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Thermoelasticity with Finite Wave Speeds$

Józef Ignaczak and Martin Ostoja-Starzewski

Print publication date: 2009

Print ISBN-13: 9780199541645

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780199541645.001.0001

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Domain of Influence Theorems

Domain of Influence Theorems

Chapter:
(p.51) 4 Domain of Influence Theorems
Source:
Thermoelasticity with Finite Wave Speeds
Author(s):

Józef Ignaczak

Martin Ostoja‐Starzewski

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199541645.003.0004

Abstract and Keywords

The first part of this chapter concerns a domain of influence theorem for a potential‐temperature problem, which is a particular form of a displacement‐temperature problem in the L‐S theory. The second part of this chapter concerns an analogous theorem for the G‐L theory. In the third part we formulate a domain of influence theorem for a natural stress‐heat flux problem in the L‐S theory. In the fourth part we present a domain of influence theorem for a natural stress‐temperature problem in the G‐L theory. Parts 1–4 are restricted to the setting of a homogeneous and isotropic thermoelastic body. Finally, in the fifth part we formulate a number of domain of influence theorems for a nonhomogeneous anisotropic thermoelastic solid in the L‐S and G‐L theories.

Keywords:   homogeneous isotropic thermoelastic body, inhomogeneous anisotropic thermoelastic body, domain of influence theorems

In what follows, a solution to an initial‐boundary value problem associated with a thermoelastic process will also be called a thermoelastic disturbance. In particular, a solution to NDTP of the L–S (or G–L) theory is to be called a displacement–temperature disturbance of the L–S (or G–L) theory, and a solution to NSHFP of the L–S theory is to be called a stress–heat‐flux disturbance of this theory. In this chapter we shall formulate a number of theorems that imply that the thermoelastic disturbances described by the L–S and G–L theories have a character of waves propagating in B with finite wave speeds. Such theorems are called the domain of influence theorems.1

The first part of this chapter concerns a domain of influence theorem for a potential–temperature disturbance, which is a particular form of a displacement–temperature disturbance in the L–S theory. The second part of this chapter concerns an analogous theorem for the G–L theory. In the third part we formulate a domain of influence theorem for a natural stress–heat flux disturbance in the L–S theory. In the fourth part we present a domain of influence theorem for a natural stress–temperature disturbance in the G–L theory. Parts 1–4 are restricted to the setting of a homogeneous and isotropic thermoelastic body. Finally, in the fifth part we formulate a number of domain of influence theorems for a non‐homogeneous anisotropic thermoelastic solid in the L–S and G–L theories.

4.1 The potential–temperature problem in the Lord–Shulman theory

That problem represents a restriction of a NDTP of the L–S theory in which the displacement is taken as a gradient of a scalar field. Let us recall the formulation of this problem for a homogeneous and isotropic medium.2 Find a pair (u i,ϑ) satisfying the field equations

μ u i , k k + ( λ + μ ) u k , k i ρ u ¨ i ( 3 λ + 2 μ ) α ϑ , i = b i K ϑ , i i C E ( ϑ ˙ + t 0 ϑ ¨ ) ( 3 λ + 2 μ ) α θ 0 ( u ˙ k , k + t 0 u ¨ k , k ) = ( r + t 0 r ˙ ) on B × [ 0 , ) ,
(4.1.1)

(p.52) the initial conditions

u i ( · , 0 ) = u i 0 , u ˙ i ( · , 0 ) = u ˙ i 0 ϑ ( · , 0 ) = ϑ 0 , ϑ ˙ ( · , 0 ) = ϑ ˙ 0 on B ,
(4.1.2)

and the boundary conditions

u i = u i , ϑ = ϑ on B × ( 0 , ) .
(4.1.3)

Now, if we take b i = 0 on B × [0,∞) and let

u i = φ , i on B ¯ × [ 0 , ) ,
(4.1.4)

where ϕ is a scalar field defined on B × [0,∞), then the system (4.1.1) is satisfied as long as the pair (ϕ,ϑ) satisfies the system of equations

2 φ ρ λ + 2 μ φ ¨ 3 λ + 2 μ λ + 2 μ α ϑ = 0 2 ϑ C E k ( ϑ ˙ + t 0 ϑ ¨ ) on B × [ 0 , ) . 3 λ + 2 μ k α θ 0 2 ( φ ˙ + t 0 φ ¨ ) = 1 k ( r + t 0 r ˙ )
(4.1.5)

At this point, let us introduce the notations3

x ˆ 0 = k C E C 1 , t ˆ 0 = k C E C 1 2 ,
(4.1.6)

where

1 C 1 2 = ρ λ + 2 μ ,

and

φ ˆ 0 = ( 3 λ + 2 μ ) α θ 0 λ + 2 μ x ˆ 0 2 , ϑ ˆ 0 = θ 0 , r ˆ 0 = k θ 0 x ˆ 0 2 .
(4.1.7)

Clearly, x̂: 0 and t̂: 0 have the dimensions of length and time, respectively, while ϕ̂:0, ϑ̂:0 and r̂: 0 have the dimensions of potential, temperature and heat source. Taking these parameters as units of reference for respective quantities appearing in eqns (4.1.5) and keeping there the same notations for dimensionless quantities, we pass to the dimensionless form of eqns (4.1.5)

2 φ φ ¨ ϑ = 0 2 ϑ L ϑ ˙ 2 L φ ˙ = ( r + t 0 r ˙ ) on B × [ 0 , ) ,
(4.1.8)

