Nonlinear Hermitian generalized hygrothermoelastic stress and wave propagation analyses of thick FGM spheres exhibiting temperature, moisture, and strain-rate material dependencies

Composite Structures 229 (2019) 111364

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Nonlinear Hermitian generalized hygrothermoelastic stress and wave propagation analyses of thick FGM spheres exhibiting temperature, moisture, and strain-rate material dependencies

T

M. Shariyat , S. Jahanshahi, H. Rahimi ⁎

Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran 19991-43344, Iran

ARTICLE INFO

ABSTRACT

Keywords: Nonlinear generalized hygrothermoelasticity Functionally graded hollow sphere Lord-Shulman Temperature-dependency Wave propagation Moisture

The present article is dedicated to the dynamic stress and displacement distributions and hygrothermoelastic wave propagation and reflection responses of FG hollow spheres subjected to thermomechanical shocks. The strain-rate and temperature dependencies of the material properties are accounted for. Hence, the material properties of the sphere are dependent on coordinates, time, loading rate, temperature, and ambient humidity. Furthermore, material degradation due to moisture absorption is considered. It is the first time that such a complex and more realistic combination is taken into account in the wave propagation analysis. The nonlinear coupled Lord-Shulman-type generalized hygrothermoelasticity equations that are developed by the inclusion of the moisture absorption state variable into the free energy function, are solved by using a Galerkin-type finite element method, iterative solution algorithm, and a second-order Runge-Kutta time integration procedure. Results are extracted by using the C1-continuous Hermitian rather than common C0-continuous Lagrangian elements to guarantee exact continuity of the stresses at the mutual boundaries of the elements and preclude the numerical locking phenomenon. Comprehensive sensitivity analyses including the effects of various factors are performed. Results reveal the significant effects of the temperature, strain-rate, and moisture absorption on both the material properties and constitutive law and consequently, on the transient stress distribution and the hygrothermoelastic wave propagation/reflection phenomenon. Furthermore, the results confirm that in the FG structures, the thermal and stress wavefronts travel with variable and time-dependent speeds.

1. Introduction A wide variety of pressure vessels are prone to thermal, mechanical or thermomechanical shocks. The spherical heads of the CNG highpressure tanks, cylindrical/spherical hydropuls pressure containers, and cylindrical combustion chambers, are among these vessels. The resulting stress distributions and thermoelastic wave propagation at the early times of the shock loading/unloading (e.g., abrupt charge/discharge) are not only important from the strength consideration point of view but also the resulting permanent damages in the microstructure that follow the imposed loads. On the other hand, the loading rate may affect the resulting stresses; as the loading rate affects the elastic modulus of the material in shock loading [1,2], especially in composite and non-metallic materials [3–7]. Furthermore, fewer researches have considered the temperature-dependence of the material properties in the thermoelastic wave propagation analyses.

⁎

Majority of the thermoelastic analyses on the hollow cylinders and spheres have been carried out for steady-state conditions. It is evident that these conditions are not as critical as the time-dependent rapid/ shock loading or unloading conditions. Some authors used the classical theory of thermoelasticity to study the thermoelastic behaviors of the cylindrical/spherical pressure vessels. Shariyat et al. [8,9] studied the nonlinear thermoelastic stress distribution and wave propagation in thick temperature-dependent functionally graded (FG) cylinders. Hata [10] studied the dynamic stresses caused by the wave propagation and interference due to sudden heating of the isotropic homogeneous hollow spheres. Xi and Wei [11] obtained a closed-form solution for distributions of dynamic thermoelastic stresses of transversely isotropic hollow spheres subjected to rapid heating shocks. Ding et al. [12] solved the dynamic thermoelasticity of the transversely isotropic hollow spheres by the separation of variables method. Tsai and Hung [13] studied dynamic thermoelastic responses of a bi-layered composite

Corresponding author. E-mail address: [email protected] (M. Shariyat). URL: https://wp.kntu.ac.ir/shariyat/publications.html (M. Shariyat).

https://doi.org/10.1016/j.compstruct.2019.111364 Received 7 June 2019; Received in revised form 17 August 2019; Accepted 4 September 2019 Available online 12 September 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.

