Work per unit volume

The stiffness matrix, Cy, has 36 constants in Equation (2.1). However, less than 36 of the constants can be shown to actually be independent for elastic materials when important characteristics of the strain energy are considered. Elastic materials for which an elastic potential or strain energy density function exists have incremental work per unit volume of... 

Figure 6.31. Expansion work per unit volume of ethane, propane, and isobutane.

The volume of the vapor is 0.10 x 22.7 = 2.27 m. The explosion energy of the vapor can be calculated by multiplying the expansion work per unit volume by the vapor volume ... 

Explosion energy can be calculated by employing a slight variation on Eq. (6.3.26), by multiplying expansion work per unit volume by fluid volume, instead of multiplying expansion work per unit mass by fluid mass. Both propane and butane must be considered. This gives, for example, for vapor energy for the 50% fill-ratio case ... 

The presented explanation for the existence of the fracture envelope will be used in formulating a fracture criterion for polymer fibres. Let us suppose a hypothetical polymer fibre with chains having a single orientation angle in the unloaded state. The shape of the fracture envelope is now calculated by taking into account the shear deformation of the chains only. For this case the work per unit volume up to fracture is given by... 

Due to the fact that [78] the energy input was split into mass-related (E/pV) and kinetic (E n) energies, an earlier paper concerning the emulsification process should also be referred to [79]. As emulsifyer, a teeth-rimed rotor-stator machine was used and the results (dso) were correlated with both power per unit volume (P/V) and work per unit volume (Pt/m3). This paper is also of special interest because the evaluation is performed in a dimensional-analytical manner. The dimensionally formulated result reads. 

Figure 10.6. Top Sehematie view of the foree-deformation relationships of marshmallows. Notiee the absenee of a shoulder. After Kaletune et al. (1992). Bottom. Sehematie view of how the degree of elastieity ean be assessed from a eompression-deeompression eurve. (Notiee that the area under the stress-stress strain eurve has work per unit volume units and that the degree of elastieity ean be defined as the reeoverable/total work ratio.)...

Mechanical work due to application of a force is defined as the integral of fdl (technically f dl ). Here we will consider work per unit volume. Thus... 

Example 13.3 The mechanical properties of nylon 6,6 vary with its moisture content. A nylon specimen with a moisture content (MC) of 2.5% has an elastic modulus of 1.2 GN/m, while that for a sample of moisture content of 0.2% is 2.8 GN/m. Calculate the elastic energy or work per unit volume in each sample subjected to a tensile strain of 10%. 

Integration of this equation shows that for the elastic component of the work per unit volume, there is no net energy lost or gained. The viscous component becomes... 

The work that must be expended in order to create a cavity in a liquid measures its stiffness. Such a cavity is required in order to accommodate a molecule of the liquid itself, condensing into it from the vapour, or of a solute particle. This work per unit volume of the cavity can be obtained from the thermodynamic quantities characterizing the liquid its molar enthalpy of vapourization AyH, its molar volume V, its isobaric expansibility ap, its isothermal compressibility kj, and its vapour pressure pc, all at the temperature of interest T. These yield the difference between the cohesive energy density ced = = [AyH — RT]/V and the internal pressure... 

Figure 15. The change of the induced apparent internal work per unit volume, Sw vs. Ljj o by moisture adsorption for PAN-PSt-WPC (irradiation-injection) system.

The extent to which a material absorbs energy without fracture. A measure of the ability of a material to absorb energy. The actual work per unit volume or unit mass of... 

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 ... 

Vy + vf)/2 gives the following equation for work per unit volume exerted by... 

Taking Eq. (3.74) per unit volume, denoting work per unit volume by W and... 

When in a dielectric body the material is polarized by the application of an electric field work is done by electric forces. As in the analogous case of a deformed elastic body where the work per unit volume may be interpreted as an elastic energy density called deformation energy, the work of the electric forces may be understood as increasing the electrostatic field energy density. For an infinitesimal increment this takes the form... 

If the plastic response remains essentially isotropic, then there must be an equivalent plastic strain measure, having the form of some function of the principal values of the plastic strain rate tensor, such that the Mises stress times the equivalent plastic strain rate is the local rate of plastic work per unit volume of material. In terms of the principal plastic strain rates e, 2 3, this equivalent plastic strain rate is defined as... 

It is implicit in such descriptions of small-strain elasticity that the stress is unaffected by the deformation, which justifies the assumption of linearity. For the large deformations associated with rubber elasticity and shear yielding in glassy and semicrystalline polymers, however, this is no longer valid, and it is more usual to express deformations in terms of the deformation ratios, 2, such that the point x, Y,z) is transformed to (Aix, 2y,hz). If the coordinate system is defined with respect to the deformed system, it is always possible to choose the axes so that they are not rotated by the deformation. The shear components of the deformation are then equal to zero, which permits considerable simplification. For example, the incremental work per unit volume of undeformed material is given by Eq. (10), where the fi are the applied forces along the axes of a unit cube of undeformed material [5]. 

As mentioned, the area inside the hysteresis loop represents of the energy lost or dissipated during cyclic deformation. The dissipation can be shown to be proportional to the loss modulus using the basic relationships between work and energy. Recall that the work per unit volume of a stressed material is given by... 

There is a second characteristic property which all cases have in common One always deals with a pair of energy conjugated variables, that is to say, the displacement caused by the field results in work. More specifically, if a field gives rise to a displacement dx, then the work per unit volume is... 

Equation (5.145) gives an expression for w which is the work per unit volume. The volume of the specimen is Aq/q and so... 

In addition to dimensional recoveiy, work recoveiy often is studied for tensile deformation. While stretching a fiber, the total work done is either stored in chemical bonds or is lost, typically in the form of heat. The work stored in the chemical bonds is recoverable, but the work lost is not. In Figure 15.23, the area under the stretch curve is the total work per unit volume done during stretching. The area under the recovery curve is the work per unit volume returned during recovery. Work recovery can then be defined as ... 

---