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Thermo-hyperelastic models for nanostructured materials

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Abstract

In the framework of continuum thermodynamics, the present paper presents the thermo-hyperelastic models for both the surface and the bulk of nanostructured materials, in which the residual stresses are taken into account. Due to the existence of residual stresses, different configuration descriptions of the surface (or the bulk) thermo-hyperelastic constitutive equations are not the same even in the cases of infinitesimal deformation. As an example, the effective thermal expansion coefficient of spherical nanoparticles is analyzed.

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Abbreviations

1 :

identity tensor in 3D Euclidean space

A i :

covariant base vectors of the reference configuration

C, C*, Ĉ :

Green deformation tensor defined from the reference to the current configurations, from the stress-free to the reference configurations and from the stress-free to the current configurations, respectively

C sE , C E :

surface and bulk specific heat at constant volume

E, E*, Ê :

Green stain tensor defined from the reference to the current configurations, from the stress-free to the reference configurations and from the stress-free to the current configurations, respectively

E s :

surface Green strain tensor

F, F*, \( \hat F \) :

deformation gradient defined from the reference to the current configurations, from the stress-free to the reference configurations and from the stress-free to the current configurations, respectively

F s , F −1 s :

surface deformation gradient and its inverse

F (o) s , F −1(o) s :

out-plane terms of F s and F −1 s

i :

index ranging over the integers 1, 2 and 3

I 0 :

identity tensor on the tangent plane of the surface in the reference configurations

J, J*:

the determinants of F and F*, respectively

J 1, J 2 :

two invariants of the surface right stretch tensor

k s , k b :

surface and bulk thermal conductivities

L :

elasticity tensor defined in the reference configuration

L 1 :

stress-temperature modulus defined in the reference configuration

P i :

covariant base vectors of the stress-free configuration

q 0 b :

heat flows entering the surface from the bulk defined in the reference configurations

q 0 s , q 0 b :

surface and bulk heat fluxes defined in the reference configuration

\( \hat q_b^R \) R b :

initial value of q 0 b

r 0, \( \bar r \) :

radii of nanoparticles in the reference and the stress-free configurations

r 0 s , r 0 b :

surface and bulk heat supply defined in the reference configuration

S :

first kind Piola-Kirchhoff stress

S s :

first kind Piola-Kirchhoff stress of the surface

T R :

bulk residual stress in the reference configuration

T s :

second kind Piola-Kirchhoff stress of the surface

T s , T b :

surface and bulk temperature, respectively

T 0 s , T 0 b :

surface and bulk reference temperature, respectively

u :

displacement measured from the reference configuration

U s :

surface right stretch tensor

w 0 s :

surface free energy per unit area defined in the reference configuration

α e :

effective thermal expansion coefficient of nanoparticles

β s , β b :

surface and bulk coefficients related to thermal expansion

γ :

surface energy per unit area defined in the current configuration

γ 0*, γ 1*, γ 1 :

surface tension, surface Lamé constants

ɛ 0 s :

surface internal energy per unit area defined in the reference configuration

η 0 s :

surface entropy per unit area defined in the reference configurations

η :

bulk entropy per unit volume

η R :

residual value of η in the reference configuration

λ, μ :

bulk Lamé constants

θ s , θ b :

surface and bulk temperature change, respectively

σ, σ s :

Cauchy stress and surface Cauchy stress, respectively

∇0, ∇0s :

bulk and surface gradient operators defined in the reference configuration

(∧):

quantity defined in the stress-free configuration

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Authors and Affiliations

  1. State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

    ZhiQiao Wang & YaPu Zhao

  2. School of Engineering and Technology, China University of Geosciences, Beijing, 100083, China

    ZhiQiao Wang

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  1. ZhiQiao Wang
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  2. YaPu Zhao
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Correspondence to YaPu Zhao.

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Wang, Z., Zhao, Y. Thermo-hyperelastic models for nanostructured materials. Sci. China Phys. Mech. Astron. 54, 948–956 (2011). https://doi.org/10.1007/s11433-011-4299-8

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  • DOI: https://doi.org/10.1007/s11433-011-4299-8

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