Modulation of elastic instabilities via particle interaction-induced stiffening in soft particulate composites

We investigate purely elastic flow instabilities in the almost ideal planar stagnation point elongational flow field generated by a microfluidic optimized-shape cross-slot extensional rheometer (OSCER). We use time-resolved flow velocimetry and full-field birefringence microscopy to study the behavior of a series of well-characterized viscoelastic polymer solutions under conditions of low fluid inertia and over a wide range of imposed deformation rates. At low deformation rates the flow is steady and symmetric and appears Newtonian-like, while at high deformation rates we observe the onset of a flow asymmetry resembling the purely elastic instabilities reported in standard-shaped cross-slot devices. However, for intermediate rates, we observe a new type of elastic instability characterized by a lateral displacement and time-dependent motion of the stagnation point. At the onset of this new instability, we evaluate a well-known dimensionless criterion M that predicts the onset of elastic instabilities based on geometric and rheological scaling parameters. The criterion yields maximum values of M which compare well with critical values of M for the onset of elastic instabilities in viscometric torsional flows. We conclude that the same mechanism of tension acting along curved streamlines governs the onset of elastic instabilities in both extensional (irrotational) and torsional (rotational) viscoelastic flows.

In this study, we combine the elastic instability and non-linear rate-dependent phenomena to achieve microstructure tunability in soft layered materials. In these soft composites, elastic instabilities give rise to formation of wrinkles or wavy patterns. In elastic materials, the critical wavelength as well as amplitude at a particular strain level are exclusively defined by the composite microstructure and contrast in the elastic moduli of the phases. Here, we propose to use rate-dependent soft constituents to increase the admissible range of tunable microstructures. Through the experiments on 3D printed soft laminates, and through the numerical simulation of the visco-hyperelastic composites, we demonstrate the existence of various instability-induced wavy patterns corresponding to the identical deformed state of the identical soft composites.

Elastic instabilities can trigger dramatic microstructure transformations giving rise to unusual behavior in soft matter. Motivated by this phenomenon, we study instability-induced pattern formations in soft magnetoactive elastomer (MAE) composites deforming in the presence of a magnetic field. We show that identical MAE composites with periodically distributed particles can switch to a variety of new patterns with different periodicity upon developments of instabilities. The newly formed patterns and postbuckling behavior of the MAEs are dictated by the magnitude of the applied magnetic field. We identify the particular levels of magnetic fields that give rise to strictly doubled or multiplied periodicity upon the onset of instabilities in the periodic particulate soft MAE. Thus, the predicted phenomenon can be potentially used for designing new reconfigurable soft materials with tunable material microstructures remotely controlled by a magnetic field.

We determine both experimentally and numerically the onset of elastic flow instabilities in viscoelastic polymer solutions with different levels of shear thinning. Previous experiments realized in microfluidic serpentine channels using dilute polymeric solutions showed that the onset of elastic instabilities strongly depends on the channel curvature. The scaling dependence is well captured by the general instability scaling criterion proposed by Pakdel and McKinley [Phys. Rev. Lett., 1996, 76, 2459:1-4]. We determine here the influence of fluid shear thinning on the onset of such purely-elastic flow instabilities. By testing a set of polyethylene oxide solutions of high molecular weight at different polymer concentrations in microfluidic serpentine channels we observe that shear thinning has a stabilizing effect on the microfluidic flow. Three-dimensional numerical simulations performed using the White-Metzner model predict similar trends, which are not captured by a simple scaling analysis using the Pakdel-McKinley criterion.