(p.53) where

L = 1 + t 0 t ,
(4.1.9)
= ( 3 λ + 2 μ ) 2 α 2 θ 0 ( λ + 2 μ ) C E .
(4.1.10)

We now formulate the potential–temperature problem (PTP) in the L–S theory, with no heat sources (r = 0) and body forces (b i = 0). Find a pair (ϕ,ϑ) satisfying the field equations

2 φ φ ¨ ϑ = 0 2 ϑ L ϑ ˙ 2 L φ ˙ = 0 on B × [ 0 , ) ,
(4.1.11)

the initial conditions

φ ( · , 0 ) = φ 0 , φ ˙ ( · , 0 ) = φ ˙ 0 ϑ ( · , 0 ) = ϑ 0 , ϑ ˙ ( · , 0 ) = ϑ ˙ 0 on B ,
(4.1.12)

and the boundary conditions

φ , k n k = f , ϑ = g on B × ( 0 , ) ·
(4.1.13)

The pair (ϕ,ϑ) describes a thermoelastic process caused by a thermomechanical loading (ϕ0,ϕ̇00,ϑ̇0,f,g). It is evident that the initial conditions (4.1.12) are consistent with the conditions (4.1.2) and the hypothesis (4.1.4) for u i0 = ϕ0,i and u˙i0 = ϕ̇0,i. Moreover, the boundary conditions (4.1.3) and (4.1.13) are identical if eqn (4.1.3) 1 is replaced by the conditions for the normal component of the displacement vector, and g = ϑ.

The system of equations (4.1.11) is called a central system of equations in the L–S theory, while the problem described by eqns (4.1.11–13) is called a central problem of that theory.4 The role of the central problem of the L–S theory in solving a general problem of the theory is similar to that of an initial‐boundary value problem for the classical wave equation in solving a general problem of linear isothermal elastodynamics.

Definition 4.1 Let t ∈ (0,∞) be a fixed time. The set

D 0 ( t ) = { x B ¯ : ( 1 ) i f x B , t h e n φ 0 0 o r φ ˙ 0 0 o r ϑ 0 0 o r ϑ ˙ 0 0 , ( 2 ) i f ( x , τ ) B × [ 0 , t ] , t h e n f ( x , τ ) 0 o r g ( x , τ ) 0 }
(4.1.14)

is called the support of a thermomechanical load of PTP in the L–S theory.

(p.54) This definition is a generalization of the concept of the support of a function.5 It is apparent that, if the domain B fills the entire E 3 space, then in the central problem (4.1.11–13) there are no functions f and g. In such a case the PTP of the L–S theory is a Cauchy problem for an unbounded space and the set D 0(t) does not depend on time.

Definition 4.2 Let υ > 0 be a number satisfying the inequality

υ max ( 2 , 1 + , t 0 1 ) ,
(4.1.15)

and let Σ υ t(x) be an open ball of radius υ t, centered at x. The domain of influence for the thermomechanical load at time t for the PTP (4.1.11–13) is the set

D ( t ) = { x B ¯ : D 0 ( t ) Σ υ t ( x ) ¯ }
(4.1.16)

Clearly, D(t) is a set of all the points of B that may be reached by the thermomechanical disturbances propagating from D 0(t) with a speed not greater than υ.

We shall now formulate a theorem stating that the thermomechanical load restricted to the interval [0,t] does not influence the points outside the domain D(t).

Theorem 4.1 (On the domain of influence for a PTP in the L–S theory6) If the pair (ϕ,ϑ) is a smooth solution of PTP (4.1.11–13) and if D(t) is the domain of influence for the thermomechanical load at time t, then

φ = ϑ = 0 o n { B ¯ D ( t ) } × [ 0 , t ] .
(4.1.17)

The proof of this theorem is based on the following lemma:

Lemma 4.1 Let (ϕ,ϑ) be a solution to eqns (4.1.11–13), and let pC 1(B¯) denote a scalar field such that the set

E 0 = { x B ¯ : p ( x ) > 0 }
(4.1.18)

is bounded. Then

1 2 B [ P ( x , p ( x ) ) P ( x , 0 ) ] d υ + B 0 p ( x ) Q ( x , t ) d t d υ + B R i ( x , p ( x ) ) p , i ( x ) d υ = B 0 p ( x ) R i ( x , t ) n i ( x ) d t d a ,
(4.1.19)

(p.55) where

P ( x , t ) = ϵ ( 2 φ ˆ ) 2 + ϵ ( φ ˆ ˙ , i ) 2 + ϑ ˆ 2 + t 0 ( ϑ , i ) 2 ,
(4.1.20)
Q ( x , t ) = ( ϑ , i ) 2 ,
(4.1.21)
R i ( x , t ) = ϵ ( 2 φ ˆ ) φ ˆ ˙ , i + ϑ ˆ ( ϑ , i ϵ φ ˆ ˙ , i ) ( x , t ) B ¯ × [ 0 , ) ,
(4.1.22)

and the hut denotes the operator L7, that is

φ ˆ = L φ , ϑ ˆ = L ϑ o n B ¯ × [ 0 , ) .
(4.1.23)

Proof of Lemma 4.1 Applying the operator L to both sides of eqn (4.1.11) 1, and taking the gradient, we infer that the pair (ϕ,ϑ) satisfies the equations

( 2 φ ˆ ) , i φ ˆ ¨ ϑ ˆ , i = 0 2 ϑ ϑ ˆ ˙ ϵ 2 φ ˆ ˙ = 0 on B ¯ × [ 0 , ) .
(4.1.24)

