# American Institute of Mathematical Sciences

December  2017, 10(6): 1351-1373. doi: 10.3934/dcdss.2017072

## Two boundary value problems involving an inhomogeneous viscoelastic solid

 1 Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India 2 Department of Civil Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India 3 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA

* Corresponding author: Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA. Ph: 1 979 8624552

Dedicated to Prof. T. Roubíček on his sixtieth birthday

Received  July 2016 Revised  November 2016 Published  June 2017

Recently, a thermodynamically consistent non-linear constitutive equation has been developed to describe the large deformation cyclic response of viscoelastic polyamides (see [17]). In this paper, two boundary value problems within the context of the above model, namely the stress relaxation of a right circular annular cylinder subject to twisting, and the inflation of a sphere are studied. In addition to solving the above problems numerically, investigation of the merits and pitfalls of studying the same boundary value problem for a special class of inhomogeneous body and its homogenized counterpart is undertaken. This study finds that for moderate strains the differences in relaxation time between the actual inhomogeneous body and its homogenized counterparts may be significant.

Citation: M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072
##### References:

show all references

Dedicated to Prof. T. Roubíček on his sixtieth birthday

##### References:
Targeted boundary displacement in the inhomogeneous body
Applied boundary force on the inhomogeneous body to realize the desired boundary displacement
Variation of $\mu_{mean-hom}$ with different inner to outer radius ratio $R_i/R_o$ for different spatial variations of $\mu_1$ in the inhomogeneous model obtained by correlating the variation of the torsional moment required to engender a given angle of twist
Variation in percentage relaxation time with $R_i/R_o$ for linear variation of $\mu_1$ obtained by correlating the variation of the torsional moment required to engender a given angle of twist
Variation of $\mu_{mean-hom}$ with $R_i/R_o$ for linear variation of $\mu_1$ obtained by different correlations to engender a given angle of twist
Percentage variation in relaxation time with rate of twisting and different thicknesses of the cylinder for linear variation of $\mu_1$ obtained by correlating the variation of the torsional moment
Variation of $\mu_{mean-hom}$ with $R_i/R_o$ for different spatial variations of $\mu_1$ obtained by correlating the variation of the radial component of the normal stress applied at the inner surface of the sphere to engender a given inflation
Radial variation of the cylindrical polar components of the Cauchy stress at time $t$ $=$ $0.25$ s for different radial variations of $\mu_1$ when the annular cylinder is subjected to pure twist
Radial variation of the spherical polar components of the Cauchy stress at time $t$ $=$ $0.25$ s for different radial variations of $\mu_1$ when the annular sphere is subjected to inflation
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