Rheological and geometric scaling of purely elastic flow instabilities

A systematic theoretical study is presented of the stress-induced structural response of initially cubic single crystals to uniaxial [100] loading based on elastic stability analysis and isostress molecular-dynamics simulations through a classical description of interatomic interactions in model metallic crystals. Special emphasis is placed on the study of the atomic pattern formation characteristics in the crystal's structural response to loading at and beyond the onset of elastic instability. The instability is reached at a rigorously defined critical stress level that occurs in association with the vanishing of a shear modulus, i.e., when ${C}_{22}∕{C}_{23}\ensuremath{-}1=0$, where ${C}_{rs}$ are stress-dependent elastic moduli. Although the atomic mechanism for the onset of instability is invariant, two divergent atomic processes are found to occur beyond the onset of instability, depending on subtle differences in the elastic properties of the crystals. Our analyses and simulations of a crystal model with the relatively small initial value of ${C}_{22}∕{C}_{23}\ensuremath{-}1=0.41$ (based on the elastic moduli of copper) reveal an inhomogeneous structural transformation mechanism, through the creation of individual rotating domains that lead to formation of a new hexagonal single crystal without loss of strength. This theoretical result is consistent with what is known experimentally for metals with relatively small values of $({C}_{22}∕{C}_{23}\ensuremath{-}1)$, e.g., certain copper alloys and the alkali metals, which can undergo various cubic-to-hexagonal structural transformations. However, a crystal model based on the elastic moduli of nickel, with the larger initial value of ${C}_{22}∕{C}_{23}\ensuremath{-}1=0.73$, fails to exhibit domain rotation beyond the onset of elastic instability and, as a result, the initial destabilization of the crystal structure then leads to fracture.

Microscopic and long-wave instabilities in 3D fiber composites with non-Gaussian hyperelastic phases

Previous experimental measurements and linear stability analyses of curvilinear shearing flows of viscoelastic fluids have shown that the combination of streamwise curvature and elastic normal stresses can lead to flow destabilization. Torsional shear flows of highly elastic fluids with closed streamlines can also accumulate heat from viscous dissipation resulting in nonuniformity in the temperature profile within the flow and nonlinearity in the viscometric properties of the fluid. Recently, it has been shown by Al-Mubaiyedh et al. [Phys. Fluids 11, 3217 (1999)] that the inclusion of energetics in the linear stability analysis of viscoelastic Taylor–Couette flow can change the dominant mode of the purely elastic instability from a nonaxisymmetric and time-dependent secondary flow to an axisymmetric stationary Taylor-type toroidal vortex that more closely agrees with the stability characteristics observed experimentally. In this work, we present a detailed experimental study of the effect of viscous heating on the torsional steady shearing of elastic fluids between a rotating cone and plate and between two rotating coaxial parallel plates. Elastic effects in the flow are characterized by the Deborah number, De, while the magnitude of the viscous heating is characterized by the Nahme–Griffith number, Na. We show that the relative importance of these two competing effects can be quantified by a new dimensionless thermoelastic parameter, Θ=Na1/2/De, which is a material property of a given viscoelastic fluid independent of the rate of deformation. By utilizing this thermoelastic number, experimental observations of viscoelastic flow stability in three different fluids and two different geometries over a range of temperatures can be rationalized and the critical conditions unified into a single flow stability diagram. The thermoelastic number is a function of the molecular weight of the polymer, the flow geometry, and the temperature of the test fluid. The experiments presented here were performed using test fluids consisting of three different high molecular weight monodisperse polystyrene solutions in various flow geometries and over a large range of temperatures. By systematically varying the temperature of the test fluid or the configuration of the test geometry, the thermoelastic number can be adjusted appreciably. When the characteristic time scale for viscous heating is much longer than the relaxation time of the test fluid (Θ≪1) the critical conditions for the onset of the elastic instability are in good agreement with the predictions of isothermal linear stability analyses. As the thermoelastic number approaches a critical value, the strong temperature gradients induced by viscous heating reduce the elasticity of the test fluid and delay the onset of the instability. At even larger values of the thermoelastic parameter, viscous heating stabilizes the flow completely.

Elastic instabilities, microstructure transformations, and pattern formations in soft materials

We reveal the existence of a state in soft composites, characterized by the omni-directional negative group velocity in the vicinity of elastic instability. We show that the appearance of the negative group velocity in layered and fibrous composites foreshadows microscopic loss of the stability. In contrast with classical instability-induced pattern transformations, the transition between states with positive and negative group velocities is not accompanied by geometrical rearrangements and can be triggered by very fine variation of the compressive deformation in stable composites. Finally, we analyze the effect of the geometrical characteristics and elastic moduli of the constituents on the strain range for induced state with negative group velocities.

