Introduction to Plastic Deformation and Viscous Flow
When a material sample is subject to external loading deformations that exceed its elastic limit, or yield strength, the material will cease to deform linearly, elastically and reversibly. Instead, the sample will gradually start undergoing the phenomenon of plastic deformation, otherwise known as yielding. Contrary to the regime of elasticity, plasticity is both non-linear and permanent. This implies that its effects on the material cannot be recovered and undone once the external stress at the source of the deformation is removed. Fundamentally, plastic deformation is the consequence of large number of atoms or molecules moving relative to one another, which results in the breaking of their original bonds and the reforming of new ones. This bond-switching mechanism cannot in fact be energetically reversed upon release of the external load.
In general, the microscopic mechanisms by which such plastic deformation occurs is different for the two cases of crystalline and amorphous materials. Within the former category, plasticity is accompanied by the nucleation and propagation of crystalline defects known as dislocations. In non-crystalline solids on the other hand, plastic flow and deformation unfolds in very much the same way as the flow of liquids. This consists specifically in a viscous flow mechanism. It includes therefore the case of amorphous and semi-crystalline polymers. This is a direct result of the lack of regular atomic structure within this class of materials. Under this regime of viscous flow, the rate of deformation is typically proportional to the applied stress. For the common case where the applied stress consists in shear stress, the atoms will slide next to each other by the breaking and re-forming of interatomic bonds, in a very unpredictable fashion. The figure below illustrates the main features of viscous flow on a macroscopic scale (reproduced from Ref. ). Here, the decreasing velocity of the flow as a function of increasing distance, and therefore decreasing magnitude of the applied shear force, is clearly apparent.
Representation of the viscous flow of a liquid or fluid glass in response to an applied shear force.
The fundamental quantity defining viscous flow is called the viscosity η of the amorphous material or liquid under consideration, which is measured in units of pascal-seconds. Considering the above image, viscosity can be defined as the ratio of the applied shear stress F and the change in velocity dv of the substance with distance dy perpendicularly from the surface A across which the shear stress is applied. Therefore, the higher the viscosity, the higher essentially will be the material’s resistance to the flow induced by such shear forces. Typically, liquids have much lower viscosities than the amorphous materials for which viscous flow is relevant, such as glasses. For example, the viscosity of water at room temperature is only around 10 -3 Pa-s, whereas for solid glass the value at an equivalent temperature can have an order of magnitude as high as 10 16 Pa-s, due to their strong interatomic bonding. For the case of amorphous polymers considered here, the value of the viscosity is of order 10 8. Despite such vastly different orders of magnitude, it is important to highlight that the concept and mechanism of viscosity and viscous deformation apply equally well to both liquids and amorphous materials in their solid state. These values for the viscosity will generally decrease with increasing temperature, as a result of such bonding forces becoming weaker under thermal effects, and the flow of atoms thus being made easier.
Viscoelastic Deformation of Polymeric Materials
The mechanical behaviour of amorphous or semi-crystalline polymeric materials may be summarized as follows, depending on whether the material sample under consideration is above or below its glass transition temperature Tg and melting temperature Tm.
- Brittle elastic glass: below Tg
- Rubber-like behaviour: above Tg and below Tm
- Viscous liquid (melt): above Tm
Under the second intermediate temperature situation listed above, the polymer essentially behaves like a rubbery solid, under a regime known as viscoelasticity. Under such conditions, the sample will exhibit a combination of the mechanical response properties of both the low-temperature elastic regime, and high temperature liquid-like viscous behaviour. Hence the combined label of “viscoelasticity”.
Contrary to the instantaneous nature of elasticity, within viscous mechanical flow there is always a significant time-delay and time-dependence between application of an external loading force to the sample, and its subsequent deformation. Moreover, contrary to elasticity, viscous deformations cannot be fully recovered and reversed upon release of the external stress. It follows that in the intermediate regime of viscoelasticity, the overall response of the sample to an external load is a combination of these two behaviours: an initial instantaneous elastic strain followed by viscous and time-dependant deformation. These three different classes of mechanical response properties are summarized in the figure below, for the case of the same applied loading stress profile shown in red in the first part of the figure (figure reproduced from Ref. ):
(a) Load versus time, where load is applied instantaneously at time and released at For the load–time cycle in (a),
the strain-versustime responses are for totally elastic (b), viscoelastic (c), and viscous (d) behaviors. 
