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 Analysis Dr. Maaz akhtar
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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.