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A New Look at Linear Visco-Elasticity
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A mechanical model which simulates a sequence of micro-failures on loading has been able to replicate precisely the time-dependent mechanical behaviour of a standard linear visco-elastic solid (1). This linear visco-elastic behaviour requires a highly specialised sequence of failure and recruitment of spring-enet mechanical units which are used to incorporate the micro-failure concept into the model (figure (1)). In particular, when one spring-enet unit fails exactly one additional unit is recruited into a load-carrying capacity in order to produce a linear visco-elastic response. This recruitment can be defined by a spring-enet recruitment index qs giving the number of new spring-enet units recruited for each individual enet failure. Then an essential requirement for linear visco-elasticity is that this recruitment index is equal to unity.
Figure 1
A principle feature of the spring-enet model is that the isolated event of failure can be used to introduce associated reactions into the model which are not possible with conventional spring-dashpot alternatives in which energy dissipation occurs as a continuous process with no such discrete events. For example, the number of newly recruited units with each enet failure, as given by the recruitment index, can be varied to introduce a non-linearity into the original linear model. This non-linearity is the subject of this paper.
When the spring-enet recruitment index is greater unity, the linear micro-failure model turns into a liquid of seemingly infinite viscosity. With a value for this same index which is less that unity, the mechanical model appears to behave as an elastic solid of zero stiffness.
Therefore, it seems that the case of linear visco-elasticity for the micro-failure model is a highly specialised condition which sits between these infinitesimally elastic and infinitely viscous states.
The spring-enet constitutive model has been described in detail elsewhere (1), (2) and brief summary of the concept will be given here.
The mechanical model comprises a parallel array of pi spring-enet units which can fail and recruit new units. Typically this failure and recruitment may occur at some critical values of deformation or strain energy. As shown in figure (1), for a first time increment tr there exists a single parallel assembly of mechanical units and a portion of these units will fail as a load is applied in this first increment. Newly recruited units form a second parallel assembly and the subsequent failure of parts of the initial and second assemblies of units during the second time increment leads to the creation of a third parallel assembly of spring-enet units. This sequence continues with a newly recruited assembly of load carrying units formed at each time increment of the loading period in response to all those units that fail in the preceding interval. The non-failed or 'active' units together contribute an elastic equilibrium force, whilst the failed units temporarily contribute a decaying viscous force to comprise a visco-elastic mechanical behaviour.
Now, for the linear visco-elastic model, the number of newly recruited mechanical at any time must equal the number of unit failures, so that the total number of active units remains constant. Also, importantly, it is neither strain energy nor deformation which determines when the enet elements fail, but the portion of enets to fail a each time increment follows a simple recursive rule:-
enet fraction to fail in each assembly at time t = (d / pi) . na(0) . na(t) ............eqn. (1)
Where na(t) is the number of active units in that particular parallel assembly at time t, na(0) was the number created when the assembly was first formed and d is a constant which is between 0 and 1.
The sequence of mechanical unit failure and recruitment can be displayed as lines drawn onto a circular template in which a gradual steepening represents an increasing failure of units in the parallel assembly the line represents. The colour of each line is used to register the force contributed by the parallel assembly. The pattern above the axis represents the elastic equilibrium behaviour of the model, while below the decaying viscous force is recorded (1).
The spring-enet model behaviour given above produces a time-dependent mechanical behaviour (figs. (3b), (4b)) which precisely fits a standard linear visco-elastic behaviour as the time increment tr shrinks to zero. This convergence is demonstrated in figure (2) which shows the fit of the spring-enet model working with equation (1) to the equivalent mechanical behaviour of a standard linear visco-elastic solid.
Figure 2
In figure (2) the mechanical behaviour of the spring-enet model, comprising both the total force and the elastic equilibrium force is fitted to the equivalent behaviour of a standard linear visco-elastic solid. The precise characteristics of this solid is a 50 N/mm elastic spring in parallel with a series arrangement of a second 50 N/mm spring and a linear dashpot with a relaxation constant of 10 seconds (1). Both are deformed at a rate of 1 mm/sec for ten seconds. The maximum combined force and equilibrium force data is normalised to a unity value and the average root mean square of the difference between the spring-enet model and standard visco-elastic model through the 10 second virtual deformation is calculated as the spring-enet model time increment tr is reduced. Note that difference between the two model simulations approaches zero as tr becomes small.
