This work focuses on the formulation of a constitutive equation to predict Mullins and residual strain effects of buna-N, silicone, and neoprene rubber strings subjected to small transverse vibrations. The nonmonotone behavior exhibited by experimental data is captured by the proposed material model through the inclusion of a phenomenological non-monotonous softening function that depends on the strain intensity between loading and unloading cycles. It is shown that theoretical predictions compare well with uniaxial experimental data collected from transverse vibration tests.

It is well known that the response of rubber-like materials when subjected to cyclic loading, the load needed to stretch the material sample to a given stretch value during the second loading cycle in tension, compression, or shear deformation states, to say a few, is smaller than that required to reach the same stretch during the primary loading cycle. This reduction in the stress magnitude is known as the Mullins effect [

In the present work, we neglect healing and anisotropic material effects and use a constitutive material model to characterize softening and residual strain effects of rubber-like cords subjected to small transverse vibrations test. In particular, we characterize the nonmonotonous behavior shown by experimental data by using a nonmonotonous stress-softening function that depends on the amount of strain intensity. To consider residual strain effects, the procedure developed by Holzapfel et al. in [

The paper is organized as follows. In Section

We consider the deformation of an incompressible elastic body which in its natural configuration occupies the region

For an incompressible and isotropic elastic material, the corresponding time-independent Cauchy stress constitutive equation has the form

Based on this assumption, Elas-Zúñiga and Beatty proposed the following softening function:

Theoretical predictions provided by (

To confirm these observations, we use uniaxial tension data from a strain-controlled experiment (buna6) by Johnson on a buna-N elastomer in which, for each loading data point, there is a corresponding data point for the same amount of stretch on the unloading curve [

Normalized uniaxial extension data of buna-N elastomer strings plotted versus the normalized strain intensity ratio ^{3}. Here the color dots represent the experimental data and the dashed lines represent theoretical results obtained from (

Normalized frequency ratio experimental data of silicone elastomer cords plotted versus the normalized strain intensity ratio ^{3}. Here the color dots represent the experimental data and the dashed lines represent theoretical results obtained from (

Notice that (

Hyperelastic materials are rheologically described by employing strain energy density functions which are related to the energy stored in the material; there are several hyperelastic material models proposed to predict elastomeric material behavior, but most of them do not take into account residual strains [

Here,

Here, we follow Holzapfel et al. phenomenological model given by (

Substitution of (

If

On the primary loading path,

Therefore, the strain energy function that accounts for residual strains of an incompressible, isotropic, and hyperelastic material, in accordance with pseudo-elastic theory, has the form

In the next section, we will use (

To illustrate the application of the softening function given by (

First, let us begin with the modification of the average-stretch, full-network material model to include residual strain effects. For this material model, it is well known that the strain energy per unit volume for the loading path is given by

Substituting of (

Eliminating the pressure from (

Then, on elimination of

Recalling that for an incompressible material, the engineering stress

Here,

In equibiaxial extension, the engineering stress-softened relation is given by [

In the case of a pure shear or plane strain compression deformation state, the engineering stress-softened constitutive equation is given by

We next use Taylors’ equation [

Here, we briefly review some of the fundamental concepts that are used to describe the small amplitude transverse vibrations of rubber-like cords. Full details of the theoretical analysis can be found in [

The frequency of the fundamental mode of the small amplitude transverse vibrations about the stretched equilibrium position of an elastomeric cord with fixed ends is given by Taylor’s equation as

Of course, the normalized virgin cord frequency

To assess the precision of the derived constitutive (

We begin by using data collected from the transverse vibrations of a buna-N elastomer material cord [

Comparison of buna-N rubber strings experimental data with theoretical predictions computed from the average-stretch, full-network model for which ^{3}. Here, the color dots represent the experimental data and the solid and dashed lines represent theoretical results obtained from (

Figures

Comparison of silicone rubber strings experimental data with theoretical predictions computed from the average-stretch, full-network model for which ^{3} (experimental data adapted from [

Comparison of neoprene rubber strings experimental data with theoretical predictions computed from the average-stretch, full-network model for which ^{3} (experimental data adapted from [

In this paper, we have used a nonmonotonous softening function that accounts for the microstructural material damage upon deformation from the natural, undistorted state of the virgin material. This phenomenological softening function is based on three parameters; one is a positive softening material parameter and the other two are basically positive scaling constants that in general have the values of

The Arruda-Boyce constitutive equation for an average-stretch, full-network model of arbitrarily oriented molecular chains was modified to include the residual strain effect for the unloading paths as described by the simple constitutive relation (

Finally, we have also found, by using our theoretical model, excellent agreement with stress-softened experimental data collected for other deformation states. The results of this new work will be reported in a subsequent paper.

This work was funded by Tecnológico de Monterrey—Campus Monterrey, through the Research Chair in Nanomaterials for Medical Devices and Research Chair in Intelligent Machines and from Consejo Nacional de Ciencia y Tecnología (Conacyt), México.