Composite Structures 229 (2019) 111364

M. Shariyat, et al.

sphere due to a sudden temperature change on its outer surface, using an analytical–numerical Laplace transformation technique. Ootao et al. [14] treated transient displacements and stresses of a piezothermoelastic FG hollow sphere, using the Laplace transformation technique. Gorbanpour Arani et al. [15] used Hankel and Laplace transforms for magneto-thermoelastic stress analysis of a thick FG sphere. Povstenko [16] studied thermal stresses of a homogenous isotropic sphere with Caputo time-fractional derivatives. Shahani and Bashusqeh [17] extracted transient stresses of a thick sphere by imposing a finite Hankel transform to the classical coupled thermo-elasticity equations. In contrast to Fourier’s law assumption, the heat flux has a finite rather than infinite speed. While the stress wave propagates with the sound speed within the body, the thermal wave propagation has been imagined as a “second sound” whose speed can only be determined based on the generalized thermoelasticity theories. Limited researches have studied the thermoelastic wave propagation by the generalized (i.e., nonclassical) coupled theories. Bagri and Eslami [18] expanded the Lord–Shulman (L-S), Green-Lindsay (G-N), and Green-Naghdi (G-N) generalized coupled thermoelasticity models for the thick hollow isotropic and homogeneous spheres and cylinders and used Laplace transform and its numerical inversion techniques. Hosseini [19] studied time histories of displacements and stresses of the finite length FG thick hollow cylinders, based on the Green–Naghdi thermoelasticity theory, dividing the cylinder into many isotropic sub-cylinders and using the finite element and Newmark time-marching methods. Later, Hosseini and Abolbashari [20] applied their formulation to an infinite length cylinder and solved the resulting equations by means of the Laplace transform and its numerical inversion. Shariyat [21] investigated the nonlinear thermoelastic wave propagation and reflection in FG thick cylinders, based on the L-S, G-L, and G-N generalized thermoelasticity theories, examining various micromechanical homogenization laws. Very few researches have analyzed the hollow spheres based on the non-classic theories [18]. Bagri and Eslami [22] employed GreenLindsay theory and finite element, Laplace transformation, and numerical Laplace transform inversion techniques to study the temperature, displacement, stress, and waves propagation responses of the FG hollow spheres. Babaei [23] employed a non-Fourier heat conduction law but a classical theory for the FG spheres. Kar and Kanoria [24] determined stress, displacement, and temperature distributions of an FG orthotropic hollow sphere by the linear generalized G-N II, G-N III, and three-phase-lag theories, using Laplace transform and its inversion techniques. Later Kanoria and Ghosh [25] employed G-N theory for temperature-dependent FG spheres, using similar solution procedures. Abouelregal [26] applied a dual-phase-lag heat transfer model for a homogeneous isotropic sphere, utilizing the Laplace transform and its numerical inversion techniques. Kiani and Eslami [27] dealt with thermoelastic responses of a thick isotropic homogeneous temperatureindependent sphere, using the L-S theory, the GDQ discretization, and Newmark time marching methods. Recently, Heydarpour et al. [28] studied the thermoelastic wave propagation in spheres reinforced by different transverse gradations of the nanophases, using Lord-Shulman theory. Pressure vessels utilized in humid/diffusive environments, e.g., those used in offshore, naval/marine, submarine, and aviation facilities, as well as the fluid/steam/gas reservoirs and pipes, may encounter material degradations and subsequently, stress redistributions over time. Shariyat and his co-authors [29,30] reported the remarkable effects of the moisture absorption/diffusion on the stability, stress redistribution, and creep of the structures. Chen et al. [31] used a transfinite element method for the transient hygrothermal stress analysis of the porous and composite cylinders in the context of the coupled classical thermoelasticity. Weng [32] presented a Laplace-transformbased classical thermoelasticity formulation for double-layer cylinders subjected to hygrothermal shocks. Yang et al. [33] applied the classical thermoelasticity, Laplace transform, and the finite difference method to

the transient hygrothermal problem of a hollow cylinder. The present article is concerned with the determination of the timedependent thermoelastic stress and displacement distributions and the hygrothermoelastic wave propagation in the FG hollow spheres. The nonlinear coupled thermoelasticity equations are developed based on the coupled generalized L-S theory and solved by using Galerkin’s finite element method, iterative solution algorithm, and a second-order Runge-Kutta time integration procedure. Some of the novelties/superiorities included in the present research are:

• A Lord-Shulman generalized thermoelasticity based on a free energy • • • • • •

function that includes the moisture content (diffusion) state space is employed. A nonlinear transient hygrothermoelasticity research that includes several sources of nonlinearity is presented; thus, the Laplace transform technique, cannot be used [34,35]. Transient vibrations under thermomechanical shocks are studied in addition to wave propagation analyses. The strain-rate dependence of the material properties is considered in addition to the temperature-dependence of the material properties. The material degradation due to moisture absorption is taken into account as well. Results are extracted based on C1-continuous Hermitian rather than the common C0-continuous Lagrangian elements to guarantee continuity of the stresses at the interfaces between elements and preclude the numerical locking phenomenon. The imposed loads include both mechanical and thermal loads instead of pure thermal loads.

2. The governing equations 2.1. Definition of the geometry and material properties dependencies Let us consider the functionally graded thick hollow sphere shown in Fig. 1. The inner and outer radii of the sphere are denoted by ri and ro , respectively. The inner and outer surfaces of the vessel may be subjected to the ( pi , Ti , qi , m i ) and ( po , To, qo , m o ) pressures, temperatures, heat fluxes, and moisture concentrations, respectively. The vessel is assumed to be constructed from two metallic and

Fig. 1. The thick hollow FG sphere under the hygrothermomechanical loads. 2

Composite Structures 229 (2019) 111364

M. Shariyat, et al.

ceramic phases whose volume fractions reach 100% for either of the outer or inner layers. We use the following power law for the description of variations of the volume fraction of the material of the inner layer:

ro ro

(Vf )i =

this reason, we use the numerically more reliable generalized L-S theory that considers the internal entropy changes due to losses that have mainly been ignored by the G-N theory. It should be reminded that the range of the applicability of the generalized theories are restricted to the early extremely short times of the thermoelastic wave propagation and are not recommended for times where many wave mixings and interferences occur. For longer times, results of the generalized theories may even be less accurate than the classical theories. The governing equation of motion of the axisymmetric sphere is:

g

r ri

(1)

The subscripts i and o stand for the inner and outer radii, respectively, and the positive definite power (g) is the so-called heterogeneity index. According to Voigt’s rule of mixtures, variations of a typical material property P may be related to those of the inner and outer layers, as follows:

P (r ) = Vf i (r ) Pi + [1

r

(2)

Vf i (r )] Po

ro ro

r ri

g

ro ro

+ Po 1

r ri

g

r (r ,

(3)

(4)

(5)

M r

=

T is the absolute temperatures in Kelvin degrees. Since the moisture content of the material is a considerably small value, one may use MacLaurin’s linear expansion to express the diffusion effects on the material properties [29]:

M

=

f2 (T ) = 1 +

r

The strain-rate dependence of the material properties may be described by the following power law [3,7]: (7)

f3 ( ) = 1 +

=

de = dt

3 2

ij ij

=

ij

P = P0i (1 + i Ti + (1 +

o To

+

2 o To

+

+

3 i Ti )(1

3 o To )(1

+

+

i

o

m i)(1 + m o)(1 +

i

o

i)

o)

ro ro

r ri

1

ro ro

r

g

1 ( E r 1 [(1 E

2

(11)

) )

r]

(12)

[(1

=

[(1

=

[

+

r

(1 + )(

+

m )]

=

[

+

r

(1 + )(

+

m )]

)

r

+2

)

r

+2

(1 + )( ]

3K (

+ +

m )] m )]

(13)

are Young’s modulus and Poisson ratio, respectively,

E (1 + )(1

du , dr

=

+ P0o r ri

m (r , t )

2 )

=

, 3 (1 + ) =

E 1

2

= 3K

(14)

and K are Lame and volumetric (bulk) moduli, respectively. The strain-displacement relations are:

(8)

Combining Eqs. (3) to (7) leads to: 2 i Ti

m (r , t )

(r , t ) +

=

=

ij

t

(r , t ) +

where E and and:

For the isotropic materials, the following equation may be proposed to compute the magnitude of the strain-rate at any arbitrary material point of the body, for a very small interval of time: 3 2

t) +

(r , t ) +

or:

(6)

m

(10)

where = T T0 , m , , and are the temperature and moisture rises with respect to the stress-free state and thermal and moisture expansion coefficients, respectively, and T0 is the initial temperature associated with the stress-free condition. The Hooke’s stress-strain relations are:

m is the diffusion content. The following cubic f1 (T ) function may be used to cover a wide range of the temperature-rises: f1 (T ) = 1 + T + T 2 + T 3

M r (r , M

t) =

(r , t ) =

Properties of each material may be temperature-, moisture-, and strain rate- dependent as well; so that:

P (r , T , m , ) = P (r ) f1 (T ) f2 (m ) f3 ( )

= u¨

r

where r , , r, u , and are the radial and circumferential/meridional stress components, radius, radial displacement, and mass density, respectively. Each of the radial ( r ) and circumferential ( ) strain components is composed of mechanical and hygrothermal strains:

where Pi and Po are values of the mentioned property at the inner and outer layers, respectively. Thus, according to Eqs. (1) and (2) one has:

P = Pi

r

+2

r

=

u r

(15)

Combining Eqs. (10), (13), and (15) yields:

g

(1

r

(9)

(1

2.2. The nonlinear Lord-Shulman generalized coupled hygrothermoelasticity equations of the sphere

) 2 )

u u +2 r r u r

u r

(1 + )(

+

m)

+

2 r

= u¨

(16)

The expanded form of Eq. (16) is:

[ (1

A complete set of the governing equations comprises the motion and the energy balance equations. Since the heat conduction equation of the classical theory of thermoelasticity does not contain T¨ , it cannot be used for investigation of the thermal wave propagation. On the other hand, our previous experience and reports of Ignaczak and Ostoja-Starzewski [36] reveal that the G-L thermoelasticity theory leads to numerically very sensitive results that sometimes are not consistent. The G-L theory has been built on a transient sense of the temperature. Comparing Figs. 5, 15, and 16 respectively, with Figs. 2, 12, and 13 of Ref. [18] confirms that results of the G-L theory may sometimes be erroneous. For

[(

)] u, rr + [ ,r

+

,r

,r ) r

(1

,r

(1 + )

+

[

2 (1

,r

(1

[ (1 + ) ]( m ), r [

)

1

,r

)r

2] u

[ (1 + ) ]

(1 + ) +

(1 + )

,r

+

) r 1] u, r + 2 (1 + )

,r

,r

+

] m = u¨

,r ,r

] (17)

The comma symbol indicates a partial derivative. This equation constitutes the Navier-type hygrothermomechanical governing equation of motion. The radial derivative of each of the main material properties (such as E , , and ) is a complicated expression; so that, according to Eqs. (4) and (9): 3

Composite Structures 229 (2019) 111364

M. Shariyat, et al.

[ T , r + 2 TT , r + 3 T 2T , r ](1 +

P , r (r , T , C , ) = P0i

g ro

ri

[1 + T + T 2 + T 3](1 +

m )(1 + m )(1 +

+ P0o ([ T , r + 2 TT , r + 3 T 2T , r ](1 + ro ro

1

r ri

g

+

g ro

) + [1 + T + T 2 + T 3][ ( m ), r (1 + )

i

ro ro

m )(1 +

For the development of the energy balance equation, an amended free energy function ( ) that includes the moisture content state variable as well, must be employed. According to the foregoing equations, it may be deduced that:

=

( ij, T , m ),

ij

=

ij ( ij,

T , m ),

+2

o

ro ro

, m= 0

1 + ( 2

,r ] i

ro ro

r ri

g

r ri

+

1

C)

, r ])o

g 1

(18)

, ij | ij, , m = 0 ij

, ij kl | ij, , m = 0 ij kl

,

) + (1 +

m | ij, , m = 0

+

+2

m+

,

| ij,

,

, ij

| ij,

| ij,

+

, m=0

, m = 0 ij

, m= 0

2)

, m | ij, , m = 0

+2

, ij

+

| ij,

m

, m = 0 ij

m (20)

If the changes are mild ones, the first-order MacLaurin expansion of Eq. (20) and other quantities appeared in Eq. (19) may suffice:

(19)

qi = qi ( ij, T , T , i)

)

= | ij,

s = s ( ij, T ),

1

m)

g 1

) + [1 + T + T 2 + T 3][ ( m ), r (1 +

m )(1 +

[1 + T + T 2 + T 3](1 +

ri

r ri

) + (1 +

where s and qi are the entropy per unit mass and heat flux, respectively. As any other arbitrary function, the free energy may be expressed by a Taylor expansion around the ( ij0 = 0, T = T0, m = 0 ) stress-free condition:

= | ij,

, m= 0

++

+

, ij | ij, , m = 0 ( ij

, m | ij, , m = 0

ij0 )

+

,

| ij,

, m=0

(21)

m

0.06 0.04

u (mm)

0.02 0 -0.02

Inner point middle point

-0.04 -0.06 0

Outer point

0.2

0.4

t (sec)

0.6

0.8

1 x 10

-3

(a)

(b)

(c)

Fig. 2. Time histories of the: (a) radial displacement, (b) radial stress, and (c) hoop stress of the inner, middle, and outer layers of the moisture- and strain-rateindependent sphere. 4

Composite Structures 229 (2019) 111364

M. Shariyat, et al.

Q = T ds ij

=

ij | ij, , m = 0

+

+

ij, kl | ij, , m = 0 ( kl

ij, m | ij, , m = 0

qi = qi | ij, (T

T0) + q, T , j | ij,

+

ij,

| ij,

(22)

, T , j = 0 ( kl

, T , j = 0 [T , j

kl0 )

+ qi, T | ij,

. q = qi, i =

T0 ),

ij0

=0

ij

=

(T , j ) 0]

ij, kl kl

+

+

ij,

ij, m

=

= Cij

, ij

kl

ij

ij

ij kl

+ µ(

+

ij kl

= Cijkl = =

ij, T

c =

1 Q dT

2

, ij kl ,

m

( )

=

ij T

T ,

=

ij

ds

= T dT = Ts, T

ij

=

s , ij

s,T =

,

, ij

ij, m ij, T

s, ij =

= =

ij

=

qi (T , i ) =

1 + t0

, ij m

1 + t0

T

1 2 T0

+

qi

ij ij (

ij ij

ij ij

2

(29)

c 2T0

m) ij ij (

m)

h ( m )2 + 2T0 2 c 2 h ( m )2 2T0 2T0

2

( m)

2

qi (T , i ) =

( m)

=

dV

(31)

qi, i =

A

.q

2T

k· T

(37)

2f

=

2f f 1 2 f r2 = 2 + r2 r r r r r

(38)

t

(T

ii

+ c T)

rr ,

+

2 r

,r

+ ,r

,r

=0

(39)

qi ni dA =

V

(

qi, i ) dV

=0

u u + c T + t 0 (T + T ) u , r + +T r r

u, r +

,r T ,r +

,rr + 2

u¨ , r + ,r r

=0

u¨ r (40)

The nonlinear coupled governing Eqs. (17) and (40) may be solved by means of the finite element method. To ensure that no stress jump occurs at the mutual boundaries of the successive elements, the cubic C1-continuous Hermitian elements, rather the common C0-continuous element are used for the displacement field. In this regard, the radial domain is discretized into many nodes; however, in contrast to the finite difference and DQ procedures, variations of the material properties are not discretized, due to using the integral-type Galerkin method over the entire element. Each Hermitian element contains two nodes but four degrees of freedom:

(30)

ij T , j

V

k

3.1. The finite element form of the governing equations

(41) where H and matrices:

are the cubic Hermitian and quadratic shape function (42)

H = H1 H2 H3 H 4

where q and t 0 are the heat flux and the so-called relaxation time, spectively. On the other hand, from the balance of the rate of variations of the total heat energy of the system one may deduce:

dQ dV = dt

(36)

T ,i

3. The numerical solution scheme

The last term of Eq. (30) satisfies the Betti-Maxwell reciprocity theorem. Eq. (31) is Fourier’s heat conduction equation of the anisotropic body and predicts a nonrealistic infinite speed for the thermal wave. The Cattaneo-Vernotte modification on Fourier conduction heat transfer equation has been presented as: · q + t0 q = T (32)

V

=

Putting t 0 = 0 into Eqs. (17) and (40) gives the classical version of the nonlinear coupled hygrothermoelasticity equations of the sphere. Time variations of the mass density and specific heat of Eq. (40) are dependent on the temperature and strain-rate variations as well.

leads to the following conclusions, in the context of the temperaturemoisture analogy: ij ij

ij T , j

+ ( c + c ) T + c T¨

Combining Eqs. (21) to (28) and using expansions like:

1 = Cijkl ij kl 2 1 = ii jj + µ 2

=

or:

c T

(28)

T0

(35)

where er is the unit base vector. Therefore, according to Eqs. (34) to (38):

ij

q, ij , q, T = 0

=

ij T , j

·q =

t

f er r

f=

c is the specific heat at a constant volumetric strain. In small deformations [s ( ij0, T0) = 0 ]:

T = ln 1 + T0 T0

+ c T)

Due to the symmetry, the following identity holds in the spherical coordinates:

(27)

ln

ij ij

Therefore, from the inner product of both sides of Eq. (36) by the operator one has:

(26)

=

(T

(24)

(25)

m

ik jl )

ij,

(34)

is conduction heat transfer coefficient.

and µ are Lame moduli, ij = ij = ij E (1 2 ) is the stress-temperature coupling coefficient [Pls. see Eq. (13)], and ij is Kronecker’s delta symbol. Therefore: ij, kl

+ cT

qi

where

Cijkl =

ij ij

(23)

According to the general stress-strain equation, one may write: ij

+ s, T T ) = T

Moreover, for the isotropic materials:

,T ,j=0

It is assumed that: ij ( ij0,

ij

Eqs. (33) and (34) lead to the following conclusion:

, m=0

m

+ qi, kl | ij,

, T , j= 0

kl0 )

Q = T s = T (s, ij

is the nodal values vector and the Hermitian shape functions are:

U1 H1 = 1 3r 2 + 2r 3 U ,r1 H2 = r [r 2r 2 + r 3] U2 = , ; U ,r 2 H3 = 3r 2 2r 3 1 H 4 = r [ r 2 + r 3]

Q (33)

2

where Q , , qi , V, and A are the total heat energy, internal heat generation rate, heat flux, volume, and boundary area, respectively. For a reversible process, according to Eq. (27) one may write:

l (e )

5

r =

r

r1 l (e) (43)

is the element length. Although the moisture content may be determined based on a non-

Composite Structures 229 (2019) 111364

M. Shariyat, et al.

Table 1 Temperature-dependence coefficients of typical materials [42]. Material

Property

Base value

Zirconia (ZrO2)

E (Pa)

244.26596e9 0.2882 12.7657e−6

−1.3707e−3 1.13345e−4 −0.00149

487.34279

3.04908e−4

(K 1) (W mK ) (kg m3) c (J kgK ) Ti-6Al-4 V

E (Pa)

(K 1) (W mK ) (kg m3) c (J kgK )

1.7 5700

0.0001276 0

122.55676e9 0.28838235 7.57876e−6

−4.5864e−4 1.12136 e−4 0.00065

1.20947 4429

0.0139375 0

625.29692

−4.2239e−4

1.21393e−6 0 1. e−6

0.66485e−5 0

−6.03723e−8

0 0 0.31467e−6

0 0

7.178654e−7

−3.68138e−10 0 −0.6775e−11 0 0 0

0 0 0

0 0 0

Fig. 3. Simultaneous spatial and time distributions of the: (a) radial displacement, (b) radial stress, and (c) hoop stress of the inner, middle, and outer radii of the moisture- and strain-rate-independent sphere.

classic Fick’s law:

1 + t0

t

·J =

(45)

D

2T

D· T

The finite element governing equations may be derived after imposing Galerkin’s method:

(44)

where J and D are the diffusion flux vector and diffusion coefficient, respectively, a specified distribution will be used instead. Indeed, in contrast to the published papers on the FG cylinders and spheres, the diffusion is a long-term phenomenon (e.g., sometime takes many years in order that its effects become notable) whereas the nonclassical theories are valid for extremely short times (shock loads). In the light of Eq. (41), the system of the governing equations consisting of Eqs. (17) and (40) may be denoted by:

(46) or: (47) 3.2. The natural boundary conditions While the boundary conditions may be imposed directly or by using 6

Composite Structures 229 (2019) 111364

M. Shariyat, et al.

Fig. 4. Time histories of the: (a) radial displacement, (b) radial stress, and (c) hoop stress of the inner, middle, and outer radii of the moisture-independent but strainrate-dependent sphere.

penalty methods on Eq. (45), the finite element form of the natural boundary conditions must be assembled with the system of equations of the entire sphere, to preclude the singularity of the system of equations. Since the boundary conditions are relevant to the first and last layers of the sphere, their relevant boundary condition can be assembled respectively, with lines associated with the first and final elements, using proper weighting numbers. Sometimes, these weighting numbers have to be large enough to suppress the inertial effects in the mentioned rows of the system of equations. For example, when the sphere is subjected to dynamic pressures, the corresponding boundary conditions may be expressed as:

p )r = ri =

r )r = ri

=

(1

=

(1

p)r = ro =

r )r = ro

=

(1

=

[(1

)

) u ,r + 2

r

u r

+2

(1 + )(

2 ) H,r + H U r

) Hr , +

(1 + )(

2 H U r

+

+

following relations:

hin

+

r = ri

m)

= hout

(50)

out

i+1

=

4 ( ( t )2

i+1

i

t

i)

¨ ; i

i+1

=

2 ( t

i+1

i)

i

(52)

r = ro

The i and i + 1 subscripts correspond to the starting and finish time instants of the current time step. Therefore, Eq. (51) may be written in terms of the vector of the nodal unknown parameters of the end of the current time step. When the time step is much smaller than the time required for the thermoelastic wave to travel the sphere thickness (and hence, much less than

A conduction heat transfer at the boundary may be represented by:

=q

,r

(51)

(48)

k ,r

+

This equation may even be so condensed that only the non-derivative nodal degrees of freedom appear, without any loss in accuracy [37–41]. Using the second-order Taylor’s expansion of Runge-Kutta, one may express the nodal acceleration and velocity vectors of the end of each time step in terms of the nodal unknown parameters of the same time instant as:

¨

(1 + )(

hout

The augmented nonlinear system of equations may be represented by the following compact form:

m )]

m)

in ;

3.3. Treating the time-dependence of the augmented finite element governing equations

m) +

= hin

where h is the convection heat transfer coefficient.

r = ri

(1 + )(

,r

(49)

where q is a known value (and thus, t 0 q = 0 ). Convection heat transfer on either the internal or external layer can be satisfied by one of the 7

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Fig. 5. Spatial and time distributions of the: (a) radial displacement, (b) radial stress, and (c) hoop stress of the inner, middle, and outer radii of the moistureindependent but strain-rate-dependent sphere.

the period time of the highest notable vibration mode), one may use the following formulation to avoid redundant iterations:

3-

(53)

where

45Therefore:

(54)

6-

(55)

78-

It should be noted in contrast to the majority of the available articles, the acceleration vector is not zero at t = 0 and may be determined based on Eq. (51) as: (56)

|

3.4. Steps of the numerical solution procedure

thermoelastic (sound and second sound) waves to travel the sphere thickness. Construction of the element matrices based on the conditions of the previous step or iteration if any. In this regard, the strain-rates, temperatures, and moisture contents corresponding to the end of the previous time step may be employed. Otherwise, these quantities may be used as an initial guess. Incorporation of the boundary conditions. Determination of the nodal accelerations at t = 0 for the first time step [Eq. (56)]. Solving the resulting equations according to the second-order Runge-Kutta procedure [Eq. (53)]. Computing the nodal strain rates according to Eq. (8). If in Eq. (54), the stiffness, damping, and mass matrices belong to the i + 1 time step, updating iterations would be required to incorporate the newly computed temperatures, moisture contents, and strain-rates. The following convergence criterion may be employed in each time step:

(k + 1) i+1

(k ) i + 1|

|

(k ) i + 1|

(57)

0.0001

where k is the iteration counter. If Eq. (57) is not satisfied, the previous steps are repeated from step 3.

