
We study a mathematical model of mass points longitudinally oscillating between thermoelastoplastic springs. It is derived as a discrete version of a continuous model of longitudinal oscillations of a one-dimensional object. The problem is formulated as a system of nonlinear ordinary differential equations with Prandtl-Ishlinskii type of nonlinearity, subsequently simplified using the first integral of the energy. We show that the system is asymptotically directed to one of the many possible steady states, where all movements cease and temperatures equalize.
Citation: Jana Kopfová, Petra Nábělková. Thermoelastoplastic oscillator with Prandtl-Ishlinskii operator[J]. Mathematics in Engineering, 2025, 7(3): 264-280. doi: 10.3934/mine.2025012
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We study a mathematical model of mass points longitudinally oscillating between thermoelastoplastic springs. It is derived as a discrete version of a continuous model of longitudinal oscillations of a one-dimensional object. The problem is formulated as a system of nonlinear ordinary differential equations with Prandtl-Ishlinskii type of nonlinearity, subsequently simplified using the first integral of the energy. We show that the system is asymptotically directed to one of the many possible steady states, where all movements cease and temperatures equalize.
This paper is devoted to an open problem formulated in [1], namely, to the mathematical analysis of a model of mass points longitudinally oscillating between thermoelastoplastic springs and their properties.
In [1] we studied the model of mass points longitudinally oscillating between thermoelastic springs, see Figure 1:
m¨uj(t)=σj(t)−σj−1(t),j=1,…,n−1,σj(t)=k(uj+1(t)−uj(t))−β(θj(t)−θ∗),j=0,…,n−1,c˙θj(t)=κ(θj+1(t)−2θj(t)+θj−1(t))−βθj(t)(˙uj+1(t)−˙uj(t)),j=0,1,…,n−1, |
where uj(t) denotes the displacement of the j-th mass point from its equilibrium at time t, σj(t) is the stress of the j-th spring, θj(t) is the temperature of the j-th spring and θ∗>0 is the reference temperature. m is the weight of the mass point, k is the stiffness of the springs, β is the heat expansion coefficient of the springs, c is their specific heat capacity, κ is the heat conductivity of the mass, and the physical constants m, β, c, κ are always positive.
We showed that the system with suitable initial and boundary conditions is asymptotically directed to the unique steady state, where all movements cease and the temperatures equalize.
Here we assume that our material is elastoplastic, i.e., the stress σ depends on ux in a nonlinear way, expressed by the Prandtl-Ishlinskii operator, see [2], classically used for modelling elastoplastic materials.
Thus, we consider a system of ordinary differential equations describing a longitudinal oscillation of n−1 mass points between n thermoelastoplastic springs, namely
m¨uj(t)=σj(t)−σj−1(t),j=1,…,n−1, | (1.1) |
σj(t)=P[uj+1(t)−uj(t)]−β(θj(t)−θ∗),j=0,…,n−1, | (1.2) |
c˙θj(t)=κ(θj+1(t)−2θj(t)+θj−1(t))−βθj(t)(˙uj+1(t)−˙uj(t))+|D[uj+1(t)−uj(t)]t|,j=0,1,…,n−1, | (1.3) |
with the initial conditions for j=0,1,…,n−1
θj(0)=θ0j>0, | (1.4) |
uj(0)=u0j, | (1.5) |
˙uj(0)=u1j, | (1.6) |
and with the conditions
θ−1(t)=θ0(t), | (1.7) |
θn(t)=θn−1(t), | (1.8) |
u0(t)=un(t)=0. | (1.9) |
Here P is the Prandtl-Ishlinskii operator and D is the associated dissipation operator, (2.5), see Section 2. Let us note here and later that both the dots and the subscript t denote the time derivatives.
We show that the total energy of the system remains constant while the entropy is increasing. Furthermore, we prove that the spring temperatures remain positive and bounded and that the displacement of the mass point between two springs and its speed and acceleration are also bounded.
Using the energy's first integral, we simplify the original system of differential equations and study the stability of the new system. We show that the system is asymptotically directed to one of the many possible steady states, even though the system does not lose energy. The presence of infinitely many steady states is created by the plastic part of the springs, namely, the particular form of the Prandtl-Ishlinskii operator. This is contrary to the case of thermoelastic springs, where the system converges to a unique equilibrium, see [1].
