American Institute of Mathematical Sciences

Viscoelasticity with limiting strain

 Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  June 2019 Revised  October 2019 Published  April 2020

A self-contained review is given for the development and current state of implicit constitutive modelling of viscoelastic response of materials in the context of strain-limiting theory.

Citation: Yasemin Şengül. Viscoelasticity with limiting strain. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020330
References:

show all references

References:
Limiting strain behaviour
Experimental data for the stress-strain relationship for porcine carotid and thoracic artery tissues (cf. [43])
Left. Model A: $g(T) = \beta T + \alpha \left(1 + \frac{\gamma}{2} T^{2}\right)^{n} T$; Model B: $g(T) = \frac{T}{(1 + |T|^{r})^{1/r}}$; Model C: $g(T) = \alpha \left\{\left[1 - \exp\left(- \frac{\beta T}{1 + \delta |T|}\right)\right] + \frac{\gamma T}{1 + |T|} \right\}$; Model D: $g(T) = \alpha \left(1-\frac{1}{1 +\frac{ T}{1 + \delta |T|}}\right) + \beta \left(1 + \frac{1}{1 + \gamma T^{2}}\right)^{n} T$, where $\alpha, \beta, \gamma, \delta, n$ and $r > 0$ are constants. Right. General linear, quadratic and cubic nonlinearities
 [1] B. L. G. Jonsson. Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations. Inverse Problems & Imaging, 2009, 3 (3) : 405-452. doi: 10.3934/ipi.2009.3.405 [2] Merab Svanadze. On the theory of viscoelasticity for materials with double porosity. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2335-2352. doi: 10.3934/dcdsb.2014.19.2335 [3] Zhen Lei. Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2861-2871. doi: 10.3934/dcds.2014.34.2861 [4] Claude Vallée, Camelia Lerintiu, Danielle Fortuné, Kossi Atchonouglo, Jamal Chaoufi. Modelling of implicit standard materials. Application to linear coaxial non-associated constitutive laws. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1641-1649. doi: 10.3934/dcdss.2013.6.1641 [5] Dorin Ieşan. Strain gradient theory of porous solids with initial stresses and initial heat flux. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2169-2187. doi: 10.3934/dcdsb.2014.19.2169 [6] A. C. Eberhard, J-P. Crouzeix. Existence of closed graph, maximal, cyclic pseudo-monotone relations and revealed preference theory. Journal of Industrial & Management Optimization, 2007, 3 (2) : 233-255. doi: 10.3934/jimo.2007.3.233 [7] Donatella Donatelli, Corrado Lattanzio. On the diffusive stress relaxation for multidimensional viscoelasticity. Communications on Pure & Applied Analysis, 2009, 8 (2) : 645-654. doi: 10.3934/cpaa.2009.8.645 [8] Monica Conti, V. Pata. Weakly dissipative semilinear equations of viscoelasticity. Communications on Pure & Applied Analysis, 2005, 4 (4) : 705-720. doi: 10.3934/cpaa.2005.4.705 [9] Stephen Doty and Anthony Giaquinto. Generators and relations for Schur algebras. Electronic Research Announcements, 2001, 7: 54-62. [10] David Iglesias-Ponte, Juan Carlos Marrero, David Martín de Diego, Edith Padrón. Discrete dynamics in implicit form. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1117-1135. doi: 10.3934/dcds.2013.33.1117 [11] Victor Zvyagin, Vladimir Orlov. Weak solvability of fractional Voigt model of viscoelasticity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6327-6350. doi: 10.3934/dcds.2018270 [12] Marta Lewicka, Piotr B. Mucha. A local existence result for a system of viscoelasticity with physical viscosity. Evolution Equations & Control Theory, 2013, 2 (2) : 337-353. doi: 10.3934/eect.2013.2.337 [13] Valeria Danese, Pelin G. Geredeli, Vittorino Pata. Exponential attractors for abstract equations with memory and applications to viscoelasticity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2881-2904. doi: 10.3934/dcds.2015.35.2881 [14] Monica Conti, Elsa M. Marchini, Vittorino Pata. Semilinear wave equations of viscoelasticity in the minimal state framework. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1535-1552. doi: 10.3934/dcds.2010.27.1535 [15] Christophe Cheverry, Adrien Fontaine. Dispersion relations in cold magnetized plasmas. Kinetic & Related Models, 2017, 10 (2) : 373-421. doi: 10.3934/krm.2017015 [16] Artur Babiarz, Adam Czornik, Michał Niezabitowski, Evgenij Barabanov, Aliaksei Vaidzelevich, Alexander Konyukh. Relations between Bohl and general exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5319-5335. doi: 10.3934/dcds.2017231 [17] Evelyn Sander. Hyperbolic sets for noninvertible maps and relations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 339-357. doi: 10.3934/dcds.1999.5.339 [18] Gilles A. Francfort, Alessandro Giacomini, Alessandro Musesti. On the Fleck and Willis homogenization procedure in strain gradient plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 43-62. doi: 10.3934/dcdss.2013.6.43 [19] Ryusuke Kon. Dynamics of competitive systems with a single common limiting factor. Mathematical Biosciences & Engineering, 2015, 12 (1) : 71-81. doi: 10.3934/mbe.2015.12.71 [20] Tony Liimatainen, Mikko Salo. Nowhere conformally homogeneous manifolds and limiting Carleman weights. Inverse Problems & Imaging, 2012, 6 (3) : 523-530. doi: 10.3934/ipi.2012.6.523

2019 Impact Factor: 1.233

Tools

Article outline

Figures and Tables