Nonlinear Stress-Strain Behavior

Stress-strain behavior for metals gives linear relationship till yield point, later exhibits

nonlinear plastic deformation region. Many materials when loaded gives complete nonlinear

stress-strain behavior. Polymers such as rubbers, Elastomers etc are generally possessing

nonlinear stress-strain behavior (Figure-1).

Figure-1: Stress-strain curve for rubber-like material

Young’s modulus for materials exhibits nonlinear stress strain behavior can be determined by

dividing complete curve into small divisions such that each portion will be a straight line.

Addition of all the values and dividing by number of divisions will give the value of Young’s

modulus. Another approach which can be use to determine Young’s modulus is based on the

theory given by Gent (1996). Gent determine the elastic modulus by considering only 10% data

of total stress-strain curve where the curve is almost linear. He compared elastic modulus for

many rubbers found that by both techniques the values are very near. Hence, it is

recommended to use only 10% data of stress-strain curve to determine Young’s modulus.

Material Models for Rubberlike Materials

Elastomer can be treated as a hyperelastic material, commonly modeled as incompressible,

homogeneous, isotropic and nonlinear elastic solid. Due to long and flexible structure

elastomer has the ability to stretch several times its initial length. Elastomers at small strains

(upto 10%) have linear stress strain relation and behave like other elastic materials (Gent,

1996). In case of applications where large deformations exist, theory of large elastic

deformation should be considered. Several theories for large elastic deformation have been

developed for hyperelastic materials based on strain energy density functions. Selection of

appropriate strain energy potentials and correct determination of material coefficients are the

main factors for modeling and simulation.

Different mathematical models have been suggested for the prediction of stress-strain behavior

in elastomeric materials. Rubber elasticity theory explains the mechanical properties of a

rubber in terms of its molecular constitution. First statistical mechanics approach to describe

the force on a deforming elastomer network assumed Gaussian statistics, which assumes that a

chain never approaches its fully extended length. Researchers also suggested material models

based on non-Gaussian statistics. These are physical models based on an explanation of a

molecular chain network, phenomenological invariant-based and stretch-based continuum

mechanics approach. The distinctive feature of non-Gaussian approach is that it presumes that

a chain can attain its fully extended length.

A hyperelastic material model is a type of constitutive relation for rubberlike material in which

the stress-strain relationship is developed from a function. Most continuum mechanics

treatment of rubber elasticity begins with assuming rubbers to be hyperelastic and isotropic

material. Figure below gives a classification of different types of hyperelastic material models.

σ

ε

Stress AnalysisDr. Maaz akhtar

Figure-2: Material models for rubber-like material

Neo-Hookean Model

One major approach used to define the deformations in elastomer chains follows Gaussian

statistics. Neo-Hookean model is an example of Gaussian statistical model. In Gaussian

statistics, a chain never approaches the maximum stretch; rather it is limited to small to

moderate stretches. For a three dimensional polymer (due to large chain density) probability

distribution for any event x approaches a Gaussian distribution, given by

(1)

Standard deviation in terms of Chain density ( ) and distance between the chain ends ( ) is

given by Treloar (1975) as

(2)

Probability in x, y and z-directions are given by , and , repectively. Knowing

that , and assigning

, we get the following relation

(3)

(x0, y0, z0) is the initial (unstretched) location, while ( , , z) are the final coordinates in the

stretched condition; obviously, . According to Boltzmann general principle

of thermodynamics, entropy is proportional to the logarithm of the possible configurations

corresponding to a specified state. For small volume ( ), probability can be used to

define entropy ( ) as follows:

(4)

(5)

, (6)

where k is the Boltzmann constant. Taking change in unstretched end-to-end distance of chain

(

) and on simplification we get change in entropy

(7)

Taking summation for all chains, using

into Eq. (3.7), and on

simplification we get

(8)

(9)

(10)

Shear modulus for rubbers and elastomers is given by G= . Helmholtz free energy is given

by , hence Eq. 10 can be written as

(11)

Where is the first invariant of stretch. Eq. 11 gives the strain energy function for neo-

Hookean model (Boyce & Arruda, 2000; Treloar, 1975). As it is derived using Gaussian

statistics it gives linear response for material where elastomer chain deforms undergoes only

small to moderate stretches.

Uniaxial Tension Tests for Incompressible Hyperelastic Materials

Rubber like material (hyperelastic material) is under uniaxial tensile load as shown in Figure-3.

Load will elongate the body in axial direction while other two lateral directions it exhibits

reduction in width and height. Material is assumed to be incompressible, hence volume remain

conserved.

Figure-3: Uniaxial tension of Hyperelastic material

For uniaxial tension and

. Holzapfel (2000) gives the relationship for

determining stress as follows:

Neo-Hookean material model

Assuming rubber follows Neo-Hookean material model:

Hence, stress function can be determined by substituting ‘W’ in ‘σ’ equation,

Above expression shows that rubbery modulus or shear modulus and stretch value is required

to determine the stress generated due to uniaxial tensile load.

Mooney-Rivlin material model

Assuming rubber follows Money-Rivlin material model:

Hence, stress function can be determined by substituting ‘W’ in ‘σ’ equation,

Above expression shows that two constants and stretch value is required to determine the stress

generated due to uniaxial tensile load.

Problem

Uniaxial tensile test is conducted on incompressible Hyperelastic material ( ). If

total stretch is found to be 2, determine the amount of stress produced.

Solution