The global asymptotic stability can usually be expected in dissipative systems, i.e., systems that give energy to the environment. However, the role of the oscillating point between thermoelastoplastic springs is fundamentally different from the physical point of view. The energy of the system is preserved because the heat conductivity of the mass point causes the transformation of the mechanical energy into the thermal energy and vice versa. In agreement with the second law of thermodynamics, heat conduction causes the entropy of the system to increase, and the process is therefore irreversible. Mathematical analysis of our model of mass points longitudinally oscillating between thermoelastoplastic springs shows that the system with arbitrary initial conditions is asymptotically directed to one of the steady states, where all movements cease and the temperatures equalize, thereby simulating the principle of the heat death of the universe.
The system (1.1)–(1.3) can be further studied with non-constant material constants, namely assuming that m,β,c,κ are different for each mass point and each spring.
While there are many papers devoted to the study of the thermoelastic problems, see e.g., [3] for an extending review of published models, and other ones, mainly in the physics literature, dealing with the heat death of the universe, among all see e.g., [4,5,6]. We could not find a model similar to the one presented here in the literature.
The paper is organized as follows: Section 2 is devoted to a brief introduction to the Prandtl-Ishlinskii operator and its basic properties. In Section 3, we derive the system of ordinary differential equations for n−1 mass points as a discrete version of a continuous model of longitudinal oscillations of a one-dimensional object. The latter one we derive from the laws of continuum thermomechanics, see e.g., [7]. In Section 4, we show that the system's energy is constant, the spring temperatures are always positive, the speed of the points and the temperatures of the springs are bounded, and the entropy is increasing. In Section 5, the behavior of the system at infinity is studied.
Given a nonnegative function γ∈L1(0,∞), we define the Prandtl-Ishlinskii operator P:W1,1(0,T)→W1,1(0,T) by the integral
P[ε](t)=∫∞0γ(r)sr[ε](t)dr, | (2.1) |
where sr is the stop operator with threshold r. The modeling idea goes back to [8,9]. Let us first recall the definition of the stop operator.
Definition 2.1. Let ε∈W1,1(0,T) and r>0 be given. The variational inequality
ε(t)=σ(t)+ξ(t)∀t∈[0,T],|σ(t)|≤r∀t∈[0,T],˙ξ(t)(σ(t)−z)≥0∀|z|≤r,σ(0)=Qr(ε(0)),} | (2.2) |
where Qr is the projection of R onto the interval [−r,r], defines the stop and play operators sr and pr by the formula, see Figure 2,
σ(t)=sr[ε](t),ξ(t)=pr[ε](t). | (2.3) |
The stop and play operators were introduced in [10]. The parameter r is the memory variable, and for each given time t0, the functions r↦pr[ε](t0), r↦sr[ε](t0) represent the memory state of the system. Let us list here some basic properties of the play and stop operators. The proofs are elementary and can be found, e.g., in [2,11].
Proposition 2.2. Let ε1,ε2∈W1,1(0,T) and r>0 be given, σi=sr[εi], ξi=εi−σi=pr[εi], i=1,2. Then
(i)(σ1(t)−σ2(t))(˙ε1(t)−˙ε2(t)) ≥ 12ddt(σ1(t)−σ2(t))2for a.e.t∈[0,T];
(ii)|˙ξ1(t)−˙ξ2(t)|+ddt|σ1(t)−σ2(t)| ≤ |˙ε1(t)−˙ε2(t)|for a.e.t∈[0,T];
(iii)|σ1(t)−σ2(t)| ≤ 2max0≤τ≤t|ε1(τ)−ε2(τ)|∀t∈[0,T];
(iv)˙ξi(t)˙εi(t)=˙ξi(t)2for a.e.t∈[0,T].
