The most striking feature of the simple shear problem is that the results (76)-(78)
do not involve the shear stress. On the contrary, the shear stress is determined by the
difference of the normal stresses:
γσ 12 ¼ σ 11 À σ 22
ð79Þ
and it is determined in the same way for every homogeneous, isotropic hyperelastic
material, regardless of the form of the response functions. The formula (79) is an
example of an universal relation in the finite elasticity theory.
4 Nonlinear Viscoelasticity
4.1 Nonlinear Theory of Viscoelasticity
The theory of viscoelasticity is crucial when describing materials, such as rubber,
which exhibit time dependent stress-strain behavior.
Indeed, carbon black-filled rubber, when loaded with time-dependent external
forces, suffers a state of stress which is the superposition of two different aspects: a
time independent, long-term, behavior (sometimes improperly called
“hyperelastic”) opposed to a time dependent, short-term, response. Step-strain
relaxation tests suggest that short term stresses are larger than the long term or
quasi-static ones [117]. Moreover, oscillatory (sinusoidal) tests indicate that dissipative anelastic effects are significant, which leads to the consideration of a
constitutive relation which depends not only on the current value of the strain but
on the entire strain history. This assumption must be in accordance with some
principles which restrict the class of reliable constitutive equations. These restrictions can be classified as “physical” and “constitutive”. The former are restrictions
to which every rational physical theory must be subjected to, e.g., frame indifference. The latter, on the other hand, depends upon the material under consideration,
e.g., internal symmetries.
The principle of determinism [108] belongs to the first class. It states that: the
stress at a given material point is determined by the entire past history of the motion
in a neighborhood of the considered point.
Another basic assumption in every rational constitutive theory is the principle of
frame indifference: the response of the system must be the same for all observers
[104–108].
Then one is lead to consider constitutive restrictions. One important assumption,
proposed by [118], postulates that the material is simple, which means that the
stress at a given material point depends only on the history of the first order spatial
gradient of the deformation, in a small neighborhood of the material point. Therefore, the influence of the higher order spatial gradients is ignored. Although this
assumption is usually considered non-constitutive, it legitimately belongs to the
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
239
do not involve the shear stress. On the contrary, the shear stress is determined by the
difference of the normal stresses:
γσ 12 ¼ σ 11 À σ 22
ð79Þ
and it is determined in the same way for every homogeneous, isotropic hyperelastic
material, regardless of the form of the response functions. The formula (79) is an
example of an universal relation in the finite elasticity theory.
4 Nonlinear Viscoelasticity
4.1 Nonlinear Theory of Viscoelasticity
The theory of viscoelasticity is crucial when describing materials, such as rubber,
which exhibit time dependent stress-strain behavior.
Indeed, carbon black-filled rubber, when loaded with time-dependent external
forces, suffers a state of stress which is the superposition of two different aspects: a
time independent, long-term, behavior (sometimes improperly called
“hyperelastic”) opposed to a time dependent, short-term, response. Step-strain
relaxation tests suggest that short term stresses are larger than the long term or
quasi-static ones [117]. Moreover, oscillatory (sinusoidal) tests indicate that dissipative anelastic effects are significant, which leads to the consideration of a
constitutive relation which depends not only on the current value of the strain but
on the entire strain history. This assumption must be in accordance with some
principles which restrict the class of reliable constitutive equations. These restrictions can be classified as “physical” and “constitutive”. The former are restrictions
to which every rational physical theory must be subjected to, e.g., frame indifference. The latter, on the other hand, depends upon the material under consideration,
e.g., internal symmetries.
The principle of determinism [108] belongs to the first class. It states that: the
stress at a given material point is determined by the entire past history of the motion
in a neighborhood of the considered point.
Another basic assumption in every rational constitutive theory is the principle of
frame indifference: the response of the system must be the same for all observers
[104–108].
Then one is lead to consider constitutive restrictions. One important assumption,
proposed by [118], postulates that the material is simple, which means that the
stress at a given material point depends only on the history of the first order spatial
gradient of the deformation, in a small neighborhood of the material point. Therefore, the influence of the higher order spatial gradients is ignored. Although this
assumption is usually considered non-constitutive, it legitimately belongs to the
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
239