Multiplying eqn (4.1.24) 1 through by ϵ φ ˆ ˙ , i and using the identities

φ ˆ ˙ , i ( 2 φ ˆ ) , i = ( φ ˆ ˙ , i 2 φ ˆ ) , i 1 2 t ( 2 φ ˆ ) 2 ,
(4.1.25)
φ ˆ ˙ , i φ ˆ ¨ , i = 1 2 t ( φ ˆ ˙ , i ) 2 ,
(4.1.26)

we get

ϵ 2 t { ( 2 φ ˆ ) 2 + ( φ ˆ ˙ , i ) 2 } + ϵ φ ˆ ˙ , i ϑ ˆ , i = ϵ ( φ ˆ ˙ , i 2 φ ˆ ) , i .
(4.1.27)

Next, multiplying eqn (4.1.24) 2 through by ϑ̂: and using the relations

ϑ ˆ 2 ϑ = ( ϑ ˆ ϑ , k ) , k ϑ ˆ , k ϑ , k ,
(4.1.28)
ϑ ˆ 2 φ ˆ ˙ = ( ϑ ˆ φ ˆ ˙ , k ) , k ϑ ˆ , k φ ˆ ˙ , k ,
(4.1.29)

we arrive at

1 2 t ϑ ˆ 2 + ϑ ˆ , k ϑ , k ϵ φ ˆ ˙ , k ϑ ˆ , k = [ ϑ ˆ ( ϑ , k ϵ φ ˆ ˙ , k ) ] , k .
(4.1.30)

Now, adding eqns (4.1.27) and (4.1.30), and recalling the definition of the hut operator, we obtain

1 2 t P ( x , t ) + Q ( x , t ) = R i , i ( x , t ) ,
(4.1.31)

(p.56) where P, Q and R i are defined by eqns (4.1.20), (4.1.21) and (4.1.22), respectively. Since

0 p ( x ) R i , i ( x , t ) d t = [ 0 p ( x ) R i ( x , t ) d t ] , i R i ( x , p ( x ) ) p , i ( x ) ,
(4.1.32)

the integration of eqn (4.1.31) from t = 0 up to t = p(x), will result in

1 2 [ P ( x , p ( x ) ) P ( x , 0 ) ] + 0 p ( x ) Q ( x , t ) d t + R i ( x , p ( x ) ) p , i ( x ) = [ 0 p ( x ) R i ( x , t ) d t ] , i .
(4.1.33)
Since the set E 0 defined by eqn (4.1.18) is bounded, each term in eqn (4.1.33) has a bounded support. Therefore, integrating eqn (4.1.33) over B and using the divergence theorem, we obtain eqn (4.1.19). □

Remark 4.1 The relation (4.1.19) is called a generalized energy identity for the problem (4.1.11–13). Setting p(x) = t, we obtain the classical energy identity for that problem.8

Proof of Theorem 4.1 Let (z,λ)∈ {B−D(t)} × (0,t) be a fixed point. Let

Ω = B ¯ Σ υ λ ( z ) ¯
(4.1.34)

and let

p λ ( x ) = { λ 1 υ | x z | for x Ω , 0 for x Ω ,
(4.1.35)

where υ is a parameter defined by the inequality (4.1.15). Since λ < t, it follows from the definitions of D(t) and Ω (recall eqns (4.1.16) and (4.1.34)) that the sets D 0(t) and Ω are disjoint:

Ω D 0 ( t ) = .
(4.1.36)

Hence,

φ , i n i = 0 , ϑ = 0 on ( Ω B ) × [ 0 , t ] ,
(4.1.37)

and

φ ˆ ˙ , k n k = 0 , ϑ ˆ = 0 on ( Ω B) × [ 0 , t ] .
(4.1.38)

(p.57) Furthermore,

φ ( · , 0 ) = φ ˙ ( · , 0 ) = ϑ ˙ ( · , 0 ) = ϑ ˙ ( · , 0 ) = 0 on Ω .
(4.1.39)

Thus, in view of eqns (4.1.35), (4.1.38), and (4.1.22), we obtain

B 0 P λ ( x ) R i ( x , t ) n i ( x ) d t d a = 0.
(4.1.40)

Also, eqns (4.1.39) and (4.1.11) 1 imply that

2 φ ˆ ( · , 0 ) = ϑ ˆ ( · , 0 ) = 0 on Ω ,
(4.1.41)
φ ˆ ˙ , i ( · , 0 ) = ϑ , i ( · , 0 ) = 0 on Ω .
(4.1.42)

Thus, from the definitions of P(x,t) and p λ(x) we get

P ( x , p λ ( x ) ) P ( x , 0 ) = { P ( x , p λ ( x ) ) for x Ω , 0 for x Ω .
(4.1.43)

Upon substitution of p λ(x) into eqn (4.1.19), and using eqns (4.1.40) and (4.1.43), we find

1 2 Ω P ( x , p λ ( x ) ) d υ + Ω 0 p λ ( x ) Q ( x , t ) d t d υ = Ω R i ( x , p λ ( x ) ) p λ , i ( x ) d υ .
(4.1.44)

Since Q ≥ 0 on Ω, from eqns (4.1.35) and (4.1.44) we obtain

1 2 Ω P ( x , p λ ( x ) ) d υ 1 υ Ω | R i ( x , p λ ( x ) ) | d υ .
(4.1.45)

