Mathematical Models of Foams as Hyperelastic Bodies

Elastomers are a class of polymers having the following characteristics:

• They include natural and synthetic rubbers; they are amorphous and consist of long molecular chains (Fig. 8.10);

• The molecular chains are strongly twisted, spiral and randomly oriented in undeformed form; and

• The molecular chains during stretching get partially straightened; however, when the load stops, they go back to their original form.

Table 8.1 Types of hyperelastic K and standard T foams and their characteristics specified by the manufacturer

No.

Producer’s

signature

Density acc. to

PN-77/C-05012.03

(kg/m3)

Stiffness acc. to DIN EN ISO 3386 (kPa)

Permanent deformation acc. to PN-77/C05012.10 no more than (%)

Resiliency acc. to ISO

8307:2007 no less than (%)

1

K-2313

20.0-23.5

1.0-1.6

8

47

2

K-2518

23.5-25.5

1.3-2.2

12

45

3

K-2525

22.0-25.0

1.6-3.0

6

45

4

K-3028

27.0-31.0

2.3-3.0

5

50

5

K-3037

27.0-31.5

3.3-4.3

5

50

6

K-3530

31.5-35.0

2.4-3.2

5

56

7

K-3536

32.0-35.5

3.3-4.3

5

55

8

K-4036

37.0-41.0

3.1-3.8

6

59

9

K-4040

36.0-40.0

3.4-4.6

6

80

10

K-4542

41.0-46.0

3.3-4.8

4

60

11

T-1619

14.5-17.5

1.6-2.3

6

39

12

T-1828

16.0-20.0

2.3-3.4

8

37

13

T-2121

19.5-22.5

1.8-2.6

9

38

14

T-2130

19.5-22.5

2.7-3.6

7

38

15

T-2237

20.5-23.5

3.2-4.4

7

38

16

T-2516

23.0-25.5

1.3-1.7

5

45

17

T-2520

23.0-27.0

1.8-2.5

7

40

18

T-2538

22.0-26.0

3.3-4.6

5

40

19

T-2544

22.0-26.0

3.8-5.0

5

38

20

T-2550

22.0-26.0

4.4-5.8

5

40

21

T-2838

24.0-28.0

3.5-4.5

5

45

22

T-3030

27.5-30.0

2.4-3.4

5

50

23

T-3038

27.0-31.0

3.5-4.5

5

45

24

T-3050

27.0-31.0

4.4-5.8

5

45

25

T-3530

32.0-36.0

2.6-3.5

4

55

26

T-3543

32.0-36.0

3.8-5.0

5

45

27

T-3550

36.0-40.0

4.4-5.8

5

45

28

T-4040

36.0-40.0

3.3-4.4

5

50

29

T-4060

36.0-40.0

5.1-6.9

5

50

At the macroscopic level, the elastomer shows the following characteristics:

• may be subject to large deformations from 100 to 700 % of the initial dimension, depending on the degree of twisting of the molecular chains;

• to a minor extent, it changes the volume under load during deformation, which is why elastomers are almost incompressible; and

• the stress-strain relationship, as shown previously, is highly nonlinear.

An attempt of compression of polyurethane foams shows that load and unload curves do not overlap, creating a significant hysteresis of strains and absorbing a significant part of energy in the stress-strain cycle (Fig. 8.11). In the case of bodies of linear viscoelasticity, the shape of hysteresis residue is independent of the size of

Fig. 8.9 Stiffness of standard-type foams

the deformation, so the time of experiment has no impact on the properties of such materials being determined. The amount of energy absorbed by materials of non­linear viscoelasticity, such as polyurethane foams, depends on the size of the strains, and therefore, mechanical properties of these bodies will vary depending on the speed of the pressure and the duration of the experiment. Figure 8.12 shows hysteresis loop with a grey field of total load energy, and the light grey colour shows the amount of total unload energy. The difference between these fields is the

Fig. 8.11 Hysteresis of viscoelastic material in the stress-strain cycle

amount of absorbed energy. If the size of the strains is smaller and load and offload curves are approaching each other, then the amount of distributed energy will be minimal (Fig. 8.12b).

Foams, like other hyperelastic materials, can be described using the theory of energy of potential strains U, which determines the size of the energy accumulated in a volume unit of a material, as a function of strain at a point (Anonim 2000a, b; Hill 1978; Mills and Gilchrist 2000; Ogden 1972; Renz 1977, 1978; Storakers 1986). Before a detailed analysis of different forms of density functions of strains energy, basic concepts will be defined, such aselongation coefficient

where

ee unit strain

In addition, we also have three elongation coefficients Xь X2, X3, which allow the measurement of strain and are used to determine the density functions of strains energy. The following example illustrates strains of a rectangular element in the state of biaxial compression (Fig. 8.13). The main strains coefficients Xb X2, show the strain in a plane. In the case of the sample thickness, the coefficient X3 presents the change of material thickness (f/f0), and for an incompressible material, X3 = X 2.

Three parameters of strains are used to define the density functions of strains energy:

ii — x2 + x2 + x2,

(8.5)

i2 = x2x2 + x|x3 + x2x2,

(8.6)

i3 = x^x^

(8.7)

For an incompressible material, I3 = 1. Volume coefficient J is defined as:

V

J = xix2x3 = . 123 Vo

(8.8)

Density function of strains energy can be most often defined as W. The function of strains energy can also be the function of major strain coefficients or the function of strain parameters

Based on W, the second Piola-Kirchhoff stresses (as well as Green-Lagrange strains) can be written in the form:

Because the material is incompressible, we can separate the components of the function of strain energy into differentiable (with index d) and volume (with index b). As a result, volume strains are only a function of the volume coefficient J:

W = Wd(h, h) + Wb(J), (8.11)

W = Wd(h, k2, Xj) + Wb(J), (8.12)

where the differential of major extensions and the differential of parameters I are expressed in the form:

kp = J 3kp for p = 1,2, 3,

(8.13)

Ip = J-3Ip for p = 1,2, 3.

(8.14)

By writing I3 = J[3] [4], I3 is not used in the definition of W.

There are many forms of functions that describe the value of potential energy of strains (Anonim 2000a, b), e. g. the equations: Arruda-Boyce, Marlow, Mooney-Rivlin, neo-Hookean, Ogden, polynomial, reduced polynomial, Yeoh and van der Waals. Each of them is presented below.