Non-linear elasticity

Fig. 8.2. Stress-strain behoviour for a non-linear elastic solid. The axes are calibrated for a material such as rubber.

Figure 8.2 shows a non-linear elastic solid. Rubbers have a stress-strain curve like this, extending to very large strains (of order 5). The material is still elastic if unloaded, it follows the same path down as it did up, and all the energy stored, per unit volume, during loading is recovered on unloading - that is why catapults can be as lethal as they are. 

Other ideas proposed to explain the 3/4 power-law dependence include effects due to viscoelasticity, non-linear elasticity, partial plasticity or yielding, and additional interactions beyond simply surface forces. However, none of these ideas have been sufficiently developed to enable predictions to be made at this time. An understanding of this anomalous power-law dependence is not yet present. 

An alternative energy approach to the fracture of polymers has also been developed on the basis of non-linear elasticity. This assumes that a material without any cracks will have a uniform strain energy density (strain energy per unit volume). Let this be IIq. When there is a crack in the material this strain energy density will reduce to zero over an area as shown shaded in Fig. 2.65. This area will be given by ka where )k is a proportionality constant. Thus the loss of elastic energy due to the presence of the crack is given by... 

R.W. Ogden, Non-Linear Elastic Deformations, Halsted Press/Wiley, New York, 1984 Dover Publications, Mineola, NY, 1997, p. 227. 

It is likely that most biomaterials possess non-linear elastic properties. However, in the absence of detailed measurements of the relevant properties it is not necessary to resort to complicated non-linear theories of viscoelasticity. A simple dashpot-and-spring Maxwell model of viscoelasticity will provide a good basis to consider the main features of the behaviour of the soft-solid walls of most biomaterials in the flow field of a typical bioprocess equipment. 

FENE Finite extensible non-linear elastic potential... 

Martinson, Non Linear Elastic Effects in the Ultrasonic Assessment of Cumulative Internal Damage in Filled Polymers , JApplPhys 50 (12) (1979), 8034-37... 

Tatara analysis. For larger deformations (up to 60%) of soft solid spheres made from linear and non-linear elastic materials, Tatara (1991, 1993) proposed a model to describe the relationship between the force and displacement. For the case of linear elasticity... 

An analytical elastic membrane model was developed by Feng and Yang (1973) to model the compression of an inflated, non-linear elastic, spherical membrane between two parallel surfaces where the internal contents of the cell were taken to be a gas. This model was extended by Lardner and Pujara (1980) to represent the interior of the cell as an incompressible liquid. This latter assumption obviously makes the model more representative of biological cells. Importantly, this model also does not assume that the cell wall tensions are isotropic. The model is based on a choice of cell wall material constitutive relationships (e.g., linear-elastic, Mooney-Rivlin) and governing equations, which link the constitutive equations to the geometry of the cell during compression. 

Figure 16 Comparison of the dimensionless force Y and fractional deformation of a single 163 pm diameter ion-exchange resin particle (DOWEX 1X8-200, Sigma-Aldrich, UK) obtained by diametrical compression and by numerical simulation using the Tatara non-linear elastic model. E0n represents the initial Young s modulus at zero strain (data provided by Dr T. Liu).

E. Verron, G. Marckmann, and B. Peseux, Dynamic Inflation of Non-Linear Elastic and Viscoelastic Rubber-like Membranes, Int. J. Num. Meth. Eng. 50, 1233-1251 (2001). 

A reasonable approximation for the force between two adjacent particles is given by the so-called FENE (finitely extendable non-linear elastic) spring force law (Bird et al. 1987a)... 

In the case of solids it is evident that deformation is either linear elastic - like a Hookean solid (most solids including steel and rubber) - or non-linear elastic or viscoelastic. In the case of liquids, fluids differ between those without yield stress and those with yield stress (so-called plastic materials). Fluids without yield stress will flow if subjected to even slight shear stresses, while fluids with yield stress start to flow only above a material-specific shear stress which is indicated by o0. 

An important role in the present model is played by the strongly non-linear elastic response of the rubber matrix that transmits the stress between the filler clusters. We refer here to an extended tube model of rubber elasticity, which is based on the following fundamental assumptions. The network chains in a highly entangled polymer network are heavily restricted in their fluctuations due to packing effects. This restriction is described by virtual tubes around the network chains that hinder the fluctuation. When the network elongates, these tubes deform non-affinely with a deformation exponent v=l/2. The tube radius in spatial direction p of the main axis system depends on the deformation ratio as follows ... 

It is necessary to state more precisely and to clarify the use of the term nonlinear dynamical behavior of filled rubbers. This property should not be confused with the fact that rubbers are highly non-linear elastic materials under static conditions as seen in the typical stress-strain curves. The use of linear viscoelastic parameters, G and G", to describe the behavior of dynamic amplitude dependent rubbers maybe considered paradoxical in itself, because storage and loss modulus are defined only in terms of linear behavior. 

Previous work pursued the model analytically, for a linearly elastic [5] (or, later, non-linearly elastic [6]) material with constant thermal properties. The analytical model explained several measured fracture properties of thermoplastics the magnitude of impact fracture toughness and its dependence on impact speed [7] and the absolute magnitude of resistance to rapid crack propagation [8]. Recent results have shown that the impact fracture properties of some amorphous and crosslinked polymers show the same rate dependence [11],... 

The correlation is quite good for the SRI500 resin, while for the more ductile adhesive resin the predictions overestimate the measured failure loads. However, in the latter case an extensive damage zone develops before final failure and the non-linear elastic fracture model is no longer appropriate. It is interesting to note however, that when a fillet is left at the end of the overlap the test values are much closer to the predictions. 

This is the contour integral described by Rice wiuch is usually denoted by J for non-linear elastic materials and becomes G for the linear case given here. The result is important since, if a stress analysis for a cracked body is performed in which the stress and displacement fields are found, then G may be determined. 

Fig. 12. Equation of state (a) and phase diagram (b) of a bead-spring polymer model. Monomers interact via a truncated and shifted Lennard-Jones potential as in Fig. 6 and neighboring monomers along a molecule are bonded together via a finitely extensible non-linear elastic potential of the form iJpENE(r) = — 15e(iJo/

A simple generic bead spring model of chains can be used to study universal polymer properties that do not depend on specific chemical details. Bonds between neighbouring Lennard-Jones particles in a chain can be represented by the finite extension non-linear elastic (FENE) potential. 

In industrially interesting flows a non-linear-elastic behavior is generally observed, since the deformation y and deformation rate y are not small and the above statements do not apply. In this case the following procedure provides a solution. 

Finite-extensibility non-linear elastic (FENE) extension of dumbell model Multimode Zimm model... 

The mechanical behavior of Zr02-Ni system strongly depends on constitutional variation. The Ni-rich materials exhibit typical behavior of elasto-plastic deformation and ductile fracture similar to metallic material. The materials containing PSZ from 40-80 vol% mainly presents typically linear elastic behavior and macroscopic brittle fracture. However, the material with 60 vol% PSZ behaves as non-linear elastic behavior after the linear stage. 

The dependence of mechanical behavior on constitution in Zr02-Ni system results from the variation of microstructure and its distribution. In the regions rich in Ni or PSZ, the mechanical performance is controlled by continuous matrix component and displays elasto-plastic or linear elastic characteristics, respectively. The non-linear elastic behavior at 60 vol% PSZ is related to the connectivity transition of matrix component. 

Odgen RW (1984) Non-linear elastic deformations. Ellis Horwood, Chichester... 

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