The equivalence of linear and stochastic models
Viscoelastic materials are widely used. All plastics are viscoelastic to some degree. For athletes the viscoelastic soles of their trainers cushion the impacts of running and viscoelastic mattresses can offer them a better sleep.
Viscoelastic analysis is a highly developed theoretical approach to understand the properties of these shockabsorbing materials. Also, the concepts can be applied more generally as an analysis of a diverse group of analogous physical systems that dissipate energy. Hence viscoelasticity can provide sound foundations upon which to build the foreground content of this website.
The standard linear viscoelastic solid
Whenever energy is expended to deform a viscoelastic material, this energy is either stored within “elastic” components of the system, or dissipated away in other forms of energy such as heat by “viscous” components. In the suspension of a car, the springs are the elastic components whilst the shock absorbers fulfil a viscous function.
In conventional methods for characterising viscoelastic materials the spring is an elemental model with an ideal elastic behaviour. Springs are conservative as the stored strain energy of deformation will be used, eventually, to restore the spring to its original, unloaded state. The spring can say nothing about energy dissipation, which is beyond the conception of the simple spring model.
To include energy dissipation, conventionally, requires a viscous dashpot. Picture a pot of thick oil, sealed above with a plunger in which there is a small hole. Push down onto the plunger and slowly oil seeps out allowing sinkage. This is the action of the dashpot. As the applied force increases so does the velocity of the deformation. Dashpots are wholly wasteful. All energy is dissipated and all deformations are permanent.
Models of viscoelastic behaviour are constructed from these elemental springs and dashpots and their configuration and individual properties of stiffness and viscosity are set to simulate a specific balance of energy storage and dissipation. An example of a combination of individual elements which combine to realise an ideal viscoelastic mechanical behaviour is shown in the figure. This is the Standard Linear Viscoelastic Solid and its behaviour can be compared with the empirical, physical world to better understand the underlying processes of energy storage and dissipation operating therein. 
The Standard Linear Viscoelastic Solid 
The Standard Linear Viscoelastic Solid in above comprises an elastic spring with stiffness kv connected in series with a viscous dashpot with a viscosity cv. If a constant force F were to be applied to this top combination alone then, because they are in series, F will act through the spring causing it to extend and will also act through the dashpot resulting in a progressive extension for as long as the force is applied. Thus, the top series arrangement alone could have an elongation which could continue ad infinitum. This is, in fact, the behaviour of a viscoelastic liquid.
To confer the properties of a solid onto the arrangement, a second spring ke is placed in parallel with the series arrangement of kv and cv. The constant force F will now be shared between the component parts of this parallel assembly. In the series combination of kv and cv the deformation will continue to extend with time as described above, albeit at a slower rate as only a portion of F will now act through this upper strand of the parallel model. As the extension continues, the spring ke will become more deformed and thus will take an increasing share of the applied force. Ultimately, a point will be reached when the whole of the constant force F will be carried by the spring ke, the earlier extension of spring kv will have been totally transferred to its connecting dashpot cv so that no further force will act through this series and the dashpot will then also be inactive.
The stochastic linear viscoelastic solid
A large number of models and forms of analysis have been developed to understand the properties of viscoelastic materials.[1] However, one important consideration has led us to consider an alternative stochastic formulation. Whilst at a macroscopic scale viscoelastic behaviour might appear smooth and continuous, on a more fundamental level the transfer of stored energy into the dissipative processes should be determined by the Principle of Least Action. This energy transfer may therefore be associated with discrete events, for example microfractures within a material’s microstructure. The origins of this line of thinking were to understand the properties and failure mechanisms of biological tissues, all of which are highly viscoelastic and nonlinear[2]. It is important, however, to connect the continuous and stochastic analyses of linear viscoelasticity so that the latter may benefit from the former.
A stochastic alternative to the 
The figure here shows such a stochastic viscoelastic assembly of fundamental mechanical units in which an elastic spring of constant stiffness ko is connected to a linkage that will fail when the energy stored in the connecting spring reaches a critical value e_{1} , e_{2} , ……… , en. These values can be different numerically. Therefore, as this parallel assembly begins to be loaded, some of the fundamental mechanical units will fail earlier than others, ceasing then to be active. On subsequent loading of the assembly, the additional applied energy will be distributed only to those remaining active fundamental units, which will still contribute to the elastic behaviour. Those units that have failed will temporarily contribute a viscous force, until this contribution decays to zero.

If the load and the deformation of the above parallel assembly were to be increased continuously, a point would be reached when the last mechanical unit will have failed and the initial parallel assembly would then cease to be an integral mechanical entity able to support an applied load. However, in a generalised mechanical model, those fundamental units which earlier have failed may also have initiated the recruitment of additional units, similarly collected into additional parallel assemblies, which will continue to contribute to the load carrying capacity of the model.
For the failure of the linkage in the fundamental mechanical units above it can be assumed that inertial effects are irrelevant and the Principle of Least Action then translates into a principle of minimum stored potential energy. The system will configure itself to minimise the total potential energy stored within the spring and the linkage, so that at a critical point this criteria is satisfied by the mechanical failure of the linkage[3]. In the case of a fracture, the same threshold condition is reached where the potential in the fracture surfaces is less that the strain energy held within the deformed but intact material. In these cases the event of failure occurs at a threshold that is governed by the Principle of Least Action.
In a paper entitled “A New Look at Linear Viscoelasticity” it is shown that a precise simulation of the Standard Linear Viscoelastic behaviour occurs with a stochastic model which the number of fundamental units which fail in each assembly in the time period (t , t + t_{r }) is given by the recursive formula:
number of failures between time t and (t + tr) = f .n_{a}(t) . n_{a}(0)
where n_{a}(t) is the number of units in the particular assembly which are active at time t, n_{a}(0) is the initial number of units when that same assembly was first formed and f is a constant. Therefore, the number of units in each parallel assembly which fail during each time interval follows a sequence that is dependent only on the initial and current numbers of active fundamental units and not on any applied deformation.
This may be a special and possibly unique solution to achieve this identical matching of the standard and stochastic models of linear viscoelasticity.
Notes:
[1] Ferry, J.D. Viscoelastic Properties of Polymers. New York: John Wiley & Sons Inc., 1970.
[2] Egan J.M. A constitutive model for the mechanical behaviour of soft connective tissues. J. Biomech. 20: 681692 1987.
[3] Extending a spring tethered to the ground through a magnetic linkage gives another physical example of this general phenomenon that change occurs at a critical threshold of energy input.
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