Compressibility linear elasticity

Figure 8.1 shows the stress-strain curve of a material exhibiting perfectly linear elastic behaviour. This is the behaviour characterised by Hooke s Law (Chapter 3). All solids are linear elastic at small strains - by which we usually mean less than 0.001, or 0.1%. The slope of the stress-strain line, which is the same in compression as in tension, is of... 

When a foam is compressed, the stress-strain curve shows three regions (Fig. 25.9). At small strains the foam deforms in a linear-elastic way there is then a plateau of deformation at almost constant stress and finally there is a region of densification as the cell walls crush together. 

Linear-elasticity, of course, is limited to small strains (5% or less). Elastomeric foams can be compressed far more than this. The deformation is still recoverable (and thus elastic) but is non-linear, giving the plateau on Fig. 25.9. It is caused by the elastic... 

We have discussed the value of struts or columns in structural mechanics and described their linear elastic properties. They have another characteristic that is not quite so obvious. When columns are subject to a compressive load, they are subject to buckling. A column will compress under load until a critical load is reached. Beyond this load the column becomes unstable and lateral deformations can grow without bound. For thin columns, Euler showed that the critical force that causes a column to buckle is given by... 

Mathematical modelling of the compression of single particles 2.5.2.7 Hertz model. The mechanics of a sphere made of a linear elastic material compressed between two flat rigid surfaces have been modelled for the case of small deformations, normally less then 10% strain (Hertz, 1882). Hertz theory provides a relationship between the force F and displacement hp as follows ... 

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).

Figure 10.4. Schematic view of reversible and irreversible stress-strain relationships in a compression-deeompression cycle. Left, ideal linear elasticity (small deformation) center, nonlinear elastieity and right, a relationship showing a hysteresis loop.

The porous medium is compressible and behaves as a linear elastic solid ... 

The compaction characteristics of powders and granular materials also provide interesting insight into bulk behaviour of solids. Bulk materials in general are not linearly elastic and cannot, therefore, be characterized simply by Young s modulus and Poisson s ratio as solid materials can. If a powder is compressed, the deformation is primarily plastic and the elastic component is small in comparison. 

Linear elasticity is the most basic of all material models. Only two material parameters need to be experimentally determined the Young s modulus and the Poisson s ratio. The Young s modulus can be directly obtained from uniaxial tension or compression experiments, and typical values for a few select fluoropolymers at room temperature are presented in Table 11.2. 

Under small deformations rubbers are linearly elastic solids. Because of high modulus of bulk compression (about 2000 MN/m ) compared with the shear modulus G (about 0.2-5 MN/m ), they may be regarded as relatively incompressible. The elastic behavior under small strains can thus be described by a single elastic constant G. Poisson s ratio is effectively 1/2, and Young s modulus E is given by 3G, to good approximation. 

Soutis, Fleck, and Smith [22], proposed the use of linear elastic fracture mechanics and the principle of superposition to determine the failure strength of composites with holes, in particular, under compression. This approach essentially models the damage developing at the edge of the hole as a crack with loaded surfaces. This is a one-parameter model as the crack surface stress must be determined by tests. 

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