It is easy to see that the variational inequality (2.2) can be equivalently written in the form
˙ε(t)σ(t)=ddt(12σ2(t))+r|˙ξ(t)|, |
representing the energy balance. Indeed, ˙ε(t)σ(t) is the power supplied to the system, which is partly used for the increase of the potential 12σ2(t), and the rest r|˙ξ(t)| is dissipated. This enables us to establish the energy balance for the Prandtl-Ishlinskii operator (2.1). Indeed, if we define the Prandtl-Ishlinskii potential
V[ε](t)=12∫∞0γ(r)s2r[ε](t)dr, | (2.4) |
and the dissipation operator
D[ε](t)=∫∞0rγ(r)pr[ε](t)dr, | (2.5) |
we can write the Prandtl-Ishlinskii energy balance in the form
˙ε(t)P[ε](t)=ddtV[ε](t)+|ddtD[ε](t)| a.e. | (2.6) |
As a consequence of Proposition 2.2 (iv), we have
|ddtD[ε](t)|≤|˙ε(t)|∫∞0rγ(r)dr. | (2.7) |
We will assume throughout the whole paper the following:
Hypothesis 2.3. P is a Prandtl-Ishlinskii operator (2.1) with a measurable distribution function γ:(0,∞)→[0,∞), and
M:=∫∞0rγ(r)dr<∞. | (2.8) |
Hypothesis 2.3 enables us to estimate the dissipation from above in terms of the input velocity.
Longitudinal oscillations of a one-dimensional object of length L are considered, u(x,t) denotes the displacement of point x∈[0,L] at time t>0. The model is derived from the one-dimensional laws of continuum mechanics. From Newton's second law, the volumetric force density is proportional to the spatial derivative of the stress σ.
So we get
σx(x,t)=ρutt(x,t), | (3.1) |
where utt(x,t) is the body's acceleration and ρ is the volumetric density of the substance.
The stress for the thermoelastoplastic material is expressed as
σ(x,t)=P[ux(x,t)]−β(θ(x,t)−θ∗), | (3.2) |
where P represents the Prandtl-Ishlinskii operator, see (2.1), β is the heat expansion coefficient, θ∗ is the reference temperature and θ(x,t) is the current absolute temperature at point x and time t. With the constitutive law (3.2), we associate the free energy operator
F(ux,θ)=cθ(1−ln(θ/θ∗))+V[ux]−β(θ−θ∗)ux, | (3.3) |
where V is the Prandtl-Ishlinskii potential (2.4), and c (the specific heat capacity) is a given constant. The entropy operator S and the internal energy operator U then read
S(ux,θ)=−∂F∂θ=cln(θ/θ∗)+βux, | (3.4) |
U(ux,θ)=F(ux,θ)+θS(ux,θ)=cθ+V[ux]+βθ∗ux. | (3.5) |
We consider the first and the second principles of thermodynamics in the form
U(ux,θ)t+qx=uxtσ, | (3.6) |
S(ux,θ)t+(qθ)x≥0, | (3.7) |
where q=−κθx is the heat flux with a constant heat conductivity κ>0. Note that (3.6) is the energy conservation law, (3.7) is the Clausius-Duhem inequality. We rewrite (3.6) using (3.2), (3.5), and (2.6) as
cθt−κθxx=|D[ux]t|−βθuxt, | (3.8) |
where D[ux] is the dissipation operator (2.5). Inequality (3.7) can be formally written in the form
1θ(|D[ux]t|+κθ2xθ)≥0, |
and is clearly satisfied.
Putting together Eqs (3.1), (3.2) and (3.8) we obtain the following system of equations
ρutt(x,t)=σx(x,t),σ(x,t)=P[ux(x,t)]−β(θ(x,t)−θ∗),cθt(x,t)=κθxx(x,t)−βθ(x,t)uxt(x,t)+|D[ux(x,t)]t|. |
We discretize this system in space, and assuming the weight is equally distributed, we get
m¨uj(t)=σj(t)−σj−1(t),j=1,…,n−1,σj(t)=P[uj+1(t)−uj(t)]−β(θj(t)−θ∗),j=0,…,n−1,c˙θj(t)=κ(θj+1(t)−2θj(t)+θj−1(t))−βθj(t)(˙uj+1(t)−˙uj(t))+|D[uj+1(t)−uj(t)]t|,j=0,…,n−1. |
This is the system (1.1)–(1.3), that describes our problem.
