Fig. (1): Standard Linear
Visco-Elastic Solid
Some of this information has now been published in:-
Egan JM. A new look at linear visco-elasticity. Materials Letters 31: 351-357 1997.
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The aim here is to introduce a novel graphical insight into the mechanical behaviour of a visco-elastic material. It is hoped that this graphic fingerprint of a mechanical behaviour may be a useful descriptor, or at least offer a different look at a material, from that found in empirical plots of force and deformation behaviour.
The mechanical behaviour of a material is implicitly determined its ability to store or dissipate energy applied during a deformation. Linear elastic springs and viscous dashpots are fundamental components which can be assembled to model these processes of energy storage and dissipation and thus simulate a mechanical behaviour for a visco-elastic material.
Combinations of springs and dashpots have been used to simulate specific mechanical behaviours and linear visco-elastic theory has been widely developed and utilised in a broad range of rheological applications (1,2 Refs.).
Here an alternative idea, which will be referred to as the enet model, is presented which differs from linear visco-elasticity in one important aspect. In linear visco-elasticity, dashpots dissipate strain energy in serially connected springs as a continuous process. In the enet model, this dissipation occurs only at critical points of energy input as defined by failure criteria of enet elements which replace the dashpots in the mechanical network.
A highly specific form of the enet model has been found to match precisely the mechanical properties of the standard linear visco-elastic solid. This enables their relationship to be explored. Furthermore, the enet model enables a new graphical template to be created which gives a visualisation of the 'strategy' a material deploys to handle applied mechanical energy. The technique has been used to investigate the mechanical behaviour of the standard linear visco-elastic solid and its characteristic linearity is clearly evident in a graphical template obtained using the enet model.
The standard linear visco-elastic solid as shown in figure (1) is an elementary combination of a spring with stiffness ke in parallel with a series combination of a spring with stiffness kv and a dashpot with a co-efficient of viscosity cv . It provides the most general linear visco-elastic relationship to include load, deformation and their first derivatives (2 Refs. ).
Fig. (1): Standard Linear
Visco-Elastic Solid
When the standard linear visco-elastic solid is loaded, both springs will be extended and will combine their forces to give a typical force-deformation behaviour. With time, the strain energy in spring kv will be dissipated through its connecting dashpot, so that the force contributed by this arm of the standard solid will decay. As a consequence, if the deformation of the model were to be conducted infinitely slowly allowing time for the dashpot to completely 'relax' the force in spring kv , the resulting force will be that due only to spring ke and this is termed the 'equilibrium force' .
When the standard linear visco-elastic solid is subject to a constant rate of extension, the concept of superposition can be employed to calculate the force F necessary to reach a deformation x at time t thus :-
Equation (1) reveals the total force-deformation (F - x) characteristics and equilibrium force-deformation characteristics (Feq - x) for the standard linear visco-elastic solid where Feq = ke . x
The fundamental mechanical unit of the enet model comprises an elastic spring connected in series with an enet element which remains dormant mechanically until the deformation in the connected spring reaches a critical value at which point mechanical failure occurs. The enet model initially is constructed as a parallel array of pi fundamental mechanical units in which the springs each have the same stiffness ko and a variable enet failure deformation e1, e2, ........., en (figure (2)).
Fig. (2): Enet Model Parallel Array
The fraction of enet elements in a loaded parallel assembly which fail may be calculated as a function of the applied deformation or strain energy. Later a specific pattern of enet failure will be described which enables the model to match the mechanical behaviour of the standard linear visco-elastic solid.
An individual enet failure is associated with two subsequent events. The force in the connected spring then decays slowly in a manner comparable with the standard linear visco-elastic solid. That is, after enet failure:-
where Fv is the subsequent contribution made to the viscous force at time t by the failed fundamental unit, Ffl was the maximum force occurring at time tfl when the unit failed.
Also, in response to a fundamental unit failure, a variable number qs additional fundamental units are recruited into a load carrying capacity. These recruited units amalgamate into additional parallel assemblies which behave just as their precursors as the deformation is continued.
The following graphic displays an example of the behaviour of the enet model with qs=1, that is when one additional fundamental mechanical unit is recruited in response to each unit failure. The force due to the failed units will decrease with time and contribute temporarily to the stress relaxation behaviour of the model, whilst the force contributed by those springs with an intact enet element comprise the elastic equilibrium behaviour.