From the definition of R i (recall eqn (4.1.22)) we obtain

| R i ( x , p λ ( x ) ) | ϵ | 2 φ ˆ | | φ ˆ ˙ , i | + | ϑ ˆ | ( | ϑ , i | + ϵ | φ ˆ ˙ , i | ) ϵ 2 { ( 2 φ ˆ ) 2 + 2 ( φ ˆ ˙ , i ) 2 + ϑ ˆ 2 } + { 1 2 ϑ ˆ 2 + ( ϑ , i ) 2 } .
(4.1.46)

Therefore, from the definition of P(x,t), and in view of the inequalities (4.1.45) and (4.1.46), we arrive at

Ω { ϵ 2 ( 1 1 υ ) ( 2 φ ˆ ) 2 + ϵ 2 ( 1 2 υ ) ( φ ˆ ˙ , i ) 2 + 1 2 ( 1 1 + ϵ υ ) ϑ ˆ 2 + 1 2 ( t 0 1 υ ) ( ϑ , i ) 2 } d υ 0.
(4.1.47)

(p.58) The definition of υ (recall eqn (4.1.15)) implies that the integrand of eqn (4.1.47) is a sum of non‐negative terms. Thus, the inequality (4.1.47) implies that each of those terms must vanish in Ω. In particular, we have

2 φ ˆ ( x , p λ ( x ) ) = 0 , ϑ ˆ ( x , p λ ( x ) ) = 0 on Ω .
(4.1.48)

In view of the definition of p λ(x) (recall eqn (4.1.35)), and since the pair (ϕ,ϑ) is sufficiently smooth, we also have

2 φ ˆ ( x , p λ ( x ) ) 2 φ ˆ ( z , λ ) ϑ ˆ ( x , p λ ( x ) ) ϑ ˆ ( z , λ ) as x z .
(4.1.49)

Hence, taking the limits in eqns (4.1.48) as xz, we find

2 φ ˆ ( z , λ ) = ϑ ˆ ( z , λ ) = 0.
(4.1.50)

Since (z,λ) is an arbitrary point in {B−D(t)} × (0,t), and since (ϕ, ϑ) is sufficiently smooth, hence

2 φ ˆ = ϑ ˆ = 0 on { B ¯ D ( t ) } × ( 0 , t ) .
(4.1.51)

Thus, eqns (4.1.51) and (4.1.11) 1 imply that

φ ˆ ʹ ʹ = ϑ ˆ = 0 on { B ¯ D ( t ) } × ( 0 , t ) ,
(4.1.52)

from which

φ ¨ ( x , τ ) = φ ¨ ( · , 0 ) exp ( τ t 0 1 ) ϑ ( x , τ ) = ϑ ( · , 0 ) exp ( τ t 0 1 ) for ( x , τ ) { B ¯ D ( t ) } × ( 0 , t ) .
(4.1.53)

In view of the definition of domain D(t) and eqn (4.1.11) 1,

φ ¨ ( · , 0 ) = ϑ ( · , 0 ) = 0 on B ¯ D ( t ) ,
(4.1.54)

so that, from eqns (4.1.53) we obtain

φ ¨ = ϑ = 0 on { B ¯ D ( t ) } × [ 0 , t ] .
(4.1.55)

Finally, recalling the definition of D(t) once again, and using eqn (4.1.55), we obtain

φ = ϑ = 0 on { B ¯ D ( t ) } × [ 0 , t ] .
(4.1.56)

This theorem implies that, for a finite time t and a bounded support of the thermomechanical loading (that is, for a bounded set D 0(t), recall eqn (4.1.14)), a thermoelastic disturbance generated by the pair (ϕ,ϑ) satisfying the system (4.1.11–13) vanishes outside the bounded set D(t), which depends on the support of the load, the material constants, and the relaxation time.

In other words, the said disturbance is propagated with a finite speed, bounded from above by the speed υ. If t 0 → 0, then in view of the definition of υ, it (p.59) follows that υarrow∞. Thus, for a vanishing relaxation time, the thermoelastic disturbance described by the pair (ϕ,ϑ) attains an infinite speed, a fact that may well have been expected since the PTP of the L–S theory reduces to a potential–temperature problem of classical thermoelasticity.

4.2 The potential–temperature problem in the Green–Lindsay theory

That problem is an analog of the PTP in the L–S theory (recall eqns (4.1.11–13)) and the related theorem on the domain of influence of Section 4.1. Hence, both the formulation and proof of the theorem are similar to what was given in Section 4.1.

For a homogeneous isotropic thermoelastic body, the NDTP in the G–L theory is formulated as follows:9 Find a pair (u i,ϑ) satisfying the field equations

μ u i , k k + ( λ + μ ) u k , k i ρ u ˙ ˙ i ( 3 λ + 2 μ ) α ( ϑ + t 1 ϑ ˙ ) , i = b i k ϑ , i i C E ( ϑ ˙ + t 0 ϑ ˙ ˙ ) ( 3 λ + 2 μ ) α θ 0 u ˙ k , k = r on B × 0 , ) ,
(4.2.1)

the initial conditions

u i ( · , 0 ) = u i 0 , u ˙ i ( · , 0 ) = u ˙ i 0 ϑ ( · , 0 ) = ϑ 0 , ϑ ˙ ( · , 0 ) = ϑ ˙ 0 on B,
(4.2.2)

and the boundary conditions

u i = u i , ϑ = ϑ on B × ( 0 , ) .
(4.2.3)

Setting b i = 0 on B × 0,∞), and

u i = φ , i on B ¯ × [ 0 , ) ,
(4.2.4)

where ϕ is a potential on B × [0,∞), we conclude that eqns (4.2.1) are satisfied so long as the pair (ϕ,ϑ) satisfies the equations