The energy of the system (1.1)–(1.3) is as follows
E(t)=m2n−1∑j=1˙u2j(t)+n−1∑j=0V[uj+1(t)−uj(t)]+cn−1∑j=0θj(t), | (4.1) |
where the first term in the expression represents the kinetic energy of the points, the second term is their potential energy, and the last term represents the heat energy of the springs.
Lemma 4.1. The energy of the system (1.1)–(1.3) is positive and constant, i.e., there exists E0>0 such that E(t)=E0, for all t≥0.
Proof. We will be using the summation by parts formulas for arbitrary test sequences a1,…,an and b1,…,bn,
n∑k=1(ak−ak−1)bk=anbn−a0b1−n−1∑k=1(bk+1−bk)ak, | (4.2) |
n∑k=1(ak+1−2ak+ak−1)bk=(an−an−1)bn−(a1−a0)b1−n−1∑k=1(ak+1−ak)(bk+1−bk). | (4.3) |
We write Eqs (1.1) and (1.2) together as
m¨uj(t)=P[uj+1(t)−uj(t)]−P[uj(t)−uj−1(t)]+β(θj−1(t)−θj(t)), | (4.4) |
for j=1,…,n−1, and test (4.4) by ˙uj(t)
m¨uj(t)˙uj(t)=(P[uj+1(t)−uj(t)]−P[uj(t)−uj−1(t)])˙uj(t)+β(θj−1(t)−θj(t))˙uj(t). |
Using (1.3), (1.7), (1.9), (4.2) and (4.3) and summing for j=0,1,…,n−1, we have
mn−1∑j=1¨uj(t)˙uj(t)+cn−1∑j=0˙θj(t)=−n−1∑j=0[P[uj+1(t)−uj(t)](˙uj+1(t)−˙uj(t))]+n−1∑j=0|D[uj+1(t)−uj(t)]t|, | (4.5) |
and after rewriting and using (2.6)
ddt[m2n−1∑j=1˙u2j(t)+n−1∑j=0V[uj+1(t)−uj(t)]+cn−1∑j=0θj(t)]=0. |
Thus, the energy of the system (4.1) must be constant, which means that there exists E0 such that E(t)=E0, ∀t∈[0,∞). Accordingly also E(0)=E0, i.e.,
E(0)=m2n−1∑j=1(u1j)2+n−1∑j=0V[u0j+1−u0j]+cn−1∑j=0θ0j=E0, |
which implies using (1.4) and (2.4) that E0>0.
We rewrite the second order equation (1.1) with (1.2) as a first order system and together with (1.3), we get the following system
˙uj(t)=vj(t),˙vj(t)=1m(P[uj+1(t)−uj(t)]−P[uj(t)−uj−1(t)])+βm(θj−1(t)−θj(t)),c˙θj(t)=κ(θj+1(t)−2θj(t)+θj−1(t))−βθj(t)(˙uj+1(t)−˙uj(t))+|D[uj+1(t)−uj(t)]t|, | (4.6) |
for j=1,…n−1 in the first two equations and j=0,…,n−1 in the last one.