Through years of evolution, biological soft fibrous tissues have developed remarkable functional properties, unique hierarchical architectures, and -most notably, an unparalleled and extremely efficient deformation ability. Whereas the structure-function relationship is well-studied in natural hard materials, soft materials are not getting similar attention, despite their high prevalence in nature. These soft materials are usually constructed as fiber-reinforced composites consisting of diverse structural motifs that result in an overall unique mechanical behavior with large deformations. Biomimetics of their mechanical behavior is currently a significant bioengineering challenge. The unique properties of soft fibrous tissues stem from their structural complexity, which, unfortunately, also hinders our ability to generate adequate synthetic analogs, such that autografts remain the “gold standard” materials for soft-tissue repair and replacement. This review seeks to understand the structural and deformation mechanisms of soft collagenous tissues, with a particular emphasis on tendon and ligaments, the annulus fibrosus (AF) in the intervertebral disc (IVD), skin, and blood vessels. We examined and compared different mechanical and structural motifs in these different tissue types, which are subjected to complex and varied mechanical loads, to isolate the mechanisms of their deformation behavior. Herein, we focused on their composite structure from a perspective of the different building blocks, architecture, crimping patterns, fiber orientation, organization and their structure-function relationship. In the second part of the review, we presented engineered soft composite applications that used these structural motifs to mimic the structural and mechanical behavior of soft fibrous tissues. Moreover, we demonstrated new methodologies and materials that use biomimetic principles as a guide. These novel architectural materials have tailor-designed J-shaped large deformations behavior. Structural motifs in soft composites hold valuable insights that could be exploited to generate the next generation of materials. They actually have a two-fold effect: 1) to get a better understanding of the complex structure-function relationship in a simple material system using reverse biomimetics and 2) to develop new and efficient materials. These materials could revolutionize the future tailor-designed soft composite materials together with various soft-tissue repair and replacement applications that will be mechanically biocompatible with the full range of native tissue behaviors.

Elastic instabilities are identified as flow instabilities occurring in the presence of low inertial effects, induced by the combination of strong elastic forces with nonlinearities of the flow. In continuous flow laminar mixing applications, the onset of these instabilities is likely to occur in the window of applied flow rates; therefore, it is important to understand the effects of their onset on the process efficiency. In this work, we investigated experimentally the onset of elastic instabilities in two tubular static mixers with different geometric features, i.e., the Kenics helical mixer and the SMB-R mixer, the latter characterized by a double X-shaped bar geometry. We obtained concentration maps at various mixer lengths by means of planar laser induced fluorescence techniques. To deduce a generalized effect of the fluid elasticity on the mixing patterns, we tested three fluids with different rheological behavior—a Boger fluid and two shear-thinning fluids. For all cases, we observed deviations from the Newtonian benchmark as soon as the Deborah number exceeded unity, even though different transitions occurred as the mean flow rate increased. The effect of the instability on the mixing patterns strongly depended on the different kinematics induced by the two geometries: for the helical mixer, the typical lamellar structure is not recovered and the two liquid streams remain unmixed, while for the SMB-R, the concentration maps are strongly unsteady, showing temporally and spatially chaotic fluctuations of the mass fraction. In both cases, the instabilities worsen the mixing efficiency compared to the Newtonian case.