2 φ ρ λ + 2 μ φ ¨ 3 λ + 2 μ λ + 2 μ α ( ϑ + t 1 ϑ ˙ ) = 0 2 ϑ C E k ( ϑ ˙ + t 0 ϑ ¨ ) 3 λ + 2 μ k α θ 0 2 φ ˙ = r k on B × [ 0 , ) .
(4.2.5)

Transforming eqns (4.2.5) into a dimensionless form, in a way similar as in Section 4.1, and keeping the same notations for dimensionless quantities, we obtain

2 φ φ ¨ ( ϑ + t 1 ϑ ˙ ) = 0 2 ϑ ( ϑ ˙ + t 0 ϑ ¨ ) ϵ 2 φ ˙ = r on B × [ 0 , ) .
(4.2.6)

(p.60) The potential–temperature problem in the G–L theory with null body forces and null heat sources is now formulated as follows: Find a pair (ϕ, ϑ) satisfying the field equations

2 φ φ ¨ ( ϑ + t 1 ϑ ˙ ) = 0 2 ϑ ( ϑ ˙ + t 0 ϑ ¨ ) ϵ 2 φ ˙ = 0 on B × [ 0 , ) .
(4.2.7)

the initial conditions

φ ( · , 0 ) = φ 0 , φ ˙ ( · , 0 ) = φ ˙ 0 ϑ ( · , 0 ) = ϑ 0 , ϑ ˙ ( · , 0 ) = ϑ ˙ 0 on B,
(4.2.8)

and the boundary conditions

φ , k n k = f , ϑ = g on B × ( 0 , ) .
(4.2.9)
All the symbols here have the analogous meaning as in Section 4.1.

Relations (4.2.7) represent a central system of equations of the G–L theory, while a PTP described by the eqns (4.2.7–9) is a central problem of that theory. Clearly, the set D 0(t) of Section 4.1 is a support of the thermomechanical load for that problem (see eqn (4.1.14)), while the concept of a domain of influence for that problem is contained in the following definition.

Definition 4.3 Let υ > 0 be a number satisfying the inequality

υ max { 2 , ( 1 + ϵ ) t 1 t 0 , 1 t 1 } ,
(4.2.10)

and let Συ t(x) be a ball of radius υ t and center at point x. The domain of influence of a thermomechanical loading at time t for the central problem (4.2.7–9) is the set

D ( t ) = { x B ¯ : D 0 ( t ) Σ υ t ( x ) ¯ } .
(4.2.11)

Here, D 0(t) is given by the formula (4.1.14). Using this definition, we shall now prove

Theorem 4.2 (On the domain of influence for a PTP in the G–L theory)10 If the pair (ϕ,ϑ) is a smooth solution of PTP (4.2.7–9) and if D(t) is the domain of influence for the thermomechanical load at time t, then

φ = ϑ = 0 o n { B ¯ D ( t ) } × [ 0 , t ] .
(4.2.12)

The proof of this theorem is analogous to the proof of Theorem 4.1 of Section 4.1, and is based on the following lemma:

(p.61) Lemma 4.2 Let (ϕ,ϑ) be a solution of (4.2.7–9) and let p be a scalar field of Lemma 4.1. Then, the following generalized energy identity holds true

1 2 B [ P ( x , p ( x ) ) P ( x , 0 ) d υ + B 0 p ( x ) Q ( x , t ) d t d υ + B R i ( x , p ( x ) ) p , i ( x ) d υ = B 0 p ( x ) R i ( x , t ) n i ( x ) d t d a ,
(4.2.13)

where

P ( x , t ) = ϵ ( 2 φ ) 2 + ϵ ( φ ˙ , i ) 2 + t 1 ( ϑ , i ) 2 + t 0 t 1 [ ( ϑ + t 1 ϑ ˙ ) 2 + ( t 1 t 0 1 ) ϑ 2 ] ( x , t ) B × [ 0 , ) ,
(4.2.14)
Q ( x , t ) = ( ϑ , i ) 2 + ( t 1 t 0 ) ϑ ˙ 2 ( x , t ) B × [ 0 , ) ,
(4.2.15)
R i ( x , t ) = ( ϑ + t 1 ϑ ˙ ) ϑ , i + ϵ [ 2 φ ( ϑ + t 1 ϑ ˙ ) ] φ ˙ , i ( x , t ) B × [ 0 , ) .
(4.2.16)

Similar to what was done in Section 4.1, for convenience we have dropped (x,t) in the right‐hand sides of eqns (4.2.14–16).

Proof. By assumption, the pair (ϕ,ϑ) satisfies the system (4.2.7). Taking the gradient on both sides of eqn (4.2.7) 1 and multiplying through by ϕ̇,i, we obtain

φ ˙ , i 2 φ , i φ ˙ , i φ ¨ , i φ ˙ , i ( ϑ + t 1 ϑ ˙ ) , i = 0 ,
(4.2.17)

from which

1 2 t ( φ ˙ , i ) 2 + 1 2 t ( 2 φ ) 2 + φ ˙ , i ( ϑ + t 1 ϑ ˙ ) , i + ( t 1 t 0 ) φ ˙ , i ϑ ˙ , i = ( φ ˙ , i 2 φ ) , i .
(4.2.18)