We now simplify the system (4.6). Instead of the equations for the temperatures of the springs θj,j=0,...,n−1, we formulate the equations for the temperature differences
wj(t):=θj−1(t)−θj(t),j=1,…,n−1, |
by using the energy equality E(t)=E0 in the form
n−1∑j=0θj(t)=1c[E0−m2n−1∑j=1˙u2j(t)−n−1∑j=0V[uj+1(t)−uj(t)]], |
and we introduce, for simplicity, the new constants notation:
q:=βm, r:=βE0nc2, g:=κc. |
The system (4.6) is therefore as follows
˙uj(t)=vj(t),˙vj(t)=qwj(t)+Pj(t),˙wj(t)=r(vj+1(t)−2vj(t)+vj−1(t))+g(wj+1(t)−2wj(t)+wj−1(t))+˜hj(t), | (4.7) |
with w0=wn=0, u0=un=0, v0=vn=0,
Pj(t):=1m(P[uj+1(t)−uj(t)]−P[uj(t)−uj−1(t)]), |
j=1,…,n−1, and ˜hj(t), j=1,…,n−1, are given as follows
˜hj=βc[vj−1Aj+vjBj+vj+1Cj]+1c(|D[uj(t)−uj−1(t)]t|−|D[uj+1(t)−uj(t)]t|), |
where
Aj=1n[−m2cn−1∑j=1v2j−1cn−1∑j=0V[uj+1(t)−uj(t)]]−j−1∑i=1inwi+n−1∑i=jn−inwi, |
for j=2,…,n−1,
Bj=−2n[−m2cn−1∑j=1v2j−1cn−1∑j=0V[uj+1(t)−uj(t)]]+j−1∑i=12inwi−n−2jnwj−n−1∑i=j+12(n−i)nwi, |
for j=1,…,n−1,
Cj=1n[−m2cn−1∑j=1v2j−1cn−1∑j=0V[uj+1(t)−uj(t)]]−j∑i=1inwi+n−1∑i=j+1n−inwi, |
for j=1,…,n−2.
Let us note that we only need Aj for j>1, because v0=0 and Cj for j<n−1, because vn=0. We can write the system (4.7) in a matrix form as
˙U(t)=AU(t)+f(U(t)), | (4.8) |
where the matrix A is a 3(n−1)×3(n−1) constant matrix consisting of nine square block matrices As, s=1,…,9, of order n−1, i.e.,
(A1A2A3A4A5A6A7A8A9), |
where A1=A3=A4=A5=A7 are null matrices, A2=I is the unit matrix, A6=qI and A8, A9 are symmetric tridiagonal matrices in the form
As=(−2aaa−2aa⋱⋱⋱⋱⋱⋱aa−2a), |
s=8,9 and a=r,g respectively.
U(t)=(u1(t),…,un−1(t),v1(t),…,vn−1(t),w1(t),…,wn−1(t))T |
and
f(U(t))=(0,…,0,P1(t),…,Pn−1(t),˜h1(t),…,˜hn−1(t))T. |
The initial conditions (1.4)–(1.6) are transformed to
U(0)=(u01(t),…,u0n−1(t),u11(t),…,u1n−1(t),0,…,0)T. | (4.9) |
Lemma 4.2. System (4.8) coupled with the initial condition (4.9) has a unique continuously differentiable solution which exists on the interval [0,t∗) for some t∗∈R+.
Proof. The proof follows from the classical results e.g., [12, Chap. II, Th. 1.1] or [13], since the function f(U(t)) is Lipschitz continuous with respect to U.
Lemma 4.3. The functions θj(t), j=0,…,n−1, are positive for all t∈[0,t∗).
Proof. Assume now that [0,t∗∗] is some interval on which the temperatures are all positive. We consider Eq (1.3) on the interval [0,t∗∗], we multiply it by 1θj, sum over j=0,...,n−1 and using the boundary conditions (1.9), we rewrite it in the following form
cddt[ln(n−1∏j=0θj(t))]=cddt(n−1∑j=0lnθj(t))=cn−1∑j=0˙θj(t)θj(t)=κn−1∑j=0(θj+1(t)−θj(t)θj(t)−θj(t)−θj−1(t)θj(t))+n−1∑j=0|D[uj+1(t)−uj(t)]t|θj(t)≥κn−1∑j=1(θj(t)−θj−1(t)θj−1(t)−θj(t)−θj−1(t)θj(t))=κn−1∑j=1θj(t)(θj(t)−θj−1(t))−θj−1(t)(θj(t)−θj−1(t))θj−1(t)θj(t)=κn−1∑j=1(θj−1(t)−θj(t))2θj−1(t)θj(t)≥0. |
This means that the ln(∏n−1j=0θj(t)) is a non-decreasing function in t, consequently ∏n−1j=0θj(t) is non-decreasing in t, and taking into account that on [0,t∗∗] the temperatures θj(t) are all positive, the statement of our lemma easily follows.
Lemma 4.4. The functions ˙uj(t), ¨uj(t) for all j=1,…,n−1 and θj(t) for all j=0,…,n−1 are bounded on [0,t∗). Consequently, the solution can be extended and all the bounds remain valid for all t≥0.
