Materials Physics and Mechanics 46 (2020) 27-41 Received: June 27, 2020

ON THE DEFORMATION OF A COMPOSITE ROD IN THE

FRAMEWORK OF GRADIENT THERMOELASTICITY
А.О. Vatulyan*, S.А. Nesterov
South Mathematical Institute - branch of VSC RAS, 362027, Vladikavkaz, Vatutin str., 53, Russia
*e-mail: 1079@list.ru

Abstract. The gradient thermoelasticity problem for a composite rod based on the applied
one-parameter model is investigated. To find the Cauchy stresses, the Vishik-Lyusternik
asymptotic approach is used, taking into account the presence of boundary-layer solutions in
the vicinity of the rods' boundaries and interface. A new dimensionless parameter equal to the
ratio of the second rod length and the gradient parameter are introduced. Simplified formulas
are constructed in order to find the distribution of the Cauchy stresses depending on the new
parameter. After finding the Cauchy stresses distribution, moment stresses, total stresses,
displacements, and deformations are further calculated. The dependence of the Cauchy stress
jump on the ratio of the rods' physical characteristics and the scale parameter is investigated.
The analysis of the results provided is performed.
Keywords: composite rod, gradient model, thermoelasticity, Cauchy stresses, moment
stresses, asymptotic approach, boundary layer

1. Introduction
An interest in studying the stress-strain state (SSS) of composite structures of small sizes is
associated with the prospects for the development of microelectronics, nanostructures,
aerospace systems, and highly sensitive equipment. In such structures, especially in coatings,
the sizes of the studied elements may become comparable with the characteristic sizes of the
material's microstructure. In addition, large stress concentrations can occur at the interface
between dissimilar materials, which greatly affects the product strength. In classical
mechanics, the constitutive equations do not include any scale parameters; therefore, they
cannot be used in modeling scale effects. At present, gradient elasticity theories are
commonly used to model the effects observed in ultrathin structures, as well as in
nanostructured materials, geomaterials, and biomaterials which include length dimension
parameters in the constitutive relations.
The gradient elasticity theory is a generalization of the classical theory of elasticity. It
was formulated in the 60s of the last century in the works of Toupin [1] and Mindlin [2,3].
Subsequently, a large number of researchers dealt with the development of the gradient
elasticity theory [4-27]. The scientific schools of Aifantis [4-9] and Lurie [13-20] made a
particularly large contribution to the development of gradient mechanics. In the gradient
elasticity theory, the strain energy density depends not only on the strain but also on the first
strain gradient. The mathematical formulation of the gradient theory of elasticity in the
general case is completely determined by the variational Lagrange principle. The equilibrium
equations of the gradient theory with respect to displacements or stresses have a higher order
of differential equations compared to the classical theory, and in order to construct a solution,
it is required to satisfy additional boundary conditions. Note that the practical use of the

