3. Graphs of orientation angle Xref of the reassembled fibers as a function of R for each of the six paths depicted in Fig. 2.

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_

_

Fig. 4. Covering of the (kz, w2) plane by constant RACT curves and Xref lines. The separation increment for constant Xref solid lines is 5° and the separation _

increment for constant RACT solid curves is 0.05Ro. The dotted lines of constant Xref provide another 2.5° of separation and the dotted curves of constant RACT provide another 0.025Ro of separation.

All of the graphs in Fig. 3 are continuous. This is due to the fact that traversing each of the deformation paths P2(s) through P6(s) in a continuous forward direction results in dissolution and reassembly at all points (kz, w2) of the deformation path within the active subregion. Alternatively stated, each and every activation curve intersects the deformation path at a unique _

point, and it is this uniqueness of intersection that ensures the continuity of Xref as a function of R. In contrast, consider again the generic deformation path of Fig. 1 where dissolution, and hence reassembly, did not occur on the curve portion from (b) _

to (d) within the active subregion. One then finds that the value of Xref for points (b) and (d) are different, giving a jump in _

Xref at the value of R associated with those two points. A specific example in which such a jump occurs is presented in the Appendix A. The relation between the morphology of the reassembled fibers and the deformation path in the active subregion is clar_

_

ified by the observation that: the values of both X and Xref depend only upon the value of kz at the instant of reassembly and, in particular, do not depend on the value of either w or RACT at the instant of reassembly. This result applies to all possible deformation paths and not just those particular paths that have been considered thus far. For the special case kz = 1 this result is apparent from (4.14) and (4.20) since the right side of those equations does not depend upon either w or RACT. The corresponding result for general kz follows from (3.18) and (3.22) by, first, replacing _

_

kz ! kz ; w ! w, and, second, by eliminating wR in favor of KACT using (4.4). This more general result is

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2K ACT kz þ K 2ACT k2z  4k3z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; tan X ¼ 2k3=2 kz ðK ACT  k2z Þ z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ 1  K ACT kz þ 1 þ 2K ACT kz þ K 2ACT k2z  4k3z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan Xref ¼ :   2 kz K ACT  k2z _

1 þ K ACT kz  2k3z þ

ð5:1Þ

ð5:2Þ _

_

It then follows that lines of constant kz in the active subregion are also lines of both constant X and constant Xref . Fig. 4 shows the active subregion covered by a non-orthogonal coordinate grid. The first set of coordinate curves in this _

_

grid consists of vertical lines of constant Xref and we here show such lines for increments of 5° in Xref . The second set of coor-

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dinate curves in the grid consists of the constant R activation curves and we here show such curves for increments of 0.05Ro. Such a coordinate grid provides a useful means for understanding how the deformation path within the active subregion correlates with both the dissolution process and the morphology of the reassembled fibers. 6. Applied torque and load on the cylinder ends For the six deformation paths P1(s) through P6(s) considered in the previous section, Fig. 3 shows precisely how the final fiber orientation morphology differs at the conclusion of the dissolution and reassembly process. It is therefore to be expected that different twisting moments Q and different axial loads P would be needed to sustain the same final deformation in these now effectively different hyperelastic materials. In order to inquire into this issue, we now consider definite forms for the energy functions Wm and Wf. A variety of forms for the matrix energy density Wm could be considered. Here we use the well known neo-Hookean form

W m ðI1 Þ ¼

l 2

ðI1  3Þ;

ð6:1Þ

where l > 0 is the standard neo-Hookean modulus. Turning to the fiber energy density Wf, a standard form is

W f ðKÞ ¼

c 2

ðK  1Þ2 ;

