Swelling-induced twisting and shearing in fiber composites: the effect of the base matrix mechanical response
Abstract
If a helical network of fibers is embedded in a swellable matrix, and if the fibers themselves resist swelling, then a change in the amount of swelling agent will cause a corresponding twisting motion in the material. This effect has recently been analyzed in highly deformable soft material tubes using the theory of hyperelasticity, suitably modified to incorporate the swelling effect. Those studies examined the effect of spiral angle and fiber-to-matrix inherent stiffness in the context of a ground state matrix material that exhibited classical neo-Hookean behavior in the absence of swelling. While such a ground state material is nonlinear in general, its shear response is linear. As we describe here, it is this shear response that governs the matrix contribution to the twist-swelling interaction. Because gels, elastomers, and even biological tissue can exhibit complex ground state behavior in shear—behavior that may depart significantly from a linear response—we then examine the effect of alternative ground state behaviors on the twist-swelling interaction. The range of behaviors considered includes materials that harden in shear, materials that soften in shear, materials that have an ultimate shear stress bound, and materials that collapse in shear. Matrix materials that either soften or collapse in shear are found to amplify the twisting effect.
1 Introduction
By embedding biassing fibers in a matrix material that is highly absorbant, it is possible to generate specialized deformation modes as the material swells. This is because the fibers naturally guide the deformation in what is essentially a prearranged way. In a previous paper [3], we analyzed this effect in tubes containing spiral patterns of fibers that lie in the plane of the tube’s cross-section. Swelling of the matrix then cause an overall tube twist to take place even though no external forces are applied. Effectively, the internal swelling acts as a body force to induce the twist.
In conjunction with other modes of deformation that can be induced by swelling, such as bending [8] and torsion [1], this suggests that a wide variety of deformation modes could be engendered by intelligent pattern design for fiber biasing [2, 4]. While fiber pattern biassing is inherently a design freedom that could be exploited, other design freedoms should not be overlooked. One of these is the qualitative mechanical behavior of the swellable base material, i.e., the matrix in which the fibers are embedded.
Here it is to be noted that previous studies often focus on matrix behavior that has an essential neo-Hookean type character and which reduces to neo-Hookean behavior in the absence of swelling. While such a ground state material is nonlinear in general, its shear response is linear. As we describe here, it is this shear response that governs the matrix contribution to the twist-swelling interaction. Because gels, elastomers, and even biological tissue can exhibit complex ground state behavior in shear — behavior that may depart significantly from a linear response — it is our purpose here to examine the effect of alternative ground state behaviors on the twist-swelling interaction. The range of behaviors considered includes materials that harden in shear, materials that soften in shear, materials that have an ultimate shear stress bound, and materials that collapse in shear.
To this end, we present the overall continuum mechanics formulation in Section 2 where the focus is on the relevant constitutive treatment for the mechanical behavior due to swelling. The overall twisting boundary value problem of interest is formulated in Section 2. It is in fact a swelling version of what is known as the boundary value problem for azimuthal shear, although for our purposes we often refer to it as twisting of a tube due to swelling. Complexities associated with this boundary value problem, while surmountable numerically as we show in later sections, then lead us to examine a simpler problem, namely swelling-induced shearing of a layer. The main advantage is that the layer problem is one of homogeneous deformation. This simpler problem has many useful analogies to the swelling-induced tube twisting problem. This analogy, and certain basic results, is described in Section 2.
Up to this point in the paper, the matrix material constitutive relation has concentrated on the neo-Hookean type response. A far more general constitutive model for the matrix, which is a swellable version of the Knowles I_{1} power law material [6], is introduced in Section 2. The layer problem for homogeneous deformation under swelling is examined thoroughly for this class of materials. We construct curves showing how the layer shears as a function of swelling. This provides certain important insights into how best to approach the tube twisting problem for the case of the power-law matrix material, and a thorough study of this problem is conducted in Section 2. Now curves of layer twisting as a function of swelling are obtained. Both the layer curves and the twisting curves have certain remarkable features. These are discussed and explained in the concluding Section 2.
2 Hyperelastic framework for treating swellable matrices and nonswelling fibers
Our examination is based on swellable hyperelastic materials. We consider materials with a single family of fibers at each material point, where the direction can vary from point to point (as in the spiral fiber patterns considered in [3]).