http://dx.doi.org/10.18149/MPM.4612020_3
© 2020, Peter the Great St. Petersburg Polytechnic University
© 2020, Institute of Problems of Mechanical Engineering RAS
28 А.О. Vatulyan, S.А. Nesterov
model [3] raises the question of identifying five additional gradient modules. To overcome
this difficulty, the applied gradient deformation models were proposed: the three-parameter
model by D.C.C. Lam [23], the one-parameter models by E.C. Aifantis [7] and
S.A. Lurie [13].
On the basis of the one-parameter model of the gradient elasticity theory, many
problems have been recently solved, both one-dimensional and two-dimensional. In [10], the
solutions of the gradient elasticity theory for a rod were obtained, both for static and dynamic
statements. In [9], the problems of the gradient theory of elasticity for composite bodies are
considered. The conditions of conjugation at the interface of materials modified in
comparison with the classical theory are obtained. For the one-dimensional problem, the exact
analytical solutions are obtained. In [16,17], a refined gradient theory on the bending of scale-
dependent hyperfine rods was constructed. In [13], the equilibrium problem of a two-layer
coating under the influence of localized normal load in the framework of the plane problem is
numerically studied. The problem statement is given on the basis of the interfacial layer
model, which is a one-parameter version of the gradient theory of elasticity. Based on the
variational formulation, the authors obtained the equilibrium equations, boundary conditions,
and conjugation conditions. The solution was carried out using the integral Fourier transform
and its numerical inversion. The dependence of the stress distribution on the layer thickness
and the gradient parameter of the model are investigated. In [26], based on a three-parameter
gradient model, the static deformation of a two-layer microplate was studied. It was found
that the Cauchy stresses break at the boundary of the layers. In [27], the problem of bending a
microbeam with a partial coating was solved. To study the scale effects, an additional scale
parameter was introduced - the ratio of the coating thickness to the gradient parameter. The
effect of a decrease in the scale parameter on changes in the distribution of displacements,
stresses, and the neutral line was studied. It is found out that gradient effects play an
important role when the scale parameter is less than unity.
Starting from the 70s of the last century, gradient theories have begun to be applied to
problems in the mechanics of coupled fields [12,14,15,18,19,21]. In [12], the formulation of
the dynamic coupled problem of gradient thermoelasticity was obtained. Further, gradient
models began to be employed to more accurately estimate the SSS of inhomogeneous
thermoelastic bodies, including the layered ones made of functionally graded materials
(FGM). FGM is a composite material manufactured by mixing different material components
(e.g., ceramic and metal ones) and is characterized by a smooth change of properties along
with the coordinate [28]. So, in [21], the SSS of a long thick-walled FGM cylinder under the
influence of thermal and mechanical load is numerically studied. The material characteristics
of the cylinder vary exponentially in the radial direction. The influence of the inhomogeneity
parameter and the gradient parameter on the distribution of stresses and displacements is
studied. In [14], the formulation of the unbound gradient thermoelasticity problem based on
the model of the interfacial layer for the coating-substrate system is presented under the
assumption of the one-dimensionality of the original problem. As a result of the numerical
solution, graphs of the distribution of stresses and strains are built, taking into account the
influence of both thermomechanical characteristics, and the gradient parameter.
In this work, we study the SSS of a composite rod under thermomechanical loading
based on the applied one-parameter model [7]. We have chosen a one-dimensional problem to
study due to the fact that for such a problem one can obtain simplified analytical solutions that
can be further used to analyze the stress state of thin coatings. The study begins with finding
the temperature distribution. Then, on the basis of the Vishik-Lyusternik asymptotic
approach, simplified analytical expressions for the Cauchy stresses in a dimensionless form
are obtained. After finding the Cauchy stresses distribution, we calculate moment stresses,
On the deformation of a composite rod in the framework of gradient thermoelasticity 29
total stresses, strains and displacements. A comparative analysis of the results obtained is
performed.

2. Constitutive relations of gradient mechanics

In 1968, R.D. Mindlin and N.N. Eshel, put forward a position that the strain energy density is
a function of not only the strain tensor but also the first strain gradient [3]. For a linear
isotropic material, the expression for the strain energy density has the form:
1
w = λεii ε jj + µεij εij + c1εij , j εik ,k + c2 εii ,k ε kj , j + c3εii ,k ε jj ,k + c4 εij ,k εij ,k + c5εij ,k ε kj ,i . (1)
2
Here λ and µ are the Lame parameters, c1 , c2 ,…, c5 are the additional gradient

