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Stress Analysis Dr. Maaz akhtar 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). σ

ε

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 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 full y extended length. A hyperelastic material model is a type of constitutive constitutive relation for rubberlike material in which the stress-strain str ess-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.

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

Probability in x, y and z -directions are given by that

, and assigning

,

and

(2) , repectively. Knowing

, we get the following relation

(3)

( x0, y0, z 0) 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

Taking summation for all chains, using simplification we get

Shear modulus for rubbers and elastomers is given by G= by , hence Eq. 10 can be written as

(7)

into Eq. (3.7), and on

(8) (9) (10)

. Helmholtz free energy is given

(11)

Where is the first invariant of stretch. Eq. 11 gives the strain energy function for neoHookean 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 ‘

’

‘

equation,

σ ’

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 ‘

’

‘

equation,

σ ’

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

). If

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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). σ

ε

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 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 full y extended length. A hyperelastic material model is a type of constitutive constitutive relation for rubberlike material in which the stress-strain str ess-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.

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

Probability in x, y and z -directions are given by that

, and assigning

,

and

(2) , repectively. Knowing

, we get the following relation

(3)

( x0, y0, z 0) 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

Taking summation for all chains, using simplification we get

Shear modulus for rubbers and elastomers is given by G= by , hence Eq. 10 can be written as

(7)

into Eq. (3.7), and on

(8) (9) (10)

. Helmholtz free energy is given

(11)

Where is the first invariant of stretch. Eq. 11 gives the strain energy function for neoHookean 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 ‘

’

‘

equation,

σ ’

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 ‘

’

‘

equation,

σ ’

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

). If

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