The swellable hyperelastic materials are treated in terms of the swelling volume constraint \(\det \mathbf {F}=v\) where F is the gradient of the deformation mapping from reference locations X to deformed locations x and v is the local swelling value. If v is equal to one everywhere, then this describes an incompressible material. Locations where v > 1 are swollen locations. For a variety of physical processes in soft materials, swelling is determined by chemical or electro-chemical influences that largely decouple from mechanical considerations. In such a case, v can be regarded as a specified field. Moreover, because \(\det \mathbf {F}=v\) is a pointwise constraint, the Cauchy stress will contain a constraint reaction pressure − pI where I is the identity tensor and p is the scalar hydrostatic pressure. The value of p is determined directly from the equations of equilibrium in conjunction with the boundary conditions. In particular, as is true for constraint reactions in general, p is not subject to a constitutive prescription.
Because we are considering a single direction of fibers at each material point, even though that direction may vary within the material due to fiber patterning, the dependence of W on C is only through the invariants I_{1},I_{2},I_{3},I_{4}, and I_{5}. The local swelling condition eliminates I_{3} because I_{3} = v^{2}. The fiber direction itself is specified by a unit vector M in the unswollen reference configuration. The deformed image of this vector is m = FM. The vector m, which is generally not a unit vector, gives the fiber direction in the deformed configuration, and its length gives the local fiber stretch.
An alternative situation, fibers that swell in a matrix that does not appreciably swell, can be described by a converse to Eq. 0.3, W = w_{m}(I_{1}) + w_{f}(I_{4},v). For example, this might describe cellulose fibers within a rubbery matrix. However, here our focus is always on Eq. 0.3.
In this paper, we retain the form used in Eq. 0.6 for w_{f} but consider alternative forms for w_{m}. Attention is restricted to the plane problem. This means that all the displacement and all of the swelling are in an (X,Y )-plane. Such a situation would arise for example if the domain is a uniform cylinder with a general cross-section Ω in the (X,Y )-plane. In the axial direction, say Z, the cylinder is then confined to be between frictionless plates, say at \(Z=\pm \frac {1}{2} L\). The fiber direction M is similarly presumed to always lie in the (X,Y )-plane in a manner that is independent of Z. The material and swelling can then vary in the (X,Y ) cross-section while being uniform in Z. This readily realizable state of affairs applies to a potential wide variety of situations and in particular describes the azimuthal shear swelling deformation examined in [3].
3 Swelling and azimuthal shear
Note that if there is no swelling at all (i.e., if v = 1 for all R) then Eq. 0.11 gives r = R making λ = 1 and hence \(I_{4}=k^{2} \cos \limits ^{2} \alpha +2 k {\cos \limits } \alpha \sin \limits \alpha +1\). In this case, k = 0 makes I_{4} = 1 which causes \({w}_{f}^{\prime }(I_{4})=0\). Thus, k = 0 solves Eq. 0.11 if v = 1 for all R. In other words, if there is no swelling, then there is no shearing.
Equation 0.24 allows for v = v(R) and α = α(R). It is also useful to note that k = 0 solves Eq. 0.24 if either α = 0 or α = π/2. Thus, locations R where the fibers are oriented purely radially or purely azimuthally are not locally sheared.
However, if 0 < α < π/2, then the possible nonlinear nature of \(w_{m}^{\prime }\) as a function I_{1}, and \({w}_{f}^{\prime }\) as a function of I_{4}, can cause Eq. 0.24 to become a highly nonlinear equation for k = k(R). For the case where w_{m} and w_{f} are given by Eqs. 0.5 and 0.6, certain analytical simplifications result, and this was exploited in the methodology of [3].
4 Local analogue: combined biaxial stretch and simple shear with swelling (a simple layer)
This motivates the consideration of plane-strain shearing of a semi-infinite layer that is also subject to swelling. The reference cross-section in the 2-D plane of interest is now \({\varOmega } = \{(X,Y): -\infty < X< \infty , 0 \le Y \le H \} \) and the fibers are oriented as in Eq. 0.26. As in the azimuthal shear problem, the third Z-direction is constrained, thus rendering the problem planar in the same way. Symmetry with respect to X requires that both v and α must be independent of X.