parameters,= ε ij
1
2
( ui, j + u j ,i ) is the tensor of small deformations of an elastic body. Note that
the practical use of this model raises the question of identifying additional modules.
To overcome this difficulty, B.S. Altan and E.C. Aifantis [7] proposed an applied one-
parameter gradient deformation model based on a simplified form of the strain energy density.
1 2
Putting in (1) c= 1 c=2 c=
5 0 , c3= ll , c4 = µl 2 , we get:
2
1 1
w= lεii ε jj + µεij εij + l 2 ( lεii ,k ε jj ,k + µεij ,k εij ,k ) . (2)
2 2
Here l is a gradient parameter with a length dimension and associated with sizes of
microstructural inhomogeneities.
The constitutive relations for the components of the Cauchy stress tensor τ ij , moment
stress tensor mijk , and total stress tensor σ ij have the form [7]:
∂w
τij = , (3)
∂εij
∂w
mijk= = l 2 τijk ,k , (4)
∂εij ,k
σij =τij − mijk ,k =(1 − l 2∇ 2 )τij . (5)
The mathematical formulation of the gradient theory of elasticity in the general case is
completely determined by the variational Lagrange principle. By varying the functional
compiled in [7], we obtain the equilibrium equation:
σij , j =
0, (6)
and the natural static boundary conditions on the surface S bounding the region V are as
follows:
mijk ,k n j nk = qi , τij n j − mijk ,k n j − (mijk ,k nk ), j + (mijk ,k n j nk ), s ns =
ti . (7)
Here ti , qi are the vectors of the given forces in the body volume and on its surface, ni
are the components of the unit normal vector to the body surface at the considered point. The
formulation of the problem is supplemented by the kinematic boundary conditions: ui = ui ,
∂u
ui ,l nl = i .
∂n
In the case of the problem of unbound thermoelasticity, according to [21], we will
replace εij with εij − γT δij in the equation (2), where γ is the temperature stress coefficient,
δij is the Kronecker symbol. In addition, the equilibrium equation (6) and the mechanical
 30 А.О. Vatulyan, S.А. Nesterov
boundary conditions (7) must be supplemented by the equation of classical thermal
conductivity:
( kijT,i ), j = 0 (8)
and thermal boundary conditions
T |S1 = 0 , T |S2 = T0 . (9)
Here S= S1 + S 2 is the body surface.
As an example, we consider the equation of equilibrium, thermal conductivity, and the
constitutive relations of gradient thermoelasticity for an inhomogeneous rod:
σ′ =0 , (10)
(k ( x)T ′)′ = 0 , (11)
σ = τ − l τ′′ ,
2
(12)
=τ E ( x)u ′ − γ ( x)T ( x) , (13)
m= l τ′ .
2
(14)
In the formulas (10) - (14), the prime sign denotes the derivative with respect to x .

3. Statement of the gradient thermoelasticity problem for a composite rod

Consider the equilibrium of a composite thermoelastic rod with a length H at the junction at
the point x = H 0 , under the influence of a combined thermo-mechanical load. One end of the
rod x = 0 is rigidly fixed and maintained at zero temperature; at the other end x = H the
force p0 acts, and the temperature T0 is maintained. The Young modulus E , the thermal
conductivity k and the thermal stress coefficient γ are piecewise continuous functions of the
coordinate x . Because the equilibrium equations in gradient theory have an increased order of
differential equations compared to the classical theory, then the additional boundary
conditions are also required. As additional boundary conditions, we take u ′(0) = 0 ,
m( H ) = 0 . In addition, according to [14,15], the interface conditions for temperature, heat
flux, displacements, strains, total stresses, and moment stresses must be satisfied at the
junction. Further in the formulas, we denote the functions and parameters corresponding to
the first and second rod by the indices "1" and "2", respectively. To simplify the calculations,
we assume that the gradient parameter is the same for each rod, i.e. l1= l2= l .
The original aim of the study was to find the distribution of the Cauchy stresses τ( x)
along the length of the composite rod. For this, we express the total stresses σ( x) , moment
stresses m( x) , and displacement gradients u ′( x) through the Cauchy stresses. Then the
formulation of the thermoelasticity problem in terms of the Cauchy stresses will take the
form:
τ1′ − l 2 τ1′′′ =0 , τ′2 − l 2 τ′′′2 =0 , (15)
(k1 ( x)T1′)′ = 0 , (k2 ( x)T2′)′ = 0 , (16)
T1 (0) = 0 , T2 ( H ) = T0 , (17)
τ1 (0) = 0 , τ′2 ( H ) = 0 , τ2 ( H ) − l 2 τ′′2 ( H ) =p0 , (18)
T1 ( H 0 ) = T2 ( H 0 ) , k1 ( H 0 )T1′( H 0 ) = k2 ( H 0 )T2′( H 0 ) , (19)
τ1 ( H 0 ) + γ1 ( H 0 )T1 ( H 0 ) τ2 ( H 0 ) + γ 2 ( H 0 )T2 ( H 0 )
= , (20)
E1 ( H 0 ) E2 ( H 0 )
τ′2 ( H 0 ) , τ1 ( H 0 ) − l 2 τ1′′( H 0 ) =τ2 ( H 0 ) − l 2 τ′′2 ( H 0 ) .
τ1′ ( H 0 ) = (21)
On the deformation of a composite rod in the framework of gradient thermoelasticity 31
Let us write out the dimensionless problem (15)-(21) by introducing the following
dimensionless parameters and functions:
x H 1 γT p γT τ m σ
x = , h0 = 0 , α = , β 0 = 0 0 , P0 = 0 , Wi = 0 i , Ωi = i , M i = i , Si = i ,
H H H E0 E0 E0 E0 E0 H E0
E k γ
si = i , ki = i , γ i = i , i = 1, 2 , k0 = max k ( x) , γ 0 = max γ ( x) , E0 = max E ( x) .
E0 k0 γ0 x∈[ 0, H ] x∈[ 0, H ] x∈[ 0, H ]