The numerical solution may be accomplished through the following steps:

9- Computing the stress components based on:

1- Definition of the material properties and boundary and initial conditions. 2- Discretization of the analysis time into many extremely small time steps that are much smaller than the time required for the

r

= * (1 = *

8

H r

2

) Hr , + r H U + H,r U

(1 + )(

(1 + )(

+

+

m)

m) (58)

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Fig. 6. Time histories of the: (a) radial displacement, (b) radial stress, and (c) hoop stress of the inner, middle, and outer radii of the moisture-dependent but strainrate-independent sphere.

10- Computing the nodal velocities and accelerations based on Eq. (52). 11- Saving the obtained nodal values and their first- and second-order time derivatives (velocities and accelerations), and the nodal strain-rates to be employed as initial values for the next time step. 12- Starting the next time step and repeating the preceding steps from step 3, until the considered final analysis time is reached.

The strain-rate dependency parameters of the materials have been extracted experimentally:

4. Results and discussions

4.1. Influence of the moisture and strain-rate dependencies of the material properties on the displacement and stress distributions, under hygromechanical shocks

Zirconia

= 0.33(wt %H2 O) 1,

Titanium

= 0(wt %H2 O)

Titanium

= 0.030572,

Titanium

= 0.269114

Magnitudes of these parameters are in a good concordance with those of some close materials [1–7,44–48].

In this section, the sensitivity of the hygrothermomechanical responses of the moisture- and temperature-dependent hollow FG sphere are studied against several parameters. Fist, the resulting dynamics stress distributions are studied for spheres under either pressure or temperature shocks, in humid ambient. Then, the influence of the strain-rate and moisture on the hygrothermoelastic wave propagation is investigated. Verification of the results has been accomplished in this second more complex part of the results. The size of the elements is so chosen that further refinements lead to negligible differences in the results. On the other hand, the time steps are adopted much smaller than the period time associated with the 10th natural frequency (about 10−6 s) in the first stage and about 10−12 s for the wave propagation analyses. Therefore, the size of the time steps is satisfactory and reasonable. On the other hand, since cubic Hermitian elements are used, the numerical or shear locking phenomena are precluded. With an emphasis on the Zirconia/Titanium ceramic mixture, the , , and coefficients are listed in Table 1 and [43]: Zirconia

0,

As a first example, consider a spherical vessel whose temperature is kept at 300 K and is subjected to a triangular pressure shock at a considerably short time:

T (r , t ) = 300K ,

pi =

2 × 105tMPa ; 0 t 0.0005s , 0; t > 0.0005s po = 0

Inner and outer radii of the vessel are: ri = 100mm , ro = 200mm . It is assumed that the innermost and outermost layers are ceramicrich (ZrO2) and metal-rich (Ti-6Al-4 V) layers, respectively. Therefore, based on Eq. (3), the whole metallic content of the vessel grows by increasing the heterogeneity index (g) and the material properties of the layers become more closer to the metallic ones. Since the g = 1 case (linear variations of the materials volume fractions) is more practical, this case is adopted for all studies. Results are plotted for three cases: (i) Moisture- and strain-rate-independent sphere.

1

9

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Fig. 7. Spatial and time distributions of the: (a) radial displacement, (b) radial stress, and (c) hoop stress of the inner, middle, and outer radii associated with the strain-rate-dependent but moisture-independent sphere.

Fig 8. Evaluation of the effects of the moisture and strain-rate dependencies of the material properties on time variations of the radial displacement.

Fig 9. Evaluation of the effects of the moisture and strain-rate dependencies of the material properties on time history of the radial stress.

(ii) Strain-rate-dependent but moisture-independent sphere. (iii) Moisture-dependent but strain-rate-independent sphere.

illustrated in Fig. 2(a) to 2(c), respectively. The responses show effects of the higher vibration modes and as may be expected, the radial vibrations following the unloading have occurred about the zero values. Effects of the higher vibration modes on the results are most observable after unloading; so that different patterns are noted in the successive

Time variations of the radial displacement, radial stress, and hoop stress of the inner, middle, and outer layers of the first case are

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displacement and stresses, clearly shows the stress waves propagations and reflections. Furthermore, the 3D plots of the radial displacement reveal that the apparent wave propagation and reflection phenomena are themselves portions of a very general overall oscillation, shown in Fig. 12; so that, the formation of the oblique waves may be seen in both the displacement and stress plots. To focus further on this hint, locations of the displacement and stress wavefronts are plotted in Figs. 13 to 15, for the three cases of dependencies of the material properties. The time instants 1 to 4 are associated with times of the shock application and the remaining ones correspond to times that follow the impact (load removal). The progress of the displacement and stress wavefronts into the vessel and the reflections may easily be observed in these figures. Comparing Figs. 13 to 15 reveals that the wave propagation speed increases by considering the strain-rate effect whereas it decreases by incorporation of the moisture content effect. Now responses of the same spherical vessel are studied under thermal shocks. It is assumed that the vessel is subjected to a temperature-rise at the inner boundary that lasts for a very small time duration:

Fig 10. Evaluation of the effects of the moisture and strain-rate dependencies of the material properties on time history of the hoop stress.

