Stiffness of Hyperelastic Polyurethane Foams

This chapter presents the results of the axial compression of polyurethane foams study (Fig. 8.17), for which a nonlinear model of Mooney-Rivlin was built and a numerical analysis of contact stresses was conducted.

By building a mathematical model of elastic foam, it was assumed that this is a model

• of isotropic and nonlinear material,

• made up of cells distributed evenly and capable of large deformations,

• capable of large deformations, over 90 % during compression and

• requiring geometrical nonlinearity during subsequent steps of strain analysis.

Stresses in the energy function of stretching for foam have been expressed in the form:

Thereby compression energy is expressed by the equation:

W = f (/1, h, /3),


і = l1 + l2 + l2,


l1l2 + l2l2+l3l2,


/3 = l2 l2l2 .


For axial compression, the stress function has the form:

Therefore, Mooney-Rivlin’s equation, appropriate for hyperelastic materials (of large deformations up to 200 %), has been written as:

For uniaxial compression or stretching, it takes the form:

Transforming this equation to the form:

the equation of a line was obtained, by means of which the coefficients C and C2, were determined, necessary for numerical analysis:

y = ax + b, (8.102)


In Fig. 8.17, the dependency stress-strain has been presented for each type of foam. As it can be seen, foams T2838 and T4060 were characterised by the greatest stiffness. The foams T2516 and T3530 were much softer. Therefore, initially it could be concluded that the foams T2516 and T3530 should be used as an outer layer of a mattress, directly in contact with the user’s body, while the foams T2838 and T4060 should be used as inner layers, to prevent greater displacements, par­ticularly at a large weight of the user.

Additionally, Table 8.3 shows that at different stages of compression of the foam, they have a variable value of Young’s modulus. For foams T2516 and T3530, the ratio E3/Ei = 0.76-0.82, while for foams T2838 and T4060, the ratio E3/ Ei = 0.43-0.58. This relevant differentiation allows greater freedom in the selection of the stiffness of foam when modelling complex systems of multi-layer mattresses.

Material constants occurring in Mooney-Rivlin’s equation are determined from the dependencies provided in Fig. 8.18. These constants represent the data necessary to build suitable numerical models.

In simulating the stress of the human thigh on the surface of the polyurethane foam mattress, in the system ABAQUS, an appropriate mesh model of the foam was

Type of foam

Young’s modulus (kPa)




















Table 8.3 Modules of linear elasticity of foams

Fig. 8.18 Functions for determining constants C1 and C2 in Mooney-Rivlin equations

Fig. 8.19 Distribution of stresses according to Mises in foams caused by operational load: a T2516, b T2838, c T 3530, d T4060

made, loading it with an analytical curve of a radius equal to the radius of the thigh of an adult man. The results of these calculations have been shown in Fig. 8.19.

An analysis of the compiled distributions of stresses according to Mises leads to interesting conclusions. The foams T2516 and T3530 are conducive to the con­centration of stresses around the sciatica bones and unequally support the user’s body, while the foams T2838 and T4060 more evenly move the stresses of the human body and ensure fuller comfort resulting from the reaction of the base.

While modelling contact of the human body with an elastic base, it is also important that in the built calculation models, the elastic properties of soft tissues of a potential user are more or less exactly presented.