On the other hand, multiplying eqn (4.2.7) 2 through by (ϑ+t 0 ϑ̇), we obtain

( ϑ + t 0 ϑ ˙ ) [ ( ϑ ϵ φ ˙ ) , i i ( ϑ ˙ + t 0 ϑ ¨ ) ] = 0
(4.2.19)

so that

1 2 t ( ϑ + t 0 ϑ ˙ ) 2 + ( ϑ + t 0 ϑ ˙ ) , i ( ϑ ϵ φ ˙ ) , i = [ ( ϑ + t 0 ϑ ˙ ) ( ϑ ϵ φ ˙ ) , i ] , i ,
(4.2.20)

or

1 2 t ( ϑ + t 0 ϑ ˙ ) 2 + t 0 2 t ( ϑ , i ) 2 + ( ϑ , i ) 2 ϵ φ ˙ , i ( ϑ + t 0 ϑ ˙ ) , i = [ ( ϑ + t 0 ϑ ˙ ) ( ϑ ϵ φ ˙ ) , i ] , i .
(4.2.21)

(p.62) Then, multiplying eqn (4.2.7) 2 through by ϑ̇, we get

ϑ ˙ [ ( ϑ ϵ φ ˙ ) , i i ( ϑ ˙ + t 0 ϑ ¨ ) ] = 0 ,
(4.2.22)

so that

t 0 2 t ϑ ˙ 2 + 1 2 t ( ϑ , i ) 2 + ϑ ˙ 2 ϵ φ , i ϑ ˙ , i = [ ϑ ˙ ( ϑ ϵ φ ˙ ) , i ] , i .
(4.2.23)

Now, adding eqn (4.2.18) multiplied through by ε with eqns (4.2.21) and (4.2.23) multiplied through by (t 1t 0), we obtain

1 2 t { ϵ [ ( φ ˙ , i ) 2 + ( 2 φ ) 2 ] + ( ϑ + t 0 ϑ ˙ ) 2 + t 0 ( ϑ , i ) 2 + ( t 1 t 0 ) [ t 0 ϑ ˙ 2 + ( ϑ , i ) 2 ] } + ( ϑ , i ) 2 + ( t 1 t 0 ) ϑ ˙ 2 = { ( ϑ + t 1 ϑ ˙ ) ϑ , i + ϵ [ 2 φ ( ϑ + t 1 ϑ ˙ ) ] φ ˙ , i } , i .
(4.2.24)

The latter relation may also be written in the form

1 2 t P ( x , t ) + Q ( x , t ) = [ R , i ( x , t ) ] , i ,
(4.2.25)

where P, Q and R are defined by eqns (4.2.14–16), respectively. Now, integrating eqn (4.2.25) from t = 0 to t = p(x), and using eqn (4.1.32), we arrive at

1 2 [ p ( x , p ( x ) ) P ( x , 0 ) ] + 0 p ( x ) Q ( x , t ) d t = [ 0 p ( x ) R i ( x , t ) d t ] , i R i ( x , p ( x ) ) p , i ( x ) .
(4.2.26)
Finally, integrating eqn (4.2.26) over B and using the divergence theorem, we obtain eqn (4.2.13).

Proof of Theorem 4.2 Similarly to Section 4.1, we fix a point (z,λ) ∈ {B−D(t)} × (0,t) and introduce the set

Ω = B ¯ Σ υ λ ( z ) ¯ .
(4.2.27)

Moreover, we define a scalar function p λ(x) using the formula

p λ ( x ) = { λ υ 1 | x z | for x Ω , 0 for x Ω .
(4.2.28)

Then, p λ(x)>0 on Ω and

| p λ , i ( x ) | = { υ 1 on Ω , 0 on B ¯ Ω .
(4.2.29)

(p.63) Using the definitions of domains D(t) and Ω, and the inequality λ < t, we conclude that D 0(t) and Ω are disjoint, that is

Ω D 0 ( t ) = ,
(4.2.30)

so that

φ ˙ , i n i = 0 , ϑ + t 1 ϑ ˙ = 0 on ( Ω B ) × [ 0 , t ] ,
(4.2.31)
φ ( · , 0 ) = φ ˙ ( · , 0 ) = ϑ ( · , 0 ) = ϑ ˙ ( · , 0 ) = 0 on Ω .
(4.2.32)

From the definition of R i (recall eqn (4.2.16)), and from the formulas (4.2.28) and (4.2.31), we find

B 0 p λ ( x ) R i ( x , t ) n i ( x ) d t d a = 0.
(4.2.33)

Also, from the definition of P (recall eqn (4.2.14)), and from the formulas (4.2.28) and (4.2.32), we find

P ( x , p λ ( x ) ) P ( x , 0 ) = { P ( x , p λ ( x ) ) for x Ω , 0 for x Ω .
(4.2.34)

Clearly, p λ(x) satisfies the assumptions of Lemma 4.2, and so, substituting p(x) ≡ p λ(x) into eqn (4.2.13), as well as making use of eqns (4.2.33–34) and (4.2.29) 2, we obtain

1 2 Ω P ( x , p λ ( x ) ) d υ + Ω 0 p λ ( x ) Q ( x , t ) d t d υ = Ω R i ( x , p λ ( x ) ) p λ , i ( x ) d υ .
(4.2.35)

Since Q ≥ 0 (t 1 t 0), the relations (4.2.29)1 and (4.2.35) imply the inequality

1 2 Ω P ( x , p λ ( x ) ) 1 υ Ω | R i ( x , p λ ( x ) ) | d υ .
(4.2.36)