Boundary conditions must be specified on Y = 0 and Y = H. To put Y = 0 in correspondence with the inner circle R = A in the azimuthal shear problem, we stipulate x = X on Y = 0. For the deformation (0.25), this causes λ_{x} = 1. The upper surface Y = H is the analogue of the outer circle. However, here we temporarily allow not only expansion in this direction, but also the possibility of a shear traction τ. It is to be noted that dependence of either v or α upon Y is consistent with this scenario. This is analogous to the fact that the azimuthal shear development of the previous section allowed for the possibility of either v or α to depend upon R. For the present layer problem, if either v or α depend upon Y, then λ_{y} = v/λ_{x} = v would generally vary as well. All of these conditions would then generally make it necessary to let K = K(Y ) in order to satisfy the equations of equilibrium.
If τ = 0 then v is the sole driving agent for the amount of shear K. Increasing v in the absence of shearing then elongates the fiber. The direction of shearing that reduces this elongation is one that rotates the fibers toward the vertical. Then K > 0 gives clockwise rotation and K < 0 gives counterclockwise rotation. In order to work with K > 0 (clockwise rotation), we shall take − π/2 < α < 0, and this is the situation shown in Fig. 2. For sufficiently small swelling, this rotation can completely eliminate the fiber elongation, however that is energetically costly with respect to the matrix deformation. The optimum value of K as determined by Eq. 0.33 is one that minimizes the combined energetic cost of matrix shearing and fiber elongation.
For − π/2 < α < 0 in the layer problem, the analogous α in the azimuthal shear deformation (0.8) is the positive counterpart, i.e., 0 < α < π/2. This \(\alpha \leftrightarrow -\alpha \) correspondence is due to the fact that \((x,y,z) \leftrightarrow (\theta , r, z)\) as indicated in Eq. 0.30, with (𝜃,r,z) in that order, has effectively changed the handedness of the coordinate system.
We now focus on the mechanics of how swelling (v > 1) induces a nonzero amount of shear K without any applied shear traction (τ = 0).
The qualitative nature of the curves shown in Fig. 3 is similar to the response curves shown in Fig. 7 of [3] for the azimuthal shear problem (the twist-swelling problem). This is the anticipated result since a motivation for considering the layer problem is as a simple analogue to the twist-swelling problem. Here it is to be noted that the present X-stretch condition λ_{x} = 1 is the analogue of a condition that would take λ = r/R = 1 in the twist-swelling problem. For the twist-swelling boundary value problem, the condition λ = 1 holds on the inner radius R = A, however λ > 1 for A < R ≤ B. Thus, the layer problem is most representative of the twist-swelling problem near the inner surface.
5 Power law matrix material and the associated swelling response for the layer
We now consider the swelling layer response, again with a traction free top surface, where w_{m} and w_{f} are given by Eqs. 0.43 and 0.6. This provides a generalization of Eq. 0.37 in the form
The initial slope of the response curve is again given by Eq. 0.40 but now making use of the above values for A_{1} and A_{0}. The cubic equation also permits a large v analysis for the n = 2 response and one finds that
Thus, as \(v \rightarrow \infty \), the n = 2 response curve approaches the asymptote \(K=K_{(\infty , n=2)} \). With one limiting exception, this asymptotic value is strictly less than the value \(-\tan \alpha \) that was obtained for the n = 1 neo-Hookean case (viz. Eq. 0.41). The limiting exception is the inextensible fiber limit \(\gamma /\mu \to \infty \) in which case \(K_{(\infty , n=2)} \to -\tan \alpha \).
6 Azimuthal shear response with the power law matrix material
The governing equation for g(R) follows by substituting from Eqs. 0.43 and 0.6 into Eq. 0.24 yielding
The additional complications due to swelling and the presence of fibers motivated the local analysis of the previous section. It is thus noted that Eq. 0.49 as an ODE for \(g^{\prime }(R)\) mirrors the layer Eq. 0.45—which was a scalar equation for the amount of shear K. In particular, the special cases n = 1 and n = 2 give a cubic structure to Eq. 0.49 which facilitates the integration in those two cases.