Proof. In Lemma 4.1, we proved that E(t)=E0, ∀t≥0. Thus
0<m2n−1∑j=1˙u2j(t)+n−1∑j=0V[uj+1(t)−uj(t)]+cn−1∑j=0θj(t)=E0 |
and it follows from the positivity of temperatures that
|˙uj(t)|≤√2E0m:=B,∀j=1,…,n−1 | (4.10) |
0<θj(t)≤E0c,∀j=0,…,n−1. | (4.11) |
Furthermore from (2.1), (2.2) and (2.8)
|P[uj+1(t)−uj(t)]−P[uj(t)−uj−1(t)]|≤C, | (4.12) |
for some constant C. From Eq (4.4), it follows now that ¨uj(t) can be expressed as the sum of the bounded functions (4.11) and (4.12), therefore
|¨uj(t)|≤R,R is constant,j=1,…,n−1. |
This bounds imply that our solution can be extended for all t≥0, consequently also Lemma 4.3 holds true for t≥0, and all the bounds remain valid. This finishes the proof of the lemma.
In the discrete case the entropy of the system (1.1)–(1.3) is given as
S(t)=cn−1∑j=0(lnθj(t)θ∗)+βn−1∑j=0[(uj+1(t)−uj(t))]. |
Due to the boundary condition (1.9), we have
βn−1∑j=0[(uj+1(t)−uj(t))]=0, |
so
S(t)=cn−1∑j=0(lnθj(t)θ∗). | (4.13) |
As a consequence of the proof of Lemma 4.3, we have the following result:
Lemma 4.5. The entropy of the system (1.1)–(1.3) is increasing.
Proof. We differentiate (4.13), use (1.3) and obtain
˙S(t)=κn−1∑j=0(θj+1(t)−θj(t)θj(t)−θj(t)−θj−1(t)θj(t))+n−1∑j=0|D[uj+1(t)−uj(t)]t|θj(t)≥κn−1∑j=1(θj−1(t)−θj(t))2θj−1(t)θj(t)≥0. |
Because we have already proven that θj(t)>0, j=0,…,n−1, for all t≥0, it is clear that the derivative of the entropy is non-negative. The entropy of the system is therefore increasing, i.e., the process will proceed spontaneously and irreversibly.
Remark 4.6. We derived in the proof of Lemma 4.5 that
cn−1∑j=0˙θj(t)θj(t)=κn−1∑j=1(θj−1(t)−θj(t))2θj−1(t)θj(t)+n−1∑j=0|D[uj+1(t)−uj(t)]t|θj(t). | (4.14) |
We can rewrite the first term of the left-hand side of (4.14) as
cn−1∑j=0˙θj(t)θj(t)=cddt(∏n−1j=0θj(t))∏n−1j=0θj(t). | (4.15) |
Combining (4.14) and (4.15) we obtain
cκddt(n−1∏j=0θj(t))≥n−1∑j=1(θj−1(t)−θj(t))2θj−1(t)θj(t)n−1∏j=0θj(t). | (4.16) |
In the sequel, for simplicity of notation, we denote
G(t)=n−1∑j=1(θj−1(t)−θj(t))2θj−1(t)θj(t)n−1∏j=0θj(t). |
Lemma 5.1. The temperature differences θj−1(t)−θj(t), j=1,…,n−1, converge to 0 as t→∞, i.e.,
limt→∞(θj−1(t)−θj(t))=0,j=1,…,n−1. | (5.1) |
Proof. We integrate (4.16) from 0 to ∞ and get
cκ∫∞0[ddtn−1∏j=0θj(t)]dt≥∫∞0G(t) dt. | (5.2) |
Since the functions θj(t), j=0,…,n−1, are globally bounded, see Lemma 4.4, also their product is bounded for all t≥0. So the left-hand side of (5.2) is a finite constant (this follows from the Newton-Leibnitz formula and the definition of an improper integral) which we denote as M, i.e., M≥∫∞0G(t)dt. The function G(t) is therefore bounded, positive, and its integral is convergent. We show that ˙G(t) is bounded as well.