It is well known that inertia-free shearing flows of a viscoelastic fluid with curved streamlines, such as the torsional flow between a rotating cone and plate or the flow in a Taylor–Couette geometry, can become unstable to a three-dimensional time-dependent instability at conditions exceeding a critical Weissenberg ( $Wi$ ) number. However, the combined effects of fluid elasticity, shear thinning and finite inertia (as quantified by the Reynolds number $Re$ ) on the onset of elasto-inertial instabilities are not fully understood. Using a set of cone–plate geometries, we experimentally explore the entire $Wi$ – $Re$ phase space for a series of nonlinear viscoelastic fluids (with the dependence on shear rate $\dot{\gamma}$ quantified using a shear-thinning parameter $\beta _P(\dot {\gamma })$ ). We tune $\beta _P(\dot {\gamma })$ by varying the dissolved polymer concentration in solution. This progressively reduces shear thinning but leads to finite inertial effects before the onset of elastic instability, and thus naturally results in elasto-inertial coupling. Time-resolved rheometric measurements and flow visualization experiments allow us to investigate the effects of flow geometry, and document the combined effects of varying $Wi, Re$ and $\beta _P(\dot {\gamma })$ on the emergence of secondary motions at the onset of instability. The resulting critical state diagram quantitatively depicts the competition between the stabilizing effects of shear thinning and the destabilizing effects of inertia. We extend the curved streamline instability criterion of Pakdel & McKinley (Phys. Rev. Lett., vol. 77, no. 12, 1996, p. 2459) for the onset of purely elastic instability in curvilinear geometries by using scaling arguments to incorporate shear thinning and finite inertial effects. The augmented condition facilitates predictions of the onset of instability over a broader range of flow conditions, and thus bridges the gap between purely elastic and elasto-inertial curved streamline instabilities.

Soft materials can sustain large deformations, and, thus, can be used as a platform for tunable and switchable systems capable of wave manipulations. The fascinating elastic instability phenomenon in soft materials can be used to design new (meta-) materials with switchable microstructures, properties, and functions. Examples include the emergence of tunable band gaps at low-frequency ranges, the appearance of negative group velocity, and extreme wave slowing-down. Here, we investigate the elastic instability phenomenon in soft heterogeneous materials, and the applications of the phenomenon to design of soft acoustic metamaterials. The deformable composites typically combine soft matrix and stiffer phases (such as fibers or inclusions). We will start by considering the effect of experimentally observed wrinkling on elastic waves band-gaps in soft laminates. Next, we will discuss the emergence of the negative group velocity in 3-D fiber composite brought to the “marginally stable” state. Then, we will turn to the so-called auxetic or negative Poisson's ratio materials comprising of soft- matrix-void-stiff-inclusion systems, and illustrate the mechanisms leading to the emergence of low-frequency band gaps. Finally, a new type of instabilities giving rise to anti-symmetric domain formations will be illustrated on the examples of 3-D printed soft composites.

In the present work, a nonlinear coupled electro-magneto-elastic membrane formulation is developed for soft functional materials starting from the variational form of 3D governing equations. The resulting 2D model is applied to an internally pressurized cylindrical membrane in an azimuthal magnetic field and radial electric field. The results of our model are verified with existing literature for some special cases. The model is subsequently used to analyze mechanical and electrical limit-point instabilities and the effect of external fields on the onset of these instabilities. It is observed that the onset of mechanical limit-point instability, defined by a loss of monotonicity in the pressure versus deformation plots, is dependent on the material properties, geometrical parameters, and applied electromagnetic fields. Magnetic field-induced instability results in an initial dip in the pressure versus stretch plots, subsequently converging to the purely hyperelastic membrane behavior at larger stretches. On the other hand, applying an electric field results in an early onset of limit-point instability (i.e., at smaller stretches) compared to the hyperelastic case. In this work, we additionally observe the presence of an electrical limit-point instability, characterized by a loss of monotonicity in voltage versus stretch plots. The dependence of electrical-limit point on magnetic field and force inputs is studied. To the best of our knowledge, this effect has not been discussed in prior literature. Finally, we study the effect of Maxwell stress due to electromagnetic fields. It is observed that ignoring the Maxwell stress related traction boundary term results in an error of up to 10%, depending on the force and magnetic field inputs. In summary, our membrane model describes the interactions between the electromagnetic fields. The electric, magnetic and elastic limit points observed in these membranes under different set of loading conditions can be used towards improving the design of soft actuator and energy-harvesting devices.