Elastic work

We begin by remembering the mechanical definition of work and apply that definition to the stretching process of Fig. 3.1. Using the notation of Fig. 3.1, we can write the increment of elastic work we associated with an increment in elongation dL as... 

It is necessary to establish some conventions concerning signs before proceeding further. When the applied force is a tensile force and the distortion is one of stretching, F, dL, and dw are all defined to be positive quantities. Thus dw is positive when elastic work is done on the system. The work done by the sample when the elastomer snaps back to its original size is a negative quantity. 

Note this is the same derivation that yields the important results V = (3G/3p)y and S = (3G/3T)p when no elastic work is considered. It would be inappropriate for a book like this to digress into thermodynamics any further than this. The two relationships cited above are derived in almost every thermodynamics text the student is advised to consult a suitable reference and review this material if the treatment above is too abbreviated. 

Hoetzeldt, K., Bochmann, G., and Passler, W., Cold sprayable, low flammability and watertight molding or sealing material used for the preparation of elastic work layers for road, play or sports surfaces, German Patent, DE 4416570 A1 19,951,116, 1995. 

One can also consider the 1-dimensional (ID) analog of PV (3D) or surface tension (2D) work. In this case, consider a rubber band of length L that undergoes differential stretching dL against the tension (force) t exerted by the rubber band, as shown in Fig. 3.6. The elastic work performed on the system (the rubber band) is evidently... 

Elastic work (ID) A quantity of length (extensive) moved through a difference in tension (intensive). 

Figure 3.6 Schematic depiction of stretching a rubber band of length L against the tension (force) t to perform elastic work.

The tableting properties of materials also depend on their deformation behavior. It is apparent that the tablet tensile strength is a strong function of the plastic work required for its formation but not a function of the elastic work recovered. Consequently, it is likely that strong and ductile interparticle functions, whose formation dissipated a significant plastic work, result in strong and tough compacts [4],... 

Fiber stress at proportional limit Modulus of rupture Modulus of elasticity Work to proportional limit Work to maximum load Impact bending... 

The elastic work in cluster formation is related to the energy changes in two different kinds of chain deformations. The first kind is the work of stretching a polymer chain from a distance Xq to X, the distance corresponding to a cluster size of N, The other kind is the work of contracting a chain from a distance of Xq to 0. Accordingly, the elastic work per chain in cluster formation is given by... 

The primes associated with F and W in Equation 8 are to remind one that the numerical forms of these expressions differ from those of Equation 6. The effective dipole of a hydration shell is now attributable to the sidechain group (irrespective of neutral or charged form) plus the net dipolar arrangement of the water molecules. Hence F is different from F. The elastic work W requires moving hydration shells plus polymer chains. Thus W is not equal to W. 

When an elastomer is stretched work is done on it and its free energy is changed. The elastic work is given by fdl where / is the retractive force (Eqn. 69), and d/ is the change in length. 

Under ideal, thermodynamically reversible conditions, this mechanical work is equal to the Gibbs free energy change (AG) for the process. The numbers obtained here cannot be compared directly with the values obtained for molecules in solution using bulk thermodynamic methods (Chapter 5) because they include the additional elastic work involved in stretching the unfolded polypeptide as the tethered ends are pulled further apart than they would normally be for an unfolded protein free in solution. 

The elastic work ( -dW,) involved in stretching a one-dimensional chain a distance [dl) on applying a force (/) is... 

A = Helmholtz free energy (constant volume and temperature) dW = the elastic work... 

Elastic work Elastic part of total indentation work. 

If the virtual work theorem is applied equating the work of the inertia forces to the elastic work for the virtual displacement dv=p(x)Ay, then Equation (15.49) is obtained ... 

The gas (left) does no elastic work, the rubber (right) no volume work, as given in the two equations written... 

All of the models considered so far are based on the promotional idea. However, there are some experimental results which appear to conflict with this idea. One of these is the placement of the 4f level in -y-cerium at 0.9 eV below the Fermi level by X-ray photoemission studies (see section 4.9). If these results are correct then about 20 kcal/mole of energy would be required to lift the f level to the Fermi energy to allow transfer of electrons to the conduction band. The elastic work necessary to compress a y-Ce sample to a-Ce at room temperature is two orders of magnitude smaller. This comparison is obviously an oversimplification but it serves to emphasize the importance of the proximity of the f level to the Fermi level in any promotional theory. Also, as noted in section 4.9, a 4f level could lie 0.1 eV below the Fermi level but it would not be detected by the X-ray photoemission studies. 

The resilience, denoted and expressed in Joules (J), is the ability of a solid material to absorb elastic energy and release it when unloaded (e.g., rebound, springback). In practice, the absorbed elastic energy can be calculated from the true stress-strain plot (S - e) by integrating the surface area under the curve between the true yield strength and the origin. This area represents the amount of elastic work per unit volume that can be done on the material without causing it to rupture ... 

The description of heat, Q, in terms of the first law of thermodynamics is given by Eq. (1) of Fig. 1.2. This equation provides for the conservation of energy. The heat dQ evolved or absorbed in a process must be equal to the change in the internal energy dU diminished by the work. Equation (1) is written for the situation in which all work is volume work, -pdV. The negative sign results from the fact that for an increase in volume the system must do (lose) work. If other types of work are done, such as electrical or elastic work, additional terms must be added to Eq. (1). 

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