˙G(t)=n−1∑j=12(θj−1(t)−θj(t))(˙θj−1(t)−˙θj(t))n−1∏k=0k≠jk≠j−1θk(t)+n−1∑j=1(θj−1(t)−θj(t))2n−1∑k=0k≠jk≠j−1θk(t)n−1∏m=0m≠km≠jm≠j−1θm(t). |
From Eq (1.3), relation |D[uj+1(t)−uj(t)]t|≤M|˙uj+1(t)−˙uj(t)|, using (2.7), (2.8), and Lemma 4.4, we have that ˙G(t) is the sum of bounded functions and therefore ˙G(t) is also bounded.
Denote
∫titi−hG(t)dt=si, |
where i∈N, h=ti−ti−1 is the time step and 0=t0<t1<t2<…, and
∞∑i=1si≤M<∞. |
The series is convergent, therefore limi→∞si=0, i.e.,
limi→∞∫titi−hG(t)dt=0. | (5.3) |
Since the function ˙G(t) is bounded and (5.3) holds,
limt→∞G(t)=0. |
Hence
limt→∞(θj−1(t)−θj(t))=0,j=1,…,n−1. |
Remark 5.2. The temperature differences θj−1(t)−θj(t), j=1,…,n−1, approach zero in the limit as t→∞, temperatures of the neighbouring springs equalize over time, so all temperatures equalize over time, i.e.,
limt→∞θ0(t)=limt→∞θ1(t)=⋯=limt→∞θn−1(t)=˜T, | (5.4) |
where ˜T is a constant. Later, we show that ˜T≤E0nc, with E0 as in Lemma 4.1, cf. Lemma 5.6.
Lemma 5.3. The displacement of the k-th mass point uk(t), k=1,…,n−1, converges to some ˜uk as t→∞, i.e., for each k=1,…,n−1 there exists ˜uk such that
limt→∞uk(t)=˜uk. | (5.5) |
Proof. We consider Eq (1.3) on the interval [0,∞), we multiply it by 1θj, sum over j=0,...,k−1,1≤k≤n−1, and we rewrite it in the following form:
ddt(ck−1∑j=0lnθj(t)−βuk(t)):=gk(t)=κk−1∑j=0(θj+1(t)−θj(t)θj(t)−θj(t)−θj−1(t)θj(t))+k−1∑j=0|D[uj+1(t)−uj(t)]t|θj(t)≥κk−1∑j=0(θj+1(t)−θj(t)θj(t)−θj(t)−θj−1(t)θj(t))=κk−1∑j=1(θj−1(t)−θj(t))2θj−1(t)θj(t)≥0. | (5.6) |
Lemma 5.1 now implies that
∫∞0gk(s)ds<∞. | (5.7) |
Subsequently integrating the first line of (5.6) over the interval [0,t], t>0, we get
ck−1∑j=0lnθj(t)−βuk(t)=ck−1∑j=0lnθj(0)−βuk(0)+∫t0gk(s)ds. |
Now we take the limit on both sides as t→∞, clearly the limit of the right-hand side exists because of (5.7), and using Lemma 5.1 again we have that also the first term on the left-hand side converges as t→∞, therefore, there exists for each k=1,...,n−1,˜uk such that
limt→∞uk(t)=˜uk. |
This finishes the proof of our lemma.
Lemma 5.4. The velocity of the displacement of the j-th mass point ˙uj(t), j=1,…,n−1, approaches 0 as t→∞, i.e.,
limt→∞˙uj(t)=0. | (5.8) |
Proof. Consider now Eq (1.2) and the limit as t→∞. It follows from Lemma 5.3, the properties of the Prandtl-Ishlinskii operator, and Lemma 5.1 that the limit of the right-hand side exists. Therefore, also the limit of the left-hand side exists, namely there exists for each j=1,...,n−1,˜σj such that
limt→∞σj(t)=˜σj. |
We want to prove that there exists ˜σ such that ˜σj=˜σ for each j=1,...,n−1. Assume by contradiction that ˜σk≠˜σk−1 for some k.