ð6:2Þ

where c > 0 is a constitutive parameter indicative of the fiber stiffness. The forms (6.1) and (6.2) have been used in a number of hyperelastic models for fiber reinforcing, (see, for example, [21–23]) in which case the ratio c/l > 0 is a simple measure of the relative effect of fiber reinforcing. Observe that Wf in (6.2) obeys W 0f ð1Þ ¼ 0 which ensures that the original undeformed configuration is stress free. Here it is also to be noted from (4.2) that, prior to dissolution and reassembly, the undeformed configuration (kz = 1, w = 0, and hence I1 = 3, K = 1 for all R) gives Q = 0 regardless of any restriction on Wf. However, it follows from (4.3) that the restriction W 0f ð1Þ ¼ 0 is necessary to give P = 0 for the undeformed configuration prior to dissolution and reassembly. 6.1. Torque and load in pure torsion Consider w increasing from zero for the case of pure torsion (kz = 1) as discussed at the end of Section 4. Fig. 5 shows the torque vs. twist curves (Q vs. w) for the energy densities (6.1) and (6.2). For the purpose of this figure we again take pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KACT = 1.69 so that dissolution and reassembly again begins when Ro w ¼ K ACT  1 ¼ 0:831. For Row < 0.831 there is no dis-

Fig. 5. Graph of normalized twisting moment Q vs. w for pure torsion (kz = 1) using five different values of reinforcing strength c/l: 0, 1, 2, 5, 10 and KACT = 1.69. Fiber dissolution and reassembly occurs in the blue portion of each curve.

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Fig. 6. Monotonicity of the normalized twisting moment Q vs. w graph in pure torsion is sensitive to the value of c/l for a given KACT. This graph shows the transition ratio c/l as a function of KACT on the interval 1.2 6 KACT 6 4.

solution, so that Q as a function of w is given by (4.13) using kz = 1. The torque Q is then monotonically increasing in w for all l and c in (6.1) and (6.2). For Row > 0.831 the deformation involves an outer ring of reassembled fibers that are separated from an inner core of oripffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ginal fibers by the transformation front R ¼ RACT ¼ K ACT  1=w ¼ 0:831=w. Then Q as a function of w is given by (4.19). When a fiber dissolves and reassembles, the newly created fiber is stress-free at the instant of reassembly. Thus the material in the

Fig. 7. Graph of normalized axial force P vs. w for pure torsion (kz = 1) using five different values of reinforcing strength c/l: 0, 1, 2, 5, 10 and KACT = 1.69. Fiber dissolution and reassembly occurs in the blue portion of each curve.

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Fig. 8. Graph of normalized twisting moment Q vs. K(Ro) along the deformation paths P2(s) and P5(s) when c = l, KACT = 1.69. Fiber dissolution and reassembly occurs in the blue portion of each curve.

Fig. 9. Graph of normalized axial force P vs. K(Ro) along the deformation paths P2(s) and P5(s) when c = l, KACT = 1.69. Fiber dissolution and reassembly occurs in the blue portion of each curve.

outer ring has a decreased capacity to carry the load. There is then a decrease in the load transmitted by the torsion bar when the fibers initially reassemble. As the twist increases, the fibers stretch and again contribute to the load and so the load now increases. The qualitative behavior is sensitive to the ratio c/l as shown in Fig. 5. In particular, it is found that the curves in Fig. 5 are always monotonically increasing if c/l < 0.3624, but become monotonically decreasing for a range of w if c/l > 0.3624. The value of the ratio c/l separating monotonic and nonmonotonic behavior is sensitive to the value KACT.

