For a new class of elastic bodies, where the linearized strain tensor is given as a function of the Cauchy stress tensor, the problem of considering unsteady motions is studied. A system of partial differential equations that only depends on the stress tensor is found from the equation of motion, which is a system of six partial differential equations for the six components of the stress tensor. A simple boundary value problem is solved for a 1D bar using exact and numerical methods.
RajagopalKR. On implicit constitutive theories. Appl Math2003; 48: 279–319.
2.
RajagopalKR. The elasticity of elasticity. Z Angew Math Phys2007; 58: 309–317.
3.
RajagopalKR. Conspectus of concepts of elasticity. Math Mech Solids2011; 16: 536–562.
4.
RajagopalKRSrinivasaAR. On the response of non-dissipative solids. Proc Roy Soc Lond A2007; 463: 357–367.
5.
RajagopalKRSrinivasaAR. 2009 On a class of non-dissipative solids that are not hyperelastic. Proc Roy Soc Lond A2009; 465: 493–500.
6.
RajagopalKR. On a new class of models in elasticity. Math Computat Appl2010; 15: 506–528.
7.
BustamanteRRajagopalKR. A note on plain strain and stress problems for a new class of elastic bodies. Math Mech Solids2010; 15: 229–238.
8.
BustamanteR. Some topics on a new class of elastic bodies. Proc Roy Soc Lond A2009; 465: 1377–1392.
9.
TruesdellCANollW. In: AntmanSS (ed) The non-linear field theories of mechanics. Berlin: Springer, 2004.
10.
RajagopalKR. Non-linear elastic bodies exhibiting limiting small strain. Math Mech Solids2011; 16: 122–139.
11.
BustamanteRRajagopalKR. Solutions of some simple boundary value problems within the context of a new class of elastic materials. Int J Nonlinear Mech2011; 46: 376–386.
12.
OrtizABustamanteRRajagopalKR. A numerical study of a plate with a holte for a new class of elastic bodies. Acta Mech2012; 223(9): 1971–1981.
13.
RajagopalKRWaltonJR. Modeling fracture in the context of strain-limiting theory of elasticity: A single anti-plane crack. Int J Fracture2011; 169: 39–48.
14.
BustamanteRRajagopalKR. On the inhomogeneous shearing of a new class of elastic bodies. Math Mech Solids2012; 17: 762–778.
15.
SaitoTFurutaTHwangJH. Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism. Science2003; 300: 464–467.
16.
WitheyEJimMMinorA. The deformation of ‘Gum Metal’ in nanoindentation. Mater Sci Eng A2008; 493: 26–32.
17.
ZhangSQLiSJJiaMT. Fatigue properties of a multifunctional titanium alloy exhibiting nonlinear elastic deformation behavior. Scripta Mater2009; 60: 733–736.
18.
JohnsonPARasolfosaonPNJ. Manifestation of nonlinear elasticity in rock: Convincing evidence over large frequency and strain intervals from laboratory studies. Nonlinear Proc Geoph1996; 3: 77–88.
19.
GuyerRAJohnsonPA. Nonlinear mesoscopic elasticity: Evidence for a new class of materials. Phys Today1999; 52: 30–36.
20.
McCallKRGuyerRA. Equation of state and wave propagation in hysteretic nonlinear elastic materials. J Geophys Res1994; 99: 23887–23897.
21.
LuZ. Role of hysteresis in propagation acoustic waves in soils. Geophys Res Lett2005; 32: L14302.
22.
OstrovskyLA. Wave processes in media with strong acoustic nonlinearity. J Acoust Soc Am1991; 90: 3332–3337.
23.
Ten CateJAShanklandTJ. Slow dynamics in the nonlinear elastic response of Berea sandstone. Geophys Res Lett1996; 23: 3019–3022.
RajagopalKR. On the non-linear elastic response of bodies in the small strain range. Acta Mech2014; 255(6): 1545–1553.
26.
PopovicsSRoseJL. and Popovics, JS. The behavior of ultrasonic pulses in concrete. Cement Concrete Res1990; 20: 259–270.
27.
IacovacheM. Asupra relaţiilor dintre tensiuni într’um corp elastic în mişare. Buletin Ştiinţific, Seria1950; 2: 700–707.
28.
VâlcoviciV. Relaţiilor între tensiuni. Comunicările Academiei Republicii Populare Române1951; 4: 337–339.
29.
IgnaczakJ. A completeness problem for stress equations of motion in the linear elasticity theory. Arch Mech1963; 15: 225–234.
30.
IgnaczakJ. Direct determination of stresses from the stress equations of motion in elasticity. Arch Mech1959; 11: 671–678.
31.
ChadwickP. Continuum mechanics: Concise theory and problems. Mineola, NY: Dover Publications, 1999.
32.
TruesdellCAToupinR. The classical field theories. In FlüggeS (ed) Handbuch der Physik, Vol. III/1. Berlin: Springer, 1960, pp.226–793.
33.
RajagopalKRSaravananU. Spherical inflation of a class of compressible elastic bodies. Int J Nonlinear Mech2011; 46: 1167–1176.
34.
SfyrisDBustamanteR. Use of some theorems related with the tensor equation AX + XA = H for some classes of implicit constitutive relations. Q J Mech Appl Math2013; 66:157–163.
35.
GurtinME. The linear theory of elasticity. In TruesdellCA (ed) Mechanics of solids, Vol. II. Berlin; Heidelberg; New York; Tokyo: Springer Verlag, 1984, pp.1–295.
36.
PolyaninADZaitsevVF. Handbook of exact solutions for ordinary differential equations. Boca Raton, FL; London, New York, Washington DC: Chapman & Hall/CRC Press, 2003.
37.
Ortiz-BernardinABustamanteRRajagopalKR. A numerical study of elastic bodies that are described by constitutive equations that exhibit limited strains. Int J Sol Struct2014; 51(3–4): 875–885.
38.
Comsol Multiphysics Version 3.4, Comsol Inc.Palo Alto, CA, 2007.
39.
KannanKRajagopalKRSaccomandiG. Unsteady motions of a new class of elastic solids. Wave Motion2014; 51(5): 833–843.
40.
SaadaAS. Elasticity: Theory and application. Malabar, FL: Krieger Publishing Company, 1993.
41.
RajagopalKR. Rethinking constitutive theories. In preparation.
