When considering the homogeneity of the strain, the constant strain gradient can be written in the form:

where

a0, a(t) sizes of the angle and

h0, h(t) heights of the examined sample before and during the strain.

If the foam sample is loaded only in the direction of axis 3, then stresses in the direction of 1 and 2 do not occur. Hence, on the basis of the equation

j=1 i=1

and using the equations

If the sample is loaded with a single load K in the direction of axis 3, then the stress in the direction of 3, after taking into account the conditions of equilibrium, will amount to:

r33 — —K /(ab) = —K/a2.

Hence, the final relation for the axial load has the form:

Nl

K(k1, k3) — 2a2(k?, k3) — k—i — (ki, k3)

І—1 —І

and the next resulting relation for extensions in the directions 1 and 2:

For the case N =1 and using а1 := а, в1 := в, H := H, a dependency between k1 and k3 can be derived:

and finally,

1

k2k3 = k31+2b. (8.87)

1

The equation k^k3 = k3 1+2b allows for the separation of the parameter в, which

1

can be used for a separate analysis of fi. By application k^k3 = k31+2b, from the equation,

bj

1 + 2bj ’ or

bj = 1—2^ for j = 1 2’ •••N (8.89)

k1 for N = 1 could be eliminated. Hence, the final form of the stress-elongation relation, with the assumption that N =1, has the form as:

From the above equation, it results that the following conditions are very important:

because for в = -1/3, the value of K will always equal zero for the elongation 13. Calculations using the finite elements method

In order to verify the correctness of the constitutive model describing the mechanics of behaviour of soft foams, Schrodt et al. (2005) conducted numerical calculations using the finite elements method in the environment of the system ABAQUS. The mesh model was built using an 8-node line element of brick type. To the bottom surface of the foam, bonds were assigned to prevent displacement in the direction of the axes x, y and z. Bonds were assigned to the top surface to prevent displacement in the direction of axes x and z. Pressurers were modelled as perfectly stiff bodies. It was also assumed that the coefficient of friction between the surface of the pressurer and the surface of the foam will amount to 0.75. The sum of the actual loads applied to the sample and distribution of this value proportionally on all the nodes of the numeric model was assumed as the model load.

Parameters |
Analytical calculations (EXP) |
Numerical calculations (FEM) |
Difference FEM/EXP (%) |

ц [MPa] |
0.831 x 10-2 |
0.907 x 10-2 |
9.2 |

a |
0.198 x 102 |
0.213 x 102 |
7.6 |

в |
0.109 x 10-1 |
0.849 x 10-2 |
0.779 |

Table 8.2 Comparison of the values of material parameters of the foams which were subject to axial compression test, calculated numerically or analytically |