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Specifically, the transition value of the ratio c/l decreases with KACT. Fig. 6 provides a graph of the transition ratio c/l as a function of KACT on the interval 1.2 6 KACT 6 4. It is recalled that the Poynting effect of classical hyperelasticity in general requires a nonzero axial load P in pure torsion (kz = 1, w – 1). Fig. 7 shows P vs. w for the various cases considered previously in Fig. 5. For Row < 0.831, P is given by (4.3) _ ^ with kz = 1. For Row > 0.831, P is given by (4.12) with kz ¼ kz ¼ 1 using (4.16)–(4.18). The non-monotonicity of the curves shown in Figs. 5–9 raise the issue of stability. It is important to recall that these figures are associated with boundary value problem (S1), defined at the end of Section 3.2, in which w and kz are specified. That is, these figures illustrate the twisting moment and force required to support the specified deformation history. As the response depends on the dissolution and reassembly history of the fibers, it is expected that different curves would arise for boundary value problem (S4) when Q and P are specified. Thus, Figs. 5–9 should not be considered response curves for all conditions of loading as would be the case if the material were elastic. Nevertheless, the non-monotonicity of the curves indicate the possibility of important stability issues during the process of dissolution and reassembly of fibers. This issue has received attention in [16] and should be the subject of further research in the fiber reinforced cylinder considered here. 6.2. Torque and load for the paths P1(s)–P6(s) Similar considerations govern torque and load for the previously considered paths P1(s)–P6(s). For these paths, however, transformation is complete when kz = 1.3, whereupon the path enters the fully transformed subregion. We may graph torque Q along any of these paths. Since both w and kz vary on different path segments, it is convenient to consider graphs of Q vs. KjRo on any such path. Fig. 8 shows such a graph for the two paths P2(s) and P5(s), again using (6.1) and (6.2) for the energy densities. As in previous figures we continue to take KACT = 1.69. The ratio c/l = 1 is chosen for illustrative purposes. As required, the two curves in Fig. 8 coincide for 0 6 KjRo 6 1 þ 0:2ðp=3Þ2 corresponding to the two paths P2(s) and P5(s) being coincident over the portion P2a(s). However for KjRo > 1 þ 0:2ðp=3Þ2 , the plots diverge. This occurs because kz varies on path segment P2b(s) where Q vs. KjRo is determined using (4.11) while w varies on path segment P5a(s) where Q vs. KjRo is determined using (4.19). The remaining portions of the Q vs. KjRo graphs, separated by dots, correspond to the segments of paths P2(s) and P5(s) in the activated and fully transformed regimes. The decreasing portion of Q vs. KjRo when the path is in the active subregion can be explained by the same discussion as was given in Section 6.1. It is also important to note that the same value of RACT on the two paths will correspond to different values of KjRo . This is also shown in Fig. 8 where we indicate the location on each curve corresponding to RACT = 0.5Ro. They are (kz, w2) = (1.7766, 0.7894) and (kz, w2) = (2.0738, 1.6829), respectively on path P2 (s) and P5(s). At the conclusion of the deformation, we note that two different values of Q are needed

Fig. 10. Graphs of P vs. Q at the common endpoint (kz, w2) = (1.5, (p/3)2) of the six paths in Fig. 2. Here we consider a continuous family of such paths described via a parameter q 2 [0, 1] such that each path starts at (kz, w2) = (1, 0) and ends at (kz, w2) = (1.5, (p/3)2), and w2 = q(p/3)2 is the only horizontal segment of the path. Here five values of reinforcing strength c/l: 0, 1, 2, 5, 10 have been used and as before KACT = 1.69. Note that c = 0 corresponds to a neoHookean matrix without fibers and so gives a single value of P and Q which is independent of the path. In all other cases the final value of P and Q depends on the deformation path. The two points corresponding to the graphs in Figs. 8 and 10 are identified on the curve for c = l. The small kink in each curve corresponds to the value q (q = 0.629) that causes the deformation path to begins its horizontal segment exactly as it enters the active subregion.