The dimensionless boundary value problem of thermoelasticity (15) - (21) takes the
form:
Ω1′ − α 2 Ω1′′′ = 0 , Ω 2′ − α 2 Ω′′′2 = 0 , (22)

( k (x)W ′ )′ =
1 1 0 , ( k (x)W ′ ) =
2
′
20, (23)
W1 (0) = 0 , W2 (1) = β0 , (24)
Ω1 (0) = 0 Ω′2 (1) = 0 Ω 2 (1) − α 2 Ω′′2 (1) = P0
, , , (25)
W1 (h0 ) = W2 (h0 ) , k1 (h0 )W1′(h0 ) = k2 (h0 )W2′(h0 ) , (26)
W1 (h0 ) + γ1 (h0 )W1 (h0 ) W 2 (h0 ) + γ2 (h0 )W2 (h0 )
= , (27)
s1 (h0 ) s2 (h0 )
Ω1′ (h0 ) = Ω′2 (h0 ) Ω1 (h0 ) − α 2 Ω1′′(h0 ) = Ω 2 (h0 ) − α 2 Ω′′2 (h0 )
, . (28)

4. Solving the thermoelasticity problem for a rod

The solution of the thermoelasticity problem (22) - (28) begins with finding the temperature
distribution along the length of the composite rod based on the solution of the classical heat
conduction problem (23), (24), (26).
In case when both rods are made of inhomogeneous materials, the solution to the
problem of thermal conductivity (23), (24), (26) has the form:
x x
f1 (ξ) f1 (h0 ) + f 2 (ξ) dη dη
W1 (ξ) =β0 , W2 (ξ) =β0 , f1 (x) =∫ , f 2 (x) =∫ . (29)
f1 (h0 ) + f 2 (1) f1 (h0 ) + f 2 (1) k (η)
0 1
k (η)
h0 2

In case both rods are made of homogeneous materials, by setting in (29) k1 = const and
k2 = const , we obtain:
k2 ξ k ξ + h0 (k2 − k1 )
W1 (ξ) =β0 , W2 (ξ) =β0 1 . (30)
k1 + h0 (k2 − k1 ) k1 + h0 (k2 − k1 )
After finding the temperature distribution, further, in order to find the Cauchy stresses,
it is necessary to solve the boundary-value problem (22), (25), (27), (28). The accurate
analytical solutions were obtained in the work when both rods were made of homogeneous
materials. These solutions are cumbersome and therefore are not presented here: they are used
to evaluate the accuracy of the approximate analytical solution.
The problem (22), (25), (27), (28) contains the differential equations (22) with a small
parameter in the highest derivative and is singularly perturbed. We obtain the approximate
analytical solution to the boundary value problem (22), (25), (27), (28) based on the Vishik-
Lyusternik method [29,30].
According to the scheme of the Vishik-Lyusternik method, we construct the first
iterative process. To do this, we present solutions for each of the equations (22) in the form of
an expansion for the small parameter α in the form:
Ω1 (x, α) ≅ G1 (x, α=) g1(0) (x) + αg1(1) (x) + α 2 g1(2) (x) + ... , (31)
32 А.О. Vatulyan, S.А. Nesterov

Ω 2 (x, α) ≅ G2 (x, α= ) g 2(0) (x) + αg 2(1) (x) + α 2 g 2(2) (x) + ... . (32)
Substituting the expansions (31), (32) into (22) and performing the splitting by α
powers, we obtain the sequence of boundary value problems:
 g (0)=′ (ξ) 0, g 2(0)= ′ (ξ) 0, g 2(0) = (1) 1, g1(0)= (h0 ) g 2(0) (h0 ),
 1
 (1)′ ′ (ξ) 0, g 2(1)
 g1 = (ξ) 0, g 2(1)= = (1) 0, g1(1)= (h0 ) g 2(1) (h0 ), (33)
.................................................................................