From the definition of R i (recall eqn (4.2.16)) we get

| R , i | | ϑ + t 1 ϑ ˙ | | ϑ , i | + ϵ | 2 φ ( ϑ + t 1 ϑ ˙ ) | | φ ˙ , i | | ϑ + t 1 ϑ ˙ | | ϑ , i | + ϵ | φ ˙ , i | ( | 2 φ | + | ϑ + t 1 ϑ ˙ | ) = | ϑ + t 1 ϑ ˙ | ( | ϑ , i | + ϵ | φ ˙ , i | ) + ϵ | φ ˙ , i | | 2 φ | 1 2 { ( 1 + ϵ ) ( ϑ + t 1 ϑ ˙ ) 2 + ( ϑ , i ) 2 + 2 ϵ ( φ ˙ , i ) 2 + 2 ϵ ( 2 φ ) 2 } .
(4.2.37)

(p.64) Thus, the definition of P (recall eqn (4.2.14)) and the inequalities (4.2.36) and (4.2.37) lead to the relation

ϵ 2 ( 1 1 2 ) Ω [ ( 2 φ ) 2 + ( φ ˙ , i ) 2 ] d υ + 1 2 ( t 1 1 υ ) Ω ( ϑ , i ) 2 d υ + 1 2 ( t 0 t 1 1 + ϵ υ ) Ω ( ϑ + t 1 ϑ ˙ ) 2 d υ + 1 2 ( 1 t 0 t 1 ) Ω ϑ 2 d υ 0.
(4.2.38)

The definition of the parameter υ (recall eqn (4.2.10)) implies that the coefficients in front of the integrals in the inequality (4.2.38) are non‐negative. With these integrals being non‐negative as well, the inequality (4.2.38) implies

2 φ ( x , p λ ( x ) ) = 0 , ϑ ( x , p λ ( x ) ) = 0 x Ω .
(4.2.39)

Hence, in view of the definition of p λ(x) and the continuity of ∇2ϕ and ϑ,

2 φ ( x , p λ ( x ) ) 2 φ ( z , λ ) ϑ ( x , p λ ( x ) ) ϑ ( z , λ ) as x z .
(4.2.40)

Thus, taking the limit xz in eqns (4.2.39), we obtain

2 φ ( z , λ ) = 0 , ϑ ( z , λ ) = 0.
(4.2.41)

Since (z,λ) is an arbitrary point of the set {B−D(t)} × (0,t) and the pair (ϕ,ϑ) is sufficiently smooth on B × [0,∞), the relations (4.2.41) imply

ϑ = 2 φ = 0 on { B ¯ D ( t ) } × [ 0 , t ] .
(4.2.42)

From this and eqn (4.2.7) 1 we find

φ ¨ = 0 on { B ¯ D ( t ) } × [ 0 , t ] .
(4.2.43)

Now, since

φ ( · , 0 ) = φ ˙ ( · , 0 ) = 0 on B ¯ D ( t ) ,
(4.2.44)

the relation (4.2.43) yields

φ = 0 on { B ¯ D ( t ) } × [ 0 , t ] ,
(4.2.45)
which, in view of eqns (4.2.42) 1 and (4.2.45) gives eqn (4.2.12), thus completing the proof of Theorem 4.2. □

Theorem 4.2 implies that a potential–temperature disturbance described by eqns (4.2.7–9) propagates with a speed not greater than υ specified by eqn (4.2.10). The maximum speed of the disturbance propagating out of the domain D 0(t) depends on both relaxation times t 0 and t 1, and on the parameter of thermoelastic coupling ε. The speed υ becomes unbounded in two cases: (a) for t 1 → 0; (b) for t 1 > 0 with t 0 → 0.

(p.65) Clearly, the thermoelastic disturbances governed by eqns (4.2.7–9) are generally different from those governed by eqns (4.1.11–13). However, these disturbances have a number of common characteristics. For instance, for t 1 = t 0 > 0 the potential–temperature disturbances of the G–L theory possess the same domain of influence as the potential–temperature disturbances of the L–S theory; compare here the Definition 4.3 of Section 4.2 with the Definitions 4.1 and 4.2 of Section 4.1. Also note that, proceeding in the same manner as in Sections 4.1 and 4.2, we can formulate a number of general theorems on the domain of influence for the conventional and non‐conventional thermoelastic processes both in the L–S and the G–L theories. In particular, these general theorems may be formulated for the mixed displacement–temperature problems of Section 2.1, see (Ignaczak et al., 1986).

4.3 The natural stress–heat‐flux problem in the Lord–Shulman theory

First, we note that a NSHFP for a homogeneous isotropic body with one relaxation time involves finding a pair (S ij,q i) satisfying the field equations11

p 1 S ( i k , k j ) [ 1 2 μ ( S ¨ i j λ 3 λ + 2 μ S ¨ k k δ i j ) α 2 θ 0 C s S ¨ k k δ i j ] + α C S q ˙ k , k δ i j = F ( i j ) 1 C s ( q k , k + α θ 0 S ˙ k , k ) , i 1 k ( q ˙ i + t 0 q ¨ i ) = g i on B × [ 0 , ) ,
(4.3.1)

the initial conditions

S i j ( · , 0 ) = S i j ( 0 ) , S ˙ i j ( · , 0 ) = S ˙ i j ( 0 ) q i ( · , 0 ) = q i ( 0 ) , q ˙ i ( · , 0 ) = q ˙ i ( 0 ) on B,
(4.3.2)

and the boundary conditions

S i j n j = s i q i n i = q on B × ( 0 , ) .
(4.3.3)

In eqns (4.3.1) we have set12

F ( i j ) = ρ 1 b ( i , j ) C s 1 α r ˙ δ i j g i = C s 1 r , i on B ¯ × [ 0 , ) .
(4.3.4)
Certainly, the stress–heat‐flux thermoelastic disturbances described by the eqns (4.3.1–4) are more general than those studied in Sections 4.1 and 4.2. (p.66) Therefore, the domain of influence is also more general here than the previous ones.

