Modulus of foams

Another important rheological property of dry foams and highly concentrated emulsions is G, the shear modulus. Princen and Kiss [57] demonstrated that this property was dependent on < >, the volume fraction of the system. Previously, Stamenovic et al. [58] and, much earlier, Derjaguin and coworker [59], had derived an expression for the shear modulus of foams of volume fraction very close to unity. The value was found to depend on the surface tension of the liquid phase (in foams), for the particular case of (Jja 1. However, Princen demonstrated that the values of G obtained were overestimated by a factor of two. This error was attributed to the model used by Stamenovic and coworker, which failed to maintain the equilibrium condition that three films always meet at angles of 120° during deformation. 

Strengths and moduli of most polymers increase as the temperature decreases (155). This behavior of the polymer phase carried over into the properties of polymer foams and similar dependence of the compressive modulus of polyurethane foams on temperature has been shown (151). 

Tensile strength and modulus of rigid foams have been shown to vary with density in much the same manner as the compressive strength and modulus. General reviews of the tensile properties of rigid foams are available (22,59,60,131,156). 

Those stmctural variables most important to the tensile properties are polymer composition, density, and cell shape. Variation with use temperature has also been characterized (157). Flexural strength and modulus of rigid foams both increase with increasing density in the same manner as the compressive and tensile properties. More specific data on particular foams are available from manufacturers Hterature and in References 22,59,60,131 and 156. Shear strength and modulus of rigid foams depend on the polymer composition and state, density, and cell shape. The shear properties increase with increasing density and with decreasing temperature (157). 

Rheology. The rheology of foam is striking it simultaneously shares the hallmark rheological properties of soHds, Hquids, and gases. Like an ordinary soHd, foams have a finite shear modulus and respond elastically to a small shear stress. However, if the appHed stress is increased beyond the yield stress, the foam flows like a viscous Hquid. In addition, because they contain a large volume fraction of gas, foams are quite compressible, like gases. Thus foams defy classification as soHd, Hquid, or vapor, and their mechanical response to external forces can be very complex. 

Fig. 4. Schematic representation of a two-dimensional model to account for the shear modulus of a foam. The foam stmcture is modeled as a coUection of thin films the Plateau borders and any other fluid between the bubbles is ignored. Furthermore, aH the bubbles are taken to be uniform in size and shape.

Although aH these models provide a description of the rheological behavior of very dry foams, they do not adequately describe the behavior of foams that have more fluid in them. The shear modulus of wet foams must ultimately go to zero as the volume fraction of the bubbles decreases. The foam only attains a solid-like behavior when the bubbles are packed at a sufficiently large volume fraction that they begin to deform. In fact, it is the additional energy of the bubbles caused by their deformation that must lead to the development of a shear modulus. However, exactly how this modulus develops, and its dependence on the volume fraction of gas, is not fuHy understood. 

Now for some real numbers. Table 3.1 is a ranked list of Young s modulus of materials - we will use it later in solving problems and in selecting materials for particular applications. Diamond is at the top, with a modulus of lOOOGPa soft rubbers and foamed polymers are at the bottom with moduli as low as 0.001 GPa. You can, of course, make special materials with lower moduli - jelly, for instance, has a modulus of about 10 GPa. Practical engineering materials lie in the range 10 to 10 GPa - a... 

A sheet of chopped strand mat-reinforced polyester is 5 mm thick and 10 mm wide. If its modulus is 8 GN/m calculate its flexural stiffness when subjected to a point load of 200 N midway along a simply supported span of 300 mm. Compare this with the stiffness of a composite beam made up of two 2.5 mm thick layers of this reinforced material separated by a 10 mm thick core of foamed plastic with a modulus of 40 MN/m. ... 

Temperature dependence of the elastic modulus of the rigid urethane foam. 

Fig. 15.1 Effect of gel content and melt modulus on foam density (1808)...

Fig. 15.2 Dependence of foam density on gel content and melt modulus for samples containing varying concentrations of DCP and a TAG concentration fixed at 0.5 phr...

Figure 15.5 gives a wealth of information. It is important to understand that each series of three separate results at each nominal gel content represents an adjustment in DCP concentration to give the nominal gel content in the presence of 0, 0.5 and 2.0 phr TAC (increasing TAC concentration produces progressive increase in melt modulus and foam density). Result positively indicates that gel content is not a suitable indicator to predict foam density (as previously suggested). However, if all... 

Fig. 15.5 The dependence of foam density on melt modulus at specific gel contents (DCP, TAG ratios [phr] shown in parentheses)...

Fig. 15.8). There is a positive indieation that all data points may approximate to a mastereurve more eonelusively than those seen with melt modulus results (Fig. 15.5). The results suggested that swell ratio (itself related to crosslink density) might be a useM parameter upon which to base prediction of foaming behaviour. 

Medium to low tensile properties according to the tensile properties of the core. The tensile strength and modulus of a lOOkg/m foam are roughly as low as 3 MPa and 0.1 GPa, respectively. 

H.M. Princen and A.D. Kiss Rheology of Foams and Highly Concentrated Emulsions III. Static Shear Modulus. I. Colloid Interface Sci. 112, 427 (1986). 

For closed cell foams Gibson and Ashby predict three contributions to the Young s modulus of the foam ... 

The faces in low density LDPE foams are partly buckled or wrinkled, as a result of processing (a.l7). This affects both the bulk modulus and the Young s modulus. The foam bulk modulus Kp is predicted, using the Kelvin closed cell foam model, to be ... 

A Kelvin foam model with planar cell faces was used (a. 17) to predict the thermal expansion coefficient of LDPE foams as a function of density. The expansion of the heated gas is resisted by biaxial elastic stresses in the cell faces. However SEM shows that the cell faces are slightly wrinkled or buckled as a result of processing. This decreases the bulk modulus of the... 

Compressive modulus of various PE foam boards, in different densities and open cell content, are measured and compared with different models. Contributory elements to compressive resistance are investigated. Agreement between test results and modelling improves significantly while only considering the struts strength parallel to compressive force. 7 refs. 

The influence of adding polyfunctional monomers having different structures and functionality into a dicumyl peroxide-based crosslinking system for LDPE was investigated. Monomers employed were diallyl phthalate, trimethylolpropane trimethacrylate and triallyl cyanurate. The effects of formulation on matrix gel content and on foam density at similar gel content were examined and the dependence of foam density on melt modulus assessed. The applicability of swell ratio for estimating foam density was evaluated and the suitability of triallyl cyanurate as a crosslinking promoter for LDPE foam demonstrated. 20 refs. 

COMPRESSION MODULUS OF OPEN-CELLED POLYETHYLENE FOAM... 

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