oscillations of the displacement and stress components. This may be attributed to the superposition manner of the vibration modes of the vessel. Time histories of all points of the vessel for the whole considered analysis time are shown in Fig. 3. While the 3D plot of the hoop stress resembles that of the radial displacement, to a great extent, the 3D plot of the radial stress is a different one. Effects of the strain-rate and moisture content ( m = 0.01%) on the time and spatial distributions of the radial displacement and radial and hoop stresses may be sought in Figs. 4 to 7, respectively. Responses corresponding to the mentioned three cases are compared in Figs. 8 to 10. As may readily be seen, due to the small increase in the elastic modulus, displacements and stresses of the strain-dependent pressure vessel are smaller than those of the original sphere. On the other hand, due to the moisture-induced degradations in the material properties, displacements and stresses of the moisture-dependent vessel are larger than those of the original pressure vessel. Figs. 8 to 10 also indicate that the vibration frequencies become less and larger for the moisture-dependent and strain-rate-dependent material properties, respectively. However, since the imposed load is a mechanical rather than a thermal one, effects of the temperature-dependence of the material properties do not appear. Although the effects of the moisture and strain-rate are notable for the present results, they may be much notable in fiber-reinforced composite structures [29].

pi , po = 0,

be:

pi =

200MPa ; 0 0;

t t

100

i

=

300K ; 0 0;

t

t

100

100

The three cases defined in the previous sections are now modified to

The growth of the radial displacement and stress distributions with time is illustrated in Fig. 16, for temperature-dependent but moistureand strain-rate-independent material properties. Since the temperature of the internal layers is greater, these layers move toward and lead to tensile radial stresses. The 3D plots associated with the new three cases of material properties dependencies are depicted in Fig. 17. In contrast to the previous examples, present results reveal the interaction of the temperature-dependence of the material properties with the moisture and strain-rate dependencies. While Fig. 17 shows larger displacements for the temperature- and strain-rate-dependent material, it exhibits a larger stress release in the moisture-dependent material properties. On the other hand, the interaction of the temperature and moisture dependencies leads to exceptional converse results.

Now, the responses of the spherical vessel to an internal rectangular pressure is studied. The magnitude of the time duration of the shock is adopted much smaller than that of the previous example. Geometric and material properties of the vessel are identical with those of the previous example. Therefore, higher strain rates may be induced in the sphere. The initial and load conditions are ( is the period time associated with the first vibration mode):

T (r , t ) = 300K ,

To = 300K ,

(I) A temperature-dependent but moisture- and strain-rate-independent sphere. (II) A temperature- and strain-rate-dependent but moisture-independent sphere. (III) A temperature- and moisture-dependent but strain-rate-independent sphere.

4.2. The moisture and strain-rate effects on the stress wave propagation and reflection under hygrothermomechanical shocks

po , p (r , 0) = 0

T (r , 0) = 300K ,

4.3. Verification and sensitivity analyses against moisture and strain-rate for both the first sound (stress wave) and second sound (thermal propagation wave) results Now effects of the material properties dependencies, i.e., the temperature, moisture, and strain-rate dependencies, are investigated on both the thermal and stress wave propagation and reflection phenomena. Although some researchers have discussed the effects of the material gradation on the wave propagation phenomenon [8,9,22,49], the effect of the temperature, moisture, and strain-rate dependencies have not been studied by these authors. It is worth mentioning that like the temperature-rise term, the moisture concentration affects both the constitutive law and material properties. In this regard, we have to verify our results of the thermoelastic wave propagation, especially, the second sound (i.e., the thermal wave propagation) results, first.

,

100

The 3D plots of the spatial-time variations of the radial displacement and stress components are shown in Fig. 11, for the three cases defined in the previous example. Fig. 11 that shows both the spatial and time variations of the

11

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Fig. 11. 3D plots indicating the simultaneous radial and time distributions of the radial displacement and stress components for: (a) case (i), (b) case (ii), and (c) case (iii) of the moisture and strain-rate dependencies.

Verification is accomplished based on results of Bagri and Eslami [18], for a sphere with the following thermomechanical properties:

ro = 2, ri = 204

= 40.4GPa, W , m

2T 0

c ( + 2µ )

µ = 27GPa,

= 2707

where = (1 + ) = (3 + 2µ ) . The thermomechanical boundary conditions of the sphere are:

kg , m3

r = ri: u = 0,

= 0.02

12

=1 i

(1 + 100t ) e

100t

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Fig. 12. Oscillations of the radial displacement in longer times [case (i)].

Fig. 14. Propagation and reflection of the radial displacement and stress waves, for case (ii) of the dependencies of the material properties.

r = =

r , l

t = i

T0

;

C1 t , l

C1 t 0 , = , l T0 ( + 2µ ) u u = , l T0

t0 =

(i = r , ),

i

and the speed of the sound (stress wave) and the characteristic length l are defined as:

C1 =

r

= 0,

r

,

l=

k c C1

The t 0 = 4 dimensionless relaxation time is used. In this verification stage, no moisture-, temperature-, and strain-rate dependencies are considered. In Figs. 18 to 20, present results are compared with those of Bagri and Eslami [18]. These figures show the radial distributions of the temperature and the radial and hoop stresses, respectively, for different dimensionless time instants. As may readily be noted, there is an excellent concordance between the results, even though Ref. [18] used the transfinite element method whereas the present research has used the time-marching Hermitian element procedure. As may be seen, our figures include some additional results that belong to the t = 0.8, 1.6 time instants. Comparing Fig. 18 that clearly illustrates the gradual diffusion of the thermal energy into the sphere with Figs. 19 and 20 reveals that for the present conditions, the stress (first sound) wave travels with a velocity that is approximately twice as much as that of the thermal