Obviously, by solving each of the problems (33), it is impossible to satisfy all the
boundary and conjugation conditions. It is necessary to build on additional boundary layer
solutions that should quickly fade away with distance from the border.
According to the scheme of the Vishik-Lyusternik method, we construct the second
iterative process. For the first rod, the boundary layers are localized in the vicinity of the
attachment point x =0 and the interface point x =h0 with the second rod. For the second rod,
the boundary layers are in the vicinity of x =h0 and x =1 . We introduce the tensile
x x−h h −x x −1
coordinates in the vicinity of the boundaries η1 = , η2 = 0 , η3 = 0 , η4 = .
α α α α
Thus, the expressions for the Cauchy stresses of each rod can be represented as:
x   x − h0 
Ω1 (x, α) ≅ G1 (x, α) + Z1  , α  + Z 2  ,α , (34)
α   α 
 h −x   x −1 
Ω 2 (x, α) ≅ G2 (x, α) + Z 3  0 , α  + Z4  ,α , (35)
 α   α 
x  x x x
where Z1 = , α  z1(0)   + αz1(1)   + α 2 z1(2)   + ... ,
α  α α α
 x − h0   x − h0  (1)  x − h0  2 (2)  x − h0 
Z 2 = , α  z2(0)   + αz 2   + α z2   + ... ,
 α   α   α   α 
 h0 − x   h −x (1)  h0 − x  2 (2)  h0 − x 
Z 3 = , α  z3(0)  0  + αz3   + α z3   + ... ,
 α   α   α   α 
 x −1   x −1  (1)  x − 1  2 (2)  x − 1 
Z 4 = , α  z4(0)   + αz 4   + α z4   + ... .
 α   α   α   α 
In the expansions (34), (35) we restrict ourselves to only zero approximations. Then
approximate solutions can be represented as:
x  x − h0 
Ω1 (x, α) ≅ g1(0) (x) + z1(0)   + z2(0)  , (36)
α  α 
 h − x  (0)  x − 1 
Ω 2 (x, α) ≅ g 2(0) (x) + z3(0)  0  + z4  . (37)
 α   α 
The functions g1(0) (x) , g 2(0) (x) coincide with the solution of the problem for a
composite rod obtained on the basis of the classical model of thermoelasticity and have the
form:
g1(0) (x= ) g 2(0) (x= ) P0 . (38)
On the deformation of a composite rod in the framework of gradient thermoelasticity 33

d2 1 d2
To find the first boundary-layer solution z (η1 ) , given that 2 = 2
(0)
,
dξ α dη12
1

d4 1 d4
= , we obtain the equation
dξ 4 α 4 dη14
z1(0)′ − z1(0)′′′ =
0, (39)
with the solution
z1(0) = C1 + C2 eη1 + C3e −η1 . (40)
Since the boundary-layer solution z
(0)
1 ( η1 ) must asymptotically tend to zero for
x
−
η1 → ∞ , we assume C=
1 C=
2 0 in (40). Therefore, z1(0) = C3e α
.
To find the second boundary-layer solution z
(0)
2 ( η2 ) , we have the equation:
z2(0)′ − z2(0)′′′ =
0, (41)
which solution has the form:
z2(0) = C4 + C5eη2 + C6 e −η2 . (42)
As far as the boundary-layer solution z2 ( η2 ) also has to tend asymptotically to zero
(0)

x− h0
for η2 → −∞ , we assume in (42) C= 4 C=6 0 . So, z2(0) = C5e α . The expressions for the
Cauchy stress in the first rod will take the form:
 − 
x x− h0
Ω1 (x, =
α) P0 1 − e  + C5e α .
α
(43)
 
To find C3 , we proceed to satisfy the boundary condition on the left end of the rod
ξ =0 : Ω1 (0) = g1 (0) + z1 ( 0 ) = P0 + C3 = 0 . From this, we have C3 = − P0 . Here we take into
(0) (0)

account that when ξ =0 , only the first boundary-layer solution manifests itself z1(0) since the
value h0 is such that the influence of the second boundary layer solution z2(0) can be omitted
h0
−
due to the small size of e α .
Then the Cauchy stress in the first rod will take the form:
 − 
ξ ξ− h0
Ω1 (ξ, α) ≅ P0 1 − e  + C5e α .
α
(44)
 
