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