A classical example of viscoelastic material, which everyone can experience, is human skin tissue: when pinched, it usually takes some time to recover its original shape. Moreover, the longer the skin is pinched for, the longer it will take to recover, and conversely the more rapidly it is pinched, the less time it takes to recover. This latter case highlights another important aspect of viscoelasticity in general: its dependence on the applied stress-rate. Increasing the stress loading rate has in fact the same overall effects as lowering the temperature of the sample, by making it more elastic and less viscous.
Viscoelasticity Experimental Measurements – The Relaxation Modulus
The viscoelastic behaviour of polymeric materials can be investigated and characterized experimentally in a number of different ways. One such method consists in stress relaxation measurements. Under this approach, the polymer sample is firstly strained rapidly in tension to some fixed strain value, whilst its temperature is kept constant. The experiment then consists in measuring the stress required for maintaining this strain level across the polymer sample at its fixed and predefined value, which typically decreases with time owing to the time-dependent nature of viscoelasticity and its associated molecular relaxation processes. The ratio between this measured time-dependent stress profile and the fixed predefined value of the strain is called the relaxation modulus of the viscoelastic polymer. This quantity therefore plays the same role as the conventional elastic Young’s modulus in linear elasticity, but this time in the context of time-dependant and partially-inelastic mechanical response properties. The relaxation modulus is also typically temperature-dependant, decreasing with increasing temperature as the material becomes correspondingly softer.
Another time-dependant mechanical phenomenon which affects numerous polymers is that of viscoelastic creep. Under these circumstances, the polymer sample will gradually undergo a time-dependant deformation, when this time it is the externally-applied stress level which is fixed and predefined. A classical example of this phenomenon is when car tyres develop flat spots, when cars are left parked over long periods of time on hot asphalt surfaces during warm summer days. Similarly to the afore-mentioned relaxation modulus, we may therefore define also the time-evolving and isothermal creep modulus, as the ratio between the fixed applied stress and the resulting time-dependant strain affecting the polymer. This quantity also decreases with increasing temperature of the sample.
Computation of Polymer Viscosity
The viscosity and viscoelastic properties of polymeric materials, in both their solid state and liquid melt, can be computed through simulation techniques such as classical Molecular Dynamics (MD), as implemented for example by the LAMMPS code. MD is a computational technique by which the atomic trajectories of a system of numerous particles are generated by numerical time integration of Newton's classical equations of motion, within a given statistical ensemble. Classical MD codes have in fact proven to be effective in determining both the molecular structures and physical/chemical properties of polymer-based materials. This depends of course on a judicious choice for the interatomic potential, otherwise known as force field, governing the conduct of atomic motion and interactions during MD simulations.
The shear viscosity η of a fluid or viscoelastic amorphous polymer can, in general, be computationally evaluated in multiple alternative ways using the various options in LAMMPS. The main method implemented within the LAMMPS-based solutions offered by the Materials Square platform is based on the Green-Kubo (GK) formulae. The GK relations are a set of equations that relate the transport coefficients of thermal and mechanical transport processes in terms of integrals of different time correlation functions. The equations are based on the expectation that a flux will be linearly proportional to an applied field. This approach applies for example for the cases of electrical or thermal transport processes, where the field is given by the applied electric field and thermal gradient respectively. In our case, the shear stress is linearly proportional to the strain rate, and the proportionality constant is given by the viscosity itself. Under the GK method, the viscosity computation can in this way be accomplished through a fully equilibrated MD simulation, which is in contrast to the other alternative non-equilibrium methods for viscosity calculation described in what follows. In summary, there are basically three main steps involved in the GK calculation of the viscosity of a system. Firstly, there is the evaluation of the time correlation function. This is followed by the summing over all three dimensions, and finally the result is averaged over different time origins. For a more detailed review of the GK method, as well as some examples of its applications for viscosity computation via equilibrium MD simulations, the reader is referred to Ref. .