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to maintain this deformation at the common endpoint (kz, w2) = (1.5, (p/3)2), with a final value of Q ¼ 1:3825lR3o on path P2 (s) and a final value of Q ¼ 1:3631lR3o on path P5(s). In a similar fashion, we may consider graphs of P vs. KjRo on any of the paths P1(s)–P6(s). Fig. 9 shows such graphs for the two cases considered previously in Fig. 8. The same comments regarding the segments of these paths apply here. Now for the deformation point (kz, w2) = (1.5, (p/3)2) one finds a final value of P ¼ 4:4502lR2o for path P2(s) and a final value of P ¼ 3:6949lR2o for path P5(s). Thus the anticipated result is confirmed: the different morphologies of the reassembled fibers require different load values of Q and P to support identical states of deformation. This result can be illustrated very nicely for a general class of deformation paths. We recall that the paths P1(s) through P6(s) were originally described with the aid of a parameter q which identified the value w2, via w2 = q(p/3)2, at which the path in the (kz, w2) plane exhibited its horizontal segment. In particular, paths P1(s), P2(s), P3(s), P4(s), P5(s), P6(s) in Fig. 2 correspond to q = 0, 0.2, 0.4, 0.6, 0.8, 1, respectively. We could therefore imagine a continuous family of such paths that are parameterized by q on the interval 0 6 q 6 1. All paths end at the common value (kz, w2) = (1.5, (p/3)2) and the considerations of the above discussion indicate that the values of Q and P at this common endpoint will vary with q. In particular, we have already obtained these values for paths P2(s) and P5(s), and hence q=0.2 and 0.8 for KACT = 1.69 using (6.1) and (6.2) with ratio c/l = 1. Fig. 10 shows curves in a space of (Q, P) values. Points on each curve correspond to the value of Q and P at the endpoint (kz, w2) = (1.5, (p/3)2) on the q parameterized paths that we have just described. Each curve is swept out as q sweeps out values on the interval 0 6 q 6 1, thereby establishing that the endpoints depend on the deformation path. All curves correspond to KACT = 1.69 using (6.1) and (6.2). There are five different curves in Fig. 10 corresponding to five different values of the ratio c/l. One of the two previously obtained  curves corresponds to c/l = 1 as used previously in Figs. 8 and 9. The   ending values: ðQ ; PÞ ¼ 1:3825lR3o ; 4:4502lR2o for P2(s) corresponding to q = 0.2, and ðQ ; PÞ ¼ 1:3631lR3o ; 3:6949lR2o for P5 (s) corresponding to q = 0.8, are specifically indicated on this curve.

7. Concluding comments This work has two purposes: (1) consideration of microstructural changes that occur in composite materials consisting of elastic fibers embedded in an elastic matrix; (2) determination of the consequences of these changes for the response of structures composed of such materials. The particular microstructural process studied here is the dissolution of fibers as they undergo large stretch and their reassembly into fibers whose reference state has a new direction with respect to the matrix. In this work, fiber reassembly takes place in the direction of maximum principal stretch of the matrix. There is a range of response in which the material does not undergo microstructural changes and this is modeled within the context of anisotropic hyperelasticity. The model is extended to the regime of microstructural changes by introducing material properties that account for the processes of dissolution of the original fiber and their reassembly to form new fiber. The implications of the model have been studied in the context of a non-homogeneous deformation of a circular solid cylinder consisting of an isotropic matrix containing a distribution of helically wound fibers. The cylinder is subjected to simultaneous axial stretch and twisting about its central axis. Each fiber is stretched by an amount that depends on its radius and orientation and the imposed axial stretch and twist. As axial stretch and twist increase, more and more fibers are stretched to such an extent that they undergo dissolution and reassembly. In order to develop specific insights into the consequences of fiber dissolution and reassembly and to avoid excessive complexity, attention was confined to the special case when a single family of fibers is initially oriented in the axial direction and uniformly distributed along the radius. Moreover, the fibers were assumed to undergo instantaneous dissolution and reassembly at a special value of fiber stretch. Even with these simplifications, the process of fiber dissolution and reassembly has been shown to produce complex morphological changes in the fibrous structure. As axial stretch and twist of the cylinder increased the stretch in the fibers, the cylinder developed an inner core of original material and an outer layer of material in which the fibers have undergone dissolution and reassembly. The interface between these regions moved radially inward. Gradual reassembly of the fibers with increasing stretch and twist changed their helical orientation and their contributions to the torque and axial force. Different sequences of axial stretch and twist resulted in different helical morphologies of the reassembled fibrous structures and in different sequences of torque and axial force. Although the study presented here was carried out for a particular process of dissolution and reassembly, it provides insight into the interesting mechanical and structural changes that can occur whenever the direction of reassembly differs from that of the original fibers. Accordingly, it should be pointed out that although the dissolution and reassembly process considered here is attributed to deformation, the constitutive theory can be adapted to allow for dissolution and reassembly processes arising from non-mechanical causes such as thermal or chemical. Indeed, the constitutive theory for thermally induced microstructural changes in [24] was an outgrowth of the deformation based theory in [17], which in turn was motiovated by Tobolsky’s introduction of scission and re-cross-linking [16]. The work presented here suggests that microstructural changes arising from mechanical or non-mechanical effects can have a profound influence on the response of structures. This point is illustrated in [25–27], which consider the remodeling of the distribution of collagen fibers embedded in an extracellular matrix. The results presented here show that the constitutive theory allowing for dissolution and reformation of fibers provides a framework for analyzing such remodeling phenomena. Moreover, this suggests that the constitutive theory with dissolution and reassembly can be gainfully employed in relating microstructurally based material property changes to the alteration in the structural performance of the system.