This sequence of fundamental unit failure and recruitment is suitable for an iterative algorithmic approach. Following this algorithm, with a time increment of tr , at any time t the enet model will contain t / tr parallel assemblies of fundamental mechanical units each contributing a viscous force from those units which have failed and an equilibrium force when the enet elements remain connected. A computer model may thus calculate a (F - x) and (Feq - x) behaviour which can be compared directly to measured properties of a material or to the equivalent behaviour of the standard linear visco-elastic solid given by equation (1).
The iterative process of fundamental mechanical unit loading, failure and recruitment can rapidly lead to a numerical model of considerable complexity. It is useful, therefore, to visualise the workings of the spring-enet model graphically and a new template has been devised to do this.
In this graphical representation of enet model behaviour shown in figure (3), the forces contributed by the active and failed fundamental units are considered separately above and below a horizontal axis. Concentric circles mark the iterative intervals tr . The failure and recruitment patterns in each parallel assembly of fundamental units are represented by lines drawn onto the template. The angle of the line specifies the proportion of the assembly which has failed; a horizontal indicating no failures have occurred, a vertical line indicative of the complete failure of that assembly.
Fig. (3): Pattern of enet failure
and recruitment
Therefore, in figure (3), the proportion of the initial assembly which fail at the first time increment tr , is indicated by the angle of line a2 . The recruited second assembly is then represented by b1 and the subsequent failure of parts of the initial and second assemblies of units which occurs at the second time increment 2.tr are indicated by the angle of lines a3 and b2 while c1 represents the consequent further recruitment of units to form a third parallel assembly. As the simulation advances, further lines will be added as additional assemblies are recruited, and those earlier lines will gradually steepen as more of their constituent units will fail.
Furthermore, the colour of each line in this new type of picture can be used to register the force contributed by the assembly of fundamental units that the line represents. The pattern above the horizontal axis will represent the forces from the active units which together make up the elastic equilibrium behaviour of the model, while below the axis the contribution to the decaying viscous force of the failed fundamental units is recorded.
The computed (F - x) and (Feq - x) behaviour of the enet model has been compared with that of the standard linear visco-elastic solid using a simplex-based Gauss least-squares method to minimise the difference between the two sets of data. Comparable values of the model parameters can thus be estimated at the point of best fit. The closeness of fit between the two models has been measured using the square of the co-efficient of correlation R-squared (3 Refs. ) on both the (F - x) and ( Feq - x) measurements.
A precise simulation of the linear visco-elastic behaviour has been achieved using an enet model in which the number of fundamental units which fail in each assembly in the time period (t , t + tr ) is given by the recursive formula:-
where na(t) is the number of units in the particular assembly which are active at time t, na(0) is the initial number of units when that same assembly was first formed and d is a constant.
Therefore, the number of units in each parallel assembly which fail during a time interval follows an inductive sequence being dependent only on the initial and current numbers of active fundamental units and not on any applied deformation.
The enet model incorporating the enet element failure criteria given in equation (3) provides a (F - x) and (Feq - x) behaviour which has been fitted to comparable data from equation (1) for the standard linear visco-elastic solid. As the time interval tr decreases, this fit improves to a level at which the two models may be considered identical. For values of tr < 0.25 secs. over a 10 second simulation the R-squared value is greater than 0.99995.
At this point of convergence of fit, the estimated model parameter values become equivalent. That is, pi is such that the initial stiffness of the enet model ( pi.ko ) equals ke + kv , the values of tor for each model are the same and qs is made equal to unity in the enet model ensuring a constant number of active fundamental units and a constant stiffness throughout the simulation.
The fourth enet model parameter d, present in equation (3), defines the level of enet failure and is equivalent to the degree of viscosity in the standard linear solid, which is itself determined by the ratio of the two spring stiffnesses kv / ke . As this ratio increases, the model shown in figure (1) behaves in a more viscous manner as the dashpot is able to dissipate more of the applied energy. A higher level of enet failure is needed to replicate this increased viscosity and thus d is proportional to kv / ke as shown in figure (4).
Fig. 4: Relationship between the
enet and standard visco-elastic models
The rather simple rule in equation (3) defining a sequence of fundamental unit failure and recruitment does lead quickly to a complex numerical pattern describing the composition of a string of parallel assemblies which develop as the sequence and deformation proceed.
The equivalent enet graphical template presents a more comprehensible pattern. As the enet model is loaded, the lines sweep outward and away from the horizontal, demonstrating the failure of fundamental units. The colours show the elastic forces in each assembly rise from blue, through yellow, to a maximum red value and then fall as the proportion of units to have failed in the assembly becomes the majority. (That is, the angle the line makes to the horizontal axis exceeds 45 degrees)
Fig. 5: Linear Enet Model Pattern with
tr = 0.02 secs (10 sec simulation)
When tr is large the lines can be individually traced. However, as the time increment is shortened as in figure (5), the lines blend into a more continuous and distinctive pattern. Above the horizontal axis, the colour bands which indicate the contributions made to the elastic equilibrium force radiate outwards, linearly on the circular template. It is important to note that the individual lines which comprise this pattern are not linear but cut across the coloured linear boundaries.
