Category The History of Furniture Construction

Elastic Properties of Human Body Soft Tissues

Numerical modelling of soft tissues requires gathering of experimental data of biomechanical properties of these bodies. In conducting studies on the properties of soft tissue, on large samples of research material (pork liver), Hu and Desai (2005) assumed that the tissue is a material that is incompressible, homogenous and isotropic.

Assuming the load force of the cubic sample as F, the elongation coefficient as X and the initial contact surface on the cube Ao, stresses a according to Cauchy were written in the form:


– = k, (8.104)


while strains e in the form:

s = lnQ. (8.105)

By dividing the experimental load-strain curve (Fig. 8.20) into small subregions, the authors noted that the dependence of force on movement is linear...


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,



Results of the calculations and their comparison with the results of experimental research

Table 8.2 compares the values of material parameters of the foams determined in the course of experimental research through axial compression and calculated on the basis of the equation:

with the results of numerical calculations with the use of the finite elements method.

On the basis of the compiled values, it can be seen that the results of the numerical calculations of parameters ц and a are about 7-9 % larger in relation to the results of the analytical calculations. In the case of parameter в, numerical calculations provided a result smaller by about 22 % in relation to the analytical calculations. The presented solutions are therefore valid and can be used for analysis of stiffness of more complex multi-layer sets.


Strength-elongation relationship for the axial compression test

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


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:


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

І—1 —І

and the next resulting relation for extensions in the directio...


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...


Reduced polynomial form

This equation has the form:

N N і

U = £ C00Q1 – 3У+ Y, D J – 1)Ъ; (8.44)

i=1 i=1 1


U potential energy of strains per volume unit,

N material parameter and

Cj and D1 temperature-dependent material coefficients,

/1 = 12 + Ц +12, (8.45)


!i = J-3ki, (8.46)


J total volume coefficient,

Jel elasticity volume coefficient and Xt physical elongation.

The initial value of the figural strains modulus and the module Ko has the form:


lo = 2C10, Ko = —. (8.47)

Van der Waals equation

The equation for potential energy of strains according to van der Waals has the form:


I = (1 – b)7x + № (8.49)


g = M (8’50)

Whereby U is the potential energy of strains per unit volume, ц—the initial shear modulus, A,,,—observed elongation, a—general interaction coefficient, в—constant ...


Marlow equation

The equation for potential energy of strains according to Marlow has the form:

U = Udev(!t) + Uvol (Jel), (8.32)

where U is the potential energy of strains per volume unit, with Udev as the deformed part and Uvoi as the volume part—undeformed,

7j = k2 + k2 + k^, (8.33)


к = J-hi, (8.34)


J total volume coefficient,

Jel elasticity volume coefficient and ki main elongation.

Neo-Hookean equation

In this case, the equation for potential energy of strains has the form:

12 U = Cw(h – 3) + — J – 0 ,

where U is the potential energy of strains per unit volume, C10 and D1 temperature-dependent material coefficients:

І1 = 12 + 12 + ^2, (8-36)


1 = J-bki, (8.37)


J total volume coefficient,

Jel elasticity volume coefficient and Xi main elongation.

The initial value of th...


Ogden’s model for hyperelastic foams

This model is very similar to the incompressible material model:

w = £ a jei( if+if+ifo – 3+£ «I (yi _ f),

‘ =1 1 1 ’

where the initial shear modulus has the form:

and module Ko:

N (1 ^ jo = 53 3 + A) .

Ogden’s models are mainly used for modelling foams of deformations above 700 %.

Arruda-Boyce equation

This equation has the form:

+ D I 2


where U is the potential energy of strains per unit volume, A,,, and

D temperature-dependent material parameters,

І1 = h1 + k2 + h2, (8.28)


J total volume coefficient,

Jel elasticity volume coefficient and k physical elongation.

The initial shear modulus [io in relation to ^ is expressed by the equation:

Most often, the coefficient k„, takes the value 7, for which ko = 1.0125, and the initial value of the module ko is


Mooney-Rivlin model

There are 2-, 3-, 5- and 9-parametric Mooney-Rivlin models known. Mooney-Rivlin model with two parameters:

Mooney-Rivlin model with nine parameters, where N =3:

3 _ _ 1

W — ^ Cj(h – 3)Xh – 3)J – (Jel – 1)2. (8.18)

t+j=1 –

For all forms of the Mooney-Rivlin function, the initial value of the shear modulus we define as:

І0 — 2(C10 + C01),


C10 and C01 are the coefficients determined in experimental studies

and module к:


j — – • (8.20)

By choosing the right type of function for the tested kind of foam, one can follow the characteristics of the material stiffness a = /(e), and so:

• for a curve without point of inflection (Fig. 8.14), an equation with two parameters can be used,

• for a curve with one point of inflection (Fig. 8...


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




Density acc. to



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 (%)