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a

b

_

Fig. 11. An alteration to path P6(s) via a small detour lying within the active region and effect of such detour on the orientation angle Xref .

Appendix. A deformation path that creates a jump in fiber orientation _ pffiffiffiffiffiffiffiffiffiffi As a specific example giving a jump in Xref , consider an alteration to path P6(s) such that at some kz < K ACT on the segment P6b(s), say at kz , the path takes a temporary downward detour consisting of 3 small line segments:

    kz ; ðp=3Þ2 ! kz ; ðp=3Þ2  d1 ;     kz ; ðp=3Þ2  d1 ! kz þ d2 ; ðp=3Þ2  d1 ;     kz þ d2 ; ðp=3Þ2  d1 ! kz þ d2 ; ðp=3Þ2 ; where the positive constants d1 and d2 are sufficiently small so as to keep this detour within the active subregion. It will then be the case that RACT will remain fixed on the first of the above segments, since moving forward on this first detour segment crosses previously encountered activation curves. The activation curve for the point ðkz ; ðp=3Þ2 Þ at the start of the detour will intersect the detour later, somewhere on either the second or third detour segment depending on the values: kz , d1, d2. This _

gives the point on the detour where RACT recommences its inward motion. Then Xref at Rþ follows from (3.22) using R = RACT,  _ACT_  _ _ _   kz ¼ kz , and w ¼ p=3. Conversely, Xref at RACT follows from (3.22) using R = RACT with kz ; w as determined from the detour _

point where the motion recommences. These two values of Xref are different. Fig. 11a shows such a detour path for the case of: kz ¼ 1:1, d1 = 0.08775, d2 = 0.1. The detour starts at point ‘p’ on the acti_

vation curve for RACT = 0.694 with Xref ¼ 0:5185. As the detour segment is traversed along ‘p–q–w’, it crosses previously encountered activation curves and intersects the activation curve for RACT = 0.694 again at point ‘w’ where _

(kz, w2) = (1.121, 1.00888) and Xref ¼ 0:5002. As the detour is traversed along ‘w–r–t’, it crosses new activation curves and _

RACT decreases. Fig. 11b shows the departure of the graph of Xref vs. R due to this detour. Activation ceases at point ‘p’ on _

the activation curve for RACT = 0.694 with Xref ¼ ðp=3Þ2 and commences again at ‘w’ on the same activation curve where _

now Xref ¼ 0:5002. Proceeding on the detour along w–r, on which kz increases while w is constant, it is seen that R and _

Xref decrease as found from (3.22) and (4.4). Finally, along ‘r–t’, kz is fixed.

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