Consider now Eq (1.1). This would mean that the function ˙uj(t) is increasing or decreasing as t→∞, which is in contradiction to the results of Lemma 4.4. Therefore
limt→∞σj(t)=˜σ |
for all j and consequently
limt→∞¨uj(t)=0 |
for all j. It also follows that limt→∞˙uj(t) exists for all j and Lemma 5.3 implies that this limit must be zero, i.e.,
limt→∞˙uj(t)=0 |
for all j.
Lemma 5.5.
limt→∞P[uj+1(t)−uj(t)]=limt→∞P[uj(t)−uj−1(t)], | (5.9) |
and
limt→∞V[uj+1(t)−uj(t)]=limt→∞V[uj(t)−uj−1(t)]), |
and
limt→∞|D[uj+1(t)−uj(t)]t|=limt→∞|D[uj(t)−uj−1(t)]t|, |
j=1,…,n−1.
Proof. Taking the limit t→∞ of both sides in (4.4), we immediately get that (5.9) holds.
As a consequence of (2.6), we have
(˙uj+1(t)−˙uj(t))P[uj+1(t)−uj(t)]−(˙uj(t)−˙uj−1(t))P[uj(t)−uj−1(t)])=ddt(V[uj+1(t)−uj(t)]−V[uj(t)−uj−1])+|ddtD[uj+1(t)−uj(t)]|−|ddtD[uj(t)−uj−1(t)]| a.e. | (5.10) |
Furthermore using Proposition 2.2(ii), we get
ddt|sr[uj+1(t)−uj(t)]−sr[uj(t)−uj−1(t)]| ≤ |˙uj+1(t)−2˙uj(t)+˙uj−1(t)| |
a.e.; and using Lemma 5.4, we get that
limt→∞sr[uj+1(t)−uj(t)]=limt→∞sr[uj(t)−uj−1(t)], |
consequently from (2.4) we have
limt→∞(V[uj+1(t)−uj(t)]−V[uj(t)−uj−1(t)])=0. |
The last statement then follows from (5.10).
Lemma 5.6. There exists a constant ˜T such that the temperatures θj(t), j=1,…,n−1, converge to ˜T for t→∞, i.e.,
limt→∞θj(t)=˜T. | (5.11) |
Proof. In Eq (4.1), we take the limit as t→∞,
limt→∞(m2n−1∑j=1˙u2j(t)+n−1∑j=0V[uj+1(t)−uj(t)]+cn−1∑j=0θj(t))=E0, |
and obtain by virtue of (5.8), (5.4) and (2.4),
limt→∞θ0(t)≤E0nc. |
Moreover, due to (5.4)
limt→∞θj(t)=˜T, where ˜T≤E0nc, j=1,…,n−1. |
Remark 6.1. There is a continuum of singular points of the system (4.6). Namely, for each a∈Rn−1, points ~Va, whose first coordinate is a, next n−1 coordinates are 0 and the last n coordinates are ˜T are singular.
We can summarize the results of Section 4 regarding the system (4.6) into the following theorem.
Theorem 6.2. The nonlinear system (4.6) is asymptotically stable and converges to one of its singular points, the choice of which depends on the initial conditions.
Proof. We proved that
limt→∞uj(t)=˜uj,limt→∞˙uj(t)=limt→∞vj(t)=0,j=1,…,n−1,limt→∞θj(t)=˜T,j=0,…,n−1, |
see Lemmas 5.1–5.3. The actual value of the limit ˜uj,j=1,⋯,n−1 depends on the choice of the initial conditions. Therefore, the nonlinear system (4.6) is asymptotically stable.
We showed that the system representing a mathematical model of n−1 mass points longitudinally oscillating between n thermoelastoplastic springs is asymptotically directed to one of the many possible steady states, where all movements cease and temperatures equalize. The system can be further studied with non-constant material constants, namely assuming that m,β,c,κ are different for each mass point and each spring. The main open problem is the behavior of the system with n→∞, even for the case of thermoelastic springs.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The work of Jana Kopfová and Petra Nábělková was supported by the institutional support for the development of research organizations IČO 47813059.
All authors declare no conflicts of interest in this paper.
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