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Some results on the one-dimensional coupled nonlinear thermoviscoelastic wave propagation problem with second sound

Einige Ergebnisse zum eindimensionalen, gekoppelten, nichtlinearen Wellenausbreitungsproblem mit zweitem Schall

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Summary

One-dimensional stress and temperature fields in a suddenly loaded and/or heated semi-infinite rod of nonlinear thermoviscoelastic material are studied using coupled thermomechanical theory. The transport of heat is governed by the modified Fourier heat conduction law, and thus proceeds by wave propagation rather than by diffusion. The application of thermal and mechanical disturbances at the end of the rod gives rise to two wave fronts along which these disturbances propagate. Field solutions for the stress and temperature are obtained by numerical integration along the five characteristics of the governing equations, and results are presented for several linear and nonlinear viscoelastic models.

Zusammenfassung

Eindimensionale Spannungs- und Temperaturfelder in einem plötzlich belasteten und/oder erwärmten, halbunendlichen Stab aus nichtlinearem, viskoplastischen Material werden mit Hilfe der gekoppelten thermodynamischen Theorie untersucht. Der Wärmetransport gehorcht dem modifizierten Fourierschem Wärmeleitungsgesetz und erfolgt daher eher durch Wellenausbreitung als durch Diffusion. Die Aufbringung thermischer oder mechanischer Störungen am Ende des Stabs läßt zwei Wellenfronten entstehen, mit welchen die Störungen fortschreiten. Feldlösungen für Spannungen und Temperatur werden durch numerische Integration längs der fünf Charakteristiken der beschreibenden Gleichungen erhalten und die Resultate für einige lineare und nichtlineare viskoelastische Modelle angegeben.

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Abbreviations

C σ :

specific heat at constant stress

E :

Young's modulus

J :

integer as an indication of position

K :

integer for the condensation of computer storage, Eq. (42)

k :

isotropic thermal conductivity

\(k_i = \frac{1}{{\tau _i \mu _i ^{q_i } }}\) :

nondimensional material constants

M :

number of nonlinear memory integrals, Eq. (2)

N :

number of components of strain, Eq. (7)

n :

steady creep power, Eq. (2)

Q :

one-dimensional heat flux, Eq. (1)

q i :

transient creep powers, Eq. (2)

T :

temperature

T 0 :

constant reference temperature

t :

time

\(t_s = (\lambda /E)^n /\bar \sigma _0 ^{n - 1} \) :

time scale for nondimensionalization

V 1,V 2 :

coupled wave speeds, Eq. (6a)

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

uncoupled elastic mechanical wave speed

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

uncoupled thermal wave speed

v :

particle velocity

x :

space coordinate

α:

coefficient of thermal expansion

γ, δ:

positive nondimensional quantities governing the wave speeds, Eqs. (5)

ε:

one-dimensional strain

ε1 :

linear elastic strain, Eq. (7b)

ε2 :

steady creep strain, Eq. (7c)

ε i ,i=3, ...,N :

transient creep strains, Eq. (7d)

ε T :

thermal strain, Eq. (7e)

θ=T−T 0 :

temperature increment relative to constant reference temperatureT 0

θ 0 :

input temperature discontinuity

λ,µ i :

material constants, Eq. (2)

ϱ:

mass density

σ:

one-dimensional stress

σ i :

input stress discontinuity

τ:

relaxation time of heat conduction, Eq. (1)

τ i :

retardation time of transient creep, Eq. (2)

[] j ,j=1, 2:

indicates a discontinuity across the leading and lagging wave fronts respectively

(−):

indicates nondimensional variables and parameters (drooped after Eq. (9))

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Authors and Affiliations

  1. State University of New York at Buffalo, Buffalo, NY, USA

    C. Lee & F. A. Cozzarelli

  2. National Taiwan University, Taipei, Taiwan, China

    W. P. Chang

Authors
  1. C. Lee
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  2. W. P. Chang
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  3. F. A. Cozzarelli
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With 12 Figures

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

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Lee, C., Chang, W.P. & Cozzarelli, F.A. Some results on the one-dimensional coupled nonlinear thermoviscoelastic wave propagation problem with second sound. Acta Mechanica 37, 111–129 (1980). https://doi.org/10.1007/BF01441248

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  • DOI: https://doi.org/10.1007/BF01441248

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