For the linear model above the value of the recruitment index qs must be unity. Making qs take a value other than unity introduces a non-linearity into the model. To observe the effects of this change we will examine a case when the linear spring-enet model with an initial stiffness of 100 N/mm (pi = 1 and d = 0.346) is 'deformed' at a rate of 1mm/sec for 10 seconds and consider how the total force and the equilibrium force component of the visco-elastic behaviour varies with qs.
For values of qs which are less than unity, the total force and equilibrium force behaviour is in shown in figure (3a) and the corresponding graphical template is shown in figure (4a) for the case when qs = 0.8 and tr = 0.05 seconds.
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Figure (3 a,b)
Comparable results for values of qs which are greater than unity are shown in figures (3b) and (4c) for the case of qs = 1.2 and tr = 0.05 seconds. Here figure (4b) shows the linear case where qs = 1.0 and the white area in (4c) is where the viscous force exceeds the maximum red limit set for the elastic forces.
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Figure (4 a,b,c)
To understand the spring-enet model properties as this matches the behaviour of a standard linear visco-elastic solid, it is necessary to consider the effect as the time increment tr converges to zero. To do this, the total force and equilibrium force values at the end of the 10 second model simulation above are followed as tr is reduced for qs = 0.5 in figure (5a) and qs = 1.1 in figure (5b).
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Figure (5 a,b)
A fundamental weakness of the approach employed here is that the time increment tr must take some finite value for practical computing purposes, so that the spring-enet model cannot be taken to full convergence to its equivalent linear visco-elastic behaviour. With the smallest time increment of 0.001 seconds reported here, there were 10,000 assemblies of spring-enet units contributing to the mechanical behaviour at 10 seconds. Despite this, figure (5) shows the model behaviour is some distance from its convergent limit. There must be an element of doubt as to what will occur as the value of tr moves closer to zero. However, if one may extrapolate the trends which appear in figure (5), the observations to show the following:-
For the case of qs < 1: Both the total force and equilibrium force are reduced from their unity value. As tr is reduced, these force values fall further and the viscous force, which is the difference between the total and equilibrium forces, decreases in magnitude. The limit to this reduction appears to be a case where the total force and equilibrium force are equal. There is no viscous force, so that the model in this limit is behaving as an elastic solid. However, this convergence appears as a low and possibly zero force value indicating an infinitesimal elastic stiffness.
For the case of qs > 1: The total force increases from its unity value, slowly at first up to a point where the growth of force becomes enormous, rapidly approaching infinite values. At this point the equilibrium force falls towards zero. This critical point is clearly apparent in the graphical pattern in figure (4c). As tr is reduced, the point at which the viscous force is launched to extremely large values occurs closer to the start of the simulation. The limit of this trend appears to be where the total force comprises only an infinite viscous force at the start of the simulation. There is no elastic equilibrium force, so that the model in this limit is behaving as a liquid of infinite viscosity.
The above observations are inevitable consequences of equation (1). For qs < 1 there will be a gradual diminution in the total number of active units as the failure and recruitment sequence proceeds.
For qs > 1 there will be an increase in the total number of active units during the deformation process. A point will inevitably be reached when (d/pi).na(0).na(0)>1, whereupon all newly recruited units will fail during their first increment of loading. There will be an exponential growth in the number of these soon-to-fail units which is precisely what is seen at the critical point of figure (4c).
There is something special about equation (1) in that it appears to provide a linear visco-elastic behaviour at the zero limit of the spring-enet model time increment tr. In this case, the model shows that a linear visco-elastic behaviour is only possible for a unity value of the recruitment index qs . On either side of this special value are mechanical behaviours which do not resemble any physical phenomena. This may simply be a trick of numbers or it could reflect some real constraint to which physical linearly acting systems are subject.
1. A New Look at Linear Visco-Elasticity
See also: Egan J.M. Mat. Letters 31: 351-357, 1997.
2. Egan J.M. A constitutive model for the mechanical behaviour of soft connective tissues. J. Biomech. 20: 681-692 1987.