Fig. 13. Propagation and reflection of the radial displacement and stress waves, for case (i) of the dependencies of the material properties. Numbers indicate the wave location at different time instants.

r = ro:

( + 2µ )

=0

The results are expressed in terms of the following dimensionless quantities:

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0.05

0

5

25 27 28 29 30 31 32 33 37 38 39 40 41 42 43 44 1

100

24 23 22 21 20

0 26 34 35 36

2

3

120

4

7 5 6

140

r (mm)

8

160

u (mm)

u (mm)

0.1

1819 17 16 15 14 13 12 11 10 9

180

-5 -10 -15

x 10

-6

2 345 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

-20 100

200

120

140 160 r (mm)

180

200

180

200

(a)

(a) Radial stress (MPa)

0.06 0.04 0.02 0

22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

-0.02 100

120

140 160 r (mm)

(b)

Fig. 16. Growth in the radial distributions of the: (a) radial displacement and (b) radial stress, for the new case (I) of the dependencies of the material properties.

(b) Fig. 15. Propagation and reflection of the radial displacement and stress waves, for case (iii) of the dependencies of the material properties.

linear media, collapse of the smooth wave can lead to formation of the shock wave front at which the spatial derivatives have discontinuities [50–60].

(second sound) wave. On the other hand, due to the mentioned hint, it can be deduced from Figs. 19 and 20 that a significant (even more that 100%) radial stress relief but a hoop stress growth has occurred in regions behind the thermal wave front. Now, the simultaneous effects of the temperature, moisture and strain-rate dependencies on the material properties and the constitutive law are taken account to present more realistic and practical results. Indeed, since the principle of superposition does not hold for nonlinear systems, studying the individual effects of the parameters may not help. In other words, some of these effects may cancel a portion of the effects of the other parameters. The temperature distribution has almost remained unchanged; so that the resulting changes were ignorable. But the resulting stress distributions and the predicted hygrothermomechanical wave propagation and reflection phenomena were quite different, as Figs. 21 and 22 show. In addition to different slopes, the resulting stresses were slightly lower; a fact that indicates a lower rigidity. Furthermore, the speed of the stress wave has become less. Although for the present material properties and boundary conditions neglecting the material properties dependencies and the relevant changes in the constitutive law (the nonactual case) has led to an overdesigned vessel, the resulting design may not be on the safe side, for other boundary conditions or material properties. As may be seen in Figs. 21 and 22, the differences between the two types of results grow with time; so that predictions of the simplified model become unreliable, even for the very short times following the impact. It should be reminded that in propagation of acoustic waves in non-

5. Conclusions A Hermitian finite-element-based time marching procedure is employed to ascertain formation and evolutions of the temperature, displacement, and stress distributions in a thick hollow functionally graded sphere at the early times following the hygrothermal shock. In this regard, a nonlinear Lord-Shulman-type generalized hygrothermoelasticy formulation is developed based on the amendment of both constitutive law and the free energy function. The presented model can consider effects of the temperature, moisture absorption, and strain-rate on both the constitutive law and material properties. Apart from the scientific novelties that are listed in the Introduction section, the following practical conclusions may be drawn based on the obtained numerical results:

• The temperature-dependence of the material properties may have a significant influence on the results of the displacements and stresses. • Elastic wave propagations/reflections may lead to oblique waves formation. • Generally, wave propagation and reflection phenomena are themselves portions of a more general overall oscillation/mixing. • Temperature-dependency of the material properties may also affect •

14

various parameters of the wave propagation phenomenon in the thermal environments. The strain-rate dependence leads to a small increase in the elastic

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Fig. 17. 3D plots indicating the influence of the interactions between effects of the temperature-dependence and other dependencies of the material properties, for the new: (a) case (I), (b) case (II), and (c) case (III) of the material properties dependencies combinations.

modulus and consequently, smaller displacements and stresses.

propagation and reflection processes.

• In mechanical loading, the moisture-induced degradations in the material properties lead to larger displacements and stresses. • The vibration frequencies become less and larger for the moisture• • •

• Since the principle of superposition does not hold for the nonlinear

dependent and strain-rate-dependent material properties, respectively. The wave propagation speed increases by considering the strain-rate effect and decreases by incorporation of the moisture content effect. In thermal loading, larger displacements happen in the temperatureand strain-rate-dependent vessel but a larger stress release occurs in the moisture-dependent sphere. Generally, the speed of the stress (first sound) wave is larger than that of the thermal (second sound) wave. This hint leads to stress relief and growth in the distributions of the radial and hoop stresses, respectively, in regions behind the thermal wave front, both in wave

• •

15

thermoelasticity, studying the individual effects of the parameters may not help; so that, some of these effects may partially cancel the effects of each other. Although the strain-rate and moisture have ignorable effects on the temperature distribution, they significantly affect (reduce) the stresses and the hygrothermomechanical wave propagation and reflection speeds. For the present material properties and boundary conditions neglecting the dependencies of the material properties has led to an overdesigned vessel. The resulting design may not be on the safe side, for other boundary conditions or material properties.

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Fig. 21. Influence of the actual moisture concentration, strain-rate, and temperature-dependence factors on the shock stress distribution and wave propagation and reflection of the radial stress.

Fig. 18. Comparative and new results to verify present results with results of Bagri and Eslami [18], for various dimensionless time instants.

Fig. 19. Comparative and new radial stress results to verify present results with results of Bagri and Eslami [18], for various dimensionless time instants.

Fig. 22. Influence of the actual moisture concentration, strain-rate, and temperature-dependence factors on the shock stress distribution and wave propagation and reflection of the hoop stress.

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Fig. 20. Comparative and new hoop stress results to verify present results with results of Bagri and Eslami [18], for various dimensionless time instants.

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