The set

D 0 ( t ) = { x B ¯ : ( 1 ) If x B then S i j ( 0 ) 0 o r S ˙ i j ( 0 ) 0 or q i ( 0 ) 0 or q ˙ i ( 0 ) 0 ; ( 2 ) If ( x , τ ) B × [ 0 , t ] , then F ( i j ) 0 or g i 0 ; ( 3 ) If ( x , τ ) B × [ 0 , t ] , then s i 0 or q 0 }
(4.3.5)
is called a support of the thermomechanical loading at time t for the problem (4.3.1–3).

The domain of influence of the thermomechanical loading at time t for the problem (4.3.1–3) is the set

D ( t ) = { x B ¯ : D 0 ( t ) Σ υ t ( x ) ¯ φ } ,
(4.3.6)

where υ is a parameter with dimension of velocity, satisfying the inequality13

υ max ( υ 1 , υ 2 , υ 3 ) ,
(4.3.7)

where

υ 1 = ( 2 μ ρ ) 1 2 ,
(4.3.8)
υ 2 = { 3 λ + 2 μ ρ C S C E [ 1 ( 1 C E C S ) 1 2 ] 1 } 1 2 ,
(4.3.9)
υ 3 = { k t 0 1 C S [ 1 + C S C E ( 1 C E C S ) 1 2 ] } 1 2 .
(4.3.10)

The following theorem holds true:

Theorem 4.3 (On the domain of influence for a NSHFP in the L–S theory)14 If the pair (S ij,q i) is a smooth solution of NSHFP described by eqns (4.3.1–3) and if D(t) is given by the formula (4.3.6), then

S i j = 0 , q i = 0 on { B ¯ D ( t ) } × [ 0 , t ] .
(4.3.11)

The proof of the theorem is based on the following lemma of Zaremba type:

(p.67) Lemma 4.3 Let (S ij,q i) be a solution of eqns (4.3.1–3) and let p be a scalar field of Lemma 4.1 of Section 4.1. Then, the following generalized energy identity holds true for the problem (4.1.1–3)15

1 2 B [ P ( x , p ( x ) ) P ( x , 0 ) ] d υ + B 0 p ( x ) Q ( x , t ) d t d υ + B R i ( x , p ( x ) ) p , i ( x ) d υ = B 0 p ( x ) S ( x , t ) d t d υ + B 0 p ( x ) R i ( x , t ) n i ( x ) d t d a ,
(4.3.12)

where

P ( x , t ) = ρ 1 S i k , k S i j , j + 1 2 μ ( S ˙ i j S ˙ i j λ 3 λ + 2 μ S ˙ k k 2 ) α 2 θ 0 C S S ˙ k k 2 + 1 θ 0 C S ( q k , k ) 2 + t 0 θ 0 k ( q ˙ i ) 2 o n B ¯ × [ 0 , ) ,
(4.3.13)
Q ( x , t ) = 1 θ 0 k ( q ˙ i ) 2 o n B ¯ × [ 0 , ) ,
(4.3.14)
R i ( x , t ) = p 1 S ˙ i j S j k , k + C S 1 ( θ 0 1 q k , k + α S ˙ k k ) q ˙ i o n B ¯ × [ 0 , ) ,
(4.3.15)
S ( x , t ) = F ( i j ) S ˙ i j + θ 0 1 g i q ˙ i o n B ¯ × [ 0 , ) .
(4.3.16)

Proof of Lemma 4.3 Multiplying eqn (4.3.1) 1 through by Ṡij and eqn (4.3.1) 2 through by θ0 −1 q˙i, and adding the results, we obtain

1 2 t P ( x , t ) + Q ( x , t ) = [ R i ( x , t ) ] , i + S ( x , t ) ,
(4.3.17)
where P, Q, R i, S are given by formulas (4.3.13–16). Next, we integrate eqn (4.3.17) from t = 0 to t = p(x) with the use of formula (4.1.32). Finally, integrating the result over B and using the divergence theorem, we obtain the required identity (4.3.12). □

Proof of Theorem 4.3 Proceeding in a way similar to that employed in the proof of Theorem 4.1 of Section 4.1, we fix a point (z,λ) ∈ {B−D(t)} × (0,t), and introduce the set

Ω = B ¯ Σ υ λ ( z ) ¯ .
(4.3.18)

Moreover, we define a scalar function p λ(x) using the formula

p λ ( x ) = { λ υ 1 | x z | for x Ω , 0 for x Ω .
(4.3.19)

(p.68) where υ is specified by eqn (4.3.7). Then, p λ(x) >0 on Ω and

Ω D 0 ( t ) = φ .
(4.3.20)

From this,

S ˙ i j n i = 0 , q ˙ i n i = 0 on ( Ω B ) × ( 0 , t ) ,
(4.3.21)
F ( i j ) = 0 , g i = 0 on Ω × ( 0 , t ) ,
(4.3.22)
S i j ( · , 0 ) = S ˙ i j ( · , 0 ) = q i ( · , 0 ) = q ˙ i ( · , 0 ) = 0 on Ω ,
(4.3.23)

as well as

B 0 p λ ( x ) R i ( x , t ) n i ( x ) d t d a = 0 ,
(4.3.24)
B