A thermodynamically consistent non-linear mathematical model for thermoviscoelastic plates/shells with finite deformation and finite strain based on classical continuum mechanics

Surana, Karan S.; Mathi, S. S. C.

doi:10.1016/j.ijnonlinmec.2020.103565

Cited by 5 publications

(22 citation statements)

References 57 publications

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“…In a recent paper Surana et al [13] presented thermodynamically and mathematically consistent mathematical model in Lagrangian description using CCM for thermoviscoelastic solids without memory using contravariant second Piola-Kirchhoff stress tensor, Green's strain tensor (a covariant measure) and rates of Green's strain tensor up to order n. This mathematical model was used to derive a space-time decoupled finite element formulation for plates and shells, thus incorporating finite deformation, finite strain and rates of finite strain up to order n with formulations. In this formulation, stiffness as well as damping matrices are up to quadratic functions of the dofs, hence are nonlinear.…”

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confidence: 95%

“…The resulting mathematical model cannot be supported by CBL of CCM, hence is thermodynamically inconsistent. In reference [13] authors consider the same 1D constitutive theory purely in time as used in reference [9]. The authors express 2) and (3) in reference [10]?…”

Section: )mentioning

confidence: 99%

“…In another recent paper by Surana et al [14] various aspects of finite deformation, finite strain nonlinear dynamics and dynamic bifurcation in TVES without memory are considered in detail using currently used 1D phenomenological models as well as using the thermodynamically and mathematically consistent mathematical models based on CCM presented in reference [13]. The work presented in this paper is an extension, modification and application of many concepts and methodologies presented by Surana et al in reference [13], hence we summarize significant aspects of the findings and the conclusion of reference [13] in the following. 1) Considering the mathematical model for TVEs without memory based on CCM, if we use space-time decoupled finite element method in conjunction with GM/WF in space, and then we obtain a system of nonlinear ODEs in time.…”

Section: )mentioning

confidence: 99%

“…It is obvious a single ODE in time as in (13) cannot possibly describe the same physics as the system of ODEs (9), hence also can describe the nonlinear dynamics physics in (7) -( 9). This mathematical model (10) has been discussed in ref [14] and is in widespread use in the nonlinear dynamics studies in many published works.…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

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Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

Surana,

Mathi

2024

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This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy inequality and the representation theorem for thermoviscoelastic solids (TVES) with rheology. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics and are based on contravariant deviatoric second Piola-Kirchhoff stress tensor and its work conjugate covariant Green's strain tensor and their material derivatives of up to order m and n respectively.

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Section: )mentioning

confidence: 95%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Introduction and Literature Reviewmentioning

confidence: 99%

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Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

Surana,

Mathi

2024

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“…Quadratic elements (e.g., the quad elements with eight or nine nodes) have additional degrees of freedom for the sides that allow them to curve and represent the actual deformation of the element much better [44] and thus, manifest no shear/membrane/bending locking or mechanism formation [45]. For the same reasoning, some researchers have precisely used the nine-node elements [23,46] for finite deformation analyses.…”

Section: F I G U R E 2 the Employed Nine-node Biquadratic Quadrilater...mentioning

confidence: 99%

An accurate hyperelasticity‐based plate theory and nonlinear energy‐based micromechanics for impact and shock analyses of compliant particle‐reinforced FG hyperelastic plates

Shariyat

Abedi

2022

Z Angew Math Mech

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The current article is devoted to the nonlinear dynamic and impact response analyses of rectangular incompressible neo‐Hookean hyperelastic plates with uniform/nonuniform transverse distributions of stiff elastic reinforcing particles. The nonlinear finite element governing equations are derived by using the principle of minimum total potential energy and solved by an updating scheme that uses the Runge‐Kutta time integration and penalty methods. The impact energy expressions are written for the integrated plate‐indenter system. The novelties of the article are: (i) realistic modeling the energy absorption of a particle‐reinforced nonlinear hyperelastic matrix based on the volume fractions of the constituent phases instead of using wrong Voigt‐type models or impractical predefined gradation coefficients, (ii) considering non‐uniform distributions of the reinforcing particles, (iii) using a highly accurate asymmetric (all the available theories are symmetric) infinite‐order plate theory, (iv) including the inherent hyperelastic nature of the material in the definition of the parameters of the plate theory, (v) considering the thickness changes of the plate, (vi) including von Karman's kinematic nonlinearity, (vii) analyzing both the dynamic and impact behaviors, (viii) proposing an adaptive algorithm to match the description of the displacement field with the load/stress variations, (ix) updating not only the stiffness but also the mass matrix within the time steps, (x) incorporation of the nonlinear boundary conditions by the penalty method, and (xi) performing time‐dependent analyses instead of the natural frequency analyses. Results reveal the drastic effects of the volume fraction of the reinforcing particles on the dynamic responses and report the impact responses of the FG hyperelastic plates, for the first time.

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Thermodynamic theory of the most energy-efficient natural repose angle

Zegzulka

Maršík

Nečas

et al. 2023

Continuum Mech. Thermodyn.

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A thermodynamically consistent non-linear mathematical model for thermoviscoelastic plates/shells with finite deformation and finite strain based on classical continuum mechanics

Cited by 5 publications

References 57 publications

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

An accurate hyperelasticity‐based plate theory and nonlinear energy‐based micromechanics for impact and shock analyses of compliant particle‐reinforced FG hyperelastic plates

Thermodynamic theory of the most energy-efficient natural repose angle

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