Given the physical meaning, we similarly determine the third and fourth boundary-layer
h0 −x x−1
solutions in the form z (0)
3 = С7 e α
, z (0)
4 = С8e α
. From the condition Ω′2 (1) =
0 follows the
h0 −1
relationship between the constants C7 and C8 in the form С8 = С7 e α . The expressions for
the stress in the second rod will take the form:
 h0α−x h0 −1 x−1

Ω 2 (x, α) ≅ P0 + C7  e +e α e α . (45)
 
The Unknowns C5 and C7 are determined from the boundary conditions (27), (28),
h0
−
assuming 1 − e α
≅ 1 in the calculations.
34 А.О. Vatulyan, S.А. Nesterov
1− h
Then the expressions for the dimensionless Cauchy stresses, denoting by δ0 = 0 the
α
ratio of the length of the second rod and the gradient parameter α , can be represented as:
 − 
ξ ξ− h0

(
Ω1 (ξ, α) ≅ P0 1 − e α  + Κ 1 − e −2 δ0 e α ,) (46)
 
 h0 −ξ ξ−1

Ω 2 (ξ, α) ≅ P0 − Κ  e α + e −δ0 e α  , (47)
 
where
P ( s (h ) − s2 (h0 ) ) + W1 (h0 ) ( s1 (h0 ) γ2 (h0 ) − s2 (h0 ) γ1 (h0 ) )
Κ= 0 1 0 . (48)
s1 (h0 ) + s2 (h0 ) + ( s1 (h0 ) − s2 (h0 ))e −2 δ0
If we put in (46), (47) α =0 , then we obtain the expressions for stresses corresponding
to the classical thermoelasticity: Ω1 =Ω 2 =P0 .
From the formulas (46), (47) it follows that at the point ξ =h0 there is a stress jump.
h0
−
α
Neglecting the magnitude P0 e compared with 2K , we get the expression for the stress
jump:
∆Ω = Ω1 − Ω 2 ≅ 2K . (49)
The value of the stress jump, according to (48), (49), is determined by the mechanical
stress P0 , the temperature W1 (h0 ) , and the relation between thermoelastic characteristics and
the parameter δ0 . From the formula (48) it follows that if a continuous change in the
thermomechanical characteristics through the junction of the rods is ensured, then there will
be no Cauchy stress jump.
If δ0 ≤ 1 (the relative length of the second rod is comparable to or less than the value of
the gradient parameter α ), then scale effects will appear, consisting in the dependence of the
Cauchy stress jump on the value of the parameter δ0 . When δ0 =0 the Cauchy stress jump is
minimal; with the increase δ0 from 0 to δ0  1 comes the exponential increase of ∆Ω . At
δ0 =0 , the value ∆Ω for a rod made of homogeneous parts, in the case of mechanical
loading, is determined by the formula
 s 
∆Ω ≅ P0 1 − 2  . (50)
 s1 

If the elastic modulus of the first rod is greater than that of the second one, we have the
following in the dimensionless form: s1 = 1 , s2 ∈ [0,1) . The maximum stress jump, equal to
∆Ω ≅ P0 , will be at s2 = 0 . If the elastic modulus of the first rod is less than the second one,
we have: s2 = 1 , s1 ∈ [0,1) . The absolute value of the maximum stress jump | ∆Ω |→ +∞ will
be at s1 → 0 .
At δ0  1 (the relative length of the second rod is much larger than the gradient
parameter α ), the exponents e −δ0 are very small quantities. Then in the expressions (46)-(48)
ξ−1
one can put 1 − e ≅ 1 , e e ≅ 0 , s1 (h0 ) + s2 (h0 ) + ( s1 (h0 ) − s2 (h0 ))e −2 δ0 ≅ s1 (h0 ) + s2 (h0 ) .
−2 δ0 −δ0 α