An alternative method is to perform a non-equilibrium MD (NEMD) simulation by shearing and deforming the simulation box. For example, one or more moving walls can be implemented to shear the fluid in between them. The velocity profile established within the fluid through this procedure can then be computed, together with the relevant component of the shear stress, from which the value of the viscosity can finally be evaluated.
A final method available for viscosity computation using MD techniques consists in the periodic perturbation method, which is also a non-equilibrium MD method. However, instead of measuring the momentum flux in response to an applied velocity gradient, it measures the velocity profile in response to an applied stress directly.
For complete instructions on how to compute the viscosity using LAMMPS, the reader is referred to the relevant page on the code’s official documentation manual, which can be found under Ref. .
Industrial and Commercial Applications of Viscoelastic Polymers
Viscoelastic polymers, and viscoelastic materials in general, are appreciated in real commercial applications particularly for their ability to creep, undergo stress relaxation and absorb energy upon impact. One example of their application is polymer foams used in seat cushions or matrasses. As a result of their creeping process, these cushions will progressively adapt and conform to the body shape, thus making them ergonomic and comfortable. However, after some months they typically have to be replaced, as a result of becoming stiffer with time.
An additional example of use of viscoelastic polymers are guitar strings, which are typically composed of Nylon. Normally, such guitar strings will go out of tune with time when they are left at constant strain and length, due to stress relaxation. This reduction in tensile stress in fact implies a decrease in the pitch and frequency of the string. Of course, the solution to tune the strings again is then to tighten them, thus increasing their tensile stress. Some thicker guitar strings can also be made of metallic materials instead of Nylon, and these metallic strings typically do not go out of tune as easily, as a result of metals being on average less viscoelastic than polymers.
Furthermore, viscoelastic materials and polymers are often employed as dampeners in the construction of skyscrapers, when such tall buildings are for example subject to significant vibrations caused by wind or earthquakes. When acting as dampeners, viscoelastic materials have in fact the capacity to absorb such vibrational energy. Another application of viscoelastic polymers as vibrational dampeners is in the insultation of helicopter fuselages from the noise made by the main rotor.
Viscoelastic materials are also very useful as impact absorbers, reducing the impact force of a projectile on a surface, for instance, by up to a factor of two compared to a purely elastic coating. A typical example of such highly viscoelastic polymers employed as impact absorbers are the elastomers, of which there is a great synthetic variety (e.g. Sorbothane, Implus and Noene), although on average they all behave similarly to natural rubber. As a result of such impact-absorbing abilities, viscoelastic polymers are for example employed in car bumpers, as protection barriers in race tracks and highways, on computer drives to protect them from shock, as internal padding in military helmets, and in sports equipment such as protective mats (in case of fall) and shock-absorbing running shoes.
: “Materials Science and Engineering” (10th edition). William D. Callister, David G. Rethwisch. Wiley, 2020 (https://fac.ksu.edu.sa/sites/default/files/ch15.pdf )
: LAMMPS viscosity computation manual page (https://docs.lammps.org/Howto_viscosity.html)
: “Examples and Applications of Viscoelastic Materials”, Auckland University Engineering Department (http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_I/BookSM_Part_I/10_Viscoelasticity/10_Viscoelasticity_02_Examples.pdf)
: “Introduction to Polymer Viscoelasticity”, Montgomery T. Shaw and William J. MacKnight, Wiley-Interscience, 3rd edition (2005) (https://structures.dhu.edu.cn/_upload/article/files/f6/62/f5c6159f4c86ae7a86fbd6b48811/e1e8047e-a204-4e77-9d90-f243755b1a50.pdf)
: “Introduction to the Viscoelastic Response in Polymers”, María L. Cerrada (https://ruc.udc.es/dspace/bitstream/handle/2183/11487/CC-80%20art%2011.pdf;sequence=1)
: “Accurate computation of shear viscosity from equilibrium molecular dynamics simulations”, D. Nevins and F.J. Spera (https://magma.geol.ucsb.edu/papers/mol_sim_NaCl_viscosity.pdf)
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