At each time increment, that is, along every circular segment drawn above the horizontal axis, the colours retain a constant angular relationship. As the deformation proceeds, there appears to be a constant composition of assemblies which contribute at the maximum red level and those, formed both earlier and later, which add to the equilibrium force to a lesser degree.
The value of any constitutive model must be determined by the insight the model provides into the material it is set to represent. The patterns on the graphical template derived here may be descriptive of the strategy a material employs to handle the energy of deformation and it is hoped that the concept may provide a useful tool in the analysis of structure-function relationships in the material.
The scope of the present paper however, is limited to forging a link between the enet model and the theoretical foundations of linear visco-elastic theory. This is convenient because the behaviour of the standard linear visco-elastic solid is free from noise and any secondary and tertiary influences which combine in the mechanical behaviour of a real material. Therefore, a perfect matching of the linear visco-elastic and enet models has proved possible and at this point of perfect fit, the graphical template obtained from the enet model does show a characteristic linearity consistent with its theoretically linear behaviour.
Various combinations of spring and dashpot components have been used to enable linear visco-elastic theory to simulate the mechanical behaviour of specific materials, although a precise representation often requires the extension of this model to infinite arrays of components as defined through the relaxation spectra of linear visco-elasticity (1). Also, it has been shown that linear visco-elasticity does provide a generalised representation of thermodynamic phenomena occurring within the vicinity of an equilibrium (4 Refs. ). However, non-linear materials, such as polymers and biological tissues, do require non-linear constitutive formulae to characterise their mechanical properties (5 Refs. ).
In linear visco-elasticity the conditions for energy dissipation are narrowly defined by the assumed Newtonian behaviour of the dashpot: that its rate of deformation is linearly proportional to the applied force. This has necessitated the incorporation of complex non-Newtonian dissipative mechanisms into visco-elastic theory (1 Refs. ).
This tightly restricted behaviour of the standard linear visco-elastic solid is matched by a highly specific definition of enet failure criteria in the matched model given in equation (3). Other formulae and statistical failure criteria (uniform, Gaussian, exponential, Weibull and gamma) have been applied without convergence to the properties of the linear visco-elastic solid, leading the author to believe that equation (3) represents a special and possibly unique solution to achieve this identical matching.
What is surprising is that the failure criteria defined by equation (3) are independent of deformation and include only an inductive numerical sequence to control the enet failure events.
Rowse (6 Refs. ) used a 'bead-spring' model to characterise the interactions which occur along a flexible polymer chain and thereby derive a mechanical behaviour. This stochastic model can similarly be made analogous to a linear visco-elastic behaviour (1,7 Refs. ). The enet model uses a similar approach to characterise the conditions under which strain energy stored within the elastic structures of a loaded material is transferred or dissipated.
This transfer of energy from elastic structures into whatever dissipative mechanism exist within a material must act under the constraint of Hamilton's principle to reduce the total potential energy in the system. If this energy transfer occurs with a disruption of some interactions within the material micro-structure, then it is likely that such interactions will fail in a stochastic manner when loaded to a critical value (8 Refs. ).
The methodology described here has also been applied to characterise non-linear visco-elastic behaviour. Non-linearities may be introduced into the numerical model either by associating statistical distribution to the failure characteristics of the enet population, or by allowing the recruitment parameter qs to take a value other than unity. This latter step may reflect a gradual recruitment of load bearing components within a material micro-structure as is understood to occur when biological tissues are deformed. The graphical template may then provide a useful tool to understand the energy dependent consequences of these micro-structural non-linearities.
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(2) Fung, Y.C. A First Course in Continuum Mechanics. New York: Prentice-Hall Inc., 1977.
(3) Weatherburn, C.E. A First Course in Mathematical Statistics. Cambridge: Cambridge University Press, 1968
(4) Biot, M.A. Theory of stress-strain relations in anisotropic visco-elasticity and relaxation phenomena. App. Phys. 25: 1385-91 1954.
(5) Lockett, F.S. Nonlinear Visco-elastic Solids. London: Academic Press, 1972.
(6) Rouse, P.E. A theory of linear visco-elastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21: 1272-1280 1953.
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