In this case, the value of the stress jump ∆Ω is independent of the specific parameter
value δ0 , and it is determined by the ratio of thermomechanical characteristics. The value ∆Ω
On the deformation of a composite rod in the framework of gradient thermoelasticity 35

for a rod made of homogeneous parts, in the case of mechanical loading, at δ0  1 , is

determined by the formula
s −s
∆Ω ≅ 2 P0 1 2 . (51)
s1 + s2
The absolute value of the maximum stress jump in this case is | ∆Ω |≅ 2P0 and will be
reached for s1 = 1 , s2 = 0 or s2 = 1 , s1 = 0 .
After finding the laws of distribution of the dimensionless Cauchy stresses along the
coordinate ξ , we further calculate the dimensionless moment stresses M i =α 2 Ω′i , and the
1
dimensionless strains Ε= i ( Wi + γiWi ) , i = 1, 2 . The total stresses Si =Ωi − M i′ , based on
si
boundary conditions (25), (28), are the same and equal S1 (x)= S1 (x)= P0 regardless of the
material and gradient characteristics of the rods. The displacements distributions U i , i = 1, 2
by the coordinate ξ are found by integrating the expressions for strains Εi , given the
boundary condition U1 (0) = 0 and the conjugation condition U1 (h0 ) = U 2 (h0 ) .

5. Computation results
This section presents the results of calculations on finding the distribution of dimensionless
Cauchy stresses, moment stresses, total stresses, strains, and coordinate displacements for
both mechanical and thermal loading.
Example 1. Consider the case of mechanical loading of a composite rod (β =0 ,
P0 = 0.1) , the parts of which are made of homogeneous materials with the following
characteristics: δ0 =5 , s1 = 0.5 , s2 = 1 . The influence of the gradient parameter α magnitude
on the accuracy of the calculation of the dimensionless Cauchy stresses by the asymptotic
formulas (46), (47) is studied. During the calculations, it was found that the error in the
approximate calculation of the dimensionless Cauchy stresses does not exceed 1% at
α ≤ 0.02 .

a) b)
Fig. 1. Distribution graphs along the coordinate ξ : a) dimensionless Cauchy stresses;
b) dimensionless moment stresses under mechanical loading
36 А.О. Vatulyan, S.А. Nesterov

а) b)
Fig. 2. Distribution graphs along the coordinate ξ : a) dimensionless strains; b) dimensionless
displacements during mechanical loading

Figures 1 and 2 show images of the distribution of the dimensionless Cauchy stresses
(Fig. 1a), the moment stresses (Fig. 1b), the strains (Fig. 2a), and the displacements (Fig. 2b)
at α =0.01 . The value h0 is determined from the expression h0 = 1 − δ0 α .
From Figure 1a it follows that the Cauchy stresses: 1) near the end face x =0 decay
exponentially to zero in accordance with the boundary condition Ω1 (0) = 0 ; 2) experience a
jump at the point ξ =h0 , which values, according to (51), is determined by the ratio of the
elastic modulus of the rods. From Figure 1b it follows that the moment stresses equal to zero,
except for the vicinity of the fixing and conjugation points, and reach a peak at the point of
contact of the rods.
Figure 2a depicts the strains and displacements.
In the case of mechanical loading, we study the dependence of the jump of the Cauchy
stresses ∆Ω at the point ξ =h0 , calculated by the formula (49), on the value of the parameter
δ0 at P0 = 0.1 , α =0.01 and various ratios of the elastic modulus. Figure 3 presents the results
of calculations of the dependence of the stress jump on the parameter δ0 for: 1) s1 = 1 ,
s2 = 0.5 (Fig. 3а); 2) s1 = 0.5 , s2 = 1 (Fig. 3b). In this case, the solid line shows the
dependence ∆Ω(δ0 ) , obtained in the course of the exact analytical solution, and the dots – on
the basis of the formula (49).
On the deformation of a composite rod in the framework of gradient thermoelasticity 37

а) b)
Fig. 3. Graph of the dependence of the jump in the dimensionless Cauchy stresses on the
parameter δ0 under mechanical loading

From Figure 3 it follows that the minimum Cauchy stress jump occurs when δ0 0 1 , i.e.
when the length of the second rod is much less than the gradient parameter α . As you
increase δ0 , the stress jump increases exponentially. Starting from δ0 > 3 , i.e. when the length
of the second rod becomes 3 times greater than the gradient parameter α , ∆Ω almost reaches
a stationary value.
Example 2. Consider the case of thermal loading of a composite rod ( β0 =0.1 , P0 = 0 ),
the parts of which are made of homogeneous materials with the following characteristics:
α =0.01 , δ0 =8 , s1 = 1 , s2 = 1 , k1 = 1 , k2 = 0.25 , γ1 =0.5 , γ2 =1 .To find the temperature at
the point h0 , we use the first formula (30).