6.1 Linear response in shear (the neo-Hookean type response, n = 1)
The location of maximum outer twist ϕ_{o} vs. fiber winding angle α (both in degrees) of the five twist-swelling response curves in Fig. 7 (for the n = 1 neo-Hookean type matrix)
v | α | ϕ_{o} |
---|---|---|
1.5 | 34.26 | 9.80 |
2 | 40.97 | 16.07 |
8/3 | 46.41 | 22.02 |
4 | 53.17 | 30.16 |
5 | 56.26 | 34.65 |
This bounding curve is also shown in Fig. 7. As v increases, the various curves in Fig. 7 approach this upper bound value, albeit in a highly nonuniform fashion in view of “pinning condition” ϕ_{o} ≡ 0 at α = π/2. In other words, circumferential fibers (α = π/2) give no shearing tendency upon swelling. However, highly inclined fibers (α = π/2 − 𝜖 with 𝜖 > 0 very small but nonzero) become formally unbounded in length as 𝜖 → 0. Straightening these long fibers thus gives a formally unbounded twist ϕ_{o} in the limit as 𝜖 → 0.
6.2 Stiffening in shear (n = 2)
Other values of n (other than 1 and 2) when used in Eq. 0.49 give highly nonlinear equations for k (with embedded radical expressions) that require numerical root finding procedures (as considered shortly). However, this n = 2 case permits an explicit solution for k and hence \(g^{\prime }\). Numerical integration starting from g(A) = 0 then provides g(R).
The location of maximum outer twist ϕ_{o} vs. fiber winding angle α (both in degrees) for the five twist-swelling response curves in Fig. 11
v | α | ϕ_{o} |
---|---|---|
1.5 | 34.31 | 9.30 |
2 | 39.60 | 14.06 |
8/3 | 43.26 | 17.32 |
4 | 46.04 | 19.97 |
5 | 46.86 | 20.74 |
6.3 Other values of n in Eq. 0.43
When n is different than 1 or 2, the governing Eq. 0.49 is not readily solved for k by algebraic means. Thus a numerical root finding procedure is utilized prior to the numerical integration to obtain g(R). In particular, as g(A) = 0, a recursive tangent line approximations can be used to approximate g(R) for A ≤ R ≤ B. For this purpose, the interval [A,B] is divided into some number of equal subintervals where the size of each interval is dR.
7 Discussion and conclusions
Whereas previous studies have concentrated on the effect of fiber stiffness and patterning on eliciting complex deformation as a material swells, the present work focuses specific attention on the effect of the base matrix response, i.e., the mechanical behavior of the material in which the fibers are embedded. We show that the base matrix material response also has a significant effect on both simple layer shearing and circular tube twisting as the material swells.
For circular tube twisting, we consider the relative rotation of the tube cross-section when the tube wall contains spirally patterned fibers that proceed from the inner to the outer radius. Because the local deformation is one of simple shear superposed on biaxial deformation, the tube twist findings of Section 2 are anticipated on the basis of certain key understandings of the matrix-fiber-swelling mechanics obtained in Sections 2 and 2 for the more straight forward problem of a simple material layer that swells. For the simpler layer problem, the fibers proceed through the layer cross-section along straight lines. Both horizontally aligned fibers and vertically aligned fibers provide no shearing tendency as the layer swells. However, inclined fibers will cause shearing. In all cases throughout this study, the fibers were modeled on the basis of w_{f} as given by Eq. 0.6
Section 2 restricted attention to the matrix material model w_{m} given by Eq. 0.5 with q = 0 and provided the most intuitive key understanding, namely that the generation of shear under simple swelling is due to the presence of fibers, and the effect is magnified as the fibers become relatively stiffer with respect to the matrix. In this case, the ratio γ/μ provided a complete characterization of this relative stiffness. This led to the shear-swelling behavior shown in Fig. 3 with curves for different γ/μ approaching a common asymptote as v becomes large. The rate of approach increased with γ/μ and the common asymptotic value has magnitude \(\tan \alpha \). This asymptote is associated with a shearing that places the fibers in the minimal length position, i.e., along the Y axis. Small values of α mean the originally unswollen state involves nearly vertical fibers, and so there is minimal total shearing capacity. Values of α near ± π/2 mean the originally unswollen state involves nearly horizontal fibers and so there is a large shearing capacity and in fact the asymptote’s magnitude \(|\tan \alpha |\) becomes unboundedly large because the fibers become unboundedly long as they proceed from Y = 0 to Y = H in the limit as α →±π/2. However, if α = ±π/2 then the fibers are purely horizontal, and there is no shearing tendency because the fibers no longer traverse the layer thickness.
