Of Polyurethane Foams

The optimisation of the construction of mattresses and/or seats is very important in the use of furniture for sleeping and relaxation, motor vehicles, aircrafts or reha­bilitation medical equipment. Descriptions of the mechanics of hard foams are known on the basis of articles of Renz (1977, 1978). Czysz (1986) described the behaviour of soft polyurethane foams as an elastic Hooke’s body. Using the function of strains energy built in the system ABAQUS (Anonim 2000a, b), Mills and Gilchrist (2000) conducted calculations for soft foams under compression. In this research, only the main parameters were compared with the experimental results, without making a thorough comparative analysis of detailed parameters, mainly the parameter Д The purpose of the research carried out by Schrodt et al. (2005) was the use of the standard function of strains energy “hyperfoam” in the system ABAQUS to describe mechanical properties of the foams. For the research, the authors used polyurethane foams type SAF 6060, with the dimensions 200 x 200 mm and a height of 50 mm.

As previous experimental research has shown, soft polyurethane foams behave like viscoelastic materials. Therefore, for their description, constitutive equations of a viscoelastic body are mostly used. In general models, Schrodt et al. (2005) broke up the stress tensor S into the part comprising the equations of stresses in terms of elasticity SG and SOV part of the stresses representing properties of the material memory. Therefore, the stress tensor was written as:

S = Sg + Sov. (8.60)

Hyperelastic material, as an example of Cauchy’s flexible material, was char­acterised by the function of strains energy. The stress tensor can be obtained by the differentiation of the function of strains energy, which reflects the strain tensor. Therefore, basing on the mechanical energy equation:

d = JS • D whereby J = det F, (8.61)


m function of strains energy, F strain gradient,

S Cauchy stress tensor and D gradient of strains tensor

D = – F-TCF- 2


C the right Cauchy-Green tensor, the dot above the symbol means differentiation after time.

According to the assumptions of Schrodt et al. (2005), m is a scalar, a non-negative function tensor of the right extension of the tensor U or the right Cauchy-Green tensor:

x = x(U)= d(C) = { = 0 for C = I. (8.63)

By entering the above equation to

d = JS • D, (8.64)

by reference to

the general structure of the constitutive equation for nonlinear, hyperelastic and anisotropic body is obtained, in the form:

S = 2J-1F FT. (8.66)

The function of strains energy for highly compressible polymers

For the description of mechanical behaviour of highly compressible polymers, the function of strains energy has the form given by Hill (1978) and Storakes (1986):


^і, аi material parameters and

f(J) volume function, which fulfils the condition f(1) = 0

By using this equation, we obtain the constitutive equation form:


Xi value of the right elongation of the tensor U and ni value of the left elongation of the tensor V.

The probable form of the volume function f(J) was given by Storakes (1986):

f (J)= b – 0. (8.69)


Pj additional material parameter.

Hence the number 3 N of material coefficients aj, в and ^ (j = 1, 2, …N) was obtained, which should be determined in the course of experimental research. In addition, the initial value of the shear modulus and compression module has been defined (Anonim 2000a, b):

In this way, also the relation between Poisson’s ratio uj and the parameter Д – was obtained:



hence, respectively:

bj = for j = 1’ 2—N. (8.72)

For individual cases в = : в = const, v is equal to the standard value of Poisson’s ratio. According to Hill (1978) and Storakes (1986), it is justified to use the fol­lowing equations:

Sa > 0(i = 1.2,.. V) and b > — ^ (8.73)