а) b)
Fig. 4. Distribution graphs along the coordinate ξ : a) dimensionless Cauchy stresses;
b) dimensionless moment stresses under thermal loading
38 А.О. Vatulyan, S.А. Nesterov

а) b)
Fig. 5. Distribution graphs along the coordinate ξ : a) dimensionless strains; b) dimensionless
displacements during mechanical loading

Figures 4-5 show images of the distribution of the dimensionless functions: the Cauchy
stresses (Fig. 4a), the moment stresses (Fig. 4b), the strains(Fig. 5a), and the displacements
(Fig. 5b). The error in calculating the distribution of the dimensionless functions was less
than 1%.
From Figure 4a, it follows that the Cauchy stresses are equal to zero, with the exception
of the vicinity of the junction of the rods, where a stress jump occurs, due to the difference in
the coefficients of the thermal stresses of the rods. From Figure 4b it follows that the moment
stresses are equal to zero, with the exception of the vicinity of the junction of the rods and
reach a peak at the junction point of the rods.
The magnitude of the Cauchy stress jump during the thermal way of loading the rod
made of homogeneous parts is:
W (h ) ( s γ − s γ )
∆W ≅ 2 1 0 1 2 2 −12 δ0 . (52)
s1 + s2 + ( s1 − s2 )e
For the case of thermal loading, we study the dependence of the Cauchy stress jump
∆Ω at a point ξ =h0 on the parameter δ0 at β0 =0.1 , k1 = 1 , k2 = 0.25 , s1 = 0.5 , s2 = 1 ,
γ1 =1 , γ2 =0.8 , α =0.01 . The value h0 , necessary to find the temperature, is determined
from the expression h0 = 1 − δ0 α . In Figure 6, the solid line shows the dependence obtained in
the course of the exact analytical solution, and the dots – on the basis of approximate
formulas.
On the deformation of a composite rod in the framework of gradient thermoelasticity 39

Fig. 6. Graph of the dependence of the jump in the dimensionless Cauchy stresses on the
parameter δ0 under thermal loading

From Figure 6 it follows that in the case of thermal loading, the maximum rate of
change of the function ∆Ω(δ0 ) is observed at δ0 < 1 , i.e. when the length of the second rod is
less than the gradient parameter α . For δ0 > 1 , with an increase of δ0 , a smooth change in
function ∆Ω(δ0 ) is observed.

6. Conclusion
A statement of the gradient thermoelasticity problem for a composite rod based on the one-
parameter Aifantis model is given. After finding the temperature distribution from the
solution of the classical heat conduction problem, simplified analytical expressions for finding
the Cauchy stresses are obtained on the basis of the asymptotic Vishik-Lyusternik approach.
The cases of thermal and mechanical loading of the rod are considered. A new scale
parameter is introduced, equal to the ratio of the length of the second rod and the gradient
parameter. After finding the distribution of the Cauchy stresses, moment stresses, total
stresses, displacements, and deformations are calculated. It was revealed that, within the
framework of the gradient theory, the deformations are continuous at the point of contact of
the rods. This fact explains the jump in the Cauchy stresses in the vicinity of the point of the
rod conjugation. The magnitude of the Cauchy stress jump depends on both the ratio of
thermomechanical characteristics and the value of the scale parameter. The dependence of the
Cauchy stress jump on the scale parameter is investigated. It was found out that the stress
jump function changes most rapidly at values of the scale parameter less than the length of the
second rod. The moment stresses are continuous, equal to zero, except for the vicinity of the
fixing and conjugation points, and reach a peak at the rods' conjugation point. The total
stresses, which are a combination of the Cauchy stresses and the first gradient of the moment
stresses, are continuous in each rod and equal to the value of the mechanical load at the
rod's end.

Acknowledgements. The authors are grateful to Professor Zubov L.M. for valuable comments.
The work is supported by the South Mathematical Institute - branch of VSC RAS, Vladikavkaz.
40 А.О. Vatulyan, S.А. Nesterov
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