All of these results are mirrored in the tube problem for swelling-induced twist where ϕ_{o} is the analogue of K and the curves of ϕ_{o} vs. v are the analogue of the curves of K vs. v for the layer problem. The maximum twisting capacity is now given by \(\phi _{o}^{dfsr}\) from Eq. 0.50, which not only involves dependence on α but also on the tube aspect ratio B/A. This is reflective of the fact that the deformed fiber orientation under swelling varies with radius, and every location must have its fibers brought into radial alignment before the twisting capacity is formally exhausted. As in the case of the layer, if w_{f} and w_{m} are given by Eqs. 0.6 and 0.5 then the asymptote \(\phi _{o}^{dfsr}\) is independent of γ/μ, although it is approached more rapidly as γ/μ becomes large. The asymptotic value is formally zero (no twisting capacity) if the fibers in the unswollen state are already radial (α = 0), whereas the asymptotic value tends to infinity as α →±π/2 because now the unswollen state involves unboundedly large spirals of fiber paths that are nearly circular. However, if α = ±π/2 then the unswollen spirals degenerate to closed perfect circles, and there is no longer any twist under pure swelling, just as the purely horizontal fibers caused the layer problem to lose its shearing tendency.
The above summary results are for w_{m} given by Eq. 0.5 which means that the matrix has a straightforward linear relation in simple shear between shear stress and the amount of shearing. Polymers, gels, and the ground substance in biological tissue may exhibit far more complex behavior in simple shear, specifically some hardening or softening is not uncommon. Furthermore, under softening, the shear traction may or may not have an ultimate value. Finally, softening may be so severe as to elicit a collapse in shear phenomenon as might occur in phase transition or damage scenarios—either permanent or temporary (reversible/healable).
To investigate these effects, w_{m} was replaced with Eq. 0.43 where now the power-law exponent n allows a simple distinction between all of these cases. In addition, the special value n = 1 recovers the original model (0.5). The results of Section 2 show these significant effects in the context of the layer analysis. Here it is immediate that matrix hardening (n > 1), at least as it impacts upon the relative fiber-to-matrix stiffness, would correspond to fiber softening. The converse also holds. Thus, increasing n could be expected to have qualitatively similar effects as decreasing γ/μ, and vice versa. This is indeed the case; however, the layer analysis shows that the effect of varying n is in fact more profound. Here the analytical result (0.46) for n = 2 showed that the asymptote itself, not just the rate of approach to the asymptote, was dependent on γ/μ. Moreover, the full shearing capacity — meaning a situation where the asymptotic value of the K vs. v curves for the layer has magnitude\(| \tan \alpha |\) — is only achieved as γ/μ becomes unboundedly large. For finite γ/μ, the asymptote height remains below that associated with exhausting this capacity.
This effect is mirrored in the swelling tube findings of Section 2, where Fig. 12 clearly shows how the asymptotes of the various ϕ_{o} vs. v curves are below the upper bound provided by \(\phi _{o}^{dfsr}\). For the n = 2 tube twist problem, because it was the solution to a boundary value problem, we were unable to determine this asymptotic value by pure analysis and so found it numerically. This is in contrast to the layer problem, where the relative simplicity of the homogeneous deformation allowed us to extract the asymptotic value \(K_{(\infty , n=2)}<-\tan \alpha \) for finite values of γ/μ.
For values of n different from 1 and 2, we appealed to more extended computational procedures for both the layer and the tube problem. Because n = 1 already gave asymptotes that fully exhausted either the shearing capacity or the twisting capacity, a similar full use of this deformation capacity was found for any n ≤ 1 because by softening the matrix it relatively stiffens the fibers. This can only increase the asymptotes beyond their n = 1 heights. However, since the n = 1 asymptotes already involved full use of the shearing/twisting capacity, such full usage is maintained for n < 1. Thus all the softening cases made use of the full capacity. For this reason, the n ≤ 1 curves in both Figs. 5 and 17 show common upper bound (full usage capacity) asymptotes for the n ≤ 1 cases. Finally, the n = 2 and n = 3 curves in Figs. 5 and 17 also show similar commonality. Now however the n = 2 and n = 3 asymptotes are below that associated with full usage of either layer shearing capacity (Fig. 5) or tube twisting capacity (Fig. 17) as the swelling becomes large.
Notes
Acknowledgments
Open Access funding provided by the Qatar National Library. HD and TP acknowledges the NPRP grant #8-2424-1-477 from the Qatar National Research Fund (a member of the Qatar Foundation). The statements made herein are solely the responsibility of the authors.
Compliance with Ethical Standards
Conflict of interest
The authors declare that they have no conflict of interest.
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