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On the thermodynamics of nonlinear single integral representations for thermoviscoelastic materials with applications to one-dimensional wave propagation

Zur Beschreibung des nichtlinearen Verhaltens thermoviskoelastischer Stoffe durch einfache Integrale mit Anwendung auf eindimensionale Wellenausbreitung

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Summary

Thermodynamic theory is used to develop single integral constitutive relations for the nonlinear thermoviscoelastic response to arbitrary stress and temperature histories; the thermomechanically coupled energy equation is also obtained. The thermorheologically simple material, modified superposition and the isotropic stress power law are discussed in detail. A modified Fourier heat conduction law is employed to ensure that the propagation of thermal disturbances takes place at a finite velocity. Using the nonlinear thermoviscoelastic stress power law along with the linearized energy equation and modified Fourier law, one-dimensional wave front solutions are obtained.

Zusammenfassung

Mit Hilfe der Thermodynamik werden einfache Integrale enthaltende Werkstoffbeziehungen für das nichtlineare thermoviskoelastische Verhalten unter beliebigen Spannungs- und Temperaturverläufen entwickelt und die thermomechanisch gekoppelte Energiegleichung wird angegeben. Im Detail werden der thermodynamisch-einfache Werkstoff, die modifizierte Überlagerung und das isotrope Spannungs-Potenzgesetz diskutiert. Damit thermische Störungen sich mit endlicher Geschwindigkeit ausbreiten, wird ein modifiziertes Fouriersches Wärmeleitgesetz verwendet. Unter Verwendung des nichtlinearen thermoviskoelastischen Spannungs-Potenzgesetzes, der linearisierten Energiegleichung und des modifizierten Wärmeleitgesetzes werden Lösungen der eindimensionalen Wellenfrontausbreitung erhalten.

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Abbreviations

C ijkl :

linear elastic compliance, Equation (29)

C s ,C ts ,m s ,m tk ,v s ,v tk :

inelastic material constant, Equation (38)

C ε :

specific heat at constant strain, Equation (57)

C σ :

specific heat at constant stress, Equation (29)

E :

Young's modulus, Equation (34)

F kl :

tensor functions of stress, Equation (28)

f (1) mn ,f (2) kl :

tensor functions of the stress and temperature, Equation (8)

f[σ(t)]:

monotonically increasing function of stress, Equation (1)

G :

Gibbs free energy, Equation (2)

\(\bar G\) :

initial Gibbs free energy, Equation (29)

\(\bar G^E \) :

Gibbs free energy due to the instantaneous elastic response of the material, Equation (7)

\(\bar G^M \) :

Gibbs free energy due to memory, Equation (7)

ĝ i :

temperature gradient\(\left. {\frac{{\partial \Theta }}{{\partial x_i }}} \right|_t (x_i ,\xi )\), Equation (20)

H :

Helmholtz free energy, Equation (2)

J 1 :

first invariant of the stress tensor, Equation (38)

I 1 :

second invariant of the stress deviator tensor, Equation (38)

J(t) :

creep compliance function, Equation (1)

J ijkl (t) :

temperature independent material property, Equation (8)

J s :

steady creep compliance function, Equation (38)

J tk :

transient creep compliance function, Equation (38)

J I :

shear creep compliance function, Equation (34)

J II :

bulk creep compliance function, Equation (34)

K :

isotropic thermal conductivity, Equation (42a)

K ij :

thermal conductivity tensor, Equation (3)

M :

number of nonlinear memory integrals, Equation (36)

N=M+2:

number of components of strain, Equation (49)

n :

steady creep power, Equation (45)

Q :

one-dimensional heat flux vector, Equation (48 b)

Q i :

heat flux vector, Equation (3)

q i :

transient creep powers, Equation (46)

S :

entropy per unit mass, Equation (4)

\(\bar S\) :

initial entropy density, Equation (29)

s ij :

stress deviator tensor

T :

temperature

T 0 :

constant reference temperature, Equation (4)

t :

time

V 1,V 2 :

wave speeds

\(V_e = \sqrt {E/\varrho } \) :

uncoupled elastic mechanical wave speed

\(V_T = \sqrt {K/\varrho C_\sigma \tau } \) :

uncoupled thermal speed

x i :

space coordinate

α:

coefficient of thermal expansion, Equation (34)

α ij :

thermal strain coefficient, Equation (24)

γ, δ:

positive quantities in base characteristics equation, Equation (55)

ε:

one-dimensional strain

ε ij :

strain tensor

ε 1 :

linear elastic strain, Equation (49)

ε 2 :

steady creep strain, Equation (49)

ε 1,i=3,...,N :

transient creep strains, Equation (49)

ε T :

thermal strain, Equation (49)

\(\bar \varepsilon _{ij} \) :

initial strain, Equation (29)

η:

reciprocal of the isotropic conductivity, Equation (41)

Θ :

reciprocal of the conductivity

ϑ:

temperature difference betweenT and a constant reference temperature, Equation (3)

Θ 0 :

initial temperature discontinuity, Equation (73)

λ,µ i :

material constants, Equation (45), (46)

λ 1,λ 2,λ 3 :

functions of the three invariants of the stress tensor, Equation (35)

\(\bar \lambda , \bar \mu \) :

Lamé constants, Equation (57)

A :

rate of energy dissipation, Equation (13)

ν:

elastic Poisson's ratio, Equation (34)

ζ:

reduced time, Equation (17)

ϱ:

mass density, Equation (2)

σ:

one-dimensional stress

σ ij :

stress tensor

τ:

relaxation time of heat conduction, Equation (3)

τ i :

retardation time in transient creep, Equation (39)

ϕ:

shift factor, Equation (37)

[ ] j ,j=1, 2:

indicates a discontinuity across the leading and lagging wave fronts respectively

\(\widehat{( )}\) :

designates dependent variables in (x i , ζ) space

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  1. State University of New York at Buffalo, Buffalo, N.Y., USA

    W. P. Chang &  F. A. Cozzarelli

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  1. W. P. Chang
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This research was supported in part by the Office of Naval Research under Contract No. N00014-75-C-0302.

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Chang, W.P., Cozzarelli, F.A. On the thermodynamics of nonlinear single integral representations for thermoviscoelastic materials with applications to one-dimensional wave propagation. Acta Mechanica 25, 187–206 (1977). https://doi.org/10.1007/BF01376991

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